Coherence was originally conceived in connection with Thomas Young's double-slit experiment in optics but is now used in any field that involves waves, such as acoustics, electrical engineering, neuroscience, and quantum mechanics. Coherence describes the statistical similarity of a field (electromagnetic field, quantum wave packet etc.) at two points in space or time. The property of coherence is the basis for commercial applications such as holography, the Sagnac gyroscope, radio antenna arrays, optical coherence tomography and telescope interferometers (astronomical optical interferometers and radio telescopes).


A precise definition is given in the article on degree of coherence.


The coherence function between two signals formula_1 and formula_2 is defined as


The coherence varies in the interval formula_11. If formula_12 it means that the signals are perfectly correlated or linearly related and if formula_13 they are totally uncorrelated. If a linear system is being measured, formula_1 being the input and formula_2 the output, the coherence function will be unitary all over the spectrum. However, if non-linearities are present in the system the coherence will vary in the limit given above.


These states are unified by the fact that their behavior is described by a wave equation or some generalization thereof.


In most of these systems, one can measure the wave directly. Consequently, its correlation with another wave can simply be calculated. However, in optics one cannot measure the electric field directly as it oscillates much faster than any detector's time resolution. Instead, one measures the intensity of the light. Most of the concepts involving coherence which will be introduced below were developed in the field of optics and then used in other fields. Therefore, many of the standard measurements of coherence are indirect measurements, even in fields where the wave can be measured directly.


Temporal coherence is the measure of the average correlation between the value of a wave and itself delayed by τ, at any pair of times. Temporal coherence tells us how monochromatic a source is. In other words, it characterizes how well a wave can interfere with itself at a different time. The delay over which the phase or amplitude wanders by a significant amount (and hence the correlation decreases by significant amount) is defined as the coherence time "τc". At a delay of τ=0 the degree of coherence is perfect, whereas it drops significantly as the delay passes "τ=τc". The coherence length "Lc" is defined as the distance the wave travels in time τc.


One should be careful not to confuse the coherence time with the time duration of the signal, nor the coherence length with the coherence area (see below).


The relationship between coherence time and bandwidth.


It can be shown that the larger the range of frequencies Δf a wave contains, the faster the wave decorrelates (and hence the smaller τc is). Thus there is a tradeoff:


Formally, this follows from the convolution theorem in mathematics, which relates the Fourier transform of the power spectrum (the intensity of each frequency) to its autocorrelation.


We consider four examples of temporal coherence.


The high degree of monochromaticity of lasers implies long coherence lengths (up to hundreds of meters). For example, a stabilized and monomode helium–neon laser can easily produce light with coherence lengths of 300 m. Not all lasers have a high degree of monochromaticity, however (e.g. for a mode-locked Ti-sapphire laser, Δλ ≈ 2 nm - 70 nm). LEDs are characterized by Δλ ≈ 50 nm, and tungsten filament lights exhibit Δλ ≈ 600 nm, so these sources have shorter coherence times than the most monochromatic lasers.


Holography requires light with a long coherence time. In contrast, optical coherence tomography, in its classical version, uses light with a short coherence time.


Holography requires temporally and spatially coherent light. Its inventor, Dennis Gabor, produced successful holograms more than ten years before lasers were invented. To produce coherent light he passed the monochromatic light from an emission line of a mercury-vapor lamp through a pinhole spatial filter.


In February 2011 it was reported that helium atoms, cooled to near absolute zero / Bose–Einstein condensate state, can be made to flow and behave as a coherent beam as occurs in a laser.


Waves of different frequencies (in light these are different colours) can interfere to form a pulse if they have a fixed relative phase-relationship (see Fourier transform). Conversely, if waves of different frequencies are not coherent, then, when combined, they create a wave that is continuous in time (e.g. white light or white noise). The temporal duration of the pulse formula_19 is limited by the spectral bandwidth of the light formula_20 according to:


which follows from the properties of the Fourier transform and results in Küpfmüller's uncertainty principle (for quantum particles it also results in the Heisenberg uncertainty principle).


If the phase depends linearly on the frequency (i.e. formula_22) then the pulse will have the minimum time duration for its bandwidth (a "transform-limited" pulse), otherwise it is chirped (see dispersion).


Measurement of the spectral coherence of light requires a nonlinear optical interferometer, such as an intensity optical correlator, frequency-resolved optical gating (FROG), or spectral phase interferometry for direct electric-field reconstruction (SPIDER).


Light also has a polarization, which is the direction in which the electric field oscillates. Unpolarized light is composed of incoherent light waves with random polarization angles. The electric field of the unpolarized light wanders in every direction and changes in phase over the coherence time of the two light waves. An absorbing polarizer rotated to any angle will always transmit half the incident intensity when averaged over time.


If the electric field wanders by a smaller amount the light will be partially polarized so that at some angle, the polarizer will transmit more than half the intensity. If a wave is combined with an orthogonally polarized copy of itself delayed by less than the coherence time, partially polarized light is created.


The polarization of a light beam is represented by a vector in the Poincaré sphere. For polarized light the end of the vector lies on the surface of the sphere, whereas the vector has zero length for unpolarized light. The vector for partially polarized light lies within the sphere


Coherent superpositions of "optical wave fields" include holography. Holographic objects are used frequently in daily life in television and credit card security.


Further applications concern the coherent superposition of "non-optical wave fields". In quantum mechanics for example one considers a probability field, which is related to the wave function formula_23 (interpretation: density of the probability amplitude). Here the applications concern, among others, the future technologies of quantum computing and the already available technology of quantum cryptography. Additionally the problems of the following subchapter are treated.


Coherence is used to check the quality of the transfer functions (FRFs) being measured. Low coherence can be caused by poor signal to noise ratio, and/or inadequate frequency resolution.


According to quantum mechanics, all objects can have wave-like properties (see de Broglie waves). For instance, in Young's double-slit experiment electrons can be used in the place of light waves. Each electron's wave-function goes through both slits, and hence has two separate split-beams that contribute to the intensity pattern on a screen. According to standard wave theory these two contributions give rise to an intensity pattern of bright bands due to constructive interference, interlaced with dark bands due to destructive interference, on a downstream screen. This ability to interfere and diffract is related to coherence (classical or quantum) of the waves produced at both slits. The association of an electron with a wave is unique to quantum theory.


When the incident beam is represented by a quantum pure state, the split beams downstream of the two slits are represented as a superposition of the pure states representing each split beam. The quantum description of imperfectly coherent paths is called a mixed state. A perfectly coherent state has a density matrix (also called the "statistical operator") that is a projection onto the pure coherent state and is equivalent to a wave function, while a mixed state is described by a classical probability distribution for the pure states that make up the mixture.


Macroscopic scale quantum coherence leads to novel phenomena, the so-called macroscopic quantum phenomena. For instance, the laser, superconductivity and superfluidity are examples of highly coherent quantum systems whose effects are evident at the macroscopic scale. The macroscopic quantum coherence (off-diagonal long-range order, ODLRO) for superfluidity, and laser light, is related to first-order (1-body) coherence/ODLRO, while superconductivity is related to second-order coherence/ODLRO. (For fermions, such as electrons, only even orders of coherence/ODLRO are possible.) For bosons, a Bose–Einstein condensate is an example of a system exhibiting macroscopic quantum coherence through a multiple occupied single-particle state.


The classical electromagnetic field exhibits macroscopic quantum coherence. The most obvious example is the carrier signal for radio and TV. They satisfy Glauber's quantum description of coherence.


Recently M. B. Plenio and co-workers constructed an operational formulation of quantum coherence as a resource theory. They introduced coherence monotones analogous to the entanglement monotones. Quantum coherence has been shown to be equivalent to quantum entanglement in the sense that coherence can be faithfully described as entanglement, and conversely that each entanglement measure corresponds to a coherence measure.


In analytic geometry, spatial transformations in the 3-dimensional Euclidean space formula_1 are distinguished into active or alibi transformations, and passive or alias transformations. An active transformation is a transformation which actually changes the physical position (alibi, elsewhere) of a point, or rigid body, which can be defined in the absence of a coordinate system; whereas a passive transformation is merely a change in the coordinate system in which the object is described (alias, other name) (change of coordinate map, or change of basis). By "transformation", mathematicians usually refer to active transformations, while physicists and engineers could mean either. Both types of transformation can be represented by a combination of a translation and a linear transformation.


Put differently, a "passive" transformation refers to description of the "same" object in two different coordinate systems.


On the other hand, an "active transformation" is a transformation of one or more objects with respect to the same coordinate system. For instance, active transformations are useful to describe successive positions of a rigid body. On the other hand, passive transformations may be useful in human motion analysis to observe the motion of the tibia relative to the femur, that is, its motion relative to a ("local") coordinate system which moves together with the femur, rather than a ("global") coordinate system which is fixed to the floor.


As an example, let the vector formula_2, be a vector in the plane. A rotation of the vector through an angle θ in counterclockwise direction is given by the rotation matrix:


which can be viewed either as an "active transformation" or a "passive transformation" (where the above matrix will be inverted), as described below.


Spatial transformations in the Euclidean space formula_1.


In general a spatial transformation formula_5 may consist of a translation and a linear transformation. In the following, the translation will be omitted, and the linear transformation will be represented by a 3×3-matrix formula_6.


As an active transformation, formula_6 transforms the initial vector formula_8 into a new vector formula_9.


If one views formula_10 as a new basis, then the coordinates of the new vector formula_11 in the new basis are the same as those of formula_12 in the original basis. Note that active transformations make sense even as a linear transformation into a different vector space. It makes sense to write the new vector in the unprimed basis (as above) only when the transformation is from the space into itself.


On the other hand, when one views formula_6 as a passive transformation, the initial vector formula_8 is left unchanged, while the coordinate system and its basis vectors are transformed in the opposite direction, that is, with the inverse transformation formula_15.


This gives a new coordinate system XYZ with basis vectors:


The new coordinates formula_17 of formula_18 with respect to the new coordinate system XYZ are given by:


From this equation one sees that the new coordinates are given by


As a passive transformation formula_6 transforms the old coordinates into the new ones.


Note the equivalence between the two kinds of transformations: the coordinates of the new point in the active transformation and the new coordinates of the point in the passive transformation are the same, namely


A point particle (ideal particle or point-like particle, often spelled pointlike particle) is an idealization of particles heavily used in physics. Its defining feature is that it lacks spatial extension; being dimensionless, it does not take up space. A point particle is an appropriate representation of any object whenever its size, shape, and structure are irrelevant in a given context. For example, from far enough away, any finite-size object will look and behave as a point-like object. A point particle can also be referred in the case of a moving body in terms of physics.


In the theory of gravity, physicists often discuss a ', meaning a point particle with a nonzero mass and no other properties or structure. Likewise, in electromagnetism, physicists discuss a ', a point particle with a nonzero charge.


Sometimes, due to specific combinations of properties, extended objects behave as point-like even in their immediate vicinity. For example, spherical objects interacting in 3-dimensional space whose interactions are described by the inverse square law behave in such a way as if all their matter were concentrated in their centers of mass. In Newtonian gravitation and classical electromagnetism, for example, the respective fields outside a spherical object are identical to those of a point particle of equal charge/mass located at the center of the sphere.


In quantum mechanics, the concept of a point particle is complicated by the Heisenberg uncertainty principle, because even an elementary particle, with no internal structure, occupies a nonzero volume. For example, the atomic orbit of an electron in the hydrogen atom occupies a volume of ~10−30 m3. There is nevertheless a distinction between elementary particles such as electrons or quarks, which have no known internal structure, versus composite particles such as protons, which do have internal structure: A proton is made of three quarks.


Elementary particles are sometimes called "point particles", but this is in a different sense than discussed above.


When a point particle has an additive property, such as mass or charge, concentrated at a single point in space, this can be represented by a Dirac delta function.


Point mass (pointlike mass) is the concept, for example in classical physics, of a physical object (typically matter) that has nonzero mass, and yet explicitly and specifically is (or is being thought of or modeled as) infinitesimal (infinitely small) in its volume or linear dimensions.


A common use for point mass lies in the analysis of the gravitational fields. When analyzing the gravitational forces in a system, it becomes impossible to account for every unit of mass individually. However, a spherically symmetric body affects external objects gravitationally as if all of its mass were concentrated at its center.


A point mass in probability and statistics does not refer to mass in the sense of physics, but rather refers to a finite nonzero probability that is concentrated at a point in the probability mass distribution, where there is a discontinuous segment in a probability density function. To calculate such a point mass, an integration is carried out over the entire range of the random variable, on the probability density of the continuous part. After equating this integral to 1, the point mass can be found by further calculation.


A point charge is an idealized model of a particle which has an electric charge. A point charge is an electric charge at a mathematical point with no dimensions.


The fundamental equation of electrostatics is Coulomb's law, which describes the electric force between two point charges. The electric field associated with a classical point charge increases to infinity as the distance from the point charge decreases towards zero making energy (thus mass) of point charge infinite.


Earnshaw's theorem states that a collection of point charges cannot be maintained in an equilibrium configuration solely by the electrostatic interaction of the charges.


In quantum mechanics, there is a distinction between an elementary particle (also called "point particle") and a composite particle. An elementary particle, such as an electron, quark, or photon, is a particle with no internal structure. Whereas a composite particle, such as a proton or neutron, has an internal structure (see figure).


However, neither elementary nor composite particles are spatially localized, because of the Heisenberg uncertainty principle. The particle wavepacket always occupies a nonzero volume. For example, see atomic orbital: The electron is an elementary particle, but its quantum states form three-dimensional patterns.


Nevertheless, there is good reason that an elementary particle is often called a point particle. Even if an elementary particle has a delocalized wavepacket, the wavepacket can be represented as a quantum superposition of quantum states wherein the particle is exactly localized. Moreover, the "interactions" of the particle can be represented as a superposition of interactions of individual states which are localized. This is not true for a composite particle, which can never be represented as a superposition of exactly-localized quantum states. It is in this sense that physicists can discuss the intrinsic "size" of a particle: The size of its internal structure, not the size of its wavepacket. The "size" of an elementary particle, in this sense, is exactly zero.


For example, for the electron, experimental evidence shows that the size of an electron is less than 10−18 m. This is consistent with the expected value of exactly zero. (This should not be confused with the classical electron radius, which, despite the name, is unrelated to the actual size of an electron.)


In a relativistic theory of physics, a Lorentz scalar is an expression, formed from items of the theory, which evaluates to a scalar, invariant under any Lorentz transformation. A Lorentz scalar may be generated from e.g., the scalar product of vectors, or from contracting tensors of the theory. While the components of vectors and tensors are in general altered under Lorentz transformations, Lorentz scalars remain unchanged.


A Lorentz scalar is not always immediately seen to be an invariant scalar in the mathematical sense, but the resulting scalar value is invariant under any basis transformation applied to the vector space, on which the considered theory is based. A simple Lorentz scalar in Minkowski spacetime is the "spacetime distance" ("length" of their difference) of two fixed events in spacetime. While the "position"-4-vectors of the events change between different inertial frames, their spacetime distance remains invariant under the corresponding Lorentz transformation. Other examples of Lorentz scalars are the "length" of 4-velocities (see below), or the Ricci curvature in a point in spacetime from General relativity, which is a contraction of the Riemann curvature tensor there.


In special relativity the location of a particle in 4-dimensional spacetime is given by


where formula_2 is the position in 3-dimensional space of the particle, formula_3 is the velocity in 3-dimensional space and formula_4 is the speed of light.


The "length" of the vector is a Lorentz scalar and is given by


where formula_6 is the proper time as measured by a clock in the rest frame of the particle and the Minkowski metric is given by


Often the alternate signature of the Minkowski metric is used in which the signs of the ones are reversed.


In the Minkowski metric the space-like interval formula_9 is defined as


We use the space-like Minkowski metric in the rest of this article.


The velocity in spacetime is defined as


The magnitude of the 4-velocity is a Lorentz scalar,


The inner product of acceleration and velocity.


The 4-acceleration is always perpendicular to the 4-velocity


Therefore, we can regard acceleration in spacetime as simply a rotation of the 4-velocity. The inner product of the acceleration and the velocity is a Lorentz scalar and is zero. This rotation is simply an expression of energy conservation:


where formula_17 is the energy of a particle and formula_18 is the 3-force on the particle.


Energy, rest mass, 3-momentum, and 3-speed from 4-momentum.


where formula_20 is the particle rest mass, formula_21 is the momentum in 3-space, and


Measurement of the energy of a particle.


Consider a second particle with 4-velocity formula_23 and a 3-velocity formula_24. In the rest frame of the second particle the inner product of formula_23 with formula_26 is proportional to the energy of the first particle


where the subscript 1 indicates the first particle.


Since the relationship is true in the rest frame of the second particle, it is true in any reference frame. formula_28, the energy of the first particle in the frame of the second particle, is a Lorentz scalar. Therefore,


in any inertial reference frame, where formula_30 is still the energy of the first particle in the frame of the second particle .


Measurement of the rest mass of the particle.


In the rest frame of the particle the inner product of the momentum is


Therefore, the rest mass (m) is a Lorentz scalar. The relationship remains true independent of the frame in which the inner product is calculated. In many cases the rest mass is written as formula_32 to avoid confusion with the relativistic mass, which is formula_33


Measurement of the 3-momentum of the particle.


The square of the magnitude of the 3-momentum of the particle as measured in the frame of the second particle is a Lorentz scalar.


Measurement of the 3-speed of the particle.


The 3-speed, in the frame of the second particle, can be constructed from two Lorentz scalars


Scalars may also be constructed from the tensors and vectors, from the contraction of tensors (such as formula_36), or combinations of contractions of tensors and vectors (such as formula_37).


In electromagnetism, the Lorenz gauge condition or Lorenz gauge, for Ludvig Lorenz, is a partial gauge fixing of the electromagnetic vector potential by requiring formula_1 The name is frequently confused with Hendrik Lorentz, who has given his name to many concepts in this field. The condition is Lorentz invariant. The condition does not completely determine the gauge: one can still make a gauge transformation formula_2 where formula_3 is a harmonic scalar function (that is, a scalar function satisfying formula_4 the equation of a massless scalar field). The Lorenz condition is used to eliminate the redundant spin-0 component in the representation theory of the Lorentz group. It is equally used for massive spin-1 fields where the concept of gauge transformations does not apply at all.


In electromagnetism, the Lorenz condition is generally used in calculations of time-dependent electromagnetic fields through retarded potentials. The condition is


where formula_6 is the four-potential, the comma denotes a partial differentiation and the repeated index indicates that the Einstein summation convention is being used. The condition has the advantage of being Lorentz invariant. It still leaves substantial gauge degrees of freedom.


In ordinary vector notation and SI units, the condition is


where formula_8 is the magnetic vector potential and formula_9 is the electric potential; see also gauge fixing.


A quick justification of the Lorenz gauge can be found using Maxwell's equations and the relation between the magnetic vector potential and the magnetic field:


Since the curl is zero, that means there is a scalar function formula_13 such that


This gives the well known equation for the electric field,


This result can be plugged into the Ampère–Maxwell equation,


To have Lorentz invariance, the time derivatives and spatial derivatives must be treated equally (i.e. of the same order). Therefore, it is convenient to choose the Lorenz gauge condition, which gives the result


A similar procedure with a focus on the electric scalar potential and making the same gauge choice will yield


These are simpler and more symmetric forms of the inhomogeneous Maxwell's equations. Note that the Coulomb gauge also fixes the problem of Lorentz invariance, but leaves a coupling term with first-order derivatives.


is the vacuum velocity of light, and formula_21 is the d'Alembertian operator. These equations are not only valid under vacuum conditions, but also in polarized media, if formula_22 and formula_23 are source density and circulation density, respectively, of the electromagnetic induction fields formula_24 and formula_25 calculated as usual from formula_13 and formula_27 by the equations


The explicit solutions for formula_13 and formula_8 – unique, if all quantities vanish sufficiently fast at infinity – are known as retarded potentials.


Time translation symmetry or temporal translation symmetry (TTS) is a mathematical transformation in physics that moves the times of events through a common interval. Time translation symmetry is the hypothesis that the laws of physics are unchanged, (i.e. invariant) under such a transformation. Time translation symmetry is a rigorous way to formulate the idea that the laws of physics are the same throughout history. Time translation symmetry is closely connected, via the Noether theorem, to conservation of energy. In mathematics, the set of all time translations on a given system form a Lie group.


There are many symmetries in nature besides time translation, such as spatial translation or rotational symmetries. These symmetries can be broken and explain diverse phenomena such as crystals, superconductivity, and the Higgs mechanism. However, it was thought until very recently that time translation symmetry could not be broken. Time crystals, a state of matter first observed in 2017, break time translation symmetry.


Symmetries are of prime importance in physics and are closely related to the hypothesis that certain physical quantities are only relative and unobservable. Symmetries apply to the equations that govern the physical laws (e.g. to a Hamiltonian or Lagrangian) rather than the initial conditions, values or magnitudes of the equations themselves and state that the laws remain unchanged under a transformation. If a symmetry is preserved under a transformation it is said to be "invariant". Symmetries in nature lead directly to conservation laws, something which is precisely formulated by the Noether theorem.


To formally describe time translation symmetry we say the equations, or laws, that describe a system at times formula_1 and formula_2 are the same for any value of formula_1 and formula_4.


One finds for its solutions formula_6 the combination:


The invariance of a Hamiltonian formula_10 of an isolated system under time translation implies its energy does not change with the passage of time. Conservation of energy implies, according to the Heisenberg equations of motion, that formula_11.


Where formula_14 is the time translation operator which implies invariance of the Hamiltonian under the time translation operation and leads to the conservation of energy.


In many nonlinear field theories like general relativity or Yang–Mills theories, the basic field equations are highly nonlinear and exact solutions are only known for ‘sufficiently symmetric’ distributions of matter (e.g. rotationally or axially symmetric configurations). Time translation symmetry is guaranteed only in spacetimes where the metric is static: that is, where there is a coordinate system in which the metric coefficients contain no time variable. Many general relativity systems are not static in any frame of reference so no conserved energy can be defined.


Time crystals, a state of matter first observed in 2017, break discrete time translation symmetry.


In physics and mathematics, an ansatz (; , meaning: "initial placement of a tool at a work piece", plural Ansätze ; ) is an educated guess or an additional assumption made to help solve a problem, and which may later be verified to be part of the solution by its results.


An ansatz is the establishment of the starting equation(s), the theorem(s), or the value(s) describing a mathematical or physical problem or solution. It typically provides an initial estimate or framework to the solution of a mathematical problem, and can also take into consideration the boundary conditions (in fact, an ansatz is sometimes thought of as a "trial answer" and an important technique in solving differential equations).


After an ansatz, which constitutes nothing more than an assumption, has been established, the equations are solved more precisely for the general function of interest, which then constitutes a confirmation of the assumption. In essence, an ansatz makes assumptions about the form of the solution to a problem so as to make the solution easier to find.


It has been demonstrated that machine learning techniques can be applied to provide initial estimates similar to those invented by humans and to discover new ones in case no ansatz is available.


Given a set of experimental data that looks to be clustered about a line, a linear ansatz could be made to find the parameters of the line by a least squares curve fit. Variational approximation methods use ansätze and then fit the parameters.


Another example could be the mass, energy, and entropy balance equations that, considered simultaneous for purposes of the elementary operations of linear algebra, are the "ansatz" to most basic problems of thermodynamics.


Another example of an ansatz is to suppose the solution of a homogeneous linear differential equation to take an exponential form, or a power form in the case of a difference equation. More generally, one can guess a particular solution of a system of equations, and test such an ansatz by directly substituting the solution into the system of equations. In many cases, the assumed form of the solution is general enough that it can represent arbitrary functions, in such a way that the set of solutions found this way is a full set of all the solutions.


In mathematics, a tensor is an algebraic object that describes a (multilinear) relationship between sets of algebraic objects related to a vector space. Objects that tensors may map between include vectors and scalars, and even other tensors. Tensors can take several different forms – for example: scalars and vectors (which are the simplest tensors), dual vectors, multilinear maps between vector spaces, and even some operations such as the dot product. Tensors are defined independent of any basis, although they are often referred to by their components in a basis related to a particular coordinate system.


Tensors are important in physics because they provide a concise mathematical framework for formulating and solving physics problems in areas such as mechanics (stress, elasticity, fluid mechanics, moment of inertia, ...), electrodynamics (electromagnetic tensor, Maxwell tensor, permittivity, magnetic susceptibility, ...), or general relativity (stress–energy tensor, curvature tensor, ...) and others. In applications, it is common to study situations in which a different tensor can occur at each point of an object; for example the stress within an object may vary from one location to another. This leads to the concept of a tensor field. In some areas, tensor fields are so ubiquitous that they are often simply called "tensors".


Tensors were conceived in 1900 by Tullio Levi-Civita and Gregorio Ricci-Curbastro, who continued the earlier work of Bernhard Riemann and Elwin Bruno Christoffel and others, as part of the "absolute differential calculus". The concept enabled an alternative formulation of the intrinsic differential geometry of a manifold in the form of the Riemann curvature tensor.


Although seemingly different, the various approaches to defining tensors describe the same geometric concept using different language and at different levels of abstraction. For example, tensors are defined and discussed for statistical and machine learning applications.


Just as the components of a vector change when we change the basis of the vector space, the components of a tensor also change under such a transformation. Each type of tensor comes equipped with a "transformation law" that details how the components of the tensor respond to a change of basis. The components of a vector can respond in two distinct ways to a change of basis (see covariance and contravariance of vectors), where the new basis vectors formula_1 are expressed in terms of the old basis vectors formula_2 as,


Here "R"" j""i" are the entries of the change of basis matrix, and in the rightmost expression the summation sign was suppressed: this is the Einstein summation convention, which will be used throughout this article. The components "v""i" of a column vector v transform with the inverse of the matrix "R",


where the hat denotes the components in the new basis. This is called a "contravariant" transformation law, because the vector components transform by the "inverse" of the change of basis. In contrast, the components, "w""i", of a covector (or row vector), w transform with the matrix "R" itself,


This is called a "covariant" transformation law, because the covector components transforms by the "same matrix" as the change of basis matrix. The components of a more general tensor transform by some combination of covariant and contravariant transformations, with one transformation law for each index. If the transformation matrix of an index is the inverse matrix of the basis transformation, then the index is called "contravariant" and is conventionally denoted with an upper index (superscript). If the transformation matrix of an index is the basis transformation itself, then the index is called "covariant" and is denoted with a lower index (subscript).


As a simple example, the matrix of a linear operator with respect to a basis is a rectangular array formula_6 that transforms under a change of basis matrix formula_7 by formula_8. For the individual matrix entries, this transformation law has the form formula_9 so the tensor corresponding to the matrix of a linear operator has one covariant and one contravariant index: it is of type (1,1).


Combinations of covariant and contravariant components with the same index allow us to express geometric invariants. For example, the fact that a vector is the same object in different coordinate systems can be captured by the following equations, using the formulas defined above:


where formula_11 is the Kronecker delta, which functions similarly to the identity matrix, and has the effect of renaming indices ("j" into "k" in this example). This shows several features of the component notation: the ability to re-arrange terms at will (commutativity), the need to use different indices when working with multiple objects in the same expression, the ability to rename indices, and the manner in which contravariant and covariant tensors combine so that all instances of the transformation matrix and its inverse cancel, so that expressions like formula_12 can immediately be seen to be geometrically identical in all coordinate systems.


Similarly, a linear operator, viewed as a geometric object, does not actually depend on a basis: it is just a linear map that accepts a vector as an argument and produces another vector. The transformation law for how the matrix of components of a linear operator changes with the basis is consistent with the transformation law for a contravariant vector, so that the action of a linear operator on a contravariant vector is represented in coordinates as the matrix product of their respective coordinate representations. That is, the components formula_13 are given by formula_14. These components transform contravariantly, since


The transformation law for an order tensor with "p" contravariant indices and "q" covariant indices is thus given as,


Here the primed indices denote components in the new coordinates, and the unprimed indices denote the components in the old coordinates. Such a tensor is said to be of order or "type" . The terms "order", "type", "rank", "valence", and "degree" are all sometimes used for the same concept. Here, the term "order" or "total order" will be used for the total dimension of the array (or its generalization in other definitions), in the preceding example, and the term "type" for the pair giving the number of contravariant and covariant indices. A tensor of type is also called a -tensor for short.


This discussion motivates the following formal definition:


to each basis of an "n"-dimensional vector space such that, if we apply the change of basis


then the multidimensional array obeys the transformation law


The definition of a tensor as a multidimensional array satisfying a transformation law traces back to the work of Ricci.


An equivalent definition of a tensor uses the representations of the general linear group. There is an action of the general linear group on the set of all ordered bases of an "n"-dimensional vector space. If formula_23 is an ordered basis, and formula_24 is an invertible formula_25 matrix, then the action is given by


Let "F" be the set of all ordered bases. Then "F" is a principal homogeneous space for GL("n"). Let "W" be a vector space and let formula_27 be a representation of GL("n") on "W" (that is, a group homomorphism formula_28). Then a tensor of type formula_27 is an equivariant map formula_30. Equivariance here means that


When formula_27 is a tensor representation of the general linear group, this gives the usual definition of tensors as multidimensional arrays. This definition is often used to describe tensors on manifolds, and readily generalizes to other groups.


A downside to the definition of a tensor using the multidimensional array approach is that it is not apparent from the definition that the defined object is indeed basis independent, as is expected from an intrinsically geometric object. Although it is possible to show that transformation laws indeed ensure independence from the basis, sometimes a more intrinsic definition is preferred. One approach that is common in differential geometry is to define tensors relative to a fixed (finite-dimensional) vector space "V", which is usually taken to be a particular vector space of some geometrical significance like the tangent space to a manifold. In this approach, a type tensor "T" is defined as a multilinear map,


where "V"∗ is the corresponding dual space of covectors, which is linear in each of its arguments. The above assumes "V" is a vector space over the real numbers, ℝ. More generally, "V" can be taken over an arbitrary field of numbers, "F" (e.g. the complex numbers) with a one-dimensional vector space over "F" replacing ℝ as the codomain of the multilinear maps.


By applying a multilinear map "T" of type to a basis {e"j"} for "V" and a canonical cobasis {ε"i"} for "V"∗,


a -dimensional array of components can be obtained. A different choice of basis will yield different components. But, because "T" is linear in all of its arguments, the components satisfy the tensor transformation law used in the multilinear array definition. The multidimensional array of components of "T" thus form a tensor according to that definition. Moreover, such an array can be realized as the components of some multilinear map "T". This motivates viewing multilinear maps as the intrinsic objects underlying tensors.


In viewing a tensor as a multilinear map, it is conventional to identify the double dual "V"∗∗ of the vector space "V", i.e., the space of linear functionals on the dual vector space "V"∗, with the vector space "V". There is always a natural linear map from "V" to its double dual, given by evaluating a linear form in "V"∗ against a vector in "V". This linear mapping is an isomorphism in finite dimensions, and it is often then expedient to identify "V" with its double dual.


For some mathematical applications, a more abstract approach is sometimes useful. This can be achieved by defining tensors in terms of elements of tensor products of vector spaces, which in turn are defined through a universal property. A type tensor is defined in this context as an element of the tensor product of vector spaces,


A basis of and basis of naturally induce a basis of the tensor product . The components of a tensor are the coefficients of the tensor with respect to the basis obtained from a basis for and its dual basis , i.e.


Using the properties of the tensor product, it can be shown that these components satisfy the transformation law for a type tensor. Moreover, the universal property of the tensor product gives a -to- correspondence between tensors defined in this way and tensors defined as multilinear maps.


Tensor products can be defined in great generality – for example, involving arbitrary modules over a ring. In principle, one could define a "tensor" simply to be an element of any tensor product. However, the mathematics literature usually reserves the term "tensor" for an element of a tensor product of any number of copies of a single vector space and its dual, as above.


In many applications, especially in differential geometry and physics, it is natural to consider a tensor with components that are functions of the point in a space. This was the setting of Ricci's original work. In modern mathematical terminology such an object is called a tensor field, often referred to simply as a tensor.


In this context, a coordinate basis is often chosen for the tangent vector space. The transformation law may then be expressed in terms of partial derivatives of the coordinate functions,


This table shows important examples of tensors on vector spaces and tensor fields on manifolds. The tensors are classified according to their type , where "n" is the number of contravariant indices, "m" is the number of covariant indices, and gives the total order of the tensor. For example, a bilinear form is the same thing as a -tensor; an inner product is an example of a -tensor, but not all -tensors are inner products. In the -entry of the table, "M" denotes the dimensionality of the underlying vector space or manifold because for each dimension of the space, a separate index is needed to select that dimension to get a maximally covariant antisymmetric tensor.


Raising an index on an -tensor produces an -tensor; this corresponds to moving diagonally down and to the left on the table. Symmetrically, lowering an index corresponds to moving diagonally up and to the right on the table. Contraction of an upper with a lower index of an -tensor produces an -tensor; this corresponds to moving diagonally up and to the left on the table.


Assuming a basis of a real vector space, e.g., a coordinate frame in the ambient space, a tensor can be represented as an organized multidimensional array of numerical values with respect to this specific basis. Changing the basis transforms the values in the array in a characteristic way that allows to "define" tensors as objects adhering to this transformational behavior. For example, there are invariants of tensors that must be preserved under any change of the basis, thereby making only certain multidimensional arrays of numbers a tensor. Compare this to the array representing formula_40 not being a tensor, for the sign change under transformations changing the orientation.


Because the components of vectors and their duals transform differently under the change of their dual bases, there is a covariant and/or contravariant transformation law that relates the arrays, which represent the tensor with respect to one basis and that with respect to the other one. The numbers of, respectively, (contravariant indices) and dual (covariant indices) in the input and output of a tensor determine the "type" (or "valence") of the tensor, a pair of natural numbers , which determine the precise form of the transformation law. The "" of a tensor is the sum of these two numbers.


The collection of tensors on a vector space and its dual forms a tensor algebra, which allows products of arbitrary tensors. Simple applications of tensors of order , which can be represented as a square matrix, can be solved by clever arrangement of transposed vectors and by applying the rules of matrix multiplication, but the tensor product should not be confused with this.


There are several notational systems that are used to describe tensors and perform calculations involving them.


Ricci calculus is the modern formalism and notation for tensor indices: indicating inner and outer products, covariance and contravariance, summations of tensor components, symmetry and antisymmetry, and partial and covariant derivatives.


The Einstein summation convention dispenses with writing summation signs, leaving the summation implicit. Any repeated index symbol is summed over: if the index is used twice in a given term of a tensor expression, it means that the term is to be summed for all . Several distinct pairs of indices may be summed this way.


Penrose graphical notation is a diagrammatic notation which replaces the symbols for tensors with shapes, and their indices by lines and curves. It is independent of basis elements, and requires no symbols for the indices.


The abstract index notation is a way to write tensors such that the indices are no longer thought of as numerical, but rather are indeterminates. This notation captures the expressiveness of indices and the basis-independence of index-free notation.


A component-free treatment of tensors uses notation that emphasises that tensors do not rely on any basis, and is defined in terms of the tensor product of vector spaces.


There are several operations on tensors that again produce a tensor. The linear nature of tensor implies that two tensors of the same type may be added together, and that tensors may be multiplied by a scalar with results analogous to the scaling of a vector. On components, these operations are simply performed component-wise. These operations do not change the type of the tensor; but there are also operations that produce a tensor of different type.


The tensor product takes two tensors, "S" and "T", and produces a new tensor, , whose order is the sum of the orders of the original tensors. When described as multilinear maps, the tensor product simply multiplies the two tensors, i.e.


which again produces a map that is linear in all its arguments. On components, the effect is to multiply the components of the two input tensors pairwise, i.e.


If is of type and is of type , then the tensor product has type .


Tensor contraction is an operation that reduces a type tensor to a type tensor, of which the trace is a special case. It thereby reduces the total order of a tensor by two. The operation is achieved by summing components for which one specified contravariant index is the same as one specified covariant index to produce a new component. Components for which those two indices are different are discarded. For example, a -tensor formula_43 can be contracted to a scalar through formula_44. Where the summation is again implied. When the -tensor is interpreted as a linear map, this operation is known as the trace.


The contraction is often used in conjunction with the tensor product to contract an index from each tensor.


The contraction can also be understood using the definition of a tensor as an element of a tensor product of copies of the space "V" with the space "V"∗ by first decomposing the tensor into a linear combination of simple tensors, and then applying a factor from "V"∗ to a factor from "V". For example, a tensor formula_45 can be written as a linear combination


The contraction of "T" on the first and last slots is then the vector


In a vector space with an inner product (also known as a metric) "g", the term contraction is used for removing two contravariant or two covariant indices by forming a trace with the metric tensor or its inverse. For example, a -tensor formula_48 can be contracted to a scalar through formula_49 (yet again assuming the summation convention).


When a vector space is equipped with a nondegenerate bilinear form (or "metric tensor" as it is often called in this context), operations can be defined that convert a contravariant (upper) index into a covariant (lower) index and vice versa. A metric tensor is a (symmetric) (-tensor; it is thus possible to contract an upper index of a tensor with one of the lower indices of the metric tensor in the product. This produces a new tensor with the same index structure as the previous tensor, but with lower index generally shown in the same position of the contracted upper index. This operation is quite graphically known as "lowering an index".


Conversely, the inverse operation can be defined, and is called "raising an index". This is equivalent to a similar contraction on the product with a -tensor. This "inverse metric tensor" has components that are the matrix inverse of those of the metric tensor.


If a particular surface element inside the material is singled out, the material on one side of the surface will apply a force on the other side. In general, this force will not be orthogonal to the surface, but it will depend on the orientation of the surface in a linear manner. This is described by a tensor of type, in linear elasticity, or more precisely by a tensor field of type , since the stresses may vary from point to point.


Applications of tensors of order > 2.


The concept of a tensor of order two is often conflated with that of a matrix. Tensors of higher order do however capture ideas important in science and engineering, as has been shown successively in numerous areas as they develop. This happens, for instance, in the field of computer vision, with the trifocal tensor generalizing the fundamental matrix.


The field of nonlinear optics studies the changes to material polarization density under extreme electric fields. The polarization waves generated are related to the generating electric fields through the nonlinear susceptibility tensor. If the polarization P is not linearly proportional to the electric field E, the medium is termed "nonlinear". To a good approximation (for sufficiently weak fields, assuming no permanent dipole moments are present), P is given by a Taylor series in E whose coefficients are the nonlinear susceptibilities:


Here formula_51 is the linear susceptibility, formula_52 gives the Pockels effect and second harmonic generation, and formula_53 gives the Kerr effect. This expansion shows the way higher-order tensors arise naturally in the subject matter.


The vector spaces of a tensor product need not be the same, and sometimes the elements of such a more general tensor product are called "tensors". For example, an element of the tensor product space is a second-order "tensor" in this more general sense, and an order- tensor may likewise be defined as an element of a tensor product of different vector spaces. A type tensor, in the sense defined previously, is also a tensor of order in this more general sense. The concept of tensor product can be extended to arbitrary modules over a ring.


The notion of a tensor can be generalized in a variety of ways to infinite dimensions. One, for instance, is via the tensor product of Hilbert spaces. Another way of generalizing the idea of tensor, common in nonlinear analysis, is via the multilinear maps definition where instead of using finite-dimensional vector spaces and their algebraic duals, one uses infinite-dimensional Banach spaces and their continuous dual. Tensors thus live naturally on Banach manifolds and Fréchet manifolds.


Suppose that a homogeneous medium fills , so that the density of the medium is described by a single scalar value in . The mass, in kg, of a region is obtained by multiplying by the volume of the region , or equivalently integrating the constant over the region:


where the Cartesian coordinates are measured in m. If the units of length are changed into cm, then the numerical values of the coordinate functions must be rescaled by a factor of 100:


The numerical value of the density must then also transform by formula_56 to compensate, so that the numerical value of the mass in kg is still given by integral of formula_57. Thus formula_58 (in units of ).


More generally, if the Cartesian coordinates undergo a linear transformation, then the numerical value of the density must change by a factor of the reciprocal of the absolute value of the determinant of the coordinate transformation, so that the integral remains invariant, by the change of variables formula for integration. Such a quantity that scales by the reciprocal of the absolute value of the determinant of the coordinate transition map is called a scalar density. To model a non-constant density, is a function of the variables (a scalar field), and under a curvilinear change of coordinates, it transforms by the reciprocal of the Jacobian of the coordinate change. For more on the intrinsic meaning, see Density on a manifold.


A tensor density transforms like a tensor under a coordinate change, except that it in addition picks up a factor of the absolute value of the determinant of the coordinate transition:


Here "w" is called the weight. In general, any tensor multiplied by a power of this function or its absolute value is called a tensor density, or a weighted tensor. An example of a tensor density is the current density of electromagnetism.


Under an affine transformation of the coordinates, a tensor transforms by the linear part of the transformation itself (or its inverse) on each index. These come from the rational representations of the general linear group. But this is not quite the most general linear transformation law that such an object may have: tensor densities are non-rational, but are still semisimple representations. A further class of transformations come from the logarithmic representation of the general linear group, a reducible but not semisimple representation, consisting of an with the transformation law


The transformation law for a tensor behaves as a functor on the category of admissible coordinate systems, under general linear transformations (or, other transformations within some class, such as local diffeomorphisms.) This makes a tensor a special case of a geometrical object, in the technical sense that it is a function of the coordinate system transforming functorially under coordinate changes. Examples of objects obeying more general kinds of transformation laws are jets and, more generally still, natural bundles.


Succinctly, spinors are elements of the spin representation of the rotation group, while tensors are elements of its tensor representations. Other classical groups have tensor representations, and so also tensors that are compatible with the group, but all non-compact classical groups have infinite-dimensional unitary representations as well.


The concepts of later tensor analysis arose from the work of Carl Friedrich Gauss in differential geometry, and the formulation was much influenced by the theory of algebraic forms and invariants developed during the middle of the nineteenth century. The word "tensor" itself was introduced in 1846 by William Rowan Hamilton to describe something different from what is now meant by a tensor. The contemporary usage was introduced by Woldemar Voigt in 1898.


Tensor calculus was developed around 1890 by Gregorio Ricci-Curbastro under the title "absolute differential calculus", and originally presented by Ricci-Curbastro in 1892. It was made accessible to many mathematicians by the publication of Ricci-Curbastro and Tullio Levi-Civita's 1900 classic text "Méthodes de calcul différentiel absolu et leurs applications" (Methods of absolute differential calculus and their applications).


In the 20th century, the subject came to be known as "tensor analysis", and achieved broader acceptance with the introduction of Einstein's theory of general relativity, around 1915. General relativity is formulated completely in the language of tensors. Einstein had learned about them, with great difficulty, from the geometer Marcel Grossmann. Levi-Civita then initiated a correspondence with Einstein to correct mistakes Einstein had made in his use of tensor analysis. The correspondence lasted 1915–17, and was characterized by mutual respect:


Tensors were also found to be useful in other fields such as continuum mechanics. Some well-known examples of tensors in differential geometry are quadratic forms such as metric tensors, and the Riemann curvature tensor. The exterior algebra of Hermann Grassmann, from the middle of the nineteenth century, is itself a tensor theory, and highly geometric, but it was some time before it was seen, with the theory of differential forms, as naturally unified with tensor calculus. The work of Élie Cartan made differential forms one of the basic kinds of tensors used in mathematics.


From about the 1920s onwards, it was realised that tensors play a basic role in algebraic topology (for example in the Künneth theorem). Correspondingly there are types of tensors at work in many branches of abstract algebra, particularly in homological algebra and representation theory. Multilinear algebra can be developed in greater generality than for scalars coming from a field. For example, scalars can come from a ring. But the theory is then less geometric and computations more technical and less algorithmic. Tensors are generalized within category theory by means of the concept of monoidal category, from the 1960s.


Frame-dragging is an effect on spacetime, predicted by Albert Einstein's general theory of relativity, that is due to non-static stationary distributions of mass–energy. A stationary field is one that is in a steady state, but the masses causing that field may be non-static — rotating, for instance. More generally, the subject that deals with the effects caused by mass–energy currents is known as gravitoelectromagnetism, which is analogous to the magnetism of classical electromagnetism.


The first frame-dragging effect was derived in 1918, in the framework of general relativity, by the Austrian physicists Josef Lense and Hans Thirring, and is also known as the Lense–Thirring effect. They predicted that the rotation of a massive object would distort the spacetime metric, making the orbit of a nearby test particle precess. This does not happen in Newtonian mechanics for which the gravitational field of a body depends only on its mass, not on its rotation. The Lense–Thirring effect is very small – about one part in a few trillion. To detect it, it is necessary to examine a very massive object, or build an instrument that is very sensitive.


In 2015, new general-relativistic extensions of Newtonian rotation laws were formulated to describe geometric dragging of frames which incorporates a newly discovered antidragging effect.


Rotational frame-dragging (the Lense–Thirring effect) appears in the general principle of relativity and similar theories in the vicinity of rotating massive objects. Under the Lense–Thirring effect, the frame of reference in which a clock ticks the fastest is one which is revolving around the object as viewed by a distant observer. This also means that light traveling in the direction of rotation of the object will move past the massive object faster than light moving against the rotation, as seen by a distant observer. It is now the best known frame-dragging effect, partly thanks to the Gravity Probe B experiment. Qualitatively, frame-dragging can be viewed as the gravitational analog of electromagnetic induction.


Another interesting consequence is that, for an object constrained in an equatorial orbit, but not in freefall, it weighs more if orbiting anti-spinward, and less if orbiting spinward. For example, in a suspended equatorial bowling alley, a bowling ball rolled anti-spinward would weigh more than the same ball rolled in a spinward direction. Note, frame dragging will neither accelerate nor slow down the bowling ball in either direction. It is not a "viscosity". Similarly, a stationary plumb-bob suspended over the rotating object will not list. It will hang vertically. If it starts to fall, induction will push it in the spinward direction.


Linear frame dragging is the similarly inevitable result of the general principle of relativity, applied to linear momentum. Although it arguably has equal theoretical legitimacy to the "rotational" effect, the difficulty of obtaining an experimental verification of the effect means that it receives much less discussion and is often omitted from articles on frame-dragging (but see Einstein, 1921).


Static mass increase is a third effect noted by Einstein in the same paper. The effect is an increase in inertia of a body when other masses are placed nearby. While not strictly a frame dragging effect (the term frame dragging is not used by Einstein), it is demonstrated by Einstein that it derives from the same equation of general relativity. It is also a tiny effect that is difficult to confirm experimentally.


In 1976 Van Patten and Everitt proposed to implement a dedicated mission aimed to measure the Lense–Thirring node precession of a pair of counter-orbiting spacecraft to be placed in terrestrial polar orbits with drag-free apparatus. A somewhat equivalent, cheaper version of such an idea was put forth in 1986 by Ciufolini who proposed to launch a passive, geodetic satellite in an orbit identical to that of the LAGEOS satellite, launched in 1976, apart from the orbital planes which should have been displaced by 180 deg apart: the so-called butterfly configuration. The measurable quantity was, in this case, the sum of the nodes of LAGEOS and of the new spacecraft, later named LAGEOS III, LARES, WEBER-SAT.


The Gravity Probe B experiment was a satellite-based mission by a Stanford group and NASA, used to experimentally measure another gravitomagnetic effect, the Schiff precession of a gyroscope, to an expected 1% accuracy or better. Unfortunately such accuracy was not achieved. The first preliminary results released in April 2007 pointed towards an accuracy of 256–128%, with the hope of reaching about 13% in December 2007.


In 2008 the Senior Review Report of the NASA Astrophysics Division Operating Missions stated that it was unlikely that Gravity Probe B team will be able to reduce the errors to the level necessary to produce a convincing test of currently untested aspects of General Relativity (including frame-dragging).


On May 4, 2011, the Stanford-based analysis group and NASA announced the final report, and in it the data from GP-B demonstrated the frame-dragging effect with an error of about 19 percent, and Einstein's predicted value was at the center of the confidence interval.


NASA published claims of success in verification of frame dragging for the GRACE twin satellites and Gravity Probe B, both of which claims are still in public view. A research group in Italy, USA, and UK also claimed success in verification of frame dragging with the Grace gravity model, published in a peer reviewed journal. All the claims include recommendations for further research at greater accuracy and other gravity models.


In the case of stars orbiting close to a spinning, supermassive black hole, frame dragging should cause the star's orbital plane to precess about the black hole spin axis. This effect should be detectable within the next few years via astrometric monitoring of stars at the center of the Milky Way galaxy.


By comparing the rate of orbital precession of two stars on different orbits, it is possible in principle to test the no-hair theorems of general relativity, in addition to measuring the spin of the black hole.


Relativistic jets may provide evidence for the reality of frame-dragging. Gravitomagnetic forces produced by the Lense–Thirring effect (frame dragging) within the ergosphere of rotating black holes combined with the energy extraction mechanism by Penrose have been used to explain the observed properties of relativistic jets. The gravitomagnetic model developed by Reva Kay Williams predicts the observed high energy particles (~GeV) emitted by quasars and active galactic nuclei; the extraction of X-rays, γ-rays, and relativistic e−– e+ pairs; the collimated jets about the polar axis; and the asymmetrical formation of jets (relative to the orbital plane).


The Lense–Thirring effect has been observed in a binary system that consists of a massive white dwarf and a pulsar.


Frame-dragging may be illustrated most readily using the Kerr metric, which describes the geometry of spacetime in the vicinity of a mass "M" rotating with angular momentum "J", and Boyer–Lindquist coordinates (see the link for the transformation):


and where the following shorthand variables have been introduced for brevity


In the non-relativistic limit where "M" (or, equivalently, "r""s") goes to zero, the Kerr metric becomes the orthogonal metric for the oblate spheroidal coordinates


We may rewrite the Kerr metric in the following form


This metric is equivalent to a co-rotating reference frame that is rotating with angular speed Ω that depends on both the radius "r" and the colatitude "θ"


In the plane of the equator this simplifies to:


Thus, an inertial reference frame is entrained by the rotating central mass to participate in the latter's rotation; this is frame-dragging.


An extreme version of frame dragging occurs within the ergosphere of a rotating black hole. The Kerr metric has two surfaces on which it appears to be singular. The inner surface corresponds to a spherical event horizon similar to that observed in the Schwarzschild metric; this occurs at


where the purely radial component "grr" of the metric goes to infinity. The outer surface can be approximated by an oblate spheroid with lower spin parameters, and resembles a pumpkin-shape with higher spin parameters. It touches the inner surface at the poles of the rotation axis, where the colatitude "θ" equals 0 or π; its radius in Boyer-Lindquist coordinates is defined by the formula


where the purely temporal component "gtt" of the metric changes sign from positive to negative. The space between these two surfaces is called the ergosphere. A moving particle experiences a positive proper time along its worldline, its path through spacetime. However, this is impossible within the ergosphere, where "gtt" is negative, unless the particle is co-rotating with the interior mass "M" with an angular speed at least of Ω. However, as seen above, frame-dragging occurs about every rotating mass and at every radius "r" and colatitude "θ", not only within the ergosphere.


The Lense–Thirring effect inside a rotating shell was taken by Albert Einstein as not just support for, but a vindication of Mach's principle, in a letter he wrote to Ernst Mach in 1913 (five years before Lense and Thirring's work, and two years before he had attained the final form of general relativity). A reproduction of the letter can be found in Misner, Thorne, Wheeler. The general effect scaled up to cosmological distances, is still used as a support for Mach's principle.


Inside a rotating spherical shell the acceleration due to the Lense–Thirring effect would be


for "MG" ≪ "Rc"2 or more precisely,


The spacetime inside the rotating spherical shell will not be flat. A flat spacetime inside a rotating mass shell is possible if the shell is allowed to deviate from a precisely spherical shape and the mass density inside the shell is allowed to vary.


A node is a point along a standing wave where the wave has minimum amplitude. For instance, in a vibrating guitar string, the ends of the string are nodes. By changing the position of the end node through frets, the guitarist changes the effective length of the vibrating string and thereby the note played. The opposite of a node is an anti-node, a point where the amplitude of the standing wave is at maximum. These occur midway between the nodes.


Standing waves result when two sinusoidal wave trains of the same frequency are moving in opposite directions in the same space and interfere with each other. They occur when waves are reflected at a boundary, such as sound waves reflected from a wall or electromagnetic waves reflected from the end of a transmission line, and particularly when waves are confined in a resonator at resonance, bouncing back and forth between two boundaries, such as in an organ pipe or guitar string.


In a standing wave the nodes are a series of locations at equally spaced intervals where the wave amplitude (motion) is zero (see animation above). At these points the two waves add with opposite phase and cancel each other out. They occur at intervals of half a wavelength (λ/2). Midway between each pair of nodes are locations where the amplitude is maximum. These are called the antinodes. At these points the two waves add with the same phase and reinforce each other.


In cases where the two opposite wave trains are not the same amplitude, they do not cancel perfectly, so the amplitude of the standing wave at the nodes is not zero but merely a minimum. This occurs when the reflection at the boundary is imperfect. This is indicated by a finite standing wave ratio (SWR), the ratio of the amplitude of the wave at the antinode to the amplitude at the node.


In resonance of a two dimensional surface or membrane, such as a drumhead or vibrating metal plate, the nodes become nodal lines, lines on the surface where the surface is motionless, dividing the surface into separate regions vibrating with opposite phase. These can be made visible by sprinkling sand on the surface, and the intricate patterns of lines resulting are called Chladni figures.


In transmission lines a voltage node is a current antinode, and a voltage antinode is a current node.


Nodes are the points of zero displacement, not the points where two constituent waves intersect.


Where the nodes occur in relation to the boundary reflecting the waves depends on the end conditions or boundary conditions. Although there are many types of end conditions, the ends of resonators are usually one of two types that cause total reflection:


A sound wave consists of alternating cycles of compression and expansion of the wave medium. During compression, the molecules of the medium are forced together, resulting in the increased pressure and density. During expansion the molecules are forced apart, resulting in the decreased pressure and density.


The number of nodes in a specified length is directly proportional to the frequency of the wave.


The characteristic sound that allows the listener to identify a particular instrument is largely due to the relative magnitude of the harmonics created by the instrument.


In quantum field theory, a contact term is a radiatively induced point-like interaction.


These typically occur when the vertex for the emission of a massless particle such as a photon, a graviton, or a gluon, is proportional to formula_1 (the invariant momentum of the radiated particle).


This factor cancels the formula_2 of the Feynman propagator, and causes the exchange of the massless particle to produce a point-like


formula_3-function effective interaction, rather than the usual


formula_4 long-range potential. A notable example occurs in the weak interactions where a W-boson radiative correction to a gluon vertex produces a formula_1 term, leading to


what is known as a "penguin" interaction. The contact term then generates a correction to the full action of the theory.


Contact terms occur in gravity when there are non-minimal interactions,


The non-minimal couplings are quantum equivalent to an "Einstein frame," with a pure Einstein-Hilbert action,


owing to gravitational contact terms. These arise classically from graviton exchange interactions.


The contact terms are an essential, yet hidden, part of the action and, if they are ignored, the Feynman diagram loops in different frames yield different results. At the leading order in formula_9 including the contact terms is equivalent to performing a Weyl Transformation to remove the non-minimal couplings and taking the theory to the Einstein-Hilbert form. In this sense, the Einstein-Hilbert form of the action is unique and "frame ambiguities" in loop calculations do not exist.


Unification of the observable fundamental phenomena of nature is one of the primary goals of physics. The two great unifications to date are Isaac Newton’s unification of gravity and astronomy, and James Clerk Maxwell’s unification of electromagnetism; the latter has been further unified with the concept of electroweak interaction. This process of "unifying" forces continues today, with the ultimate goal of finding a theory of everything.


The "first great unification" was Isaac Newton's 17th century unification of gravity, which brought together the understandings of the observable phenomena of gravity on Earth with the observable behaviour of celestial bodies in space.


Unification of Magnetism, Electricity, Light and related radiation.


The "second great unification" was James Clerk Maxwell's 19th century unification of electromagnetism. It brought together the understandings of the observable phenomena of magnetism, electricity and light (and more broadly, the spectrum of electromagnetic radiation). This was followed in the 20th century by Albert Einstein's unification of space and time, and of mass and energy. Later, quantum field theory unified quantum mechanics and special relativity.


The ancient Chinese observed that certain rocks (lodestone and magnetite) were attracted to one another by an invisible force. This effect was later called magnetism, which was first rigorously studied in the 17th century. But even before the Chinese discovered magnetism, the ancient Greeks knew of other objects such as amber, that when rubbed with fur would cause a similar invisible attraction between the two. This was also first studied rigorously in the 17th century and came to be called electricity. Thus, physics had come to understand two observations of nature in terms of some root cause (electricity and magnetism). However, further work in the 19th century revealed that these two forces were just two different aspects of one force—electromagnetism.


This process of "unifying" forces continues today, and electromagnetism and the weak nuclear force are now considered to be two aspects of the electroweak interaction.


Unification of the remaining fundamental forces: Theory of Everything.


In mathematics and classical mechanics, the Poisson bracket is an important binary operation in Hamiltonian mechanics, playing a central role in Hamilton's equations of motion, which govern the time evolution of a Hamiltonian dynamical system. The Poisson bracket also distinguishes a certain class of coordinate transformations, called "canonical transformations", which map canonical coordinate systems into canonical coordinate systems. A "canonical coordinate system" consists of canonical position and momentum variables (below symbolized by formula_1 and formula_2, respectively) that satisfy canonical Poisson bracket relations. The set of possible canonical transformations is always very rich. For instance, it is often possible to choose the Hamiltonian itself formula_3 as one of the new canonical momentum coordinates.


In a more general sense, the Poisson bracket is used to define a Poisson algebra, of which the algebra of functions on a Poisson manifold is a special case. There are other general examples, as well: it occurs in the theory of Lie algebras, where the tensor algebra of a Lie algebra forms a Poisson algebra; a detailed construction of how this comes about is given in the universal enveloping algebra article. Quantum deformations of the universal enveloping algebra lead to the notion of quantum groups.


All of these objects are named in honor of Siméon Denis Poisson.


Given two functions f and g that depend on phase space and time, their Poisson bracket formula_4 is another function that depends on phase space and time. The following rules hold for any three functions formula_5 of phase space and time:


Also, if a function formula_10 is constant over phase space (but may depend on time), then formula_11 for any formula_12.


In canonical coordinates (also known as Darboux coordinates) formula_13 on the phase space, given two functions formula_14 and formula_15, the Poisson bracket takes the form


The Poisson brackets of the canonical coordinates are


Hamilton's equations of motion have an equivalent expression in terms of the Poisson bracket. This may be most directly demonstrated in an explicit coordinate frame. Suppose that formula_19 is a function on the solution’s trajectory-manifold. Then from the multivariable chain rule,


Further, one may take formula_21 and formula_22 to be solutions to Hamilton's equations; that is,


Thus, the time evolution of a function formula_12 on a symplectic manifold can be given as a one-parameter family of symplectomorphisms (i.e., canonical transformations, area-preserving diffeomorphisms), with the time formula_26 being the parameter: Hamiltonian motion is a canonical transformation generated by the Hamiltonian. That is, Poisson brackets are preserved in it, so that "any time formula_26" in the solution to Hamilton's equations,


can serve as the bracket coordinates. "Poisson brackets are canonical invariants".


The operator in the convective part of the derivative, formula_30, is sometimes referred to as the Liouvillian (see Liouville's theorem (Hamiltonian)).


An integrable dynamical system will have constants of motion in addition to the energy. Such constants of motion will commute with the Hamiltonian under the Poisson bracket. Suppose some function formula_31 is a constant of motion. This implies that if formula_32 is a trajectory or solution to Hamilton's equations of motion, then


where, as above, the intermediate step follows by applying the equations of motion and we supposed that formula_12 does not explicitly depend on time. This equation is known as the Liouville equation. The content of Liouville's theorem is that the time evolution of a measure (or "distribution function" on the phase space) is given by the above.


If the Poisson bracket of formula_12 and formula_37 vanishes (formula_38), then formula_12 and formula_37 are said to be in involution. In order for a Hamiltonian system to be completely integrable, formula_41 independent constants of motion must be in mutual involution, where formula_41 is the number of degrees of freedom.


Furthermore, according to Poisson's Theorem, if two quantities formula_43 and formula_44 are explicitly time independent (formula_45) constants of motion, so is their Poisson bracket formula_46. This does not always supply a useful result, however, since the number of possible constants of motion is limited (formula_47 for a system with formula_41 degrees of freedom), and so the result may be trivial (a constant, or a function of formula_43 and formula_44.)


Let formula_51 be a symplectic manifold, that is, a manifold equipped with a symplectic form: a 2-form formula_52 which is both closed (i.e., its exterior derivative formula_53 vanishes) and non-degenerate. For example, in the treatment above, take formula_51 to be formula_55 and take


If formula_57 is the interior product or contraction operation defined by formula_58, then non-degeneracy is equivalent to saying that for every one-form formula_59 there is a unique vector field formula_60 such that formula_61. Alternatively, formula_62. Then if formula_63 is a smooth function on formula_51, the Hamiltonian vector field formula_65 can be defined to be formula_66. It is easy to see that


The Poisson bracket formula_68 on ("M", ω) is a bilinear operation on differentiable functions, defined by formula_69; the Poisson bracket of two functions on "M" is itself a function on "M". The Poisson bracket is antisymmetric because:


Here "Xgf" denotes the vector field "Xg" applied to the function "f" as a directional derivative, and formula_71 denotes the (entirely equivalent) Lie derivative of the function "f".


If α is an arbitrary one-form on "M", the vector field Ωα generates (at least locally) a flow formula_72 satisfying the boundary condition formula_73 and the first-order differential equation


The formula_72 will be symplectomorphisms (canonical transformations) for every "t" as a function of "x" if and only if formula_76; when this is true, Ωα is called a symplectic vector field. Recalling Cartan's identity formula_77 and "d"ω = 0, it follows that formula_78. Therefore, Ωα is a symplectic vector field if and only if α is a closed form. Since formula_79, it follows that every Hamiltonian vector field "Xf" is a symplectic vector field, and that the Hamiltonian flow consists of canonical transformations. From above, under the Hamiltonian flow "XH",


This is a fundamental result in Hamiltonian mechanics, governing the time evolution of functions defined on phase space. As noted above, when "{f,H} = 0", "f" is a constant of motion of the system. In addition, in canonical coordinates (with formula_81 and formula_82), Hamilton's equations for the time evolution of the system follow immediately from this formula.


It also follows from that the Poisson bracket is a derivation; that is, it satisfies a non-commutative version of Leibniz's product rule:


The Poisson bracket is intimately connected to the Lie bracket of the Hamiltonian vector fields. Because the Lie derivative is a derivation,


Thus if "v" and "w" are symplectic, using formula_84, Cartan's identity, and the fact that formula_85 is a closed form,


Thus, the Poisson bracket on functions corresponds to the Lie bracket of the associated Hamiltonian vector fields. We have also shown that the Lie bracket of two symplectic vector fields is a Hamiltonian vector field and hence is also symplectic. In the language of abstract algebra, the symplectic vector fields form a subalgebra of the Lie algebra of smooth vector fields on "M", and the Hamiltonian vector fields form an ideal of this subalgebra. The symplectic vector fields are the Lie algebra of the (infinite-dimensional) Lie group of symplectomorphisms of "M".


It is widely asserted that the Jacobi identity for the Poisson bracket,


follows from the corresponding identity for the Lie bracket of vector fields, but this is true only up to a locally constant function. However, to prove the Jacobi identity for the Poisson bracket, it is sufficient to show that:


where the operator formula_90 on smooth functions on "M" is defined by formula_91 and the bracket on the right-hand side is the commutator of operators, formula_92. By , the operator formula_90 is equal to the operator "Xg". The proof of the Jacobi identity follows from because the Lie bracket of vector fields is just their commutator as differential operators.


The algebra of smooth functions on M, together with the Poisson bracket forms a Poisson algebra, because it is a Lie algebra under the Poisson bracket, which additionally satisfies Leibniz's rule . We have shown that every symplectic manifold is a Poisson manifold, that is a manifold with a "curly-bracket" operator on smooth functions such that the smooth functions form a Poisson algebra. However, not every Poisson manifold arises in this way, because Poisson manifolds allow for degeneracy which cannot arise in the symplectic case.


Given a smooth vector field formula_94 on the configuration space, let formula_95 be its conjugate momentum. The conjugate momentum mapping is a Lie algebra anti-homomorphism from the Poisson bracket to the Lie bracket:


This important result is worth a short proof. Write a vector field formula_94 at point formula_98 in the configuration space as


where the formula_100 is the local coordinate frame. The conjugate momentum to formula_94 has the expression


where the formula_2 are the momentum functions conjugate to the coordinates. One then has, for a point formula_104 in the phase space,


The above holds for all formula_106, giving the desired result.


Poisson brackets deform to Moyal brackets upon quantization, that is, they generalize to a different Lie algebra, the Moyal algebra, or, equivalently in Hilbert space, quantum commutators. The Wigner-İnönü group contraction of these (the classical limit, ) yields the above Lie algebra.


To state this more explicitly and precisely, the universal enveloping algebra of the Heisenberg algebra is the Weyl algebra (modulo the relation that the center be the unit). The Moyal product is then a special case of the star product on the algebra of symbols. An explicit definition of the algebra of symbols, and the star product is given in the article on the universal enveloping algebra.


In physics and thermodynamics, the ergodic hypothesis says that, over long periods of time, the time spent by a system in some region of the phase space of microstates with the same energy is proportional to the volume of this region, i.e., that all accessible microstates are equiprobable over a long period of time.


Liouville's theorem states that, for Hamiltonian systems, the local density of microstates following a particle path through phase space is constant as viewed by an observer moving with the ensemble (i.e., the convective time derivative is zero). Thus, if the microstates are uniformly distributed in phase space initially, they will remain so at all times. But Liouville's theorem does "not" imply that the ergodic hypothesis holds for all Hamiltonian systems.


The ergodic hypothesis is often assumed in the statistical analysis of computational physics. The analyst would assume that the average of a process parameter over time and the average over the statistical ensemble are the same. This assumption—that it is as good to simulate a system over a long time as it is to make many independent realizations of the same system—is not always correct. (See, for example, the Fermi–Pasta–Ulam–Tsingou experiment of 1953.)


Assumption of the ergodic hypothesis allows proof that certain types of perpetual motion machines of the second kind are impossible.


Systems that are ergodic are said to have the property of ergodicity; a broad range of systems in geometry, physics and stochastic probability theory are ergodic. Ergodic systems are studied in ergodic theory.


In macroscopic systems, the timescales over which a system can truly explore the entirety of its own phase space can be sufficiently large that the thermodynamic equilibrium state exhibits some form of ergodicity breaking. A common example is that of spontaneous magnetisation in ferromagnetic systems, whereby below the Curie temperature the system preferentially adopts a non-zero magnetisation even though the ergodic hypothesis would imply that no net magnetisation should exist by virtue of the system exploring all states whose time-averaged magnetisation should be zero. The fact that macroscopic systems often violate the literal form of the ergodic hypothesis is an example of spontaneous symmetry breaking.


However, complex disordered systems such as a spin glass show an even more complicated form of ergodicity breaking where the properties of the thermodynamic equilibrium state seen in practice are much more difficult to predict purely by symmetry arguments. Also conventional glasses (e.g. window glasses) violate ergodicity in a complicated manner. In practice this means that on sufficiently short time scales (e.g. those of parts of seconds, minutes, or a few hours) the systems may behave as "solids", i.e. with a positive shear modulus, but on extremely long scales, e.g. over millennia or eons, as "liquids", or with two or more time scales and "plateaux" in between.


The "commutative property" (or "commutative law") is a property generally associated with binary operations and functions. If the commutative property holds for a pair of elements under a certain binary operation then the two elements are said to "commute" under that operation.


The term "commutative" is used in several related senses.


Two well-known examples of commutative binary operations:


Subtraction is noncommutative, since formula_10. However it is classified more precisely as anti-commutative, since formula_11.


Some truth functions are noncommutative, since the truth tables for the functions are different when one changes the order of the operands. For example, the truth tables for and are


Function composition of linear functions from the real numbers to the real numbers is almost always noncommutative. For example, let formula_12 and formula_13. Then


This also applies more generally for linear and affine transformations from a vector space to itself (see below for the Matrix representation).


Matrix multiplication of square matrices is almost always noncommutative, for example:


The vector product (or cross product) of two vectors in three dimensions is anti-commutative; i.e., "b" × "a" = −("a" × "b").


Records of the implicit use of the commutative property go back to ancient times. The Egyptians used the commutative property of multiplication to simplify computing products. Euclid is known to have assumed the commutative property of multiplication in his book "Elements". Formal uses of the commutative property arose in the late 18th and early 19th centuries, when mathematicians began to work on a theory of functions. Today the commutative property is a well-known and basic property used in most branches of mathematics.


The first recorded use of the term "commutative" was in a memoir by François Servois in 1814, which used the word "commutatives" when describing functions that have what is now called the commutative property. The word is a combination of the French word "commuter" meaning "to substitute or switch" and the suffix "-ative" meaning "tending to" so the word literally means "tending to substitute or switch". The term then appeared in English in 1838 in Duncan Farquharson Gregory's article entitled "On the real nature of symbolical algebra" published in 1840 in the Transactions of the Royal Society of Edinburgh.


In truth-functional propositional logic, "commutation", or "commutativity" refer to two valid rules of replacement. The rules allow one to transpose propositional variables within logical expressions in logical proofs. The rules are:


where "formula_19" is a metalogical symbol representing "can be replaced in a proof with".


"Commutativity" is a property of some logical connectives of truth functional propositional logic. The following logical equivalences demonstrate that commutativity is a property of particular connectives. The following are truth-functional tautologies.


In group and set theory, many algebraic structures are called commutative when certain operands satisfy the commutative property. In higher branches of mathematics, such as analysis and linear algebra the commutativity of well-known operations (such as addition and multiplication on real and complex numbers) is often used (or implicitly assumed) in proofs.


The associative property is closely related to the commutative property. The associative property of an expression containing two or more occurrences of the same operator states that the order operations are performed in does not affect the final result, as long as the order of terms does not change. In contrast, the commutative property states that the order of the terms does not affect the final result.


Most commutative operations encountered in practice are also associative. However, commutativity does not imply associativity. A counterexample is the function


which is clearly commutative (interchanging "x" and "y" does not affect the result), but it is not associative (since, for example, formula_25 but formula_26). More such examples may be found in commutative non-associative magmas.


Some forms of symmetry can be directly linked to commutativity. When a commutative operator is written as a binary function then the resulting function is symmetric across the line formula_27. As an example, if we let a function "f" represent addition (a commutative operation) so that formula_28 then formula_29 is a symmetric function, which can be seen in the adjacent image.


For relations, a symmetric relation is analogous to a commutative operation, in that if a relation "R" is symmetric, then formula_30.


In quantum mechanics as formulated by Schrödinger, physical variables are represented by linear operators such as formula_31 (meaning multiply by formula_31), and formula_33. These two operators do not commute as may be seen by considering the effect of their compositions formula_34 and formula_35 (also called products of operators) on a one-dimensional wave function formula_36:


According to the uncertainty principle of Heisenberg, if the two operators representing a pair of variables do not commute, then that pair of variables are mutually complementary, which means they cannot be simultaneously measured or known precisely. For example, the position and the linear momentum in the formula_31-direction of a particle are represented by the operators formula_31 and formula_40, respectively (where formula_41 is the reduced Planck constant). This is the same example except for the constant formula_42, so again the operators do not commute and the physical meaning is that the position and linear momentum in a given direction are complementary.


In geometry, the center of curvature of a curve is found at a point that is at a distance from the curve equal to the radius of curvature lying on the normal vector. It is the point at infinity if the curvature is zero. The osculating circle to the curve is centered at the centre of curvature. Cauchy defined the center of curvature "C" as the intersection point of two infinitely close normal lines to the curve. The locus of centers of curvature for each point on the curve comprise the evolute of the curve. This term is generally used in physics regarding to study of lenses and mirrors.


It can also be defined as the spherical distance between the point at which all the rays falling on a lens or mirror either seems to converge to (in the case of convex lenses and concave mirrors) or diverge from (in the case of concave lenses or convex mirrors) and the lens/mirror itself.


A variable structure system, or VSS, is a discontinuous nonlinear system of the form


where formula_2 is the state vector, formula_3 is the time variable, and formula_4 is a "piecewise continuous" function. Due to the "piecewise" continuity of these systems, they behave like different continuous nonlinear systems in different regions of their state space. At the boundaries of these regions, their dynamics switch abruptly. Hence, their "structure" "varies" over different parts of their state space.


The development of variable structure control depends upon methods of analyzing variable structure systems, which are special cases of hybrid dynamical systems.


The covariant formulation of classical electromagnetism refers to ways of writing the laws of classical electromagnetism (in particular, Maxwell's equations and the Lorentz force) in a form that is manifestly invariant under Lorentz transformations, in the formalism of special relativity using rectilinear inertial coordinate systems. These expressions both make it simple to prove that the laws of classical electromagnetism take the same form in any inertial coordinate system, and also provide a way to translate the fields and forces from one frame to another. However, this is not as general as Maxwell's equations in curved spacetime or non-rectilinear coordinate systems.


This article uses the classical treatment of tensors and Einstein summation convention throughout and the Minkowski metric has the form diag(+1, −1, −1, −1). Where the equations are specified as holding in a vacuum, one could instead regard them as the formulation of Maxwell's equations in terms of "total" charge and current.


For a more general overview of the relationships between classical electromagnetism and special relativity, including various conceptual implications of this picture, see Classical electromagnetism and special relativity.


Lorentz tensors of the following kinds may be used in this article to describe bodies or particles:


The signs in the following tensor analysis depend on the convention used for the metric tensor. The convention used here is , corresponding to the Minkowski metric tensor:


The electromagnetic tensor is the combination of the electric and magnetic fields into a covariant antisymmetric tensor whose entries are B-field quantities.


and the result of raising its indices is


where E is the electric field, B the magnetic field, and "c" the speed of light.


The four-current is the contravariant four-vector which combines electric charge density "ρ" and electric current density j:


The electromagnetic four-potential is a covariant four-vector containing the electric potential (also called the scalar potential) "ϕ" and magnetic vector potential (or vector potential) A, as follows:


The differential of the electromagnetic potential is


In the language of differential forms, which provides the generalisation to curved spacetimes, these are the components of a 1-form formula_16 and a 2-form formula_17 respectively. Here, "formula_18" is the exterior derivative and formula_19 the wedge product.


The electromagnetic stress–energy tensor can be interpreted as the flux density of the momentum four-vector, and is a contravariant symmetric tensor that is the contribution of the electromagnetic fields to the overall stress–energy tensor:


where "formula_21" is the electric permittivity of vacuum, "μ"0 is the magnetic permeability of vacuum, the Poynting vector is


and the Maxwell stress tensor is given by