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The stress–energy tensor, sometimes called the stress–energy–momentum tensor or the energy–momentum tensor, is a
tensor In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tensor ...
physical quantity that describes the
density Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematical ...
and flux of
energy In physics, energy (from Ancient Greek: ἐνέργεια, ''enérgeia'', “activity”) is the quantitative property that is transferred to a body or to a physical system, recognizable in the performance of work and in the form of hea ...
and momentum in
spacetime In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differ ...
, generalizing the stress tensor of
Newtonian physics Classical mechanics is a physical theory describing the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars, and galaxies. For objects governed by classical mec ...
. It is an attribute of
matter In classical physics and general chemistry, matter is any substance that has mass and takes up space by having volume. All everyday objects that can be touched are ultimately composed of atoms, which are made up of interacting subatomic part ...
, radiation, and non-gravitational force fields. This density and flux of energy and momentum are the sources of the gravitational field in the
Einstein field equations In the general theory of relativity, the Einstein field equations (EFE; also known as Einstein's equations) relate the geometry of spacetime to the distribution of matter within it. The equations were published by Einstein in 1915 in the form ...
of
general relativity General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
, just as mass density is the source of such a field in Newtonian gravity.


Definition

The stress–energy tensor involves the use of superscripted variables (''not'' exponents; see
tensor index notation In mathematics, Ricci calculus constitutes the rules of index notation and manipulation for tensors and tensor fields on a differentiable manifold, with or without a metric tensor or connection. It is also the modern name for what used to be c ...
and
Einstein summation notation In mathematics, especially the usage of linear algebra in Mathematical physics, Einstein notation (also known as the Einstein summation convention or Einstein summation notation) is a notational convention that implies summation over a set of i ...
). If Cartesian coordinates in SI units are used, then the components of the position
four-vector In special relativity, a four-vector (or 4-vector) is an object with four components, which transform in a specific way under Lorentz transformations. Specifically, a four-vector is an element of a four-dimensional vector space considered as a ...
are given by: , , , and , where ''t'' is time in seconds, and ''x'', ''y'', and ''z'' are distances in meters. The stress–energy tensor is defined as the
tensor In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tensor ...
''T''''αβ'' of order two that gives the flux of the ''α''th component of the momentum
vector Vector most often refers to: *Euclidean vector, a quantity with a magnitude and a direction *Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism Vector may also refer to: Mathematic ...
across a surface with constant ''x''''β''
coordinate In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is sign ...
. In the theory of relativity, this momentum vector is taken as the four-momentum. In general relativity, the stress–energy tensor is symmetric, :T^ = T^. In some alternative theories like Einstein–Cartan theory, the stress–energy tensor may not be perfectly symmetric because of a nonzero
spin tensor In mathematics, mathematical physics, and theoretical physics, the spin tensor is a quantity used to describe the rotational motion of particles in spacetime. The tensor has application in general relativity and special relativity, as well as qu ...
, which geometrically corresponds to a nonzero
torsion tensor In differential geometry, the notion of torsion is a manner of characterizing a twist or screw of a moving frame around a curve. The torsion of a curve, as it appears in the Frenet–Serret formulas, for instance, quantifies the twist of a cur ...
.


The components of the stress-energy tensor

Because the stress–energy tensor is of order 2, its components can be displayed in 4 × 4 matrix form: : (T^)_ = \begin T^ & T^ & T^ & T^ \\ T^ & T^ & T^ & T^ \\ T^ & T^ & T^ & T^ \\ T^ & T^ & T^ & T^ \end. In the following, and range from 1 through 3: In solid state physics and
fluid mechanics Fluid mechanics is the branch of physics concerned with the mechanics of fluids ( liquids, gases, and plasmas) and the forces on them. It has applications in a wide range of disciplines, including mechanical, aerospace, civil, chemical and ...
, the stress tensor is defined to be the spatial components of the stress–energy tensor in the
proper frame A proper frame, or comoving frame, is a frame of reference that is attached to an object. The object in this frame is stationary within the frame, which is useful for many types of calculations. For example, a freely falling elevator is a proper fr ...
of reference. In other words, the stress–energy tensor in
engineering Engineering is the use of scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engineering encompasses a broad range of more speciali ...
''differs'' from the relativistic stress–energy tensor by a momentum-convective term.


Covariant and mixed forms

Most of this article works with the contravariant form, of the stress–energy tensor. However, it is often necessary to work with the covariant form, :T_ = T^ g_ g_, or the mixed form, :T^\mu_\nu = T^ g_, or as a mixed tensor density :\mathfrak^\mu_\nu = T^\mu_\nu \sqrt \,. This article uses the spacelike
sign convention In physics, a sign convention is a choice of the physical significance of signs (plus or minus) for a set of quantities, in a case where the choice of sign is arbitrary. "Arbitrary" here means that the same physical system can be correctly describ ...
(−+++) for the metric signature.


Conservation law


In special relativity

The stress–energy tensor is the conserved Noether current associated with
spacetime In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differ ...
translation Translation is the communication of the meaning of a source-language text by means of an equivalent target-language text. The English language draws a terminological distinction (which does not exist in every language) between ''transla ...
s. The divergence of the non-gravitational stress–energy is zero. In other words, non-gravitational energy and momentum are conserved, :0 = T^_ = \nabla_\nu T^. \! When gravity is negligible and using a Cartesian coordinate system for spacetime, this may be expressed in terms of partial derivatives as :0 = T^_ = \partial_ T^. \! The integral form of this is :0 = \int_ T^ \mathrm^3 s_ \! where ''N'' is any compact four-dimensional region of spacetime; \partial N is its boundary, a three-dimensional hypersurface; and \mathrm^3 s_ is an element of the boundary regarded as the outward pointing normal. In flat spacetime and using Cartesian coordinates, if one combines this with the symmetry of the stress–energy tensor, one can show that
angular momentum In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational analog of linear momentum. It is an important physical quantity because it is a conserved quantity—the total angular momentum of a closed syst ...
is also conserved: :0 = (x^ T^ - x^ T^)_ . \!


In general relativity

When gravity is non-negligible or when using arbitrary coordinate systems, the divergence of the stress–energy still vanishes. But in this case, a coordinate-free definition of the divergence is used which incorporates the covariant derivative :0 = \operatorname T = T^_ = \nabla_ T^ = T^_ + \Gamma^_T^ + \Gamma^_ T^ where \Gamma^_ is the
Christoffel symbol In mathematics and physics, the Christoffel symbols are an array of numbers describing a metric connection. The metric connection is a specialization of the affine connection to surfaces or other manifolds endowed with a metric, allowing dist ...
which is the gravitational force field. Consequently, if \xi^ is any Killing vector field, then the conservation law associated with the symmetry generated by the Killing vector field may be expressed as :0 = \nabla_\nu \left(\xi^ T_^\right) = \frac \partial_\nu \left(\sqrt\ \xi^ T_^\right) The integral form of this is :0 = \int_ \sqrt \ \xi^ T_^ \ \mathrm^3 s_ = \int_ \xi^ \mathfrak_^ \ \mathrm^3 s_


In special relativity

In
special relativity In physics, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between space and time. In Albert Einstein's original treatment, the theory is based on two postulates: # The laws ...
, the stress–energy tensor contains information about the energy and momentum densities of a given system, in addition to the momentum and energy flux densities. Given a Lagrangian Density \mathcal that is a function of a set of fields \phi_ and their derivatives, but explicitly not of any of the spacetime coordinates, we can construct the tensor by looking at the total derivative with respect to one of the generalized coordinates of the system. So, with our condition :\frac = 0 By using the chain rule, we then have :\frac = \partial^\mathcal = \frac\frac + \frac\frac Written in useful shorthand, :\partial^\mathcal = \frac\partial^\partial_\phi_ + \frac\partial^\phi_ Then, we can use the Euler–Lagrange Equation: :\partial_\left(\frac\right) = \frac And then use the fact that partial derivatives commute so that we now have :\partial^\mathcal = \frac\partial_\partial^\phi_ + \partial_\left(\frac\right)\partial^\phi_ We can recognize the right hand side as a product rule. Writing it as the derivative of a product of functions tells us that :\partial^\mathcal = \partial_\left frac\partial^\phi_\right/math> Now, in flat space, one can write \partial^\mathcal = \partial_g^\mathcal. Doing this and moving it to the other side of the equation tells us that :\partial_\left frac\partial^\phi_\right- \partial_\left(g^\mathcal\right) = 0 And upon regrouping terms, :\partial_\left frac\partial^\phi_ - g^\mathcal\right= 0 This is to say that the divergence of the tensor in the brackets is 0. Indeed, with this, we define the stress–energy tensor: :T^ \equiv \frac\partial^\phi_ - g^\mathcal By construction it has the property that :\partial_T^ = 0 Note that this divergenceless property of this tensor is equivalent to four
continuity equations A continuity equation or transport equation is an equation that describes the transport of some quantity. It is particularly simple and powerful when applied to a conserved quantity, but it can be generalized to apply to any extensive quantity. ...
. That is, fields have at least four sets of quantities that obey the continuity equation. As an example, it can be seen that T^_0 is the energy density of the system and that it is thus possible to obtain the Hamiltonian density from the stress–energy tensor. Indeed, since this is the case, observing that \partial_T^ = 0, we then have : \frac + \nabla\cdot\left(\frac\dot_\right) = 0 We can then conclude that the terms of \frac\dot_ represent the energy flux density of the system.


Trace

Note that the trace of the stress–energy tensor is defined to be T^_, where :T^_ = T^g_ . When we use the formula for the stress–energy tensor found above, :T^_ = \fracg_\partial^\phi_-g_g^\mathcal . Using the raising and lowering properties of the metric and that g^g_ = \delta^_, :T^_ = \frac\partial_\phi_-\delta^_\mathcal . Since \delta^_ = 4, :T^_ = \frac\partial_\phi_-4\mathcal .


In general relativity

In
general relativity General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
, the
symmetric Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definiti ...
stress–energy tensor acts as the source of spacetime curvature, and is the current density associated with
gauge transformation In physics, a gauge theory is a type of field theory in which the Lagrangian (and hence the dynamics of the system itself) does not change (is invariant) under local transformations according to certain smooth families of operations (Lie group ...
s of gravity which are general curvilinear
coordinate transformation In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is sign ...
s. (If there is
torsion Torsion may refer to: Science * Torsion (mechanics), the twisting of an object due to an applied torque * Torsion of spacetime, the field used in Einstein–Cartan theory and ** Alternatives to general relativity * Torsion angle, in chemistry Bi ...
, then the tensor is no longer symmetric. This corresponds to the case with a nonzero
spin tensor In mathematics, mathematical physics, and theoretical physics, the spin tensor is a quantity used to describe the rotational motion of particles in spacetime. The tensor has application in general relativity and special relativity, as well as qu ...
in Einstein–Cartan gravity theory.) In general relativity, the
partial derivatives In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). Pa ...
used in special relativity are replaced by covariant derivatives. What this means is that the continuity equation no longer implies that the non-gravitational energy and momentum expressed by the tensor are absolutely conserved, i.e. the gravitational field can do work on matter and vice versa. In the classical limit of Newtonian gravity, this has a simple interpretation: kinetic energy is being exchanged with
gravitational potential energy Gravitational energy or gravitational potential energy is the potential energy a massive object has in relation to another massive object due to gravity. It is the potential energy associated with the gravitational field, which is released (conver ...
, which is not included in the tensor, and momentum is being transferred through the field to other bodies. In general relativity the Landau–Lifshitz pseudotensor is a unique way to define the ''gravitational'' field energy and momentum densities. Any such stress–energy pseudotensor can be made to vanish locally by a coordinate transformation. In curved spacetime, the spacelike
integral In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along wit ...
now depends on the spacelike slice, in general. There is in fact no way to define a global energy–momentum vector in a general curved spacetime.


Einstein field equations

In general relativity, the stress-energy tensor is studied in the context of the Einstein field equations which are often written as :R_ - \tfrac R\,g_ + \Lambda g_ = T_, where R_ is the Ricci tensor, R is the Ricci scalar (the
tensor contraction In multilinear algebra, a tensor contraction is an operation on a tensor that arises from the natural pairing of a finite-dimensional vector space and its dual. In components, it is expressed as a sum of products of scalar components of the tens ...
of the Ricci tensor), g_\, is the metric tensor, is the
cosmological constant In cosmology, the cosmological constant (usually denoted by the Greek capital letter lambda: ), alternatively called Einstein's cosmological constant, is the constant coefficient of a term that Albert Einstein temporarily added to his field eq ...
(negligible at the scale of a galaxy or smaller), and G is the universal gravitational constant.


Stress–energy in special situations


Isolated particle

In special relativity, the stress–energy of a non-interacting particle with rest mass ''m'' and trajectory \mathbf_\text(t) is: :T^(\mathbf, t) = \frac\;\, \delta\left(\mathbf - \mathbf_\text(t)\right) = \frac\; v^(t) v^(t)\;\, \delta(\mathbf - \mathbf_\text(t)) where \left(v^\right)_ \! is the velocity vector (which should not be confused with
four-velocity In physics, in particular in special relativity and general relativity, a four-velocity is a four-vector in four-dimensional spacetimeTechnically, the four-vector should be thought of as residing in the tangent space of a point in spacetime, spacet ...
, since it is missing a \gamma) : \left(v^\right)_ = \left(1, \frac(t) \right) \,, \delta is the Dirac delta function and E = \sqrt is the
energy In physics, energy (from Ancient Greek: ἐνέργεια, ''enérgeia'', “activity”) is the quantitative property that is transferred to a body or to a physical system, recognizable in the performance of work and in the form of hea ...
of the particle. Written in language of classical physics, the stress-energy tensor would be (relativistic mass, momentum, the
dyadic product In mathematics, specifically multilinear algebra, a dyadic or dyadic tensor is a second order tensor, written in a notation that fits in with vector algebra. There are numerous ways to multiply two Euclidean vectors. The dot product takes in two v ...
of momentum and velocity) :\left( \frac , \, \mathbf , \, \mathbf \, \mathbf \right).


Stress–energy of a fluid in equilibrium

For a
perfect fluid In physics, a perfect fluid is a fluid that can be completely characterized by its rest frame mass density \rho_m and ''isotropic'' pressure ''p''. Real fluids are "sticky" and contain (and conduct) heat. Perfect fluids are idealized models in whi ...
in
thermodynamic equilibrium Thermodynamic equilibrium is an axiomatic concept of thermodynamics. It is an internal state of a single thermodynamic system, or a relation between several thermodynamic systems connected by more or less permeable or impermeable walls. In the ...
, the stress–energy tensor takes on a particularly simple form :T^ \, = \left(\rho + \right)u^u^ + p g^ where \rho is the mass–energy density (kilograms per cubic meter), p is the hydrostatic pressure ( pascals), u^ is the fluid's
four-velocity In physics, in particular in special relativity and general relativity, a four-velocity is a four-vector in four-dimensional spacetimeTechnically, the four-vector should be thought of as residing in the tangent space of a point in spacetime, spacet ...
, and g^ is the matrix inverse of the metric tensor. Therefore, the trace is given by :T^_ = g_ T^ = 3p - \rho c^2 \,. The
four-velocity In physics, in particular in special relativity and general relativity, a four-velocity is a four-vector in four-dimensional spacetimeTechnically, the four-vector should be thought of as residing in the tangent space of a point in spacetime, spacet ...
satisfies :u^ u^ g_ = - c^2 \,. In an
inertial frame of reference In classical physics and special relativity, an inertial frame of reference (also called inertial reference frame, inertial frame, inertial space, or Galilean reference frame) is a frame of reference that is not undergoing any acceleration. ...
comoving with the fluid, better known as the fluid's
proper frame A proper frame, or comoving frame, is a frame of reference that is attached to an object. The object in this frame is stationary within the frame, which is useful for many types of calculations. For example, a freely falling elevator is a proper fr ...
of reference, the four-velocity is :(u^)_ = (1, 0, 0, 0) \,, the matrix inverse of the metric tensor is simply : (g^)_ \, = \left( \begin - \frac & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end \right) \, and the stress–energy tensor is a diagonal matrix : (T^)_ = \left( \begin \rho & 0 & 0 & 0 \\ 0 & p & 0 & 0 \\ 0 & 0 & p & 0 \\ 0 & 0 & 0 & p \end \right).


Electromagnetic stress–energy tensor

The Hilbert stress–energy tensor of a source-free electromagnetic field is : T^ = \frac \left( F^ g_ F^ - \frac g^ F_ F^ \right) where F_ is the electromagnetic field tensor.


Scalar field

The stress–energy tensor for a complex scalar field \phi that satisfies the Klein–Gordon equation is :T^ = \frac \left(g^ g^ + g^ g^ - g^ g^\right) \partial_\bar\phi \partial_\phi - g^ m c^2 \bar\phi \phi , and when the metric is flat (Minkowski in Cartesian coordinates) its components work out to be: :\begin T^ & = \frac \left(\partial_0 \bar \partial_0 \phi + c^2 \partial_k \bar \partial_k \phi \right) + m \bar \phi, \\ T^ = T^ & = - \frac \left(\partial_0 \bar \partial_i \phi + \partial_i \bar \partial_0 \phi \right),\ \mathrm \\ T^ & = \frac \left(\partial_i \bar \partial_j \phi + \partial_j \bar \partial_i \phi \right) - \delta_ \left(\frac \eta^ \partial_\alpha \bar \partial_\beta \phi + m c^2 \bar \phi\right). \end


Variant definitions of stress–energy

There are a number of inequivalent definitions of non-gravitational stress–energy:


Hilbert stress–energy tensor

The Hilbert stress–energy tensor is defined as the functional derivative :T_ = \frac\frac = \frac\frac = -2 \frac + g_ \mathcal_\mathrm, where S_ is the nongravitational part of the
action Action may refer to: * Action (narrative), a literary mode * Action fiction, a type of genre fiction * Action game, a genre of video game Film * Action film, a genre of film * ''Action'' (1921 film), a film by John Ford * ''Action'' (1980 fil ...
, \mathcal_ is the nongravitational part of the
Lagrangian Lagrangian may refer to: Mathematics * Lagrangian function, used to solve constrained minimization problems in optimization theory; see Lagrange multiplier ** Lagrangian relaxation, the method of approximating a difficult constrained problem with ...
density, and the Euler-Lagrange equation has been used. This is symmetric and gauge-invariant. See Einstein–Hilbert action for more information.


Canonical stress–energy tensor

Noether's theorem Noether's theorem or Noether's first theorem states that every differentiable symmetry of the action of a physical system with conservative forces has a corresponding conservation law. The theorem was proven by mathematician Emmy Noether ...
implies that there is a conserved current associated with translations through space and time. This is called the canonical stress–energy tensor. Generally, this is not symmetric and if we have some gauge theory, it may not be
gauge invariant In physics, a gauge theory is a type of field theory in which the Lagrangian (and hence the dynamics of the system itself) does not change (is invariant) under local transformations according to certain smooth families of operations (Lie group ...
because space-dependent
gauge transformation In physics, a gauge theory is a type of field theory in which the Lagrangian (and hence the dynamics of the system itself) does not change (is invariant) under local transformations according to certain smooth families of operations (Lie group ...
s do not commute with spatial translations. In
general relativity General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
, the translations are with respect to the coordinate system and as such, do not transform covariantly. See the section below on the gravitational stress–energy pseudo-tensor.


Belinfante–Rosenfeld stress–energy tensor

In the presence of spin or other intrinsic angular momentum, the canonical Noether stress–energy tensor fails to be symmetric. The Belinfante–Rosenfeld stress–energy tensor is constructed from the canonical stress–energy tensor and the spin current in such a way as to be symmetric and still conserved. In general relativity, this modified tensor agrees with the Hilbert stress–energy tensor.


Gravitational stress–energy

By the equivalence principle gravitational stress–energy will always vanish locally at any chosen point in some chosen frame, therefore gravitational stress–energy cannot be expressed as a non-zero tensor; instead we have to use a pseudotensor. In general relativity, there are many possible distinct definitions of the gravitational stress–energy–momentum pseudotensor. These include the Einstein pseudotensor and the Landau–Lifshitz pseudotensor. The Landau–Lifshitz pseudotensor can be reduced to zero at any event in spacetime by choosing an appropriate coordinate system.


See also

* Electromagnetic stress–energy tensor * Energy condition * Energy density of electric and magnetic fields *
Maxwell stress tensor The Maxwell stress tensor (named after James Clerk Maxwell) is a symmetric second-order tensor used in classical electromagnetism to represent the interaction between electromagnetic forces and mechanical momentum. In simple situations, such as ...
* Poynting vector * Ricci calculus * Segre classification


Notes and references

*


External links


Lecture, Stephan Waner


— A simple discussion of the relation between the Stress–Energy tensor of General Relativity and the metric {{DEFAULTSORT:Stress-energy tensor Tensor physical quantities Density