Quadrant Model of Reality Book 18 Science Physics

Similar to my previous Quadrant Model books, this Quadrant Model book will be organized and designed around the four fields of inquiry.

Science

Physics

Chemistry

Biology

Psychology

Sociology

Religion

Buddhism

Christianity

Islam

Hinduism

Judaism

Other

Art

Painting

Music

Dance

Literature

Cinema

Philosophy

Science chapter

Physics chapter

QMRGauss's law for magnetism, which is one of the four Maxwell's equations, states that the total magnetic flux through a closed surface is equal to zero. (A "closed surface" is a surface that completely encloses a volume(s) with no holes.) This law is a consequence of the empirical observation that magnetic monopoles have never been found.

In other words, Gauss's law for magnetism is the statement:

\Phi_B=\,\!\oiint\scriptstyle S\mathbf{B} \cdot d\mathbf S = 0

for any closed surface S.

Maxwell's equations (sometimes called the Maxwell equations) are the set of four equations, attributed to James Clerk Maxwell, that describe the behavior of both the electric and magnetic fields, as well as their interactions with matter.

QMRMaxwell's equations[edit]

Maxwell's equations of electromagnetism relate the electric and magnetic fields to each other and to the motions of electric charges. The standard equations provide for electric charges, but they posit no magnetic charges. Except for this difference, the equations are symmetric under the interchange of the electric and magnetic fields.[20] In fact, symmetric Maxwell's equations can be written when all charges (and hence electric currents) are zero, and this is how the electromagnetic wave equation is derived.

Fully symmetric Maxwell's equations can also be written if one allows for the possibility of "magnetic charges" analogous to electric charges.[21] With the inclusion of a variable for the density of these magnetic charges, say ρm, there will also be a "magnetic current density" variable in the equations, jm.

If magnetic charges do not exist – or if they do exist but are not present in a region of space – then the new terms in Maxwell's equations are all zero, and the extended equations reduce to the conventional equations of electromagnetism such as ∇⋅B = 0 (where ∇⋅ is divergence and B is the magnetic B field).

In Gaussian cgs units[edit]

The extended Maxwell's equations are as follows, in Gaussian cgs units:[24]

Maxwell's equations and Lorentz force equation with magnetic monopoles: Gaussian cgs units

Name Without magnetic monopoles With magnetic monopoles

Gauss's law \nabla \cdot \mathbf{E} = 4 \pi \rho_{\mathrm e}

Gauss's law for magnetism \nabla \cdot \mathbf{B} = 0 \nabla \cdot \mathbf{B} = 4 \pi \rho_{\mathrm m}

Faraday's law of induction -\nabla \times \mathbf{E} = \frac{1}{c}\frac{\partial \mathbf{B}} {\partial t} -\nabla \times \mathbf{E} = \frac{1}{c}\frac{\partial \mathbf{B}} {\partial t} + \frac{4 \pi}{c}\mathbf{j}_{\mathrm m}

Ampère's law (with Maxwell's extension) \nabla \times \mathbf{B} = \frac{1}{c}\frac{\partial \mathbf{E}}{\partial t} + \frac{4 \pi}{c} \mathbf{j}_{\mathrm e}

Lorentz force law[24][25] \mathbf{F}=q_{\mathrm e}\left(\mathbf{E}+\frac{\mathbf{v}}{c}\times\mathbf{B}\right) \mathbf{F}=q_{\mathrm e}\left(\mathbf{E}+\frac{\mathbf{v}}{c}\times\mathbf{B}\right) + q_{\mathrm m}\left(\mathbf{B}-\frac{\mathbf{v}}{c}\times\mathbf{E}\right)

In these equations ρm is the magnetic charge density, jm is the magnetic current density, and qm is the magnetic charge of a test particle, all defined analogously to the related quantities of electric charge and current; v is the particle's velocity and c is the speed of light. For all other definitions and details, see Maxwell's equations. For the equations in nondimensionalized form, remove the factors of c.

In SI units[edit]

In SI units, there are two conflicting units in use for magnetic charge qm: webers (Wb) and ampere·meters (A·m). The conversion between them is qm(Wb) = μ0qm(A·m), since the units are 1 Wb = 1 H·A = (1 H·m−1)·(1 A·m) by dimensional analysis (H is the henry – the SI unit of inductance).

Maxwell's equations then take the following forms (using the same notation above):[26]

Maxwell's equations and Lorentz force equation with magnetic monopoles: SI units

Name Without magnetic monopoles Weber convention Ampere·meter convention

Gauss's Law \nabla \cdot \mathbf{E} = \frac{\rho_{\mathrm e}}{\epsilon_0}

Gauss's Law for magnetism \nabla \cdot \mathbf{B} = 0 \nabla \cdot \mathbf{B} = \rho_{\mathrm m} \nabla \cdot \mathbf{B} = \mu_0\rho_{\mathrm m}

Faraday's Law of induction -\nabla \times \mathbf{E} = \frac{\partial \mathbf{B}} {\partial t} -\nabla \times \mathbf{E} = \frac{\partial \mathbf{B}} {\partial t} + \mathbf{j}_{\mathrm m} -\nabla \times \mathbf{E} = \frac{\partial \mathbf{B}} {\partial t} + \mu_0\mathbf{j}_{\mathrm m}

Ampère's Law (with Maxwell's extension) \nabla \times \mathbf{B} = \mu_0 \epsilon_0 \frac{\partial \mathbf{E}} {\partial t} + \mu_0 \mathbf{j}_{\mathrm e}

Lorentz force equation \mathbf{F}=q_{\mathrm e}\left(\mathbf{E}+\mathbf{v}\times\mathbf{B}\right)

\mathbf{F}=q_{\mathrm e}\left(\mathbf{E}+\mathbf{v}\times\mathbf{B}\right)

+ \frac{q_{\mathrm m}}{\mu_0}\left(\mathbf{B}-\mathbf{v}\times \frac{\mathbf{E}}{c^2}\right)

\mathbf{F}=q_{\mathrm e}\left(\mathbf{E}+\mathbf{v}\times\mathbf{B}\right)

+ q_{\mathrm m}\left(\mathbf{B}-\mathbf{v}\times\frac{\mathbf{E}}{c^2}\right)

magnetic monopole wiki

Tensor formulation[edit]

Maxwell's equations in the language of tensors makes Lorentz covariance clear. The generalized equations are:[27][28]

Maxwell equations Gaussian units SI units (Wb) SI units (A⋅m)

Faraday-Gauss law \partial_\alpha F^{\alpha\beta} = \frac{4\pi}{c}J^\beta_{\mathrm e} \partial_\alpha F^{\alpha\beta} = \mu_0 J^\beta_{\mathrm e} \partial_\alpha F^{\alpha\beta} = \mu_0 J^\beta_{\mathrm e}

Ampère-Gauss law \partial_\alpha {\tilde F^{\alpha\beta}} = \frac{4\pi}{c} J^\beta_{\mathrm m} \partial_\alpha {\tilde F^{\alpha\beta}} = \frac{\mu_0}{c} J^\beta_{\mathrm m} \partial_\alpha {\tilde F^{\alpha\beta}} = \frac{1}{c} J^\beta_{\mathrm m}

Lorentz force law \frac{dp_\alpha}{d\tau} = \left[ q_{\mathrm e} F_{\alpha\beta} + q_{\mathrm m} {\tilde F_{\alpha\beta}} \right] \frac{v^\beta}{c} \frac{dp_\alpha}{d\tau} = \left[ q_{\mathrm e} F_{\alpha\beta} + q_{\mathrm m} {\tilde F_{\alpha\beta}} \right] v^\beta \frac{dp_\alpha}{d\tau} = \left[ q_{\mathrm e} F_{\alpha\beta} + \frac{q_{\mathrm m}}{\mu_0} {\tilde F_{\alpha\beta}} \right] v^\beta

where

Fαβ is the electromagnetic tensor,

~

F

αβ =

1

/

2

εαβγδFγδ is the dual electromagnetic tensor,

for a particle with electric charge qe and magnetic charge qm; v is the four-velocity and p the four-momentum,

for an electric and magnetic charge distribution; Je = (ρe, je) is the electric four-current and Jm = (ρm, jm) the magnetic four-current.

For a particle having only electric charge, one can express its field using a four-potential, according to the standard covariant formulation of classical electromagnetism:

F_{\alpha \beta} = \partial_{\alpha} A_{\beta} - \partial_{\beta} A_{\gamma} \,

However, this formula is inadequate for a particle that has both electric and magnetic charge, and we must add a term involving another potential P.[29][30]

F_{\alpha \beta} = \partial_{\alpha} A_{\beta} - \partial_{\beta} A_{\alpha} \ +\partial^{\mu}(\varepsilon_{\alpha\beta\mu\nu}P^{\nu}),

This formula for the fields is often called the Cabibbo-Ferrari relation, though Shanmugadhasan proposed it earlier.[30] The quantity εαβγδ is the Levi-Civita symbol, and the indices (as usual) behave according to the Einstein summation convention.

QMRNeighbourhoods definition[edit]

This axiomatization is due to Felix Hausdorff. Let X be a set; the elements of X are usually called points, though they can be any mathematical object. We allow X to be empty. Let N be a function assigning to each x (point) in X a non-empty collection N(x) of subsets of X. The elements of N(x) will be called neighbourhoods of x with respect to N (or, simply, neighbourhoods of x). The function N is called a neighbourhood topology if the axioms below[2] are satisfied; and then X with N is called a topological space.

If N is a neighbourhood of x (i.e., N ∈ N(x)), then x ∈ N. In other words, each point belongs to every one of its neighbourhoods.

If N is a subset of X and contains a neighbourhood of x, then N is a neighbourhood of x. I.e., every superset of a neighbourhood of a point x in X is again a neighbourhood of x.

The intersection of two neighbourhoods of x is a neighbourhood of x.

Any neighbourhood N of x contains a neighbourhood M of x such that N is a neighbourhood of each point of M.

The first three axioms for neighbourhoods have a clear meaning. The fourth axiom has a very important use in the structure of the theory, that of linking together the neighbourhoods of different points of X.

The fourth is always different

QMRA hierarchy of mathematical spaces: The inner product induces a norm. The norm induces a metric. The metric induces a topology.

QMrA metric space is an ordered pair (M,d) where M is a set and d is a metric on M, i.e., a function

d \colon M \times M \to \mathbb{R}

such that for any x, y, z \in M, the following holds:[1]

1. d(x,y) \ge 0 non-negativity

2. d(x,y) = 0 \iff x = y\, identity of indiscernibles

3. d(x,y) = d(y,x) symmetry

4. d(x,z) \le d(x,y) + d(y,z) triangle inequality

QMRIn 1933, Enrico Fermi proposed the first theory of the weak interaction, known as Fermi's interaction. He suggested that beta decay could be explained by a four-fermion interaction, involving a contact force with no range.[3][4]

However, it is better described as a non-contact force field having a finite range, albeit very short.[citation needed] In 1968, Sheldon Glashow, Abdus Salam and Steven Weinberg unified the electromagnetic force and the weak interaction by showing them to be two aspects of a single force, now termed the electro-weak force.[citation needed]

The existence of the W and Z bosons was not directly confirmed until 1983.

QMRAccording to the electroweak theory, at very high energies, the universe has four massless gauge boson fields similar to the photon and a complex scalar Higgs field doublet. However, at low energies, gauge symmetry is spontaneously broken down to the U(1) symmetry of electromagnetism (one of the Higgs fields acquires a vacuum expectation value). This symmetry breaking would produce three massless bosons, but they become integrated by three photon-like fields (through the Higgs mechanism) giving them mass. These three fields become the W+, W− and Z bosons of the weak interaction, while the fourth gauge field, which remains massless, is the photon of electromagnetism.[18]

qMRIn particle physics, Fermi's interaction (also the Fermi theory of beta decay) is an explanation of the beta decay, proposed by Enrico Fermi in 1933.[1][2][3][4] The theory posits four fermions directly interacting with one another, at one vertex.

Fermi's four-fermion theory describes the weak interaction remarkably well. Unfortunately, the calculated cross-section grows as the square of the energy \sigma \approx G_{\rm F}^2 E^2 , making it unlikely that the theory is valid at energies much higher than about 100 GeV. The solution is to replace the four-fermion contact interaction by a more complete theory (UV completion)—an exchange of a W or Z boson as explained in the electroweak theory.

In the original theory, Fermi assumed that the form of interaction is a contact coupling of two vector currents. Subsequently, it was pointed out by Lee and Yang that nothing prevented the appearance of an axial, parity violating current, and this was confirmed by experiments carried out by Chien-Shiung Wu.[8][9]

MuonFermiDecay.gif

Fermi's interaction showing the 4-point fermion vector current, coupled under Fermi's Coupling Constant GF. Fermi's Theory was the first theoretical effort in describing nuclear decay rates for Beta-Decay.

The inclusion of parity violation in Fermi's interaction was done by George Gamow and Edward Teller in the so-called Gamow-Teller Transitions which described Fermi's interaction in terms of Parity violating "allowed" decays and Parity conserving "superallowed" decays in terms of anti-parallel and parallel electron and neutrino spin states respectively. Before the advent of the electroweak theory and the Standard Model, George Sudarshan and Robert Marshak, and also independently Richard Feynman and Murray Gell-Mann, were able to determine the correct tensor structure (vector minus axial vector, V − A) of the four-fermion interaction.

QMRIn quantum field theory, fermions are described by anticommuting spinor fields. A four-fermion interaction describes a local interaction between four fermionic fields at a point. Local here means that it all happens at the same spacetime point. This might be an effective field theory or it might be fundamental.

Some examples are the following:

Fermi's theory of the weak interaction. The interaction term has a V-A (vector minus axial) form.

The Gross-Neveu model. This is a four-fermi theory of Dirac fermions without chiral symmetry and as such, it may or may not be massive.

The Thirring model. This is a four-fermi theory of fermions with a vector coupling.

The Nambu-Jona-Lasinio model. This is a four-fermi theory of Dirac fermions with chiral symmetry and as such, it has no bare mass.

A nonrelativistic example is the BCS theory at large length scales with the phonons integrated out so that the force between two dressed electrons is approximated by a contact term.

In four space-time dimensions, such theories are not renormalisable.

QMRThe Gross-Neveu model is a quantum field theory model of Dirac fermions interacting via four fermion interactions in 1 spatial and 1 time dimension. It was introduced in 1974 by David Gross and André Neveu[1] as a toy model for quantum chromodynamics, the theory of strong interactions.

It consists of N Dirac fermions, ψ1, ..., ψN. The Lagrangian density is

\mathcal{L}=\bar \psi_a \left(i\partial\!\!\!/-m \right) \psi^a + \frac{g^2}{2N}\left[\bar \psi_a \psi^a\right]^2

using the Einstein summation notation where g is the coupling constant. If the mass m is nonzero, the model is massive classically, otherwise it enjoys a chiral symmetry.

This model has an U(N) global internal symmetry. Note that it does not reduce to the massive Thirring model (which is completely integrable).

It is a 2-dimensional version of the 4-dimensional Nambu–Jona-Lasinio model (NJL), which was introduced 14 years earlier as a model of dynamical chiral symmetry breaking (but no quark confinement) modeled upon the BCS theory of superconductivity. The 2-dimensional version has the advantage that the 4-fermi interaction is renormalizable, which it is not in any higher number of dimensions.

QMRThere are three known types (flavors) of neutrinos: electron neutrino ν

e, muon neutrino ν

μ and tau neutrino ν

τ, named after their partner leptons in the Standard Model (see table at right). The current best measurement of the number of neutrino types comes from observing the decay of the Z boson. This particle can decay into any light neutrino and its antineutrino, and the more types of light neutrinos[nb 4] available, the shorter the lifetime of the Z boson. Measurements of the Z lifetime have shown that the number of light neutrino types is 3.[20] The correspondence between the six quarks in the Standard Model and the six leptons, among them the three neutrinos, suggests to physicists' intuition that there should be exactly three types of neutrino. However, actual proof that there are only three kinds of neutrinos remains an elusive goal of particle physics.

The possibility of sterile neutrinos—relatively light neutrinos which do not participate in the weak interaction but which could be created through flavor oscillation (see below)—is unaffected by these Z-boson-based measurements, and the existence of such particles is in fact hinted by experimental data from the LSND experiment. However, the currently running MiniBooNE experiment suggested, until recently, that sterile neutrinos are not required to explain the experimental data,[25] although the latest research into this area is on-going and anomalies in the MiniBooNE data may allow for exotic neutrino types, including sterile neutrinos.[26] A recent re-analysis of reference electron spectra data from the Institut Laue-Langevin[27] has also hinted at a fourth, sterile neutrino.[28]

Recently analyzed data from the Wilkinson Microwave Anisotropy Probe of the cosmic background radiation is compatible with either three or four types of neutrinos. It is hoped that the addition of two more years of data from the probe will resolve this uncertainty.[29]

QMRSolar neutrinos originate from the nuclear fusion powering the Sun and other stars. The details of the operation of the Sun are explained by the Standard Solar Model. In short: when four protons fuse to become one helium nucleus, two of them have to convert into neutrons, and each such conversion releases one electron neutrino.

QMRDark matter candidates can be divided into three classes, called cold, warm and hot dark matter.[87] These categories do not correspond to an actual temperature, but instead refer to how fast the particles were moving, thus how far they moved due to random motions in the early universe, before they slowed due to the expansion of the universe – this is an important distance called the "free streaming length". Primordial density fluctuations smaller than this free-streaming length get washed out as particles move from overdense to underdense regions, while fluctuations larger than the free-streaming length are unaffected; therefore this free-streaming length sets a minimum scale for structure formation.

Cold dark matter – objects with a free-streaming length much smaller than a protogalaxy.[88]

Warm dark matter – particles with a free-streaming length similar to a protogalaxy.

Hot dark matter – particles with a free-streaming length much larger than a protogalaxy.[89]

Though a fourth category had been considered early on, called mixed dark matter, it was quickly eliminated (from the 1990s) since the discovery of dark energy.

QMRfour currently hypothesized contributors to the energy density of the universe are curvature, matter, radiation and dark energy.[

QMRIn the Standard Model, the Higgs field is a scalar tachyonic field – 'scalar' meaning it does not transform under Lorentz transformations, and 'tachyonic' meaning the field (but not the particle) has imaginary mass and in certain configurations must undergo symmetry breaking. It consists of four components, two neutral ones and two charged component fields. Both of the charged components and one of the neutral fields are Goldstone bosons, which act as the longitudinal third-polarization components of the massive W+, W−, and Z bosons. The quantum of the remaining neutral component corresponds to (and is theoretically realised as) the massive Higgs boson,[85] this component can interact with fermions via Yukawa coupling to give them mass, as well

QMRIn the Standard Model, the Higgs field is a four-component scalar field that forms a complex doublet of the weak isospin SU(2) symmetry:

\phi=\frac{1}{\sqrt{2}}

\left(

\begin{array}{c}

\phi^1 + i\phi^2 \\ \phi^0+i\phi^3

\end{array}

\right)\;,

QMROriginally electricity and magnetism were thought of as two separate forces. This view changed, however, with the publication of James Clerk Maxwell's 1873 A Treatise on Electricity and Magnetism in which the interactions of positive and negative charges were shown to be regulated by one force. There are four main effects resulting from these interactions, all of which have been clearly demonstrated by experiments:

Electric charges attract or repel one another with a force inversely proportional to the square of the distance between them: unlike charges attract, like ones repel.

Magnetic poles (or states of polarization at individual points) attract or repel one another in a similar way and always come in pairs: every north pole is yoked to a south pole.

An electric current inside a wire creates a corresponding circular magnetic field outside the wire. Its direction (clockwise or counter-clockwise) depends on the direction of the current in the wire.

A current is induced in a loop of wire when it is moved toward or away from a magnetic field, or a magnet is moved towards or away from it; the direction of current depends on that of the movement.

QMRA Treatise on Electricity and Magnetism is a two-volume treatise on electromagnetism written by James Clerk Maxwell in 1873. Maxwell was revising the Treatise for a second edition when he died in 1879. The revision was completed by William Davidson Niven for publication in 1881. A third edition was prepared by J. J. Thomson for publication in 1892.

Preliminary. On the Measurement of Quantities.

PART I. Electrostatics.

Description of Phenomena.

Elementary Mathematical Theory of Electricity.

On Electrical Work and Energy in a System of Conductors.

General Theorems.

Mechanical Action Between Two Electrical Systems.

Points and Lines of Equilibrium.

Forms of Equipotential Surfaces and Lines of Flow.

Simple Cases of Electrification.

Spherical Harmonics.

Confocal Surfaces of the Second Degree.

Theory of Electric Images.

Conjugate Functions in Two Dimensions.

Electrostatic Instruments.

PART II. Electrokinematics.

The Electric Current.

Conduction and Resistance.

Electromotive Force Between Bodies in Contact.

Electrolysis.

Electrolytic Polarization.

Mathematical Theory of the Distribution of Electric Currents.

Conduction in Three Dimensions.

Resistance and Conductivity in Three Dimensions.

Conduction through Heterogeneous Media.

Conduction in Dielectrics.

Measurement of the Electric Resistance of Conductors.

Electric Resistance of Substances.

PART III Magnetism

Elementary Theory of Magnetism.

Magnetic Force and Magnetic Induction.

Particular Forms of Magnets.

Induced Magnetization.

Magnetic Problems.

Weber's Theory of Magnetic Induction.

Magnetic Measurements.

Terrestrial Magnetism.

Part IV. Electromagnetism.

Electromagnetic Force.

Mutual Action of Electric Currents.

Induction of Electric Currents.

Induction of a Current on Itself.

General Equations of Dynamics.

Application of Dynamics to Electromagnetism.

Electrokinetics.

Exploration of the Field by means of the Secondary Circuit.

General Equations.

Dimensions of Electric Units.

Energy and Stress.

Current-Sheets.

Parallel Currents.

Circular Currents.

Electromagnetic Instruments.

Electromagnetic Observations.

Electrical Measurement of Coefficients of Induction.

Determination of Resistance in Electromagnetic Measure.

Comparison of Electrostatic With Electromagnetic Units.

Electromagnetic Theory of Light.

Magnetic Action on Light.

Electric Theory of Magnetism.

Theories of Action at a distance.

QMRPhase transition or phase change, takes place in a thermodynamic system from one phase or state of matter to another one by heat transfer. Phase change examples are the melting of ice or the boiling of water. The Mason equation explains the growth of a water droplet based on the effects of heat transport on evaporation and condensation.

Types of phase transition occurring in the four fundamental states of matter, include:

Solid - Deposition, freezing and solid to solid transformation.

Gas - Boiling / evaporation, recombination / deionization, and sublimation.

Liquid - Condensation and melting / fusion.

Plasma - Ionization.

QMRIn physics, a state of matter is one of the distinct forms that matter takes on. Four states of matter are observable in everyday life: solid, liquid, gas, and plasma

The four fundamental states

Solid

A crystalline solid: atomic resolution image of strontium titanate. Brighter atoms are Sr and darker ones are Ti.

Main article: Solid

In a solid the particles (ions, atoms or molecules) are closely packed together. The forces between particles are strong so that the particles cannot move freely but can only vibrate. As a result, a solid has a stable, definite shape, and a definite volume. Solids can only change their shape by force, as when broken or cut.

In crystalline solids, the particles (atoms, molecules, or ions) are packed in a regularly ordered, repeating pattern. There are various different crystal structures, and the same substance can have more than one structure (or solid phase). For example, iron has a body-centred cubic structure at temperatures below 912 °C, and a face-centred cubic structure between 912 and 1394 °C. Ice has fifteen known crystal structures, or fifteen solid phases, which exist at various temperatures and pressures.[2]

Glasses and other non-crystalline, amorphous solids without long-range order are not thermal equilibrium ground states; therefore they are described below as nonclassical states of matter.

Solids can be transformed into liquids by melting, and liquids can be transformed into solids by freezing. Solids can also change directly into gases through the process of sublimation, and gases can likewise change directly into solids through deposition.

Liquid

Structure of a classical monatomic liquid. Atoms have many nearest neighbors in contact, yet no long-range order is present.

Main article: Liquid

A liquid is a nearly incompressible fluid that conforms to the shape of its container but retains a (nearly) constant volume independent of pressure. The volume is definite if the temperature and pressure are constant. When a solid is heated above its melting point, it becomes liquid, given that the pressure is higher than the triple point of the substance. Intermolecular (or interatomic or interionic) forces are still important, but the molecules have enough energy to move relative to each other and the structure is mobile. This means that the shape of a liquid is not definite but is determined by its container. The volume is usually greater than that of the corresponding solid, the best known exception being water, H2O. The highest temperature at which a given liquid can exist is its critical temperature.[3]

Gas

The spaces between gas molecules are very big. Gas molecules have very weak or no bonds at all. The molecules in "gas" can move freely and fast.

Main article: Gas

A gas is a compressible fluid. Not only will a gas conform to the shape of its container but it will also expand to fill the container.

In a gas, the molecules have enough kinetic energy so that the effect of intermolecular forces is small (or zero for an ideal gas), and the typical distance between neighboring molecules is much greater than the molecular size. A gas has no definite shape or volume, but occupies the entire container in which it is confined. A liquid may be converted to a gas by heating at constant pressure to the boiling point, or else by reducing the pressure at constant temperature.

At temperatures below its critical temperature, a gas is also called a vapor, and can be liquefied by compression alone without cooling. A vapor can exist in equilibrium with a liquid (or solid), in which case the gas pressure equals the vapor pressure of the liquid (or solid).

A supercritical fluid (SCF) is a gas whose temperature and pressure are above the critical temperature and critical pressure respectively. In this state, the distinction between liquid and gas disappears. A supercritical fluid has the physical properties of a gas, but its high density confers solvent properties in some cases, which leads to useful applications. For example, supercritical carbon dioxide is used to extract caffeine in the manufacture of decaffeinated coffee.[4]

Plasma

In a plasma, electrons are ripped away from their nuclei, forming an electron "sea". This gives it the ability to conduct electricity.

Main article: Plasma (physics)

Like a gas, plasma does not have definite shape or volume. Unlike gases, plasmas are electrically conductive, produce magnetic fields and electric currents, and respond strongly to electromagnetic forces. Positively charged nuclei swim in a "sea" of freely-moving disassociated electrons, similar to the way such charges exist in conductive metal. In fact it is this electron "sea" that allows matter in the plasma state to conduct electricity.

The plasma state is often misunderstood, but it is actually quite common on Earth, and the majority of people observe it on a regular basis without even realizing it. Lightning, electric sparks, fluorescent lights, neon lights, plasma televisions, some types of flame and the stars are all examples of illuminated matter in the plasma state.

A gas is usually converted to a plasma in one of two ways, either from a huge voltage difference between two points, or by exposing it to extremely high temperatures.

Heating matter to high temperatures causes electrons to leave the atoms, resulting in the presence of free electrons. At very high temperatures, such as those present in stars, it is assumed that essentially all electrons are "free", and that a very high-energy plasma is essentially bare nuclei swimming in a sea of electrons.

In 1924, Albert Einstein and Satyendra Nath Bose predicted the "Bose–Einstein condensate" (BEC), sometimes referred to as the fifth state of matter. In a BEC, matter stops behaving as independent particles, and collapses into a single quantum state that can be described with a single, uniform wavefunction.

In the gas phase, the Bose–Einstein condensate remained an unverified theoretical prediction for many years. In 1995, the research groups of Eric Cornell and Carl Wieman, of JILA at the University of Colorado at Boulder, produced the first such condensate experimentally. A Bose–Einstein condensate is "colder" than a solid. It may occur when atoms have very similar (or the same) quantum levels, at temperatures very close to absolute zero (−273.15 °C).

The fifth is always ultra transcendent

QMRAll of the various charges discussed above are conserved by the fact that the charge operator is best understood as the generator of a symmetry that commutes with the Hamiltonian. Thus, the eigenvalues of the various charge operators are conserved.

Absolutely conserved flavour quantum numbers are: (including the baryon number for completeness)

electric charge (Q)

weak isospin (I3)

baryon number (B)

lepton number (L)

In some theories, the individual baryon and lepton number conservation can be violated, if the difference between them (B − L) is conserved (see chiral anomaly). All other flavour quantum numbers are violated by the electroweak interactions. Strong interactions conserve all flavours.

General dynamics company four systems

QMR The visible light given off by hydrogen consists of four different colours, as shown in the picture below. The intensity of the light at different frequencies is also different. By contrast, white light consists of a continuous emission across the whole range of visible frequencies. By the end of the nineteenth century, a simple rule known as Balmer's formula had been found which showed how the frequencies of the different lines were related to each other, though without explaining why this was, or making any prediction about the intensities. The formula also predicted some additional spectral lines in ultraviolet and infrared light which had not been observed at the time. These lines were later observed experimentally, raising confidence in the value of the formula.

The fourth color line, red is a lot farther than the other three, which are closer to each other. The emission spectra of hydrogen was used by Bohr to discover the nature of the atom, and the emission spectra reflected the quadrant pattern.

Starting from only one simple assumption about the rule that the orbits must obey, the Bohr model was able to relate the observed spectral lines in the emission spectrum of hydrogen to previously known constants. In Bohr's model the electron simply wasn't allowed to emit energy continuously and crash into the nucleus: once it was in the closest permitted orbit, it was stable forever. Bohr's model didn't explain why the orbits should be quantised in that way, nor was it able to make accurate predictions for atoms with more than one electron, or to explain why some spectral lines are brighter than others.

Although some of the fundamental assumptions of the Bohr model were soon found to be wrong, the key result that the discrete lines in emission spectra are due to some property of the electrons in atoms being quantised is correct. The way that the electrons actually behave is strikingly different from Bohr's atom, and from what we see in the world of our everyday experience

QMRWithin Schrödinger's picture, each electron has four properties:

An "orbital" designation, indicating whether the particle wave is one that is closer to the nucleus with less energy or one that is farther from the nucleus with more energy;

The "shape" of the orbital, spherical or otherwise;

The "inclination" of the orbital, determining the magnetic moment of the orbital around the z-axis.

The "spin" of the electron.

The collective name for these properties is the quantum state of the electron. The quantum state can be described by giving a number to each of these properties; these are known as the electron's quantum numbers. The quantum state of the electron is described by its wave function. The Pauli exclusion principle demands that no two electrons within an atom may have the same values of all four numbers.

introduction to quantum mechanics wiki

The first property describing the orbital is the principal quantum number, n, which is the same as in Bohr's model. n denotes the energy level of each orbital. The possible values for n are integers:

n = 1, 2, 3\ldots

The next quantum number, the azimuthal quantum number, denoted l, describes the shape of the orbital. The shape is a consequence of the angular momentum of the orbital. The angular momentum represents the resistance of a spinning object to speeding up or slowing down under the influence of external force. The azimuthal quantum number represents the orbital angular momentum of an electron around its nucleus. The possible values for l are integers from 0 to n − 1:

l = 0, 1, \ldots, n-1.

The shape of each orbital has its own letter as well. The first shape is denoted by the letter s (a mnemonic being "sphere"). The next shape is denoted by the letter p and has the form of a dumbbell. The other orbitals have more complicated shapes (see atomic orbital), and are denoted by the letters d, f, and g.

The third quantum number, the magnetic quantum number, describes the magnetic moment of the electron, and is denoted by ml (or simply m). The possible values for ml are integers from −l to l:

m_l = -l, -(l-1), \ldots, 0, 1, \ldots, l.

The magnetic quantum number measures the component of the angular momentum in a particular direction. The choice of direction is arbitrary, conventionally the z-direction is chosen.

The fourth quantum number, the spin quantum number (pertaining to the "orientation" of the electron's spin) is denoted ms, with values +1⁄2 or −1⁄2.

The chemist Linus Pauling wrote, by way of example:

In the case of a helium atom with two electrons in the 1s orbital, the Pauli Exclusion Principle requires that the two electrons differ in the value of one quantum number. Their values of n, l, and ml are the same. Accordingly they must differ in the value of ms, which can have the value of +1⁄2 for one electron and −1⁄2 for the other."[36]

It is the underlying structure and symmetry of atomic orbitals, and the way that electrons fill them, that leads to the organisation of the periodic table. The way the atomic orbitals on different atoms combine to form molecular orbitals determines the structure and strength of chemical bonds between atoms.

dangerous the laser is:

Class 1 is inherently safe, usually because the light is contained in an enclosure, for example in CD players.

Class 2 is safe during normal use; the blink reflex of the eye will prevent damage. Usually up to 1 mW power, for example laser pointers.

Class 3R (formerly IIIa) lasers are usually up to 5 mW and involve a small risk of eye damage within the time of the blink reflex. Staring into such a beam for several seconds is likely to cause damage to a spot on the retina.

Class 3B can cause immediate eye damage upon exposure.

Class 4 lasers can burn skin, and in some cases, even scattered light can cause eye and/or skin damage. Many industrial and scientific lasers are in this class.

The indicated powers are for visible-light, continuous-wave lasers. For pulsed lasers and invisible wavelengths, other power limits apply. People working with class 3B and class 4 lasers can protect their eyes with safety goggles which are designed to absorb light of a particular wavelength.

Infrared lasers with wavelengths longer than about 1.4 micrometers are often referred to as "eye-safe", because the cornea tends to absorb light at these wavelengths, protecting the retina from damage. The label "eye-safe" can be misleading, however, as it applies only to relatively low power continuous wave beams; a high power or Q-switched laser at these wavelengths can burn the cornea, causing severe eye damage, and even moderate power lasers can injure the eye.

QMRThe concept of virtual particles arises in the perturbation theory of quantum field theory, an approximation scheme in which interactions (in essence, forces) between actual particles are calculated in terms of exchanges of virtual particles. Such calculations are often performed using schematic representations known as Feynman diagrams, in which virtual particles appear as internal lines. By expressing the interaction in terms of the exchange of a virtual particle with four-momentum q, where q is given by the difference between the four-momenta of the particles entering and leaving the interaction vertex, both momentum and energy are conserved at the interaction vertices of the Feynman diagram.

lagrangian points

QMRIn physics, spacetime is any mathematical model that combines space and time into a single interwoven continuum. The spacetime of our universe is "usually" interpreted from a Euclidean space perspective, which regards space as consisting of three dimensions, and time as consisting of one dimension, the "fourth dimension". By combining space and time into a single manifold called Minkowski space, physicists have significantly simplified a large number of physical theories, as well as described in a more uniform way the workings of the universe at both the supergalactic and subatomic levels. The first three squares are space. The fourth is time. The fourth is always different than the other three but as Einstein points out they are all one. The unity and division is the nature of the quadrant model pattern.

QMRSpacetime in literature[edit]

Incas regarded space and time as a single concept, referred to as pacha (Quechua: pacha, Aymara: pacha).[3][4] The peoples of the Andes maintain a similar understanding.[5]

The idea of a unified spacetime is stated by Edgar Allan Poe in his essay on cosmology titled Eureka (1848) that "Space and duration are one". In 1895, in his novel The Time Machine, H. G. Wells wrote, "There is no difference between time and any of the three dimensions of space except that our consciousness moves along it", and that "any real body must have extension in four directions: it must have Length, Breadth, Thickness, and Duration".

Marcel Proust, in his novel Swann's Way (published 1913), describes the village church of his childhood's Combray as "a building which occupied, so to speak, four dimensions of space—the name of the fourth being Time".

Mathematical concept[edit]

In Encyclopedie under the term dimension Jean le Rond d'Alembert speculated that duration (time) might be considered a fourth dimension if the idea was not too novel.[6]

Another early venture was by Joseph Louis Lagrange in his Theory of Analytic Functions (1797, 1813). He said, "One may view mechanics as a geometry of four dimensions, and mechanical analysis as an extension of geometric analysis".[7]

The ancient idea of the cosmos gradually was described mathematically with differential equations, differential geometry, and abstract algebra. These mathematical articulations blossomed in the nineteenth century as electrical technology stimulated men like Michael Faraday and James Clerk Maxwell to describe the reciprocal relations of electric and magnetic fields. Daniel Siegel phrased Maxwell's role in relativity as follows:

[...] the idea of the propagation of forces at the velocity of light through the electromagnetic field as described by Maxwell's equations—rather than instantaneously at a distance—formed the necessary basis for relativity theory.[8]

Maxwell used vortex models in his papers on On Physical Lines of Force, but ultimately gave up on any substance but the electromagnetic field. Pierre Duhem wrote:

[Maxwell] was not able to create the theory that he envisaged except by giving up the use of any model, and by extending by means of analogy the abstract system of electrodynamics to displacement currents.[9]

In Siegel's estimation, "this very abstract view of the electromagnetic fields, involving no visualizable picture of what is going on out there in the field, is Maxwell's legacy."[10] Describing the behaviour of electric fields and magnetic fields led Maxwell to view the combination as an electromagnetic field. These fields have a value at every point of spacetime. It is the intermingling of electric and magnetic manifestations, described by Maxwell's equations, that give spacetime its structure. In particular, the rate of motion of an observer determines the electric and magnetic profiles of the electromagnetic field. The propagation of the field is determined by the electromagnetic wave equation, which requires spacetime for description.

Spacetime was described as an affine space with quadratic form in Minkowski space of 1908.[11] In his 1914 textbook The Theory of Relativity, Ludwik Silberstein used biquaternions to represent events in Minkowski space. He also exhibited the Lorentz transformations between observers of differing velocities as biquaternion mappings. Biquaternions were described in 1853 by W. R. Hamilton, so while the physical interpretation was new, the mathematics was well known in English literature, making relativity an instance of applied mathematics.

The first inkling of general relativity in spacetime was articulated by W. K. Clifford. Description of the effect of gravitation on space and time was found to be most easily visualized as a "warp" or stretching in the geometrical fabric of space and time, in a smooth and continuous way that changed smoothly from point-to-point along the spacetime fabric. In 1947 James Jeans provided a concise summary of the development of spacetime theory in his book The Growth of Physical Science.[12]

Basic concepts[edit]

The basic elements of spacetime are events. In any given spacetime, an event is a unique position at a unique time. Because events are spacetime points, an example of an event in classical relativistic physics is (x,y,z,t), the location of an elementary (point-like) particle at a particular time. A spacetime itself can be viewed as the union of all events in the same way that a line is the union of all of its points, formally organized into a manifold, a space which can be described at small scales using coordinate systems.

A spacetime is independent of any observer.[13] However, in describing physical phenomena (which occur at certain moments of time in a given region of space), each observer chooses a convenient metrical coordinate system. Events are specified by four real numbers in any such coordinate system. The trajectories of elementary (point-like) particles through space and time are thus a continuum of events called the world line of the particle. Extended or composite objects (consisting of many elementary particles) are thus a union of many world lines twisted together by virtue of their interactions through spacetime into a "world-braid".

However, in physics, it is common to treat an extended object as a "particle" or "field" with its own unique (e.g., center of mass) position at any given time, so that the world line of a particle or light beam is the path that this particle or beam takes in the spacetime and represents the history of the particle or beam. The world line of the orbit of the Earth (in such a description) is depicted in two spatial dimensions x and y (the plane of the Earth's orbit) and a time dimension orthogonal to x and y. The orbit of the Earth is an ellipse in space alone, but its world line is a helix in spacetime.[14]

The unification of space and time is exemplified by the common practice of selecting a metric (the measure that specifies the interval between two events in spacetime) such that all four dimensions are measured in terms of units of distance: representing an event as (x_0,x_1,x_2,x_3) = (ct,x,y,z) (in the Lorentz metric) or (x_1,x_2,x_3,x_4) = (x,y,z,ict) (in the original Minkowski metric) where c is the speed of light.[15] The metrical descriptions of Minkowski Space and spacelike, lightlike, and timelike intervals given below follow this convention, as do the conventional formulations of the Lorentz transformation.

Spacetime intervals in flat space[edit]

In a Euclidean space, the separation between two points is measured by the distance between the two points. The distance is purely spatial, and is always positive. In spacetime, the displacement four-vector ΔR is given by the space displacement vector Δr and the time difference Δt between the events. The spacetime interval, also called invariant interval, between the two events, s2,[16] is defined as:

s^2 = \Delta r^2 - c^2\Delta t^2 \, (spacetime interval),

where c is the speed of light. The choice of signs for s^2 above follows the space-like convention (−+++).[17] Spacetime intervals may be classified into three distinct types, based on whether the temporal separation (c^2 \Delta t^2) or the spatial separation (\Delta r^2) of the two events is greater: time-like, light-like or space-like.

Certain types of world lines are called geodesics of the spacetime – straight lines in the case of Minkowski space and their closest equivalent in the curved spacetime of general relativity. In the case of purely time-like paths, geodesics are (locally) the paths of greatest separation (spacetime interval) as measured along the path between two events, whereas in Euclidean space and Riemannian manifolds, geodesics are paths of shortest distance between two points.[18][19] The concept of geodesics becomes central in general relativity, since geodesic motion may be thought of as "pure motion" (inertial motion) in spacetime, that is, free from any external influences.

Time-like interval[edit]

\begin{align} \\

c^2\Delta t^2 &> \Delta r^2\\

s^2 &< 0 \\

\end{align}

For two events separated by a time-like interval, enough time passes between them that there could be a cause–effect relationship between the two events. For a particle traveling through space at less than the speed of light, any two events which occur to or by the particle must be separated by a time-like interval. Event pairs with time-like separation define a negative spacetime interval (s^2 < 0) and may be said to occur in each other's future or past. There exists a reference frame such that the two events are observed to occur in the same spatial location, but there is no reference frame in which the two events can occur at the same time.

The measure of a time-like spacetime interval is described by the proper time interval, \Delta\tau:

\Delta\tau = \sqrt{\Delta t^2 - \frac{\Delta r^2}{c^2}} (proper time interval).

The proper time interval would be measured by an observer with a clock traveling between the two events in an inertial reference frame, when the observer's path intersects each event as that event occurs. (The proper time interval defines a real number, since the interior of the square root is positive.)

Mathematics of spacetimes[edit]

For physical reasons, a spacetime continuum is mathematically defined as a four-dimensional, smooth, connected Lorentzian manifold (M,g). This means the smooth Lorentz metric g has signature (3,1). The metric determines the geometry of spacetime, as well as determining the geodesics of particles and light beams. About each point (event) on this manifold, coordinate charts are used to represent observers in reference frames. Usually, Cartesian coordinates (x, y, z, t) are used. Moreover, for simplicity's sake, units of measurement are usually chosen such that the speed of light c is equal to 1.

A reference frame (observer) can be identified with one of these coordinate charts; any such observer can describe any event p. Another reference frame may be identified by a second coordinate chart about p. Two observers (one in each reference frame) may describe the same event p but obtain different descriptions.

Usually, many overlapping coordinate charts are needed to cover a manifold. Given two coordinate charts, one containing p (representing an observer) and another containing q (representing another observer), the intersection of the charts represents the region of spacetime in which both observers can measure physical quantities and hence compare results. The relation between the two sets of measurements is given by a non-singular coordinate transformation on this intersection. The idea of coordinate charts as local observers who can perform measurements in their vicinity also makes good physical sense, as this is how one actually collects physical data—locally.

For example, two observers, one of whom is on Earth, but the other one who is on a fast rocket to Jupiter, may observe a comet crashing into Jupiter (this is the event p). In general, they will disagree about the exact location and timing of this impact, i.e., they will have different 4-tuples (x, y, z, t) (as they are using different coordinate systems). Although their kinematic descriptions will differ, dynamical (physical) laws, such as momentum conservation and the first law of thermodynamics, will still hold. In fact, relativity theory requires more than this in the sense that it stipulates these (and all other physical) laws must take the same form in all coordinate systems. This introduces tensors into relativity, by which all physical quantities are represented.

Geodesics are said to be time-like, null, or space-like if the tangent vector to one point of the geodesic is of this nature. Paths of particles and light beams in spacetime are represented by time-like and null (light-like) geodesics, respectively.

4 by 4 is the quadrant model

Spacetime in special relativity[edit]

Main article: Minkowski space

The geometry of spacetime in special relativity is described by the Minkowski metric on R4. This spacetime is called Minkowski space. The Minkowski metric is usually denoted by \eta and can be written as a four-by-four matrix:

\eta_{ab} \, = \operatorname{diag}(1, -1, -1, -1)

where the Landau–Lifshitz time-like convention is being used. A basic assumption of relativity is that coordinate transformations must leave spacetime intervals invariant. Intervals are invariant under Lorentz transformations. This invariance property leads to the use of four-vectors (and other tensors) in describing physics.

Strictly speaking, one can also consider events in Newtonian physics as a single spacetime. This is Galilean–Newtonian relativity, and the coordinate systems are related by Galilean transformations. However, since these preserve spatial and temporal distances independently, such a spacetime can be decomposed into spatial coordinates plus temporal coordinates, which is not possible in the general case.

QMRIn mathematics, a matrix group is a group G consisting of invertible matrices over some field K, usually fixed in advance, with operations of matrix multiplication and inversion. More generally, one can consider n × n matrices over a commutative ring R. (The size of the matrices is restricted to be finite, as any group can be represented as a group of infinite matrices over any field.) A linear group is an abstract group that is isomorphic to a matrix group over a field K, in other words, admitting a faithful, finite-dimensional representation over K.

Any finite group is linear, because it can be realized by permutation matrices using Cayley's theorem. Among infinite groups, linear groups form an interesting and tractable class. Examples of groups that are not linear include all "sufficiently large" groups; for example, the infinite symmetric group of permutations of an infinite set.

They are popular in quantum mechanics. Matrices are quadrant grids

QMRIn physics, the world line of an object is the path of that object in 4-dimensional spacetime, tracing the history of its location in space at each instant in time. The concept of "world line" is distinguished from the concept of "orbit" or "trajectory" (such as an orbit in space or a trajectory of a truck on a road map) by the time dimension, and typically encompasses a large area of spacetime wherein perceptually straight paths are recalculated to show their (relatively) more absolute position states — to reveal the nature of special relativity or gravitational interactions. The idea of world lines originates in physics and was pioneered by Hermann Minkowski. The term is now most often used in relativity theories (i.e., special relativity and general relativity).

World lines as a tool to describe events[edit]

World line, worldsheet, and world volume, as they are derived from particles, strings, and branes.

A one-dimensional line or curve can be represented by the coordinates as a function of one parameter. Each value of the parameter corresponds to a point in spacetime and varying the parameter traces out a line. So in mathematical terms a curve is defined by four coordinate functions x^a(\tau),\; a=0,1,2,3 (where x^{0} usually denotes the time coordinate) depending on one parameter \tau. A coordinate grid in spacetime is the set of curves one obtains if three out of four coordinate functions are set to a constant.

Sometimes, the term world line is loosely used for any curve in spacetime. This terminology causes confusions. More properly, a world line is a curve in spacetime which traces out the (time) history of a particle, observer or small object. One usually takes the proper time of an object or an observer as the curve parameter \tau along the world line.

For example, the orbit of the Earth in space is approximately a circle, a three-dimensional (closed) curve in space: the Earth returns every year to the same point in space. However, it arrives there at a different (later) time. The world line of the Earth is helical in spacetime (a curve in a four-dimensional space) and does not return to the same point.

Spacetime is the collection of points called events, together with a continuous and smooth coordinate system identifying the events. Each event can be labeled by four numbers: a time coordinate and three space coordinates; thus spacetime is a four-dimensional space. The mathematical term for spacetime is a four-dimensional manifold. The concept may be applied as well to a higher-dimensional space. For easy visualizations of four dimensions, two space coordinates are often suppressed. The event is then represented by a point in a Minkowski diagram, which is a plane usually plotted with the time coordinate, say t, upwards and the space coordinate, say x horizontally. As expressed by F.R. Harvey

A curve M in [spacetime] is called a worldline of a particle if its tangent is future timelike at each point. The arclength parameter is called proper time and usually denoted τ. The length of M is called the proper time of the worldline or particle. If the worldline M is a line segment, then the particle is said to be in free fall.[2]

Tangent vector to a world line, four-velocity[edit]

The four coordinate functions x^a(\tau),\; a=0,1,2,3 defining a world line, are real functions of a real variable \tau and can simply be differentiated in the usual calculus. Without the existence of a metric (this is important to realize) one can speak of the difference between a point p on the curve at the parameter value \tau_0 and a point on the curve a little (parameter \tau_0+\Delta\tau) farther away. In the limit \Delta\tau\rightarrow 0, this difference divided by \Delta\tau defines a vector, the tangent vector of the world line at the point p. It is a four-dimensional vector, defined in the point p. It is associated with the normal 3-dimensional velocity of the object (but it is not the same) and therefore called four-velocity \vec{v}, or in components:

\vec{v} = (v^0,v^1,v^2,v^3) = \left( \frac{dx^0}{d\tau}\;,\frac{dx^1}{d\tau}\;, \frac{dx^2}{d\tau}\;, \frac{dx^3}{d\tau} \right)

where the derivatives are taken at the point p, so at \tau=\tau_0.

All curves through point p have a tangent vector, not only world lines. The sum of two vectors is again a tangent vector to some other curve and the same holds for multiplying by a scalar. Therefore, all tangent vectors in a point p span a linear space, called the tangent space at point p. For example, taking a 2-dimensional space, like the (curved) surface of the Earth, its tangent space at a specific point would be the flat approximation of the curved space.

QMRThe Quincunx of Time is a short science fiction novel by James Blish. It is an extended version of a short story entitled "Beep", published by Galaxy Science Fiction magazine in 1954. The novel form was first published in 1973.

I described that a quincunx is a cross made of five dots

Characters in "The Quincunx of Time"[edit]

Robin Weinbaum is Head of Security on Earth.

Thor Wald is a physicist, mathematician, and inventor of the Dirac communicator.

Dana Lje is a journalist and media star of Dutch and Indonesian ancestry, who recently exposed a major lapse of Security in the so-called "Erskine affair". She contacts Earth Security after receiving information from Interstellar Information Ltd..

"J. Shelby Stevens" is a mysterious figure, owner and sole proprietor of Interstellar Information, Ltd.. He has access to information that he could only have received via the top-secret Dirac device, but not only is this device supposed to be a secret, his communications using it cannot be picked up by the authorities, an apparent impossibility. When he makes his first and last appearance in the plot, he seems to be an old man. But is he ?

geodesic quadrants

QMRIn deriving the geodesic equation from the equivalence principle, it was assumed that particles in a local inertial coordinate system are not accelerating. However, in real life, the particles may be charged, and therefore may be accelerating locally in accordance with the Lorentz force. That is:

{d^2 X^\mu \over ds^2} = {q \over m} {F^{\mu \beta}} {d X^\alpha \over ds}{\eta_{\alpha \beta}}.

with

{\eta_{\alpha \beta}}{d X^\alpha \over ds}{d X^\beta \over ds}=-1.

The Minkowski tensor \eta_{\alpha \beta} is given by:

\eta_{\alpha \beta} = \begin{pmatrix}-1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&1\end{pmatrix}

These last three equations can be used as the starting point for the derivation of an equation of motion in General Relativity, instead of assuming that acceleration is zero in free fall.[2] Because the Minkowski tensor is involved here, it becomes necessary to introduce something called the metric tensor in General Relativity. The metric tensor g is symmetric, and locally reduces to the Minkowski tensor in free fall. The resulting equation of motion is as follows:

{d^2 x^\mu \over ds^2} =- \Gamma^\mu {}_{\alpha \beta}{d x^\alpha \over ds}{d x^\beta \over ds}\ +{q \over m} {F^{\mu \beta}} {d x^\alpha \over ds}{g_{\alpha \beta}}.

with

{g_{\alpha \beta}}{d x^\alpha \over ds}{d x^\beta \over ds}=-1.

This last equation signifies that the particle is moving along a timelike geodesic; massless particles like the photon instead follow null geodesics (replace −1 with zero on the right-hand side of the last equation). It is important that the last two equations are consistent with each other, when the latter is differentiated with respect to proper time, and the following formula for the Christoffel symbols ensures that consistency:

\Gamma^{\lambda}{}_{\alpha\beta}=\frac{1}{2}g^{\lambda \tau} \left(\frac{\partial g_{\tau\alpha}}{\partial x^\beta} + \frac{\partial g_{\tau\beta}}{\partial x^{\alpha}} - \frac{\partial g_{\alpha\beta}}{\partial x^{\tau}} \right)

minkowski space is four by four

QMRThe Einstein Cross or Q2237+030 or QSO 2237+0305 is a gravitationally lensed quasar that sits directly behind ZW 2237+030, Huchra's Lens. Four images of the same distant quasar appear around a foreground galaxy due to strong gravitational lensing.[1][2]

The quasar's redshift indicated that it is located about 8 billion light years from Earth, while the lensing galaxy is at a distance of 400 million light years. The apparent dimensions of the entire foreground galaxy are 0.87x0.34 arcminutes[citation needed], while the apparent dimension of the cross in its centre accounts for only 1.6x1.6 arcseconds.

The Einstein Cross can be found in Pegasus at 22h 40m 30.3s, +3° 21′ 31″.

Amateur astronomers are able to see some of the cross using telescopes but it requires extremely dark skies and telescope mirrors with diameters of 18 inches (46 cm) or greater.[3]

The individual images are labelled A through D (i.e. QSO 2237+0305 A), the lensing galaxy is sometimes referred to as QSO 2237+0305 G

Einstein field equations wiki

4 by 4 is a quadrant model

The Einstein field equations (EFE) may be written in the form:[1]

R_{\mu \nu} - \tfrac{1}{2}R \, g_{\mu \nu} + \Lambda g_{\mu \nu}= \frac{8 \pi G }{c^4} T_{\mu \nu}

where Rμν is the Ricci curvature tensor, R is the scalar curvature, gμν is the metric tensor, Λ is the cosmological constant, G is Newton's gravitational constant, c is the speed of light in vacuum, and Tμν is the stress–energy tensor.

The EFE is a tensor equation relating a set of symmetric 4×4 tensors. Each tensor has 10 independent components. The four Bianchi identities reduce the number of independent equations from 10 to 6, leaving the metric with four gauge fixing degrees of freedom, which correspond to the freedom to choose a coordinate system.

Although the Einstein field equations were initially formulated in the context of a four-dimensional theory, some theorists have explored their consequences in n dimensions. The equations in contexts outside of general relativity are still referred to as the Einstein field equations. The vacuum field equations (obtained when T is identically zero) define Einstein manifolds.

QMR Cartans equivalence method wiki

Cartans equivalence method has four steps

Cartan's equivalence method

From Wikipedia, the free encyclopedia

In mathematics, Cartan's equivalence method is a technique in differential geometry for determining whether two geometrical structures are the same up to a diffeomorphism. For example, if M and N are two Riemannian manifolds with metrics g and h, respectively, when is there a diffeomorphism

\phi:M\rightarrow N

such that

\phi^*h=g?

Although the answer to this particular question was known in dimension 2 to Gauss and in higher dimensions to Christoffel and perhaps Riemann as well, Élie Cartan and his intellectual heirs developed a technique for answering similar questions for radically different geometric structures. (For example see the Cartan–Karlhede algorithm.)

Cartan successfully applied his equivalence method to many such structures, including projective structures, CR structures, and complex structures, as well as ostensibly non-geometrical structures such as the equivalence of Lagrangians and ordinary differential equations. (His techniques were later developed more fully by many others, such as D. C. Spencer and Shiing-Shen Chern.)

The equivalence method is an essentially algorithmic procedure for determining when two geometric structures are identical. For Cartan, the primary geometrical information was expressed in a coframe or collection of coframes on a differentiable manifold. See method of moving frames.

Overview of Cartan's method

Specifically, suppose that M and N are a pair of manifolds each carrying a G-structure for a structure group G. This amounts to giving a special class of coframes on M and N. Cartan's method addresses the question of whether there exists a local diffeomorphism φ:M→N under which the G-structure on N pulls back to the given G-structure on M. An equivalence problem has been "solved" if one can give a complete set of structural invariants for the G-structure: meaning that such a diffeomorphism exists if and only if all of the structural invariants agree in a suitably defined sense.

Explicitly, local systems of one-forms θi and γi are given on M and N, respectively, which span the respective cotangent bundles (i.e., are coframes). The question is whether there is a local diffeomorphism φ:M→N such that the pullback of the coframe on N satisfies

\phi^*\gamma^i(y)=g^i_j(x)\theta^j(x),\ (g^i_j)\in G (1)

where the coefficient g is a function on M taking values in the Lie group G. For example, if M and N are Riemannian manifolds, then G=O(n) is the orthogonal group and θi and γi are orthonormal coframes of M and N respectively. The question of whether two Riemannian manifolds are isometric is then a question of whether there exists a diffeomorphism φ satisfying (1).

The first step in the Cartan method is to express the pullback relation (1) in as invariant a way as possible through the use of a "prolongation". The most economical way to do this is to use a G-subbundle PM of the principal bundle of linear coframes LM, although this approach can lead to unnecessary complications when performing actual calculations. In particular, later on this article uses a different approach. But for the purposes of an overview, it is convenient to stick with the principal bundle viewpoint.

The second step is to use the diffeomorphism invariance of the exterior derivative to try to isolate any other higher-order invariants of the G-structure. Basically one obtains a connection in the principal bundle PM, with some torsion. The components of the connection and of the torsion are regarded as invariants of the problem.

The third step is that if the remaining torsion coefficients are not constant in the fibres of the principal bundle PM, it is often possible (although sometimes difficult), to normalize them by setting them equal to a convenient constant value and solving these normalization equations, thereby reducing the effective dimension of the Lie group G. If this occurs, one goes back to step one, now having a Lie group of one lower dimension to work with.

The fourth step[edit]

The main purpose of the first three steps was to reduce the structure group itself as much as possible. Suppose that the equivalence problem has been through the loop enough times that no further reduction is possible. At this point, there are various possible directions in which the equivalence method leads. For most equivalence problems, there are only four cases: complete reduction, involution, prolongation, and degeneracy.

Complete reduction. Here the structure group has been reduced completely to the trivial group. The problem can now be handled by methods such as the Frobenius theorem. In other words, the algorithm has successfully terminated.

On the other hand, it is possible that the torsion coefficients are constant on the fibres of PM. Equivalently, they no longer depend on the Lie group G because there is nothing left to normalize, although there may still be some torsion. The three remaining cases assume this.

Involution. The equivalence problem is said to be involutive (or in involution) if it passes Cartan's test. This is essentially a rank condition on the connection obtained in the first three steps of the procedure. The Cartan test generalizes the Frobenius theorem on the solubility of first-order linear systems of partial differential equations. If the coframes on M and N (obtained by a thorough application of the first three steps of the algorithm) agree and satisfy the Cartan test, then the two G-structures are equivalent. (Actually, to the best of the author's knowledge, the coframes must be real analytic in order for this to hold, because the Cartan-Kähler theorem requires analyticity.)

Prolongation. This is the most intricate case. In fact there are two sub-cases. In the first sub-case, all of the torsion can be uniquely absorbed into the connection form. (Riemannian manifolds are an example, since the Levi-Civita connection absorbs all of the torsion). The connection coefficients and their invariant derivatives form a complete set of invariants of the structure, and the equivalence problem is solved. In the second subcase, however, it is either impossible to absorb all of the torsion, or there is some ambiguity (as is often the case in Gaussian elimination, for example). Here, just as in Gaussian elimination, there are additional parameters which appear in attempting to absorb the torsion. These parameters themselves turn out to be additional invariants of the problem, so the structure group G must be prolonged into a subgroup of a jet group. Once this is done, one obtains a new coframe on the prolonged space and has to return to the first step of the equivalence method. (See also prolongation of G-structures.)

Degeneracy. Because of a non-uniformity of some rank condition, the equivalence method is unsuccessful in handling this particular equivalence problem. For example, consider the equivalence problem of mapping a manifold M with a single one-form θ to another manifold with a single one-form γ such that φ*γ=θ. The zeros of these one forms, as well as the rank of their exterior derivatives at each point need to be taken into account. The equivalence method can handle such problems if all of the ranks are uniform, but it is not always suitable if the rank changes. Of course, depending on the particular application, a great deal of information can still be obtained with the equivalence method.

QMR

The fourth derivative of position is jerk. Jerk is a lot different than the other derivatives of position velocity and acceleration. The fourth square is always different.

In physics, jerk, also known as jolt, surge, or lurch, is the rate of change of acceleration; that is, the derivative of acceleration with respect to time, and as such the second derivative of velocity, or the third derivative of position. Jerk is defined by any of the following equivalent expressions:

\vec j(t)=\frac {\mathrm{d} \vec a(t)} {\mathrm{d}t}=\dot {\vec a}(t)=\frac {\mathrm{d}^2 \vec v(t)} {\mathrm{d}t^2}=\ddot{\vec v}(t)=\frac {\mathrm{d}^3 \vec r(t)} {\mathrm{d}t^3}=\overset{...}{\vec r}(t)

where

\vec a is acceleration,

\vec v is velocity,

\vec r is position,

\mathit{t} is time.

Jerk is a vector, and there is no generally used term to describe its scalar magnitude (more precisely, its norm, e.g. "speed" as the norm of the velocity vector).

According to the result of dimensional analysis of jerk, [length/time3], the SI units are m/s3 (or m·s−3). There is no universal agreement on the symbol for jerk, but j is commonly used. Newton's notation for the time derivative (\dot{ a},\;\ddot{ v},\;\overset{...}{r}) is also applied.

The fourth derivative of position, equivalent to the first derivative of jerk, is jounce.

Because of involving third derivatives, in mathematics differential equations of the form

J\left(\overset{...}{x},\ddot{x},\dot {x},x\right)=0

are called jerk equations. It has been shown that a jerk equation, which is equivalent to a system of three first order, ordinary, non-linear differential equations, is in a certain sense the minimal setting for solutions showing chaotic behaviour. This motivates mathematical interest in jerk systems. Systems involving a fourth or higher derivative are accordingly called hyperjerk systems.

Systems involving a fourth or higher derivative are accordingly called hyperjerk systems.

Animation showing a four-position external Geneva drive in operation. It looks like a quadrant

QMRBackground[edit]

The name derives from the device's earliest application in mechanical watches; Geneva in Switzerland being an important center of watchmaking. The Geneva drive is also commonly called a Maltese cross mechanism due to the visual resemblance when the driven wheel has four spokes. Since they can be made small and are able to withstand substantial mechanical stress, these mechanisms are frequently used in watches.

In the most common arrangement, the driven wheel has four slots and thus advances by one step of 90 degrees for each rotation of the drive wheel. If the driven wheel has n slots, it advances by 360°/n per full rotation of the drive wheel.

Geneva drive.svg

Because the mechanism needs to be well lubricated, it is often enclosed in an oil capsule.

Kinematics[edit]

Motion curves for one turn of the drive wheel, from top to bottom: angular position θ, angular velocity ω, angular acceleration α and angular jerk ja.

The figure shows the motions curves for an external four-slots Geneva drive, in arbitrary units. There is a discontinuity in the acceleration when the drive pin enters and leaves the slot. This generates an "infinite" peak of jerk (Dirac peak), and therefore vibrations.

dwell mechanism wiki

QMRCam-follower dwell mechanisms are used in pairs in sewing machines to operate the four motion feed dogs, with one cam moving the dog up and down, and the other cam moving the dog forwards and backwards.

QMRFeed dogs are the critical component of a "drop feed" sewing machine. A set of feed dogs typically resembles two or three short, thin metal bars, crosscut with diagonal teeth, which move back and forth in slots in a sewing machine's needle plate. Their purpose is to pull ("feed") the fabric through the machine, in discrete steps, in-between stitches.

Closeup of feed dogs showing teeth rising up through the needle plate

This arrangement is called "drop feed" in reference to the way the dogs drop below the needle plate when returning for the next stroke. Allen B. Wilson invented it during the time period 1850 to 1854,[1][2] while also developing the rotary hook. Wilson called it a "four-motion feed", in reference to the four movements the dogs perform during one full stitch: up into the fabric, back to pull the fabric along to the next stitch, down out of the fabric and below the needle plate, and then forward to return to the starting position.

Virtually all drop-feed sewing machines can vary their stitch length; this is typically controlled by a lever or dial on the front of the machine. They are usually also capable of pulling the fabric backwards, to form a backstitch.

torque tester wiki

QMRA torque transducer, similar to a load cell, is an electronic device used to convert torque into an electrical signal. This conversion is indirect and happens in two stages. Through a mechanical arrangement, the torque being sensed deforms a strain gauge. The strain gauge converts the deformation (strain) to electrical signals. A torque transducer usually consists of four strain gauges in a Wheatstone bridge configuration. Torque transducers of one or two strain gauges are also available. The electrical signal output is typically in the order of a few millivolts and usually requires amplification by an instrumentation amplifier before it can be used. The output of the transducer is plugged into an algorithm to calculate the force applied to the transducer. There are several styles available for torque transducers. Rotary, stationary (reaction), and inline are used for different calibration and audit purposes.

QMRCross-sectional view of an M4 anti-tank mine (circa 1945) showing the steel belleville spring in the fuze mechanism It is aq quadrant

QMRsillcock key sillcock key, loose key, tap key A key used to open or close sillcock valves. Many are designed the same way as a spider-type lug wrench, with four common sizes (one on each end) built into one portable tool. household

It looks like a quadrant

QMRAlternatives to general relativity are physical theories that attempt to describe the phenomena of gravitation in competition to Einstein's theory of general relativity.

There have been many different attempts at constructing an ideal theory of gravity. These attempts can be split into four broad categories:

Straightforward alternatives to general relativity (GR), such as the Cartan, Brans–Dicke and Rosen bimetric theories.

Those that attempt to construct a quantized gravity theory such as loop quantum gravity.

Those that attempt to unify gravity and other forces such as Kaluza–Klein.

Those that attempt to do several at once, such as M-theory.

The fourth is always different yet combines the previous three

Classical tests[edit]

Main article: Tests of general relativity

There are three "classical" tests (dating back to the 1910s or earlier) of the ability of gravity theories to handle relativistic effects; they are:

gravitational redshift

gravitational lensing (generally tested around the Sun)

anomalous perihelion advance of the planets (see Tests of General Relativity)

Each theory should reproduce the observed results in these areas, which have to date always aligned with the predictions of general relativity.

In 1964, Irwin I. Shapiro found a fourth test, called the Shapiro delay. It is usually regarded as a "classical" test as well.

the fourth is always different

alternatives to general relativity wiki

electromagnetic tensor wiki

QMRIn differential geometry, the four-gradient is the four-vector analogue of the gradient from Gibbs-Heaviside vector calculus.

Definition[edit]

The covariant components compactly written in index notation are:[1]

\dfrac{\partial}{\partial x^\alpha} = \left(\frac{1}{c}\frac{\partial}{\partial t}, \vec{\nabla}\right) = \left(\frac{\partial_t}{c}, \vec{\nabla}\right) = \partial_\alpha = {}_{,\alpha}

The comma in the last part above {}_{,\alpha} implies the partial differentiation with respect to x^\alpha. This is not the same as a semi-colon, used for the covariant derivative.

The contravariant components are:[2]

\mathbf{\partial} = \partial^\alpha \ = g^{\alpha \beta} \partial_\beta = \left(\frac{1}{c} \frac{\partial}{\partial t}, -\vec{\nabla} \right)= \left(\frac{\partial_t}{c}, -\vec{\nabla}\right) = \left(\frac{\partial_t}{c}, -\partial_x,-\partial_y,-\partial_z\right)

where gαβ is the metric tensor, which here has been chosen for flat spacetime with the metric signature (+,−,−,−).

Alternative symbols to \partial_\alpha are \Box and D.

Usage[edit]

The 4-Gradient is used in a number of different ways in Special Relativity:

As a 4-Divergence[edit]

The 4-Divergence of the 4-Position X^\mu gives the dimensionality of spacetime:

\mathbf{\partial} \cdot \mathbf{X} = \partial^\mu \cdot X^\nu = \partial^\mu \eta_{\mu\nu} X^\nu = (\frac{\partial_t}{c},-\vec{\nabla})\cdot (ct,\vec{x}) = \frac{\partial_t}{c}(ct) + \vec{\nabla}\cdot \vec{x} = (\partial_t t) + (\partial_x x+\partial_y y+\partial_z z) = (1) + (3) = 4

The 4-Divergence of the 4-CurrentDensity J^\mu = \rho_o U^\mu gives a conservation law - the conservation of charge:

\mathbf{\partial} \cdot \mathbf{J} = \partial^\mu \cdot J^\nu = \partial^\mu \eta_{\mu\nu} J^\nu = \partial_\nu J^\nu = (\frac{\partial_t}{c},-\vec{\nabla})\cdot (\rho c,\vec{j}) = \frac{\partial_t}{c}(\rho c) + \vec{\nabla}\cdot \vec{j} =\partial_t \rho + \vec{\nabla}\cdot \vec{j} = 0

This means that the time rate of change of the charge density must equal the negative spatial divergence of the current density \partial_t \rho = -\vec{\nabla}\cdot \vec{j}. In other words, the charge inside a box cannot just change arbitrarily, it must enter and leave the box via a current. This is a continuity equation.

As a Jacobian Matrix for the SR Metric Tensor[edit]

The 4-Gradient \partial^\mu acting on the 4-Position X^\nu gives the SR Minkowski_space metric \eta^{\mu\nu}. :

\mathbf{\partial} [\mathbf{X}] = \partial^\mu[X^\nu] = X^{\nu_,\mu} = (\frac{\partial_t}{c},-\vec{\nabla})[(ct,\vec{x})] = (\frac{\partial_t}{c},-\partial_x,-\partial_y,-\partial_z)[(ct,x,y,z)],

= \begin{bmatrix}\frac{\partial_t}{c} ct & \frac{\partial_t}{c} x & \frac{\partial_t}{c} y & \frac{\partial_t}{c} z \\ -\partial_x ct & -\partial_x x & -\partial_x y & -\partial_x z \\ -\partial_y ct & -\partial_y x & -\partial_y y & -\partial_y z \\ -\partial_z ct & -\partial_z x & -\partial_z y & -\partial_z z\end{bmatrix} = \begin{bmatrix}1 & 0 & 0 & 0\\0 & -1 & 0 & 0\\0 & 0 & -1 & 0\\0 & 0 & 0 & -1\end{bmatrix} = Diag[1,-1,-1,-1]  

\mathbf{\partial} [\mathbf{X}] = \eta^{\mu\nu}

As part of the total proper time derivative[edit]

The Scalar Product of 4-Velocity U^\mu with the 4-Gradient gives the total derivative with respect to proper time \frac{d}{d\tau}:

\mathbf{U} \cdot \mathbf{\partial} =U^\mu \cdot \partial^\nu = \gamma(c,\vec{u}) \cdot (\frac{\partial_t}{c},-\vec{\nabla}) = \gamma (c \frac{\partial_t}{c} + \vec{u} \cdot \vec{\nabla} frown emoticon \gamma (\partial_t + \frac{dx}{dt} \partial_x + \frac{dy}{dt} \partial_y + \frac{dz}{dt} \partial_z) = \gamma \frac{d}{dt} = \frac{d}{d\tau}

So, for example, the 4-Velocity U^\mu is the proper-time derivative of the 4-Position X^\mu:

\frac{d}{d\tau} \mathbf{X} = (\mathbf{U} \cdot \mathbf{\partial})\mathbf{X} = \mathbf{U} \cdot \mathbf{\partial}[\mathbf{X}] = U^\alpha \cdot \eta^{\mu\nu} = U^\alpha \eta_{\alpha \nu} \eta^{\mu\nu} = U^\alpha \delta_\alpha^\mu = U^\mu = \mathbf{U}

or

\frac{d}{d\tau} \mathbf{X} = \gamma\frac{d}{dt} \mathbf{X} = \gamma\frac{d}{dt} (ct,\vec{x}) = \gamma (\frac{d}{dt}ct,\frac{d}{dt}\vec{x}) = \gamma (c,\vec{u}) = \mathbf{U}

Another example, the 4-Acceleration A^\mu is the proper-time derivative of the 4-Velocity U^\mu:

\frac{d}{d\tau} \mathbf{U} = (\mathbf{U} \cdot \mathbf{\partial})\mathbf{U} = \mathbf{U} \cdot \mathbf{\partial}[\mathbf{U}] = U^\alpha \eta_{\alpha\mu}\partial^\mu[U^\nu]

= U^\alpha \eta_{\alpha\mu}\begin{bmatrix} \frac{\partial_t}{c} \gamma c & \frac{\partial_t}{c} \gamma \vec{u} \\ -\vec{\nabla}\gamma c & -\vec{\nabla}\gamma \vec{u} \end{bmatrix} = U^\alpha \begin{bmatrix}\ \frac{\partial_t}{c} \gamma c & 0 \\ 0 & \vec{\nabla}\gamma \vec{u} \end{bmatrix}

= \gamma (c \frac{\partial_t}{c} \gamma c , \vec{u} \cdot \nabla\gamma \vec{u} frown emoticon\gamma (c \partial_t \gamma, \frac{d}{dt}[\gamma \vec{u}] ) = \gamma (c \dot{\gamma}, \dot{\gamma} \vec{u} + \gamma \dot{\vec{u}} frown emoticon \mathbf{A}

or

\frac{d}{d\tau} \mathbf{U} =\gamma \frac{d}{dt} (\gamma c,\gamma \vec{u}) =\gamma (\frac{d}{dt}[\gamma c],\frac{d}{dt}[\gamma \vec{u}]) = \gamma (c \dot{\gamma}, \dot{\gamma} \vec{u} + \gamma \dot{\vec{u}} ) = \mathbf{A}

As

As a way to define the 4-WaveVector[edit]

The 4-WaveVector is the 4-Gradient of the negative phase, or the negative 4-Gradinet of the phase \Phi of a wave in Minkowski Space

4-WaveVector \mathbf{K} = \left(\frac{\omega}{c}, \vec{\mathbf{k}}\right) = \mathbf{\partial} [-\Phi]= -\mathbf{\partial} [\Phi]

This is mathematically equivalent to the definition of the phase of a wave as:

-\Phi = \mathbf{K} \cdot \mathbf{X} = \omega t - \vec{\mathbf{k}} \cdot \vec{\mathbf{x}}

where:

4-Position \mathbf{X} = (ct, \vec{\mathbf{x}})

As the d'Alembertian Operator[edit]

The square of \mathbf{\partial} is the Four-Laplacian, which is called the d'Alembert operator:

\mathbf{\partial} \cdot \mathbf{\partial} = \partial^\mu \cdot \partial^\nu = \eta_{\mu\nu} \partial^\mu \partial^\nu = \partial_\nu \partial^\nu = \frac{1}{c^2}\frac{\partial^2}{\partial t^2} - \nabla^2 = \left(\frac{\partial_t}{c}\right)^2 - \nabla^2.

As it is the dot product of two four-vectors, the d'Alembertian is a Lorentz invariant scalar.

Occasionally, in analogy with the 3-dimensional notation, the symbols \Box and \Box^2 are used for the 4-Gradient and d'Alembertian respectively. More commonly however, the symbol \Box is reserved for the d'Alembertian.

Some examples of the 4-Gradient as used in the d'Alembertian follow:

In the Klein-Gordon relativistic quantum wave equation for spin-0 particles (ex. Higgs_boson):

[(\mathbf{\partial} \cdot \mathbf{\partial}) + \left(\frac {m_o c}{\hbar}\right)^2]\psi = [\left(\frac{\partial_t^2}{c^2} - \vec{\nabla}^2\right) + \left(\frac {m_o c}{\hbar}\right)^2] \psi = 0

In the wave equation for the electromagnetic field:

(\mathbf{\partial} \cdot \mathbf{\partial}) A^{\alpha} = 0 {in vacuum}

(\mathbf{\partial} \cdot \mathbf{\partial}) A^\alpha = \mu_0 J^\alpha {with a 4-CurrentDensity source}

where:

4-VectorPotential A^{\alpha} = \left(\frac{\phi}{c},\vec{a}\right) is an electromagnetic potential

4-CurrentDensity J^{\alpha} = (\rho c,\vec{j}) is an electromagnetic current density

In the 4-dimensional version of Green's_function:

(\mathbf{\partial} \cdot \mathbf{\partial}) G(x-x') = \delta^4(x-x')

As a component of the Schrödinger relations in Quantum Mechanics[edit]

The 4-Gradient is connected with Quantum Mechanics. The relation between the 4-Momentum P and the 4-Gradient \partial give the Schrödinger QM relations.

\mathbf{P} = \left(\frac{E}{c},\vec{p}\right) = i\hbar \mathbf{\partial} = i\hbar \left(\frac{\partial_t}{c},-\vec{\nabla}\right)

The temporal components give: E = i\hbar \partial_t

The spatial components give: \vec{p} = -i\hbar \vec{\nabla}

Derivation[edit]

In 3 dimensions, the gradient operator maps a scalar field to a vector field such that the line integral between any two points in the vector field is equal to the difference between the scalar field at these two points. Based on this, it may appear incorrectly that the natural extension of the gradient to four dimensions should be:

\partial^\alpha \ = \left( \frac{\partial}{\partial t}, \vec{\nabla} \right) incorrect

However, a line integral involves the application of the vector dot product, and when this is extended to four-dimensional space-time, a change of sign is introduced to either the spatial co-ordinates or the time co-ordinate depending on the convention used. This is due to the non-Euclidean nature of space-time. In this article, we place a negative sign on the spatial coordinates (the time-positive Metric convention \eta^{\mu\nu}=Diag[1,-1,-1,-1]). The factor of (1/c) is to keep the correct unit dimensionality {1/[length]} for all components of the 4-vector and the (−1) is to keep the 4-gradient Lorentz covariant. Adding these two corrections to the above expression gives the correct definition of four-gradient:

\partial^\alpha \ = \left(\frac{1}{c} \frac{\partial}{\partial t}, -\vec{\nabla} \right) correct

QMRFour-tensor is a frequent abbreviation for a tensor in a four-dimensional spacetime.

Syntax[edit]

General four-tensors are usually written as A^{\mu_1,\mu_2,...,\mu_n}_{\;\nu_1,\nu_2,...,\nu_m}, with the indices taking integral values from 0 to 3. Such a tensor is said to have contravariant rank n and covariant rank m.[1]

Examples[edit]

One of the simplest non-trivial examples of a four-tensor is the four-displacement x^\mu=\left(x^0, x^1, x^2, x^3\right), a four-tensor with contravariant rank 1 and covariant rank 0. Four-tensors of this kind are usually known as four-vectors. Here the component x^0=ct gives the displacement of a body in time (time is multiplied by the speed of light c so that x^0 has dimensions of length). The remaining components of the four-displacement form the spatial displacement vector \mathbf{x}.[1]

Similarly, the four-momentum p^{\mu}=\left(E/c,p_x,p_y,p_z\right) of a body is equivalent to the energy-momentum tensor of said body. The element p^0=E/c represents the momentum of the body as a result of it travelling through time (directly comparable to the internal energy of the body). The elements p^1, p^2 and p^3 correspond to the momentum of the body as a result of it travelling through space, written in vector notation as \mathbf{p}.[1]

The electromagnetic field tensor is an example of a rank two contravariant tensor:[1]

F^{\mu\nu} = \begin{pmatrix}

0 & -E_x/c & -E_y/c & -E_z/c\\

E_x/c & 0 & -B_z & B_y\\

E_y/c & B_z & 0 & -B_x\\

E_z/c & -B_y & B_x & 0

\end{pmatrix}

QMRFour-vector formulation[edit]

Main article: Four-momentum

In the theory of special relativity, physical quantities are expressed in terms of four-vectors that include time as a fourth coordinate along with the three space coordinates. These vectors are generally represented by capital letters, for example R for position. The expression for the four-momentum depends on how the coordinates are expressed. Time may be given in its normal units or multiplied by the speed of light so that all the components of the four-vector have dimensions of length. If the latter scaling is used, an interval of proper time, τ, defined by[19]

c^2d\tau^2 = c^2dt^2-dx^2-dy^2-dz^2\,,

is invariant under Lorentz transformations (in this expression and in what follows the (+ − − −) metric signature has been used, different authors use different conventions). Mathematically this invariance can be ensured in one of two ways: by treating the four-vectors as Euclidean vectors and multiplying time by √−1; or by keeping time a real quantity and embedding the vectors in a Minkowski space.[20] In a Minkowski space, the scalar product of two four-vectors U = (U0,U1,U2,U3) and V = (V0,V1,V2,V3) is defined as

\mathbf{U} \cdot \mathbf{V} = U_0 V_0 - U_1 V_1 - U_2 V_2 - U_3 V_3\,.

In all the coordinate systems, the (contravariant) relativistic four-velocity is defined by

\mathbf{U} \equiv \frac{d \mathbf{R}}{d \tau} = \gamma \frac{d \mathbf{R}}{dt}\,,

and the (contravariant) four-momentum is

\mathbf{P} = m_0\mathbf{U}\,,

where m0 is the invariant mass. If R = (ct,x,y,z) (in Minkowski space), then

\mathbf{P} = \gamma m_0 \left(c,\mathbf{v}\right) = (m c,\mathbf{p})\,.

Using Einstein's mass-energy equivalence, E = mc2, this can be rewritten as

\mathbf{P} = \left(\frac{E}{c}, \mathbf{p}\right)\,.

Thus, conservation of four-momentum is Lorentz-invariant and implies conservation of both mass and energy.

The magnitude of the momentum four-vector is equal to m0c:

||\mathbf{P}||^2 = \mathbf{P}\cdot\mathbf{P} = \gamma^2m_0^2(c^2-v^2) = (m_0c)^2\,,

and is invariant across all reference frames.

The relativistic energy–momentum relationship holds even for massless particles such as photons; by setting m0 = 0 it follows that

E = pc\,.

In a game of relativistic "billiards", if a stationary particle is hit by a moving particle in an elastic collision, the paths formed by the two afterwards will form an acute angle. This is unlike the non-relativistic case where they travel at right angles.[21]

momentum wiki

QMRAncient philosophers as far back as Thales of Miletus c. 550 BCE had inklings of the conservation of some underlying substance of which everything is made. However, there is no particular reason to identify this with what we know today as "mass-energy" (for example, Thales thought it was water). Empedocles (490–430 BCE) wrote that in his universal system, composed of four roots (earth, air, water, fire), "nothing comes to be or perishes";[2] instead, these elements suffer continual rearrangement.

mass in special relativity wiki

QMRThe invariant mass is the ratio of four-momentum to four-velocity:[5]

p^\mu = m v^\mu\,

and is also the ratio of four-acceleration to four-force when the rest mass is constant. The four-dimensional form of Newton's second law is:

F^\mu = m A^\mu.\!

QMRFour-momentum

From Wikipedia, the free encyclopedia

(Redirected from 4-momentum)

In special relativity, four-momentum is the generalization of the classical three-dimensional momentum to four-dimensional spacetime. Momentum is a vector in three dimensions; similarly four-momentum is a four-vector in spacetime. The contravariant four-momentum of a particle with energy E and three-momentum p = (px, py, pz) = mu = γmv, where v is the particles 3-velocity and γ the Lorentz factor, is

p = (p^0 , p^1 , p^2 , p^3 ) = \left({E \over c} , p_x , p_y , p_z\right).

The quantity mv of above is ordinary non-relativistic momentum of the particle and m its rest mass. The four-momentum is useful in relativistic calculations because it is a Lorentz vector. This means that it is easy to keep track of how it transforms under Lorentz transformations.

The above definition applies under the coordinate convention that x0 = ct. Some authors use the convention x0 = t, which yields a modified definition with p0 = E/c2. It is also possible to define covariant four-momentum pμ where the sign of the energy is reversed.

Minkowski norm[edit]

Calculating the Minkowski norm of the four-momentum gives a Lorentz invariant quantity equal (up to factors of the speed of light c) to the square of the particle's proper mass:

p \cdot p = p^\mu p_\mu = \eta_{\mu\nu} p^\mu p^\nu = -{E^2 \over c^2} + |\mathbf p|^2 = -m^2c^2

where we use the convention that

\eta_{\mu\nu} = \left[\begin{smallmatrix}

-1 & 0 & 0 & 0\\

0 & 1 & 0 & 0\\

0 & 0 & 1 & 0\\

0 & 0 & 0 & 1

\end{smallmatrix}\right]

is the metric tensor of special relativity. The fact that the norm is negative reflects that the momentum is a timelike 4-vector for massive particles.

The Minkowski norm is Lorentz invariant, meaning its value is not changed by Lorentz transformations/boosting into different frames of reference. More generally, for any two 4-momenta p and q, the quantity p ⋅ q is invariant.

Relation to four-velocity[edit]

For a massive particle, the four-momentum is given by the particle's invariant mass m multiplied by the particle's four-velocity,

p^\mu = m u^\mu,

where the four-velocity u is

u = (u^0 , u^1 , u^2 , u^3 ) = \gamma (c , v_x , v_y , v_z),

and

\gamma = \frac{1}{\sqrt{1-\left(v/c\right)^2}}

is the Lorentz factor, c is the speed of light.

Derivation[edit]

There are several ways to arrive at the correct expression for 4-momentum. One way is to first define the 4-velocity u = dx/dτ and simply define p = mu, being content that it is a 4-vector with the correct units and correct behavior. Another, more satisfactory, approach is to begin with the principle of least action and use the Lagrangian framework to derive the 4-momentum, including the expression for the energy.[1] One may at once, using the observations detailed below, define 4-momentum from the action S. Given that in general for a closed system with generalized coordinates qi and canonical momenta pi,[2]

p_i = \frac{\partial S}{\partial q_i}, \quad E = -\frac{\partial S}{\partial t},

it is immediate (recalling x0 = ct, x1 = x, x2 = y, x3 = z and x0 = −x0, x1 = x1, x2 = x2, x3 = x3 in the present metric convention) that

p_\mu = -\frac{\partial S}{\partial x^\mu} = \left({E \over c} , -\mathbf p\right)

is a covariant 4-vector with the 3-vector part being the (negative of) canonical momentum.

Conservation of 4-momentum[edit]

As shown above, there are three (not independent, the last two implies the first) conservation laws:

The 4-momentum p (either covariant or contravariant) is conserved.

The total energy E = p0c is conserved.

The 3-momentum p is conserved.

Note that the invariant mass of a system of particles may be more than the sum of the particles' rest masses, since kinetic energy in the system center-of-mass frame and potential energy from forces between the particles contribute to the invariant mass. As an example, two particles with four-momenta (5 GeV/c, 4 GeV/c, 0, 0) and (5 GeV/c, −4 GeV/c, 0, 0) each have (rest) mass 3 GeV/c2 separately, but their total mass (the system mass) is 10 GeV/c2. If these particles were to collide and stick, the mass of the composite object would be 10 GeV/c2.

One practical application from particle physics of the conservation of the invariant mass involves combining the four-momenta pA and pB of two daughter particles produced in the decay of a heavier particle with four-momentum pC to find the mass of the heavier particle. Conservation of four-momentum gives pCμ = pAμ + pBμ, while the mass M of the heavier particle is given by −PC ⋅ PC = M2c2. By measuring the energies and three-momenta of the daughter particles, one can reconstruct the invariant mass of the two-particle system, which must be equal to M. This technique is used, e.g., in experimental searches for Z′ bosons at high-energy particle colliders, where the Z′ boson would show up as a bump in the invariant mass spectrum of electron–positron or muon–antimuon pairs.

If the mass of an object does not change, the Minkowski inner product of its four-momentum and corresponding four-acceleration Aμ is simply zero. The four-acceleration is proportional to the proper time derivative of the four-momentum divided by the particle's mass, so

p^{\mu} A_\mu = \eta_{\mu\nu} p^{\mu} A^\nu = \eta_{\mu\nu} p^\mu \frac{d}{d\tau} \frac{p^{\nu}}{m} = \frac{1}{2m} \frac{d}{d\tau} p \cdot p = \frac{1}{2m} \frac{d}{d\tau} (-m^2c^2) = 0 .

QMRPauli–Lubanski pseudovector

From Wikipedia, the free encyclopedia

For the notation, see Ricci calculus.

Quantum field theory

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In physics, specifically in relativistic quantum mechanics and quantum field theory, the Pauli–Lubanski pseudovector named after Wolfgang Pauli and Józef Lubański[1] is an operator defined from the momentum and angular momentum, used in the quantum-relativistic description of angular momentum.

It describes the spin states of moving particles.[2] It is the generator of the little group of the Lorentz group, that is the maximal subgroup (with four generators) leaving the eigenvalues of the four-momentum vector Pμ invariant.[3]

QMRThe vierbein or tetrad theory much used in theoretical physics is a special case of the application of Cartan connection in four-dimensional manifolds. It applies to metrics of any signature. (See metric tensor.) This section is an approach to tetrads, but written in general terms. In dimensions other than 4, words like triad, pentad, zweibein, fünfbein, elfbein etc. have been used. Vielbein covers all dimensions. (In German, vier stands for four and viel stands for many.)

For a basis-dependent index notation, see tetrad (index notation).

cartan formalism wiki

The Palatini action[edit]

In the tetrad formulation of general relativity, the action, as a functional of the vierbein e and a connection form \omega, with an associated field strength \Omega = D\omega = d\omega + \omega \wedge \omega, over a four-dimensional differentiable manifold M is given by

S\ \stackrel{\mathrm{def}}{=}\ M^2_{pl}\int_M \epsilon_{abcd}( e^{a} \wedge e^{b} \wedge \Omega^{cd}) = M^2_{pl}\int_M d^4x \epsilon^{\mu \nu \rho \sigma} \epsilon_{abcd} e^a_{\mu} e^b_{\nu} R^{cd}_{\rho \sigma}[\omega]

= M^2_{pl}\int |e| d^4 x \frac{1}{2} e^{\mu}_a e^{\nu}_b R^{ab}_{\mu \nu}  

= \frac{c^4}{16 \pi G} \int d^4x \sqrt{-g} R[g]

where \Omega_{\mu \nu} ^{ab} = R_{\mu \nu} ^{ab} is the gauge curvature 2-form, \epsilon_{abcd} is the antisymmetric Levi-Civita symbol, and that |e| = \epsilon^{\mu \nu \rho \sigma} \epsilon_{abcd} e^a_{\mu}e^b_{\nu}e^c_{\rho}e^d_{\sigma} is the determinant of e_{\mu}^a. Here we see that the differential form language leads to an equivalent action to that of the normal Einstein–Hilbert action, using the relations |e| = \sqrt{-g} and R^{\lambda \sigma}_{\mu \nu}= e^{\lambda}_a e^{\sigma}_b R^{ab}_{\mu \nu} . Note that in terms of the Planck mass, we set \hbar = c =1, whereas the last term keeps all the SI unit factors.

Note that in the presence of spinor fields, the Palatini action implies that d\omega is nonzero. So there's a non-zero torsion, i.e. that \hat{\omega}^{ab}_{\mu} = \omega^{ab}_{\mu} + K^{a b}_{\mu}. See Einstein-Cartan theory.

Example: general relativity[edit]

Main article: Frame fields in general relativity

We can describe geometries in general relativity in terms of a tetrad field instead of the usual metric tensor field. The metric tensor g_{\alpha\beta}\! gives the innerproduct in the tangent space directly:

\langle \mathbf{x},\mathbf{y} \rangle = g_{\alpha\beta} \, x^{\alpha} \, y^{\beta}.\,

The tetrad e_{\alpha}^i may be seen as a (linear) map from the tangent space to Minkowski space that preserves the inner product. This lets us find the inner product in the tangent space by mapping our two vectors into Minkowski space and taking the usual inner product there:

\langle \mathbf{x},\mathbf{y} \rangle = \eta_{ij} (e_{\alpha}^i \, x^{\alpha}) (e_{\beta}^j \, y^{\beta}).\,

Here \alpha and \beta range over tangent-space coordinates, while i and j range over Minkowski coordinates. The tetrad field e_{\alpha}^i(\mathbf{x}) defines a metric tensor field via the pullback g_{\alpha\beta}(\mathbf{x}) = \eta_{ij} \, e_{\alpha}^i(\mathbf{x}) \, e_{\beta}^j(\mathbf{x}).

QMR1905 – Annus Mirabilis papers

Main articles: Annus Mirabilis papers, Photoelectric effect, Special theory of relativity, Mass–energy equivalence and Brownian motion

The Annus Mirabilis papers are four articles pertaining to the photoelectric effect (which gave rise to quantum theory), Brownian motion, the special theory of relativity, and E = mc2 that Albert Einstein published in the Annalen der Physik scientific journal in 1905. These four works contributed substantially to the foundation of modern physics and changed views on space, time, and matter. The four papers are:

Title (translated) Area of focus Received Published Significance

On a Heuristic Viewpoint Concerning the Production and Transformation of Light Photoelectric effect 18 March 9 June Resolved an unsolved puzzle by suggesting that energy is exchanged only in discrete amounts (quanta).[123] This idea was pivotal to the early development of quantum theory.[124]

On the Motion of Small Particles Suspended in a Stationary Liquid, as Required by the Molecular Kinetic Theory of Heat Brownian motion 11 May 18 July Explained empirical evidence for the atomic theory, supporting the application of statistical physics.

On the Electrodynamics of Moving Bodies Special relativity 30 June 26 September Reconciled Maxwell's equations for electricity and magnetism with the laws of mechanics by introducing major changes to mechanics close to the speed of light, resulting from analysis based on empirical evidence that the speed of light is independent of the motion of the observer.[125] Discredited the concept of a "luminiferous ether."[126]

Does the Inertia of a Body Depend Upon Its Energy Content? Matter–energy equivalence 27 September 21 November Equivalence of matter and energy, E = mc2 (and by implication, the ability of gravity to "bend" light), the existence of "rest energy", and the basis of nuclear energy.

QMRThe Annus mirabilis papers (from Latin annus mīrābilis, "extraordinary year") are the papers of Albert Einstein published in the Annalen der Physik scientific journal in 1905. These four articles contributed substantially to the foundation of modern physics and changed views on space, time, mass, and energy. The annus mirabilis is often called the "miracle year" in English or Wunderjahr in German.

Photoelectric effect[edit]

Main article: Photoelectric effect

The article "On a Heuristic Viewpoint Concerning the Production and Transformation of Light"[einstein 1] received March 18 and published June 9, proposed the idea of energy quanta. This idea, motivated by Max Planck's earlier derivation of the law of black body radiation, assumes that luminous energy can be absorbed or emitted only in discrete amounts, called quanta. Einstein states,

Energy, during the propagation of a ray of light, is not continuously distributed over steadily increasing spaces, but it consists of a finite number of energy quanta localised at points in space, moving without dividing and capable of being absorbed or generated only as entities.

In explaining the photoelectric effect, the hypothesis that energy consists of discrete packets, as Einstein illustrates, can be directly applied to black bodies, as well.

The idea of light quanta contradicts the wave theory of light that follows naturally from James Clerk Maxwell's equations for electromagnetic behavior and, more generally, the assumption of infinite divisibility of energy in physical systems.

A profound formal difference exists between the theoretical concepts that physicists have formed about gases and other ponderable bodies, and Maxwell's theory of electromagnetic processes in so-called empty space. While we consider the state of a body to be completely determined by the positions and velocities of an indeed very large yet finite number of atoms and electrons, we make use of continuous spatial functions to determine the electromagnetic state of a volume of space, so that a finite number of quantities cannot be considered as sufficient for the complete determination of the electromagnetic state of space.

[... this] leads to contradictions when applied to the phenomena of emission and transformation of light.

According to the view that the incident light consists of energy quanta [...], the production of cathode rays by light can be conceived in the following way. The body's surface layer is penetrated by energy quanta whose energy is converted at least partially into kinetic energy of the electrons. The simplest conception is that a light quantum transfers its entire energy to a single electron [...]

Einstein noted that the photoelectric effect depended on the wavelength, and hence the frequency of the light. At too low a frequency, even intense light produced no electrons. However, once a certain frequency was reached, even low intensity light produced electrons. He compared this to Planck's hypothesis that light could be emitted only in packets of energy given by hf, where h is Planck's constant and f is the frequency. He then postulated that light travels in packets whose energy depends on the frequency, and therefore only light above a certain frequency would bring sufficient energy to liberate an electron.

Even after experiments confirmed that Einstein's equations for the photoelectric effect were accurate, his explanation was not universally accepted. Niels Bohr, in his 1922 Nobel address, stated, "The hypothesis of light-quanta is not able to throw light on the nature of radiation."

By 1921, when Einstein was awarded the Nobel Prize and his work on photoelectricity was mentioned by name in the award citation, some physicists accepted that the equation (hf = \Phi + E_k) was correct and light quanta were possible. In 1923, Arthur Compton's X-ray scattering experiment helped more of the scientific community to accept this formula. The theory of light quanta was a strong indicator of wave-particle duality, a fundamental principle of quantum mechanics.[9] A complete picture of the theory of photoelectricity was realized after the maturity of quantum mechanics.

Brownian motion[edit]

Main article: Brownian motion

The article "Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen" ("On the Motion of Small Particles Suspended in a Stationary Liquid, as Required by the Molecular Kinetic Theory of Heat"),[einstein 2] received May 11 and published July 18, delineated a stochastic model of Brownian motion.

In this paper it will be shown that, according to the molecular kinetic theory of heat, bodies of a microscopically visible size suspended in liquids must, as a result of thermal molecular motions, perform motions of such magnitudes that they can be easily observed with a microscope. It is possible that the motions to be discussed here are identical with so-called Brownian molecular motion; however, the data available to me on the latter are so imprecise that I could not form a judgment on the question...

Einstein derived expressions for the mean squared displacement of particles. Using the kinetic theory of fluids, which at the time was controversial, the article established that the phenomenon, which had lacked a satisfactory explanation even decades after it was first observed, provided empirical evidence for the reality of the atom. It also lent credence to statistical mechanics, which had been controversial at that time, as well. Before this paper, atoms were recognized as a useful concept, but physicists and chemists debated whether atoms were real entities. Einstein's statistical discussion of atomic behavior gave experimentalists a way to count atoms by looking through an ordinary microscope. Wilhelm Ostwald, one of the leaders of the anti-atom school, later told Arnold Sommerfeld that he had been convinced of the existence of atoms by Einstein's complete explanation of Brownian motion.[citation needed]

Special relativity[edit]

Main article: Special relativity

Einstein's "Zur Elektrodynamik bewegter Körper" ("On the Electrodynamics of Moving Bodies"),[einstein 3] his third paper that year, was received on June 30 and published September 26. It reconciles Maxwell's equations for electricity and magnetism with the laws of mechanics by introducing major changes to mechanics close to the speed of light. This later became known as Einstein's special theory of relativity.

The paper mentions the names of only five other scientists, Isaac Newton, James Clerk Maxwell, Heinrich Hertz, Christian Doppler, and Hendrik Lorentz. It does not have any references to any other publications. Many of the ideas had already been published by others, as detailed in history of special relativity and relativity priority dispute. However, Einstein's paper introduces a theory of time, distance, mass, and energy that was consistent with electromagnetism, but omitted the force of gravity.

At the time, it was known that Maxwell's equations, when applied to moving bodies, led to asymmetries (Moving magnet and conductor problem), and that it had not been possible to discover any motion of the Earth relative to the 'light medium'. Einstein puts forward two postulates to explain these observations. First, he applies the principle of relativity, which states that the laws of physics remain the same for any non-accelerating frame of reference (called an inertial reference frame), to the laws of electrodynamics and optics as well as mechanics. In the second postulate, Einstein proposes that the speed of light has the same value in all inertial frames of reference, independent of the state of motion of the emitting body.

Special relativity is thus consistent with the result of the Michelson–Morley experiment, which had not detected a medium of conductance (or aether) for light waves unlike other known waves that require a medium (such as water or air). Einstein may not have known about that experiment, but states,

Examples of this sort, together with the unsuccessful attempts to discover any motion of the earth relatively to the "light medium," suggest that the phenomena of electrodynamics as well as of mechanics possess no properties corresponding to the idea of absolute rest.

The speed of light is fixed, and thus not relative to the movement of the observer. This was impossible under Newtonian classical mechanics. Einstein argues,

the same laws of electrodynamics and optics will be valid for all frames of reference for which the equations of mechanics hold good. We will raise this conjecture (the purport of which will hereafter be called the "Principle of Relativity") to the status of a postulate, and also introduce another postulate, which is only apparently irreconcilable with the former, namely, that light is always propagated in empty space with a definite velocity c which is independent of the state of motion of the emitting body. These two postulates suffice for the attainment of a simple and consistent theory of the electrodynamics of moving bodies based on Maxwell's theory for stationary bodies. The introduction of a "luminiferous ether" will prove to be superfluous in as much as the view here to be developed will not require an "absolutely stationary space" provided with special properties, nor assign a velocity-vector to a point of the empty space in which electromagnetic processes take place.

The theory […] is based—like all electrodynamics—on the kinematics of the rigid body, since the assertions of any such theory have to do with the relationships between rigid bodies (systems of co-ordinates), clocks, and electromagnetic processes. Insufficient consideration of this circumstance lies at the root of the difficulties which the electrodynamics of moving bodies at present encounters.

It had previously been proposed, by George FitzGerald in 1889 and by Lorentz in 1892, independently of each other, that the Michelson-Morley result could be accounted for if moving bodies were contracted in the direction of their motion. Some of the paper's core equations, the Lorentz transforms, had been published by Joseph Larmor (1897, 1900), Hendrik Lorentz (1895, 1899, 1904) and Henri Poincaré (1905), in a development of Lorentz's 1904 paper. Einstein's presentation differed from the explanations given by FitzGerald, Larmor, and Lorentz, but was similar in many respects to the formulation by Poincaré (1905).

His explanation arises from two axioms. First, Galileo's idea that the laws of nature should be the same for all observers that move with constant speed relative to each other. Einstein writes,

The laws by which the states of physical systems undergo change are not affected, whether these changes of state be referred to the one or the other of two systems of co-ordinates in uniform translatory motion.

The second is the rule that the speed of light is the same for every observer.

Any ray of light moves in the "stationary" system of co-ordinates with the determined velocity c, whether the ray be emitted by a stationary or by a moving body.

The theory, now called the special theory of relativity, distinguishes it from his later general theory of relativity, which considers all observers to be equivalent. Special relativity gained widespread acceptance remarkably quickly, confirming Einstein's comment that it had been "ripe for discovery" in 1905. Acknowledging the role of Max Planck in the early dissemination of his ideas, Einstein wrote in 1913 "The attention that this theory so quickly received from colleagues is surely to be ascribed in large part to the resoluteness and warmth with which he [Planck] intervened for this theory". In addition, the improved mathematical formulation of the theory by Hermann Minkowski in 1907 was influential in gaining acceptance for the theory. Also, and most importantly, the theory was supported by an ever-increasing body of confirmatory experimental evidence.

Mass–energy equivalence[edit]

Main article: Mass–energy equivalence

On November 21 Annalen der Physik published a fourth paper (received September 27) "Ist die Trägheit eines Körpers von seinem Energieinhalt abhängig?" ("Does the Inertia of a Body Depend Upon Its Energy Content?"),[einstein 4] in which Einstein developed an argument for arguably the most famous equation in the field of physics: E = mc2.

Einstein considered the equivalency equation to be of paramount importance because it showed that a massive particle possesses an energy, the "rest energy", distinct from its classical kinetic and potential energies. The paper is based on James Clerk Maxwell's and Heinrich Rudolf Hertz's investigations and, in addition, the axioms of relativity, as Einstein states,

The results of the previous investigation lead to a very interesting conclusion, which is here to be deduced.

The previous investigation was based "on the Maxwell–Hertz equations for empty space, together with the Maxwellian expression for the electromagnetic energy of space ..."

The laws by which the states of physical systems alter are independent of the alternative, to which of two systems of coordinates, in uniform motion of parallel translation relatively to each other, these alterations of state are referred (principle of relativity).

The equation sets forth that energy of a body at rest (E) equals its mass (m) times the speed of light (c) squared, or E = mc2.

If a body gives off the energy L in the form of radiation, its mass diminishes by L/c2. The fact that the energy withdrawn from the body becomes energy of radiation evidently makes no difference, so that we are led to the more general conclusion that

The mass of a body is a measure of its energy-content; if the energy changes by L, the mass changes in the same sense by L/(9 × 1020), the energy being measured in ergs, and the mass in grammes.

[...]

If the theory corresponds to the facts, radiation conveys inertia between the emitting and absorbing bodies.

The mass-energy relation can be used to predict how much energy will be released or consumed by nuclear reactions; one simply measures the mass of all constituents and the mass of all the products and multiplies the difference between the two by c2. The result shows how much energy will be released or consumed, usually in the form of light or heat. When applied to certain nuclear reactions, the equation shows that an extraordinarily large amount of energy will be released, much larger than in the combustion of chemical explosives, where the mass difference is hardly measurable at all. This explains why nuclear weapons produce such phenomenal amounts of energy, as they release binding energy during nuclear fission and nuclear fusion, and also convert a much larger portion of subatomic mass to energy.

QMROther tests of Lorentz invariance[edit]

Further information: Modern searches for Lorentz violation

Figure 10. 7Li-NMR spectrum of LiCl (1M) in D2O. The sharp, unsplit NMR line of this isotope of lithium is evidence for the isotropy of mass and space.

Examples of other experiments not based on the Michelson–Morley principle, i.e. non-optical isotropy tests achieving an even higher level of precision, are Clock comparison or Hughes–Drever experiments. In Drever's 1961 experiment, 7Li nuclei in the ground state, which has total angular momentum J=3/2, were split into four equally spaced levels by a magnetic field. Each transition between a pair of adjacent levels should emit a photon of equal frequency, resulting in a single, sharp spectral line. However, since the nuclear wave functions for different MJ have different orientations in space relative to the magnetic field, any orientation dependence, whether from an aether wind or from a dependence on the large-scale distribution of mass in space (see Mach's principle), would perturb the energy spacings between the four levels, resulting in an anomalous broadening or splitting of the line. No such broadening was observed. Modern repeats of this kind of experiment have provided some of the most accurate confirmations of the principle of Lorentz invariance

QMRIn special relativity, a four-vector is an object with four in general complex components that transform in a specific way under Lorentz transformations. Specifically, a four-vector is an element of a 4-dimensional vector space considered as a representation space of the standard representation of the Lorentz group, the (½,½) representation. It differs from a Euclidean vector in how its magnitude is determined. The transformations that preserve this magnitude are the Lorentz transformations. They include spatial rotations, boosts (a change by a constant velocity to another inertial reference frame), and temporal and spatial inversions.

Four-vectors describe, for instance, position xμ in spacetime modeled as Minkowski space, a particles 4-momentum pμ, the amplitude of the electromagnetic four-potential Aμ(x) at a point x in spacetime, and the elements of the subspace spanned by the gamma matrices inside the Dirac algebra.

The Lorentz group may be represented by 4×4 matrices Λ. The action of a Lorentz transformation on a general contravariant four-vector X (like the examples above), regarded as a column vector with Cartesian coordinates with respect to an inertial frame in the entries, is given by

X^\prime = \Lambda X,

(matrix multiplication) where the components of the primed object refer to the new frame. Related to the examples above that are given as contravariant vectors, there are also the corresponding covariant vectors xμ, pμ and Aμ(x). These transform according to the rule

X^\prime = {(\Lambda^{-1})}^\mathrm T X,

where T denotes the matrix transpose. This rule is different from the above rule. It corresponds to the dual representation of the standard representation. However, for the Lorentz group the dual of any representation is equivalent to the original representation. Thus the objects with covariant indices are four-vectors as well.

For an example of a well-behaved four-component object in special relativity that is not a four-vector, see bispinor. It is similarly defined, the difference being that the transformation rule under Lorentz transformations is given by a representation other than the standard representation. In this case, the rule reads X′ = Π(Λ)X, where Π(Λ) is a 4×4 matrix other than Λ. Similar remarks apply to objects with fewer or more components that are well-behaved under Lorentz transformations. These include scalars, spinors, tensors and spinor-tensors.

The article considers four-vectors in the context of special relativity. Although the concept of four-vectors also extends to general relativity, some of the results stated in this article require modification in general relativity.

4 by 4 matrices are quadrant models

Notation[edit]

The notations in this article are: lowercase bold for three-dimensional vectors, hats for three-dimensional unit vectors, capital bold for four dimensional vectors (except for the four-gradient), and tensor index notation.

Four

Four-vectors in a real-valued basis[edit]

A four-vector A is a vector with a "timelike" component and three "spacelike" components, and can be written in various equivalent notations:[1]

\begin{align}  

\mathbf{A} & = (A^0, \, A^1, \, A^2, \, A^3) \\

& = A^0\mathbf{E}_0 + A^1 \mathbf{E}_1 + A^2 \mathbf{E}_2 + A^3 \mathbf{E}_3 \\

& = A^0\mathbf{E}_0 + A^i \mathbf{E}_i \\

& = A^\alpha\mathbf{E}_\alpha\\

\end{align}

The upper indices indicate contravariant components. Here the standard convention that Latin indices take values for spatial components, so that i = 1, 2, 3, and Greek indices take values for space and time components, so α = 0, 1, 2, 3, used with the summation convention. The split between the time component and the spatial components is a useful one to make when determining contractions of one four vector with other tensor quantities, such as for calculating Lorentz invariants in inner products (examples are given below), or raising and lowering indices.

In special relativity, the spacelike basis E1, E2, E3 and components A1, A2, A3 are often Cartesian basis and components:

\begin{align}  

\mathbf{A} & = (A_t, \, A_x, \, A_y, \, A_z) \\

& = A_t \mathbf{E}_t + A_x \mathbf{E}_x + A_y \mathbf{E}_y + A_z \mathbf{E}_z \\

\end{align}

although, of course, any other basis and components may be used, such as spherical polar coordinates

\begin{align}  

\mathbf{A} & = (A_t, \, A_r, \, A_\theta, \, A_\phi) \\

& = A_t \mathbf{E}_t + A_r \mathbf{E}_r + A_\theta \mathbf{E}_\theta + A_\phi \mathbf{E}_\phi \\

\end{align}

or cylindrical polar coordinates,

\begin{align}  

\mathbf{A} & = (A_t, \, A_r, \, A_\theta, \, A_z) \\

& = A_t \mathbf{E}_t + A_r \mathbf{E}_r + A_\theta \mathbf{E}_\theta + A_z \mathbf{E}_z \\

\end{align}

or any other orthogonal coordinates, or even general curvilinear coordinates. Note the coordinate labels are always subscripted as labels and are not indices taking numerical values. In general relativity, local curvilinear coordinates in a local basis must be used. Geometrically, a four-vector can still be interpreted as an arrow, but in spacetime - not just space. In relativity, the arrows are drawn as part of Minkowski diagram (also called spacetime diagram). In this article, four-vectors will be referred to simply as vectors.

It is also customary to represent the bases by column vectors:

\mathbf{E}_0 = \begin{pmatrix} 1 \\ 0 \\ 0 \\ 0 \end{pmatrix} \,,\quad \mathbf{E}_1 = \begin{pmatrix} 0 \\ 1 \\ 0 \\ 0 \end{pmatrix} \,,\quad \mathbf{E}_2 = \begin{pmatrix} 0 \\ 0 \\ 1 \\ 0 \end{pmatrix} \,,\quad \mathbf{E}_3 = \begin{pmatrix} 0 \\ 0 \\ 0 \\ 1 \end{pmatrix}

so that:

\mathbf{A} = \begin{pmatrix} A^0 \\ A^1 \\ A^2 \\ A^3 \end{pmatrix}

The relation between the covariant and contravariant coordinates is through the Minkowski metric tensor, η which raises and lowers indices as follows:

A_{\mu} = \eta_{\mu \nu} A^{\nu} \,,

and in various equivalent notations the covariant components are:

\begin{align}  

\mathbf{A} & = (A_0, \, A_1, \, A_2, \, A_3) \\

& = A_0\mathbf{E}^0 + A_1 \mathbf{E}^1 + A_2 \mathbf{E}^2 + A_3 \mathbf{E}^3 \\

& = A_0\mathbf{E}^0 + A_i \mathbf{E}^i \\

& = A_\alpha\mathbf{E}^\alpha\\

\end{align}

where the lowered index indicates it to be covariant. Often the metric is diagonal, as is the case for orthogonal coordinates (see line element), but not in general curvilinear coordinates.

The bases can be represented by row vectors:

\mathbf{E}^0 = \begin{pmatrix} 1 & 0 & 0 & 0 \end{pmatrix} \,,\quad \mathbf{E}^1 = \begin{pmatrix} 0 & 1 & 0 & 0 \end{pmatrix} \,,\quad \mathbf{E}^2 = \begin{pmatrix} 0 & 0 & 1 & 0 \end{pmatrix} \,,\quad \mathbf{E}^3 = \begin{pmatrix} 0 & 0 & 0 & 1 \end{pmatrix}

so that:

\mathbf{A} = \begin{pmatrix} A_0 & A_1 & A_2 & A_3 \end{pmatrix}

The motivation for the above conventions are that the inner product is a scalar, see below for details.

Lorentz transformation[edit]

Main article: Lorentz transformation

Given two inertial or rotated frames of reference, a four-vector is defined as a quantity which transforms according to the Lorentz transformation matrix Λ:

\mathbf{A}'=\boldsymbol{\Lambda}\mathbf{A}

In index notation, the contravariant and covariant components transform according to, respectively:

{A'}^\mu = \Lambda^\mu {}_\nu A^\nu \,,\quad {A'}_\mu = \Lambda_\mu {}^\nu A_\nu

in which the matrix Λ has components Λμν in row μ and column ν, and the inverse matrix Λ−1 has components Λμν in row μ and column ν.

For background on the nature of this transformation definition, see tensor. All four-vectors transform in the same way, and this can be generalized to four-dimensional relativistic tensors; see special relativity.

Pure rotations about an arbitrary axis[edit]

For two frames rotated by a fixed angle θ about an axis defined by the unit vector:

\hat{\mathbf{n}} = (\hat{n}_1,\hat{n}_2,\hat{n}_3)\,,

without any boosts, the matrix Λ has components given by:[2]

\Lambda_{00} = 1

\Lambda_{0i} = \Lambda_{i0} = 0

\Lambda_{ij} = (\delta_{ij} - \hat{n}_i \hat{n}_j) \cos\theta - \varepsilon_{ijk} \hat{n}_k \sin\theta + \hat{n}_i \hat{n}_j

where δij is the Kronecker delta, and εijk is the three-dimensional Levi-Civita symbol. The spacelike components of 4-vectors are rotated, while the timelike components remain unchanged.

For the case of rotations about the z-axis only, the spacelike part of the Lorentz matrix reduces to the rotation matrix about the z-axis:

\begin{pmatrix}

{A'}^0 \\ {A'}^1 \\ {A'}^2 \\ {A'}^3

\end{pmatrix}

=

\begin{pmatrix}

1 & 0 & 0 & 0 \\

0 & \cos\theta &-\sin\theta & 0 \\

0 & \sin\theta & \cos\theta & 0 \\

0 & 0 & 0 & 1 \\

\end{pmatrix}

\begin{pmatrix}

A^0 \\ A^1 \\ A^2 \\ A^3

\end{pmatrix}\ .

Pure boosts in an arbitrary direction[edit]

Standard configuration of coordinate systems; for a Lorentz boost in the x-direction.

For two frames moving at constant relative 3-velocity v (not 4-velocity, see below), it is convenient to denote and define the relative velocity in units of c by:

\boldsymbol{\beta} = (\beta_1,\,\beta_2,\,\beta_3) = \frac{1}{c}(v_1,\,v_2,\,v_3) = \frac{1}{c}\mathbf{v} \,.

Then without rotations, the matrix Λ has components given by:[3]

\begin{align} \Lambda_{00} & = \gamma, \\

\Lambda_{0i} & = \Lambda_{i0} = - \gamma \beta_{i}, \\

\Lambda_{ij} & = \Lambda_{ji} = ( \gamma - 1 )\dfrac{\beta_{i}\beta_{j}}{\beta^{2}} + \delta_{ij}= ( \gamma - 1 )\dfrac{v_i v_j}{v^2} + \delta_{ij}, \\

\end{align}

\,\!

where the Lorentz factor is defined by:

\gamma = \frac{1}{\sqrt{1- \boldsymbol{\beta}\cdot\boldsymbol{\beta}}} \,,

and δij is the Kronecker delta. Contrary to the case for pure rotations, the spacelike and timelike components are mixed together under boosts.

For the case of a boost in the x-direction only, the matrix reduces to;[4][5]

\begin{pmatrix}

A'^0 \\ A'^1 \\ A'^2 \\ A'^3

\end{pmatrix}

=\begin{pmatrix}

\cosh\phi &-\sinh\phi & 0 & 0 \\

-\sinh\phi & \cosh\phi & 0 & 0 \\

0 & 0 & 1 & 0 \\

0 & 0 & 0 & 1 \\

\end{pmatrix}

\begin{pmatrix}

A^0 \\ A^1 \\ A^2 \\ A^3

\end{pmatrix}

Where the rapidity ϕ expression has been used, written in terms of the hyperbolic functions:

\gamma = \cosh \phi

This Lorentz matrix illustrates the boost to be a hyperbolic rotation in four dimensional spacetime, analogous to the circular rotation above in three-dimensional space.

Properties[edit]

Linearity[edit]

Four-vectors have the same linearity properties as Euclidean vectors in three dimensions. They can be added in the usual entrywise way:

\mathbf{A}+\mathbf{B} = (A^0, A^1, A^2,A^3) + (B^0, B^1, B^2,B^3) = (A^0 + B^0, A^1 + B^1, A^2 + B^2, A^3 + B^3)  

and similarly scalar multiplication by a scalar λ is defined entrywise by:

\lambda\mathbf{A} = \lambda(A^0, A^1, A^2,A^3) = (\lambda A^0, \lambda A^1, \lambda A^2, \lambda A^3)

Then subtraction is the inverse operation of addition, defined entrywise by:

\mathbf{A}+(-1)\mathbf{B} = (A^0, A^1, A^2,A^3) + (-1)(B^0, B^1, B^2,B^3) = (A^0 - B^0, A^1 - B^1, A^2 - B^2, A^3 - B^3)  

Minkowski tensor[edit]

See also: spacetime interval

Applying the Minkowski tensor η to two four-vectors A and B, writing the result in dot product notation, we have, using Einstein notation:

\mathbf{A} \cdot \mathbf{B} = A^{\mu} \eta_{\mu \nu} B^{\nu}

It is convenient to rewrite the definition in matrix form:

\mathbf{A \cdot B} = \begin{pmatrix} A^0 & A^1 & A^2 & A^3 \end{pmatrix} \begin{pmatrix} \eta_{00} & \eta_{01} & \eta_{02} & \eta_{03} \\ \eta_{10} & \eta_{11} & \eta_{12} & \eta_{13} \\ \eta_{20} & \eta_{21} & \eta_{22} & \eta_{23} \\ \eta_{30} & \eta_{31} & \eta_{32} & \eta_{33} \end{pmatrix} \begin{pmatrix} B^0 \\ B^1 \\ B^2 \\ B^3 \end{pmatrix}

in which case ημν above is the entry in row μ and column ν of the Minkowski metric as a square matrix. The Minkowski metric is not a Euclidean metric, because it is indefinite (see metric signature). A number of other expressions can be used because the metric tensor raises and lowers the components of A and B. For contra/co-variant components of A and co/contra-variant components of B, we have:

\mathbf{A} \cdot \mathbf{B} = A_{\nu} B^{\nu} = A^{\mu} B_{\mu}

so in the matrix notation:

\mathbf{A \cdot B} = \begin{pmatrix} A_0 & A_1 & A_2 & A_3 \end{pmatrix} \begin{pmatrix} B^0 \\ B^1 \\ B^2 \\ B^3 \end{pmatrix} = \begin{pmatrix} B_0 & B_1 & B_2 & B_3 \end{pmatrix} \begin{pmatrix} A^0 \\ A^1 \\ A^2 \\ A^3 \end{pmatrix}

while for A and B each in covariant components:

\mathbf{A} \cdot \mathbf{B} = A_{\mu} \eta^{\mu \nu} B_{\nu}

with a similar matrix expression to the above.

Applying the Minkowski tensor to a four-vector A with itself we get:

\mathbf{A \cdot A} = A^\mu \eta_{\mu\nu} A^\nu

which, depending on the case, may be considered the square, or its negative, of the length of the vector.

Following are two common choices for the metric tensor in the standard basis (essentially Cartesian coordinates). If orthogonal coordinates are used, there would be scale factors along the diagonal part of the spacelike part of the metric, while for general curvilinear coordinates the entire spacelike part of the metric would have components dependent on the curvilinear basis used.

Standard basis, (+−−−) signature[edit]

In the (+−−−) metric signature, evaluating the summation over indices gives:

\mathbf{A} \cdot \mathbf{B} = A^0 B^0 - A^1 B^1 - A^2 B^2 - A^3 B^3  

while in matrix form:

\mathbf{A \cdot B} = \begin{pmatrix} A^0 & A^1 & A^2 & A^3 \end{pmatrix} \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & -1 \end{pmatrix} \begin{pmatrix} B^0 \\ B^1 \\ B^2 \\ B^3 \end{pmatrix}

It is a recurring theme in special relativity to take the expression

\mathbf{A}\cdot\mathbf{B} = A^0 B^0 - A^1 B^1 - A^2 B^2 - A^3 B^3 = C

in one reference frame, where C is the value of the inner product in this frame, and:

\mathbf{A}'\cdot\mathbf{B}' = {A'}^0 {B'}^0 - {A'}^1 {B'}^1 - {A'}^2 {B'}^2 - {A'}^3 {B'}^3 = C'

in another frame, in which C′ is the value of the inner product in this frame. Then since the inner product is an invariant, these must be equal:

\mathbf{A}\cdot\mathbf{B} = \mathbf{A}'\cdot\mathbf{B}'

that is:

C = A^0 B^0 - A^1 B^1 - A^2 B^2 - A^3 B^3 = {A'}^0 {B'}^0 - {A'}^1 {B'}^1 - {A'}^2 {B'}^2 - {A'}^3{B'}^3

Considering that physical quantities in relativity are four-vectors, this equation has the appearance of a "conservation law", but there is no "conservation" involved. The primary significance of the Minkowski inner product is that for any two four-vectors, its value is invariant for all observers; a change of coordinates does not result in a change in value of the inner product. The components of the four-vectors change from one frame to another; A and A′ are connected by a Lorentz transformation, and similarly for B and B′, although the inner products are the same in all frames. Nevertheless, this type of expression is exploited in relativistic calculations on a par with conservation laws, since the magnitudes of components can be determined without explicitly performing any Lorentz transformations. A particular example is with energy and momentum in the energy-momentum relation derived from the four-momentum vector (see also below).

In this signature we have:

\mathbf{A \cdot A} = (A^0)^2 - (A^1)^2 - (A^2)^2 - (A^3)^2

With the signature (+−−−), four-vectors may be classified as either spacelike if \mathbf{A \cdot A} < 0, timelike if \mathbf{A \cdot A} > 0, and null vectors if \mathbf{A \cdot A} = 0.

Standard basis, (−+++) signature[edit]

Some authors define η with the opposite sign, in which case we have the (−+++) metric signature. Evaluating the summation with this signature:

\mathbf{A \cdot B} = - A^0 B^0 + A^1 B^1 + A^2 B^2 + A^3 B^3

while the matrix form is:

\mathbf{A \cdot B} = \left( \begin{matrix}A^0 & A^1 & A^2 & A^3 \end{matrix} \right)

\left( \begin{matrix} -1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix} \right)

\left( \begin{matrix}B^0 \\ B^1 \\ B^2 \\ B^3 \end{matrix} \right)

Note that in this case, in one frame:

\mathbf{A}\cdot\mathbf{B} = - A^0 B^0 + A^1 B^1 + A^2 B^2 + A^3 B^3 = -C

while in another:

\mathbf{A}'\cdot\mathbf{B}' = - {A'}^0 {B'}^0 + {A'}^1 {B'}^1 + {A'}^2 {B'}^2 + {A'}^3 {B'}^3 = -C'

so that:

-C = - A^0 B^0 + A^1 B^1 + A^2 B^2 + A^3 B^3 = - {A'}^0 {B'}^0 + {A'}^1 {B'}^1 + {A'}^2 {B'}^2 + {A'}^3{B'}^3

which is equivalent to the above expression for C in terms of A and B. Either convention will work. With the Minkowski metric defined in the two ways above, the only difference between covariant and contravariant four-vector components are signs, therefore the signs depend on which sign convention is used.

We have:

\mathbf{A \cdot A} = - (A^0)^2 + (A^1)^2 + (A^2)^2 + (A^3)^2

With the signature (−+++), four-vectors may be classified as either spacelike if \mathbf{A \cdot A} > 0, timelike if \mathbf{A \cdot A} < 0, and null vectors if \mathbf{A \cdot A} = 0..

Dual vectors[edit]

Applying the Minkowski tensor is often expressed as the effect of the dual vector of one vector on the other:

\mathbf{A \cdot B} = A^*(\mathbf{B}) = A{_\nu}B^{\nu}.

Here the Aνs are the components of the dual vector A* of A in the dual basis and called the covariant coordinates of A, while the original Aν components are called the contravariant coordinates.

Four-vector calculus[edit]

Derivatives and differentials[edit]

In special relativity (but not general relativity), the derivative of a four-vector with respect to a scalar λ (invariant) is itself a four-vector. It is also useful to take the differential of the four-vector, dA and divide it by the differential of the scalar, dλ:

\underset{\text{differential}}{d\mathbf{A}} = \underset{\text{derivative}}{\frac{d\mathbf{A}}{d\lambda}} \underset{\text{differential}}{d\lambda}

where the contravariant components are:

d\mathbf{A} = (dA^0, dA^1, dA^2, dA^3)

while the covariant components are:

d\mathbf{A} = (dA_0, dA_1, dA_2, dA_3)

In relativistic mechanics, one often takes the differential of a four-vector and divides by the differential in proper time (see below).

Fundamental four-vectors[edit]

Four-position[edit]

A point in Minkowski space is a time and spatial position, called an "event", or sometimes the position 4-vector or 4-position, described in some reference frame by a set of four coordinates:

\mathbf{R}= \left(ct, \mathbf{r}\right)

where r is the three-dimensional space position vector. If r is a function of coordinate time t in the same frame, i.e. r = r(t), this corresponds to a sequence of events as t varies. The definition R0 = ct ensures that all the coordinates have the same units (of distance).[6][7][8] These coordinates are the components of the position four-vector for the event. The displacement four-vector is defined to be an "arrow" linking two events:

\Delta \mathbf{R} = \left(c\Delta t, \Delta \mathbf{r} \right)

For the differential 4-position on a world line we have, using a norm notation:

\|d\mathbf{R}\|^2 = \mathbf{dR \cdot dR} = dR^\mu dR_\mu=c^2d\tau^2=ds^2 \,,

defining the differential line element ds and differential proper time increment dτ, but this "norm" is also:

\|d\mathbf{R}\|^2 = (cdt)^2 - d\mathbf{r}\cdot d\mathbf{r} \,,

so that:

(c d\tau)^2 = (cdt)^2 - d\mathbf{r}\cdot d\mathbf{r} \,.

When considering physical phenomena, differential equations arise naturally; however, when considering space and time derivatives of functions, it is unclear which reference frame these derivatives are taken with respect to. It is agreed that time derivatives are taken with respect to the proper time τ. As proper time is an invariant, this guarantees that the proper-time-derivative of any four-vector is itself a four-vector. It is then important to find a relation between this proper-time-derivative and another time derivative (using the coordinate time t of an inertial reference frame). This relation is provided by taking the above differential invariant spacetime interval, then dividing by (cdt)2 to obtain:

\left(\frac{cd\tau}{cdt}\right)^2 = 1 - \left(\frac{d\mathbf{r}}{cdt}\cdot \frac{d\mathbf{r}}{cdt}\right) = 1 - \frac{\mathbf{u}\cdot\mathbf{u}}{c^2} = \frac{1}{\gamma(\mathbf{u})^2} \,,

where u = dr/dt is the coordinate 3-velocity of an object measured in the same frame as the coordinates x, y, z, and coordinate time t, and

\gamma(\mathbf{u}) = \frac{1}{\sqrt{1- \frac{\mathbf{u}\cdot\mathbf{u}}{c^2}}}

is the Lorentz factor. This provides a useful relation between the differentials in coordinate time and proper time:

dt = \gamma(\mathbf{u})d\tau \,.

This relation can also be found from the time transformation in the Lorentz transformations. Important four-vectors in relativity theory can be defined by dividing by this differential.

Four-gradient[edit]

Considering that partial derivatives are linear operators, one can form a four-gradient from the partial time derivative ∂/∂t and the spatial gradient ∇. Using the standard basis, in index and abbreviated notations, the contravariant components are:

\begin{align}  

\boldsymbol{\partial} & = \left(\frac{\partial }{\partial x_0}, \, -\frac{\partial }{\partial x_1}, \, -\frac{\partial }{\partial x_2}, \, -\frac{\partial }{\partial x_3} \right) \\

& = (\partial^0, \, - \partial^1, \, - \partial^2, \, - \partial^3) \\

& = \mathbf{E}_0\partial^0 - \mathbf{E}_1\partial^1 - \mathbf{E}_2\partial^2 - \mathbf{E}_3\partial^3 \\

& = \mathbf{E}_0\partial^0 - \mathbf{E}_i\partial^i \\

& = \mathbf{E}_\alpha \partial^\alpha \\

& = \left(\frac{1}{c}\frac{\partial}{\partial t} , \, - \nabla \right) \\

& = \mathbf{E}_0\frac{1}{c}\frac{\partial}{\partial t} - \nabla \\

\end{align}

Note the basis vectors are placed in front of the components, to prevent confusion between taking the derivative of the basis vector, or simply indicating the partial derivative is a component of this four-vector. The covariant components are:

\begin{align}

\boldsymbol{\partial} & = \left(\frac{\partial }{\partial x^0}, \, \frac{\partial }{\partial x^1}, \, \frac{\partial }{\partial x^2}, \, \frac{\partial }{\partial x^3} \right) \\

& = (\partial_0, \, \partial_1, \, \partial_2, \, \partial_3) \\

& = \mathbf{E}^0\partial_0 + \mathbf{E}^1\partial_1 + \mathbf{E}^2\partial_2 + \mathbf{E}^3\partial_3 \\

& = \mathbf{E}^0\partial_0 + \mathbf{E}^i\partial_i \\

& = \mathbf{E}^\alpha \partial_\alpha \\

& = \left(\frac{1}{c}\frac{\partial}{\partial t} , \, \nabla \right) \\

& = \mathbf{E}^0\frac{1}{c}\frac{\partial}{\partial t} + \nabla \\

\end{align}

Since this is an operator, it doesn't have a "length", but evaluating the inner product of the operator with itself gives another operator:

\partial^\mu \partial_\mu = \frac{1}{c^2}\frac{\partial^2}{\partial t^2} - \nabla^2

called the D'Alembert operator.

Kinematics[edit]

Four-velocity[edit]

The four-velocity of a particle is defined by:

\mathbf{U} = \frac{d\mathbf{X}}{d \tau}= \frac{d\mathbf{X}}{dt}\frac{dt}{d \tau} = \gamma(\mathbf{u})\left(c, \mathbf{u} \right),

Geometrically, U is a normalized vector tangent to the world line of the particle. Using the differential of the 4-position, the magnitude of the 4-velocity can be obtained:

\|\mathbf{U}\|^2 = U^\mu U_\mu = \frac{dX^\mu }{d\tau} \frac{dX_\mu }{d\tau}= \frac{dX^\mu dX_\mu }{d\tau^2} = c^2 \,,

in short, the magnitude of the 4-velocity for any object is always a fixed constant:

\| \mathbf{U} \|^2 = c^2 \,

The norm is also:

\|\mathbf{U}\|^2 = {\gamma(\mathbf{u})}^2 \left( c^2 - \mathbf{u}\cdot\mathbf{u} \right) \,,

so that:

c^2 = {\gamma(\mathbf{u})}^2 \left( c^2 - \mathbf{u}\cdot\mathbf{u} \right) \,,

which reduces to the definition the Lorentz factor.

Four-acceleration[edit]

The four-acceleration is given by:

\mathbf{A} =\frac{d\mathbf{U} }{d \tau} = \gamma(\mathbf{u}) \left(\frac{d{\gamma}(\mathbf{u})}{dt} c, \frac{d{\gamma}(\mathbf{u})}{dt} \mathbf{u} + \gamma(\mathbf{u}) \mathbf{a} \right).

where a = du/dt is the coordinate 3-acceleration. Since the magnitude of U is a constant, the four acceleration is orthogonal to the four velocity, i.e. the Minkowski inner product of the four-acceleration and the four-velocity is zero:

\mathbf{A}\cdot\mathbf{U} = A^\mu U_\mu = \frac{dU^\mu}{d\tau} U_\mu = \frac{1}{2} \, \frac{d}{d\tau} (U^\mu U_\mu) = 0 \,

which is true for all world lines. The geometric meaning of 4-acceleration is the curvature vector of the world line in Minkowski space.

Dynamics[edit]

Four-momentum[edit]

For a massive particle of rest mass (or invariant mass) m0, the four-momentum is given by:

\mathbf{P} = m_0 \mathbf{U} = m_0\gamma(\mathbf{u})(c, \mathbf{u}) = (E/c, \mathbf{p})

where the total energy of the moving particle is:

E = \gamma(\mathbf{u}) m_0c^2  

and the total relativistic momentum is:

\mathbf{p} = \gamma(\mathbf{u}) m_0 \mathbf{u}

Taking the inner product of the four-momentum with itself:

\|\mathbf{P}\|^2 = P^\mu P_\mu = m_0^2 U^\mu U_\mu = m_0^2 c^2

and also:

\|\mathbf{P}\|^2 = \frac{E^2}{c^2} - \mathbf{p}\cdot\mathbf{p}

which leads to the energy–momentum relation:

E^2 = c^2 \mathbf{p}\cdot\mathbf{p} + (m_0c^2)^2 \,.

This last relation is useful relativistic mechanics, essential in relativistic quantum mechanics and relativistic quantum field theory, all with applications to particle physics.

Four-force[edit]

The four-force acting on a particle is defined analogously to the 3-force as the time derivative of 3-momentum in Newton's second law:

\mathbf{F} = \frac {d \mathbf{P}} {d \tau} = \gamma(\mathbf{u})\left(\frac{1}{c}\frac{dE}{dt},\frac{d\mathbf{p}}{dt}\right) = \gamma(\mathbf{u})(P/c,\mathbf{f})

where P is the power transferred to move the particle, and f is the 3-force acting on the particle. For a particle of constant invariant mass m0, this is equivalent to

\mathbf{F} = m_0 \mathbf{A} = m_0\gamma(\mathbf{u})\left( \frac{d{\gamma}(\mathbf{u})}{dt} c, \left(\frac{d{\gamma}(\mathbf{u})}{dt} \mathbf{u} + \gamma(\mathbf{u}) \mathbf{a}\right) \right)

An invariant derived from the 4-force is:

\mathbf{F}\cdot\mathbf{U} = F^\mu U_\mu = m_0 A^\mu U_\mu = 0

from the above result.

Thermodynamics[edit]

See also: Relativistic heat conduction

Four-heat flux[edit]

The 4-heat flux vector field, is essentially similar to the 3d heat flux vector field q, in the local frame of the fluid:[9]

\mathbf{Q} = -k \boldsymbol{\partial} T = - k\left( \frac{1}{c}\frac{\partial T}{\partial t}, \nabla T\right)

where T is absolute temperature and k is thermal conductivity.

Four-baryon number flux[edit]

The flux of baryons is:[10]

\mathbf{S}= n\mathbf{U}

where n is the number density of baryons in the local rest frame of the baryon fluid (positive values for baryons, negative for antibaryons), and U the 4-velocity field (of the fluid) as above.

Four-entropy[edit]

The 4-entropy vector is defined by:[11]

\mathbf{s}= s\mathbf{S} + \frac{\mathbf{Q}}{T}

where s is the entropy per baryon, and T the absolute temperature, in the local rest frame of the fluid.[12]

Electromagnetism[edit]

Examples of four-vectors in electromagnetism include the following.

Four-current[edit]

The electromagnetic four-current is defined by

\mathbf{J} = \left( \rho c, \mathbf{j} \right)

formed from the current density j and charge density ρ.

Four-potential[edit]

The electromagnetic four-potential defined by

\mathbf{A} = \left( \phi /c, \mathbf{a} \right)

formed from the vector potential a and the scalar potential ϕ. The four-potential is not uniquely determined, because it depends on a choice of gauge.

Waves[edit]

Four-frequency[edit]

A photonic plane wave can be described by the four-frequency defined as

\mathbf{N} = \nu\left(1 , \hat{\mathbf{n}} \right)

where ν is the frequency of the wave and \hat{\mathbf{n}} is a unit vector in the travel direction of the wave. Now:

\|\mathbf{N}\| = N^\mu N_\mu = \nu ^2 \left(1 - \hat{\mathbf{n}}\cdot\hat{\mathbf{n}}\right) = 0

so the 4-frequency of a photon is always a null vector.

Four-wavevector[edit]

See also: De Broglie relation

The quantities reciprocal to time t and space r are the angular frequency ω and wave vector k, respectively. They form the components of the 4-WaveVector or wave 4-vector:

\mathbf{K} = \left(\frac{\omega}{c}, \vec{\mathbf{k}} \right) = \left(\frac{\omega}{c}, \frac{\omega}{v_p}\mathbf{\hat{n}} \right) \,.

A wave packet of nearly monochromatic light can be described by:

\mathbf{K} = \frac{2\pi}{c}\mathbf{N} = \frac{2\pi}{c} \nu(1,\hat{\mathbf{n}}) = \frac{\omega}{c}\left( 1 , \hat{\mathbf{n}} \right) \,.

The de Broglie relations then showed that 4-WaveVector applied to matter waves as well as to light waves. :

\mathbf{P} = \hbar \mathbf{K} = \left(\frac{E}{c},\vec{p}\right) = \hbar \left(\frac{\omega}{c},\vec{k} \right)\,.

yielding E = \hbar \omega and \vec{p} = \hbar \vec{k}, where ħ is the Planck constant divided by 2π.

The square of the norm is:

\| \mathbf{K} \|^2 = K^\mu K_\mu = \left(\frac{\omega}{c}\right)^2 - \mathbf{k}\cdot\mathbf{k}\,,

and by the de Broglie relation:

\| \mathbf{K} \|^2 = \frac{1}{\hbar^2} \| \mathbf{P} \|^2 = \left(\frac{m_0 c}{\hbar}\right)^2 \,,

we have the matter wave analogue of the energy–momentum relation:

\left(\frac{\omega}{c}\right)^2 - \mathbf{k}\cdot\mathbf{k} = \left(\frac{m_0 c}{\hbar}\right)^2 \,.

Note that for massless particles, in which case m0 = 0, we have:

\left(\frac{\omega}{c}\right)^2 = \mathbf{k}\cdot\mathbf{k} \,,

or ||k|| = ω/c. Note this is consistent with the above case; for photons with a 3-wavevector of modulus ω/c, in the direction of wave propagation defined by the unit vector \hat{\mathbf{n}}.

Quantum theory[edit]

4-Probability current[edit]

In quantum mechanics, the 4-probability current or probability 4-current is analogous to the electromagnetic 4-current:[13]

\mathbf{J} = (\rho c, \mathbf{j})

where ρ is the probability density function corresponding to the time component, and j is the probability current vector. In non-relativistic quantum mechanics, this current is always well defined because the expressions for density and current are positive definite and can admit a probability interpretation. In relativistic quantum mechanics and quantum field theory, it is not always possible to find a current, particularly when interactions are involved.

Replacing the energy by the energy operator and the momentum by the momentum operator in the four-momentum, one obtains the four-momentum operator, used in relativistic wave equations.

4-Spin[edit]

The four-spin of a particle is defined in the rest frame of a particle to be

\mathbf{S} = (0,\mathbf{s})

where s is the spin pseudovector. In quantum mechanics, not all three components of this vector are simultaneously measurable, only one component is. The timelike component is zero in the particle's rest frame, but not in any other frame. This component can be found from an appropriate Lorentz transformation.

The norm squared is the (negative of the) magnitude squared of the spin, and according to quantum mechanics we have

\|\mathbf{S}\|^2 = - |\mathbf{s}|^2 = -\hbar^2s(s+1)

This value is observable and quantized, with s the spin quantum number (not the magnitude of the spin vector).

Other formulations[edit]

Four-vectors in the algebra of physical space[edit]

A four-vector A can also be defined in using the Pauli matrices as a basis, again in various equivalent notations:[14]

\begin{align}  

\mathbf{A} & = (A^0, \, A^1, \, A^2, \, A^3) \\

& = A^0\boldsymbol{\sigma}_0 + A^1 \boldsymbol{\sigma}_1 + A^2 \boldsymbol{\sigma}_2 + A^3 \boldsymbol{\sigma}_3 \\

& = A^0\boldsymbol{\sigma}_0 + A^i \boldsymbol{\sigma}_i \\

& = A^\alpha\boldsymbol{\sigma}_\alpha\\

\end{align}

or explicitly:

\begin{align}  

\mathbf{A} & = A^0\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} + A^1 \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} + A^2 \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix} + A^3 \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} \\

& = \begin{pmatrix} A^0 + A^3 & A^1 -i A^2 \\ A^1 + i A^2 & A^0 - A^3 \end{pmatrix}

\end{align}

and in this formulation, the four-vector is represented as a Hermitian matrix (the matrix transpose and complex conjugate of the matrix leaves it unchanged), rather than a real-valued column or row vector. The determinant of the matrix is the modulus of the four-vector, so the determinant is an invariant:

\begin{align}  

|\mathbf{A}| & = \begin{vmatrix} A^0 + A^3 & A^1 -i A^2 \\ A^1 + i A^2 & A^0 - A^3 \end{vmatrix} \\

& = (A^0 + A^3)(A^0 - A^3) - (A^1 -i A^2)(A^1 + i A^2) \\

& = (A^0)^2 - (A^1)^2 - (A^2)^2 - (A^3)^2

\end{align}

This idea of using the Pauli matrices as basis vectors is employed in the algebra of physical space, an example of a Clifford algebra.

Four-vectors in spacetime algebra[edit]

In spacetime algebra, another example of Clifford algebra, the gamma matrices can also form a basis. (They are also called the Dirac matrices, owing to their appearance in the Dirac equation). There is more than one way to express the gamma matrices, detailed in that main article.

The Feynman slash notation is a shorthand for a four-vector A contracted with the gamma matrices:

\mathbf{A}\!\!\!\!/ = A_\alpha \gamma^\alpha = A_0 \gamma^0 + A_1 \gamma^1 + A_2 \gamma^2 + A_3 \gamma^3

The four-momentum contracted with the gamma matrices is an important case in relativistic quantum mechanics and relativistic quantum field theory. In the Dirac equation and other relativistic wave equations, terms of the form:

\mathbf{P}\!\!\!\!/ = P_\alpha \gamma^\alpha = P_0 \gamma^0 + P_1 \gamma^1 + P_2 \gamma^2 + P_3 \gamma^3 = \dfrac{E}{c} \gamma^0 - p_x \gamma^1 - p_y \gamma^2 - p_z \gamma^3

appear, in which the energy E and momentum components (px, py, pz) are replaced by their respective operators.

QMrDimensions[edit]

In general relativity, four-dimensional vectors, or four-vectors, are required. These four dimensions are length, height, width and time. A "point" in this context would be an event, as it has both a location and a time. Similar to vectors, tensors in relativity require four dimensions. One example is the Riemann curvature tensor.

electromagnetic tensor

QMRIn electromagnetism, the electromagnetic tensor or electromagnetic field tensor (sometimes called the field strength tensor, Faraday tensor or Maxwell bivector) is a mathematical object that describes the electromagnetic field of a physical system. The field tensor was first used after the 4-dimensional tensor formulation of special relativity was introduced by Hermann Minkowski. The tensor allows some physical laws to be written in a very concise form.

The electromagnetic tensor, conventionally labelled F, is defined as the exterior derivative of the electromagnetic four-potential, A, a differential 1-form:[1][2]

F \ \stackrel{\mathrm{def}}{=}\ \mathrm{d}A.

Therefore F is a differential 2-form—that is, an antisymmetric rank-2 tensor field—on Minkowski space. In component form,

F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu.

SI units for Maxwell's equations and the particle physicist's sign convention for the signature of Minkowski space (+,−,−,−), will be used throughout this article.

Relationship with the classical fields[edit]

The electromagnetic tensor is completely isomorphic to the electric and magnetic fields, though the electric and magnetic fields change with the choice of the reference frame, while the electromagnetic tensor does not. In general, the relationship is quite complicated, but in Cartesian coordinates, using the coordinate system's own reference frame, the relationship is very simple.

E_i = c F_{0i},

where c is the speed of light, and

B_i = -\frac 1 2 \epsilon_{ijk} F^{jk},

where \epsilon_{ijk} is the Levi-Civita symbol. In contravariant matrix form,

\begin{bmatrix}

0 & -E_x/c & -E_y/c & -E_z/c \\

E_x/c & 0 & -B_z & B_y \\

E_y/c & B_z & 0 & -B_x \\

E_z/c & -B_y & B_x & 0

\end{bmatrix} = F^{\mu\nu}.

The covariant form is given by index lowering,

F_{\mu\nu} = \eta_{\mu\alpha}F^{\alpha\beta}\eta_{\beta\nu} = \begin{bmatrix}

0 & E_x/c & E_y/c & E_z/c \\

-E_x/c & 0 & -B_z & B_y \\

-E_y/c & B_z & 0 & -B_x \\

-E_z/c & -B_y & B_x & 0

\end{bmatrix}.

The mixed-variance form appears in the Lorentz force equation when using the contravariant four-velocity: \frac{d p^\mu}{d \tau} = q F^{\mu}{}_{\nu} u^\nu , where

F^{\mu}{}_{\nu} = F^{\mu\beta}\eta_{\beta\nu} = \begin{bmatrix}

0 & E_x/c & E_y/c & E_z/c \\

E_x/c & 0 & B_z & -B_y \\

E_y/c & -B_z & 0 & B_x \\

E_z/c & B_y & -B_x & 0

\end{bmatrix}.

From now on in this article, when the electric or magnetic fields are mentioned, a Cartesian coordinate system is being assumed, and the electric and magnetic fields are with respect to coordinate system's own reference frame, as in the equations above.

Significance[edit]

This tensor simplifies and reduces Maxwell's equations as four vector calculus equations into two tensor field equations. In electrostatics and electrodynamics, Gauss's law and Ampère's circuital law are respectively:

\nabla \cdot \mathbf{E} = \frac{\rho}{\epsilon_0},\quad \nabla \times \mathbf{B} - \frac{1}{c^2} \frac{ \partial \mathbf{E}}{\partial t} = \mu_0 \mathbf{J}

and reduce to the inhomogeneous Maxwell equation:

\partial_{\alpha} F^{\alpha\beta} = \mu_0 J^{\beta}

where

J^{\alpha} = ( c\rho, \mathbf{J} )

is the 4-current. In magnetostatics and magnetodynamics, Gauss's law for magnetism and Maxwell–Faraday equation are respectively:

\nabla \cdot \mathbf{B} = 0,\quad \frac{ \partial \mathbf{B}}{ \partial t } + \nabla \times \mathbf{E} = 0

which reduce to Bianchi identity:

\partial_\gamma F_{ \alpha \beta } + \partial_\alpha F_{ \beta \gamma } + \partial_\beta F_{ \gamma \alpha } = 0

or using the index notation with square brackets[note 1] for the antisymmetric part of the tensor:

\partial_{ [ \alpha } F_{ \beta \gamma ] } = 0

QMRAlthough there appear to be 64 equations in Faraday-Gauss, it actually reduces to just four independent equations. Using the antisymmetry of the electromagnetic field one can either reduce to an identity (0 = 0) or render redundant all the equations except for those with λ,μ,ν = either 1,2,3 or 2,3,0 or 3,0,1 or 0,1,2.

maxwells equations in curved spacetime wiki

matrix representation for maxwells equations wiki

QMrIn electromagnetism, a branch of fundamental physics, the matrix representations of the Maxwell's equations are a formulation of Maxwell's equations using matrices, complex numbers, and vector calculus. These representations are for a homogeneous medium, an approximation in an inhomogeneous medium. A matrix representation for an inhomogeneous medium was presented using a pair of matrix equations.[1] A single equation using 4 × 4 matrices is necessary and sufficient for any homogeneous medium. For an inhomogeneous medium it necessarily requires 8 × 8 matrices.[2]

A 4 by 4 matrix is a quadrant model

Introduction[edit]

Maxwell's equations in the standard vector calculus formalism, in an inhomogeneous medium with sources, are:[3]

\begin{align}

& {\mathbf \nabla} \cdot {\mathbf D} \left({\mathbf r} , t \right) =

\rho\, \\

& {\mathbf \nabla} \times {\mathbf H} \left({\mathbf r} , t \right)

- \frac{\partial }{\partial t}

{\mathbf D} \left({\mathbf r} , t \right) =

{\mathbf J}\, \\

& {\mathbf \nabla} \times {\mathbf E} \left({\mathbf r} , t \right)

+

\frac{\partial }{\partial t}

{\mathbf B} \left({\mathbf r} , t \right) = 0\, \\

& {\mathbf \nabla} \cdot {\mathbf B} \left({\mathbf r} , t \right) = 0\,.

\end{align}

The media is assumed to be linear, that is

{\mathbf D} = \epsilon {\mathbf E}\,,\quad {\mathbf B} = \mu {\mathbf H},

where ε = ε(r, t) is the permittivity of the medium and μ = μ(r, t) the permeability of the medium (see constitutive equation). For a homogeneous medium ε and μ are constants. The speed of light in the medium is given by

v ({\mathbf r} , t) = \frac{1}{\sqrt{\epsilon ({\mathbf r} , t) \mu ({\mathbf r} , t)}}.

In vacuum, ε0 = 8.85 × 10−12 C2·N−1·m−2 and μ0 = 4π × 10−7 H·m−1

One possible way to obtain the required matrix representation is to use the Riemann-Silberstein vector [4] given by

\begin{align}

{\mathbf F}^{+} \left({\mathbf r} , t \right)

& =

\frac{1}{\sqrt{2}}

\left(

\sqrt{\epsilon ({\mathbf r} , t)} {\mathbf E} \left({\mathbf r} , t \right)

+ {\rm i} \frac{1}{\sqrt{\mu ({\mathbf r} , t)}} {\mathbf B} \left({\mathbf r} , t \right) \right) \\

{\mathbf F}^{-} \left({\mathbf r} , t \right)

& =

\frac{1}{\sqrt{2}}

\left(

\sqrt{\epsilon ({\mathbf r} , t)} {\mathbf E} \left({\mathbf r} , t \right)

- {\rm i} \frac{1}{\sqrt{\mu ({\mathbf r} , t)}} {\mathbf B} \left({\mathbf r} , t \right) \right)\,.

\end{align}

If for a certain medium ε = ε(r, t) and μ = μ(r, t) are constants (or can be treated as local constants under certain approximations), then the vectors F± (r, t) satisfy

\begin{align}

{\rm i} \frac{\partial }{\partial t} {\mathbf F}^{\pm} \left({\mathbf r} , t \right)

& =

\pm v {\mathbf \nabla} \times {\mathbf F}^{\pm} \left({\mathbf r} , t \right)

- \frac{1}{\sqrt{2 \epsilon}} ({\rm i} {\mathbf J}) \\

{\mathbf \nabla} \cdot {\mathbf F}^{\pm} \left({\mathbf r} , t \right)

& =

\frac{1}{\sqrt{2 \epsilon}} (\rho)\,.

\end{align}

Thus by using the Riemann-Silberstein vector, it is possible to reexpress the Maxwell's equations for a medium with constant ε = ε(r, t) and μ = μ(r, t) as a pair of equations.

Homogeneous medium[edit]

In order to obtain a single matrix equation instead of a pair, the following new functions are constructed using the components of the Riemann-Silberstein vector[5]

\begin{align}

\Psi^{+} ({\mathbf r} , t)

& =

\left[

\begin{array}{c}

- F_x^{+} + {\rm i} F_y^{+} \\

F_z^{+} \\

F_z^{+} \\

F_x^{+} + {\rm i} F_y^{+}

\end{array}

\right]\, \quad

\Psi^{-} ({\mathbf r} , t) =

\left[

\begin{array}{c}

- F_x^{-} - {\rm i} F_y^{-} \\

F_z^{-} \\

F_z^{-} \\

F_x^{-} - {\rm i} F_y^{-}

\end{array}

\right]\,.

\end{align}

The vectors for the sources are

\begin{align}

W^{+}

&=

\left(\frac{1}{\sqrt{2 \epsilon}}\right)

\left[

\begin{array}{c}

- J_x + {\rm i} J_y \\

J_z - v \rho \\

J_z + v \rho \\

J_x + {\rm i} J_y

\end{array}

\right]\, \quad

W^{-}

=

\left(\frac{1}{\sqrt{2 \epsilon}}\right)

\left[

\begin{array}{c}

- J_x - {\rm i} J_y \\

J_z - v \rho \\

J_z + v \rho \\

J_x - {\rm i} J_y

\end{array}

\right]\,.

\end{align}

Then,

\begin{align}

\frac{\partial}{\partial t}

\Psi^{+}

&=

- v

\left\{ {\mathbf M} \cdot {\mathbf \nabla} \right\} \Psi^{+}

- W^{+}\, \\

\frac{\partial}{\partial t}

\Psi^{-}

& =

- v

\left\{ {\mathbf M}^{*} \cdot {\mathbf \nabla} \right\} \Psi^{-}

- W^{-}\, \end{align}

where * denotes complex conjugation and the triplet, M = (Mx, My, Mz) is expressed in terms of

\Omega =

\begin{pmatrix}

{\mathbf 0} & - {\mathbf l} \\

{\mathbf l} & {\mathbf 0}

\end{pmatrix}

\, \qquad

\beta =

\begin{pmatrix}

{\mathbf l} & {\mathbf 0} \\

{\mathbf 0} & - {\mathbf l}

\end{pmatrix}

\, \qquad

{\mathbf l} =

\begin{pmatrix}

1 & 0 \\

0 & 1

\end{pmatrix}

\,.

Alternately, one may use the matrix J = −Ω. Both differ by a sign. For our purpose it is fine to use either Ω or J. However, they have a different meaning: J is contravariant and Ω is covariant. The matrix Ω corresponds to the Lagrange brackets of classical mechanics and J corresponds to the Poisson brackets. An important relation is Ω = J−1. The M-matrices are

M_x =

\begin{bmatrix}

0 & 0 & 1 & 0 \\

0 & 0 & 0 & 1 \\

1 & 0 & 0 & 0 \\

0 & 1 & 0 & 0

\end{bmatrix} =

- \beta \Omega\,

M_y =

\begin{bmatrix}

0 & 0 & - {\rm i} & 0 \\

0 & 0 & 0 & - {\rm i} \\

{\rm i} & 0 & 0 & 0 \\

0 & {\rm i} & 0 & 0

\end{bmatrix} =

{\rm i} \Omega\,

M_z =

\begin{bmatrix}

1 & 0 & 0 & 0 \\

0 & 1 & 0 & 0 \\

0 & 0 & -1 & 0 \\

0 & 0 & 0 & - 1

\end{bmatrix} =

\beta \,.

Each of the four Maxwell's equations are obtained from the matrix representation. This is done by taking the sums and differences of row-I with row-IV and row-II with row-III respectively. The first three give the y, x and z components of the curl and the last one gives the divergence conditions.

It is to be noted that the matrices M are all non-singular and all are Hermitian. Moreover, they satisfy the usual algebra of the Dirac matrices, including,

\begin{align}

M_x \beta = - \beta M_x\, \\

M_y \beta = - \beta M_y\, \\

M_x^2 = M_y^2 = M_z^2 = I\, \\

M_x M_y = - M_y M_x = {\rm i} M_z\, \\

M_y M_z = - M_z M_y = {\rm i} M_x\, \\

M_z M_x = - M_x M_z = {\rm i} M_y\,.

\end{align}

It is to be noted that the (Ψ±, M) are not unique. Different choices of Ψ± would give rise to different M, such that the triplet M continues to satisfy the algebra of the Dirac matrices. The Ψ± via the Riemann-Silberstein vector has certain advantages over the other possible choices.[6] The Riemann-Silberstein vector is well known in classical electrodynamics and has certain interesting properties and uses.[7]

In deriving the above 4 × 4 matrix representation of the Maxwell's equations, the spatial and temporal derivatives of ε(r, t) and μ(r, t) in the first two of the Maxwell's equations have been ignored. The ε and μ have been treated as local constants.

Inhomogeneous medium[edit]

In an inhomogeneous medium, the spatial and temporal variations of ε = ε(r, t) and μ = μ(r, t) are not zero. That is they are no longer local constant. Instead of using ε = ε(r, t) and μ = μ(r, t), it is advantageous to use the two derived laboratory functions namely the resistance function and the velocity function

\begin{align}

\text{ Velocity function:} \, v ({\mathbf r} , t)

& =

\frac{1}{\sqrt{\epsilon ({\mathbf r} , t) \mu ({\mathbf r} , t)}} \\

\text{Resistance function:} \, h ({\mathbf r} , t)

& =

\sqrt{\frac{\mu ({\mathbf r} , t)}{\epsilon ({\mathbf r} , t)}}\,.

\end{align}

In terms of these functions:

\varepsilon = \frac{1}{v h}\,,\quad \mu = \frac{h}{v}.

These functions occur in the matrix representation through their logarithmic derivatives;

\begin{align}

{\mathbf u} ({\mathbf r} , t)

& =

\frac{1}{2 v ({\mathbf r} , t)} {\mathbf \nabla} v ({\mathbf r} , t) =

\frac{1}{2} {\mathbf \nabla} \left\{\ln v ({\mathbf r} , t) \right\} =

- \frac{1}{2} {\mathbf \nabla} \left\{\ln n ({\mathbf r} , t) \right\} \\

{\mathbf w} ({\mathbf r} , t) &=

\frac{1}{2 h ({\mathbf r} , t)} {\mathbf \nabla} h ({\mathbf r} , t) =

\frac{1}{2} {\mathbf \nabla} \left\{\ln h ({\mathbf r} , t) \right\}\, \end{align}

where

n ({\mathbf r} , t) = \frac{c}{v ({\mathbf r} , t)}

is the refractive index of the medium.

The following matrices naturally arise in the exact matrix representation of the Maxwell's equation in a medium

\begin{align}

{\mathbf \Sigma} =

\left[

\begin{array}{cc}

{\mathbf \sigma} & {\mathbf 0} \\

{\mathbf 0} & {\mathbf \sigma}

\end{array}

\right]\, \qquad

{\mathbf \alpha} =

\left[

\begin{array}{cc}

{\mathbf 0} & {\mathbf \sigma} \\

{\mathbf \sigma} & {\mathbf 0}

\end{array}

\right]\, \qquad

{\mathbf I} =

\left[

\begin{array}{cc}

{\mathbf 1} & {\mathbf 0} \\

{\mathbf 0} & {\mathbf 1}

\end{array}

\right]\, \end{align}

where Σ are the Dirac spin matrices and α are the matrices used in the Dirac equation, and σ is the triplet of the Pauli matrices

{\mathbf \sigma} = (\sigma_x , \sigma_y , \sigma_z) =

\left[

\begin{pmatrix}

0 & 1 \\

1 & 0

\end{pmatrix}

,

\begin{pmatrix}

0 & - {\rm i} \\

{\rm i} & 0

\end{pmatrix}

,

\begin{pmatrix}

1 & 0 \\

0 & -1

\end{pmatrix}

\right]

Finally, the matrix representation is

\begin{align}

&

\frac{\partial }{\partial t}

\left[

\begin{array}{cc}

{\mathbf I} & {\mathbf 0} \\

{\mathbf 0} & {\mathbf I}

\end{array}

\right]

\left[

\begin{array}{cc}

\Psi^{+} \\

\Psi^{-}

\end{array}

\right]

-

\frac{\dot{v} ({\mathbf r} , t)}{2 v ({\mathbf r} , t)}

\left[

\begin{array}{cc}

{\mathbf I} & {\mathbf 0} \\

{\mathbf 0} & {\mathbf I}

\end{array}

\right]

\left[

\begin{array}{cc}

\Psi^{+} \\

\Psi^{-}

\end{array}

\right]

+ \frac{\dot{h} ({\mathbf r} , t)}{2 h ({\mathbf r} , t)}

\left[

\begin{array}{cc}

{\mathbf 0} & {\rm i} \beta \alpha_y \\

{\rm i} \beta \alpha_y & {\mathbf 0}

\end{array}

\right]

\left[

\begin{array}{cc}

\Psi^{+} \\

\Psi^{-}

\end{array}

\right] \\

& = - v ({\mathbf r} , t)

\left[

\begin{array}{ccc}

\left\{

{\mathbf M} \cdot {\mathbf \nabla}

+

{\mathbf \Sigma} \cdot {\mathbf u}

\right\}

& &

- {\rm i} \beta

\left({\mathbf \Sigma} \cdot {\mathbf w}\right)

\alpha_y

\\

- {\rm i} \beta

\left({\mathbf \Sigma}^{*} \cdot {\mathbf w}\right)

\alpha_y

&

\left\{

{\mathbf M}^{*} \cdot {\mathbf \nabla}

+

{\mathbf \Sigma}^{*} \cdot {\mathbf u}

\right\}

\end{array}

\right]

\left[

\begin{array}{cc}

\Psi^{+} \\

\Psi^{-}

\end{array}

\right]

- \left[

\begin{array}{cc}

{\mathbf I} & {\mathbf 0} \\

{\mathbf 0} & {\mathbf I}

\end{array}

\right]

\left[

\begin{array}{c}

W^{+} \\

W^{-}

\end{array}

\right]\,

\end{align}

The above representation contains thirteen 8 × 8 matrices. Ten of these are Hermitian. The exceptional ones are the ones that contain the three components of w(r, t), the logarithmic gradient of the resistance function. These three matrices, for the resistance function are antihermitian.

The Maxwell's equations have been expressed in a matrix form for a medium with varying permittivity ε = ε(r, t) and permeability μ = μ(r, t), in presence of sources. This representation uses a single matrix equation, instead of a pair of matrix equations. In this representation, using 8 × 8 matrices, it has been possible to separate the dependence of the coupling between the upper components (Ψ+) and the lower components (Ψ−) through the two laboratory functions. Moreover the exact matrix representation has an algebraic structure very similar to the Dirac equation.[8] It is interesting to note that the Maxwell's equations can be derived from the Fermat's principle of geometrical optics by the process of "wavization"[clarification needed] analogous to the quantization of classical mechanics.

An 8 by 8 matrix is four quadrant models

galilean transformation wiki

qMRMathematically, the Lorentz group may be described as the generalized orthogonal group O(1,3), the matrix Lie group that preserves the quadratic form

(t,x,y,z) \mapsto t^2-x^2-y^2-z^2

on R4. This quadratic form is, when put on matrix form (see classical orthogonal group), interpreted in physics as the metric tensor of Minkowski spacetime.

Lorentz group wiki

Connected components[edit]

Light cone in 2D space plus a time dimension.

Because it is a Lie group, the Lorentz group O(1,3) is both a group and admits a topological description as a smooth manifold. As a manifold, it has four connected components. Intuitively, this means that it consists of four topologically separated pieces.

The four connected components can be categorized by two transformational properties its elements have:

some elements are reversed under time-inverting Lorentz transformations, for example, a future-pointing timelike vector would be invert to a past-pointing vector

some elements have orientation reversed by improper Lorentz transformations, for example, certain vierbein (tetrads)

The subgroup of all Lorentz transformations preserving both orientation and direction of time is called the proper, orthochronous Lorentz group or restricted Lorentz group, and is denoted by SO+(1, 3). (Note that some authors refer to SO(1,3) or even O(1,3) when they actually mean SO+(1, 3).)

The set of the four connected components can be given a group structure as the quotient group O(1,3)/SO+(1,3), which is isomorphic to the Klein four-group. Every element in O(1,3) can be written as the semidirect product of a proper, orthochronous transformation and an element of the discrete group

{1, P, T, PT}

where P and T are the space inversion and time reversal operators:

P = diag(1, −1, −1, −1)

T = diag(−1, 1, 1, 1).

Thus an arbitrary Lorentz transformation can be specified as a proper, orthochronous Lorentz transformation along with a further two bits of information, which pick out one of the four connected components. This pattern is typical of finite-dimensional Lie groups.

Parabolic Lorentz transformations are often called null rotations, since they preserve null vectors, just as rotations preserve timelike vectors and boosts preserve spacelike vectors. Since these are likely to be the least familiar of the four types of nonidentity Lorentz transformations (elliptic, hyperbolic, loxodromic, parabolic), it is illustrated here how to determine the effect of an example of a parabolic Lorentz transformation on Minkowski spacetime.

The matrix given above yields the transformation

\left[ \begin{matrix} t \\ x \\ y \\ z \end{matrix} \right]

\rightarrow

\left[ \begin{matrix} t \\ x \\ y \\ z \end{matrix} \right]

+ \operatorname{Re}(\alpha) \;

\left[ \begin{matrix} x \\ t-z \\ 0 \\ x \end{matrix} \right]

+ \operatorname{Im}(\alpha) \;

\left[ \begin{matrix} y \\ 0 \\ z-t \\ y \end{matrix} \right]

+ \frac{\vert\alpha\vert^2}{2} \;

\left[ \begin{matrix} t-z \\ 0 \\ 0 \\ t-z \end{matrix} \right].

QMRGimbal lock in three dimensions[edit]

Gimbal locked airplane. When the pitch (green) and yaw (magenta) gimbals become aligned, changes to roll (blue) and yaw apply the same rotation to the airplane.

Normal situation: the three gimbals are independent

Gimbal lock: two out of the three gimbals are in the same plane, one degree of freedom is lost

Consider a case of a level sensing platform on an aircraft flying due north with its three gimbal axes mutually perpendicular (i.e., roll, pitch and yaw angles each zero). If the aircraft pitches up 90 degrees, the aircraft and platform's yaw axis gimbal becomes parallel to the roll axis gimbal, and changes about yaw can no longer be compensated for.

Solutions[edit]

This problem may be overcome by use of a fourth gimbal, intelligently driven by a motor so as to maintain a large angle between roll and yaw gimbal axes. Another solution is to rotate one or more of the gimbals to an arbitrary position when gimbal lock is detected and thus reset the device.

Modern practice is to avoid the use of gimbals entirely. In the context of inertial navigation systems, that can be done by mounting the inertial sensors directly to the body of the vehicle (this is called a strapdown system)[3] and integrating sensed rotation and acceleration digitally using quaternion methods to derive vehicle orientation and velocity. Another way to replace gimbals is to use fluid bearings or a flotation chamber.[4]

Gimbal lock on Apollo 11[edit]

A well-known gimbal lock incident happened in the Apollo 11 Moon mission. On this spacecraft, a set of gimbals was used on an inertial measurement unit (IMU). The engineers were aware of the gimbal lock problem but had declined to use a fourth gimbal.[5] Some of the reasoning behind this decision is apparent from the following quote:

"The advantages of the redundant gimbal seem to be outweighed by the equipment simplicity, size advantages, and corresponding implied reliability of the direct three degree of freedom unit."

— David Hoag, Apollo Lunar Surface Journal

They preferred an alternate solution using an indicator that would be triggered when near to 85 degrees pitch.

"Near that point, in a closed stabilization loop, the torque motors could theoretically be commanded to flip the gimbal 180 degrees instantaneously. Instead, in the LM, the computer flashed a 'gimbal lock' warning at 70 degrees and froze the IMU at 85 degrees"

— Paul Fjeld, Apollo Lunar Surface Journal

Rather than try to drive the gimbals faster than they could go, the system simply gave up and froze the platform. From this point, the spacecraft would have to be manually moved away from the gimbal lock position, and the platform would have to be manually realigned using the stars as a reference.[6]

After the Lunar Module had landed, Mike Collins aboard the Command Module joked "How about sending me a fourth gimbal for Christmas?"

Robotics[edit]

Industrial robot operating in a foundry.

In robotics, gimbal lock is commonly referred to as "wrist flip", due to the use of a "triple-roll wrist" in robotic arms, where three axes of the wrist, controlling yaw, pitch, and roll, all pass through a common point.

An example of a wrist flip, also called a wrist singularity, is when the path through which the robot is traveling causes the first and third axes of the robot's wrist to line up. The second wrist axis then attempts to spin 180° in zero time to maintain the orientation of the end effector. The result of a singularity can be quite dramatic and can have adverse effects on the robot arm, the end effector, and the process.

The importance of avoiding singularities in robotics has led the American National Standard for Industrial Robots and Robot Systems — Safety Requirements to define it as "a condition caused by the collinear alignment of two or more robot axes resulting in unpredictable robot motion and velocities".[7]

QMRThe Elysium quadrangle is one of a series of 30 quadrangle maps of Mars used by the United States Geological Survey (USGS) Astrogeology Research Program. The Elysium quadrangle is also referred to as MC-15 (Mars Chart-15).[1]

The Elysium quadrangle covers the area 180° to 225° west longitude and 0° to 30° north latitude on Mars. Elysium Planitia is in the Elysium quadrangle. The Elysium quadrangle includes a part of Lucus Planum. A small part of the Medusae Fossae Formation lies in this quadrangle. The largest craters in this quadrangle are Eddie, Lockyer, and Tombaugh. Elysium contains major volcanoes named Elysium Mons and Albor Tholus and river valleys—one of which, Athabasca Valles may be one of the youngest on Mars. On the east side is an elongated depression called Orcus Patera. A large lake may once have existed in the south near Lethe Valles and Athabasca Valles.[2]

Memnonia quadrangle

From Wikipedia, the free encyclopedia

Memnonia quadrangle

USGS-Mars-MC-16-MemnoniaRegion-mola.png

Map of Memnonia quadrangle from Mars Orbiter Laser Altimeter (MOLA) data. The highest elevations are red and the lowest are blue.

Coordinates 15°S 157.5°WCoordinates: 15°S 157.5°W

Image of the Memnonia Quadrangle (MC-16). The south includes heavily cratered highlands intersected, in the northeastern part, by Mangala Vallis. The north contains undulating wind-eroded deposits and the east contains lava flows from the Tharsis region.

The Memnonia quadrangle is one of a series of 30 quadrangle maps of Mars used by the United States Geological Survey (USGS) Astrogeology Research Program. The Memnonia quadrangle is also referred to as MC-16 (Mars Chart-16).[1]

The quadrangle is a region of Mars that covers latitude -30° to 0° and longitude 135° to 180°.[2] The western part of Memnonia is a highly cratered highland region that exhibits a large range of crater degradation.

Memnonia includes these topographical regions of Mars:

Arcadia Planitia

Amazonis Planitia

Lucus Planum

Terra Sirenum

Daedalia Planum

Terra Cimmeria

Recently, evidence of water was found in the area. Layered sedimentary rocks were found in the wall and floor of Columbus Crater. These rocks could have been deposited by water or by wind. Hydrated minerals were found in some of the layers, so water may have been involved.[3]

Many ancient river valleys Vallis including Mangala Vallis, have been found in the Memnonia quadrangle. Mangala appears to have begun with the formation of a graben, a set of faults that may have exposed an aquifer.[4] Dark slope streaks and troughts (fossae) are present in this quadrangle. Part of the Medusae Fossae Formation is found in the Memnonia quadrangle.