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{{Short description|Tensor describing the density and flux of energy in spacetime}}
{{Use American English|date=January 2019}}[[File:StressEnergyTensor contravariant.svg|right|236px|thumb|Contravariant components of the stress–energy tensor.]]
{{General relativity sidebar}}

The '''stress–energy tensor''', sometimes '''stress–energy–momentum tensor''' or '''energy–momentum tensor''', is a [[tensor]] quantity in [[physics]] that describes the [[density]] and [[flux]] of [[energy]] and [[momentum]] in [[spacetime]], generalizing the [[Cauchy stress tensor|stress tensor]] of [[Newtonian physics]]. It is an attribute of [[matter]], [[radiation]], and non-gravitational [[force field (physics)|force field]]s. The stress–energy tensor is the source of the [[gravitational field]] in the [[Einstein field equations]] of [[general relativity]], 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]] and [[Einstein notation|Einstein summation notation]]). If [[Cartesian coordinates]] in [[SI units]] are used, then the components of the position [[four-vector]] are given by: {{nowrap|1=''x''<sup>0</sup> = ''t''}}, {{nowrap|1=''x''<sup>1</sup> = ''x''}}, {{nowrap|1=''x''<sup>2</sup> = ''y''}}, and {{nowrap|1=''x''<sup>3</sup> = ''z''}}, where ''t'' is time in seconds, and ''x'', ''y'', and ''z'' are distances in meters.

The stress–energy tensor is defined as the [[tensor]] ''T''<sup>''αβ''</sup> of order two that gives the [[flux]] of the ''α''th component of the [[momentum]] [[vector (geometric)|vector]] across a surface with constant ''x''<sup>''β''</sup> [[coordinate]]. In the theory of [[general relativity|relativity]], this momentum vector is taken as the [[four-momentum]]. In general relativity, the stress–energy tensor is symmetric,<ref>On pp. 141–142 of [[Gravitation (book)|Misner, Thorne, and Wheeler]], section 5.7 "Symmetry of the Stress–Energy Tensor" begins with "All the stress–energy tensors explored above were symmetric. That they could not have been otherwise one sees as follows."</ref>
:<math>T^{\alpha \beta} = T^{\beta \alpha}.</math>

In some alternative theories like [[Einstein–Cartan theory]], the stress–energy tensor may not be perfectly symmetric because of a nonzero [[spin tensor]], which geometrically corresponds to a nonzero [[torsion tensor]].

==Identifying the components of the tensor==
Because the stress–energy tensor is of order two, its components can be displayed in 4&thinsp;×&thinsp;4 matrix form:
:<math>
(T^{\mu\nu})_{\mu,\nu=0,1,2,3} = \begin{pmatrix} T^{00} & T^{01} & T^{02} & T^{03} \\ T^{10} & T^{11} & T^{12} & T^{13} \\ T^{20} & T^{21} & T^{22} & T^{23} \\ T^{30} & T^{31} & T^{32} & T^{33} \end{pmatrix}.</math>

In the following, ''i'' and ''k'' range from 1 through 3.

The time–time component is the density of relativistic mass, i.e., the [[energy density]] divided by the speed of light squared.<ref>Charles W., Misner, Thorne, Kip S., Wheeler, John A., (1973). Gravitation. San Frandisco: W. H. Freeman and Company. {{ISBN|0-7167-0334-3}}.</ref> Its components have a direct physical interpretation. In the case of a perfect fluid this component is
:<math>T^{00} = \rho,</math>

where <math>\rho</math> is the [[relativistic mass]] per unit volume, and for an electromagnetic field in otherwise empty space this component is
:<math>T^{00} = {1 \over c^2}\left(\frac{1}{2} \epsilon_0 E^2 + \frac{1}{2\mu_0} B^2 \right),</math>

where ''E'' and ''B'' are the electric and magnetic fields, respectively.<ref>d'Inverno, R.A, (1992). Introducing Einstein's Relativity. New York: Oxford University Press. {{ISBN|978-0-19-859686-8}}.</ref>

The flux of relativistic mass across the ''x<sup>i</sup>'' surface is equivalent to the density of the ''i''th component of linear momentum,
:<math>T^{0i} = T^{i0}.</math>

The components
:<math> T^{ik}</math>
represent flux of ''i''th component of linear momentum across the ''x<sup>k</sup>'' surface.  In particular,
:<math> T^{ii}</math>
(not summed) represents [[tensile stress|normal stress]], which is called [[pressure]] when it is independent of direction. The remaining components
:<math> T^{ik} \quad i \ne k </math>
represent [[shear stress]] (compare with the [[stress (physics)|stress tensor]]).

In [[solid state physics]] and [[fluid mechanics]], the stress tensor is defined to be the spatial components of the stress–energy tensor in the [[proper frame]] of reference. In other words, the stress energy tensor in [[engineering]] ''differs'' from the relativistic stress–energy tensor by a momentum convective term.

===Covariant and mixed forms===
In most of this article we work with the contravariant form, ''T''<sup>''μν''</sup> of the stress–energy tensor. However, it is often necessary to work with the covariant form,
:<math>T_{\mu \nu} = T^{\alpha \beta} g_{\alpha \mu} g_{\beta \nu},</math>

or the mixed form,
:<math>T^\mu{}_\nu = T^{\mu \alpha} g_{\alpha \nu},</math>

or as a mixed [[tensor density]]
:<math>\mathfrak{T}^\mu{}_\nu = T^\mu{}_\nu \sqrt{-g} \,.</math>

In this article we use the spacelike [[Sign convention#Metric signature|sign convention]] (−+++) for the metric signature.

==Conservation law==

===In special relativity===
{{See also|Relativistic angular momentum|Four-momentum}}

The stress–energy tensor is the conserved [[Noether's theorem|Noether current]] associated with [[spacetime]] [[translation (physics)|translation]]s.

The divergence of the non-gravitational stress–energy is zero. In other words, non-gravitational energy and momentum are conserved,
:<math>0 = T^{\mu \nu}{}_{;\nu} = \nabla_\nu T^{\mu \nu}{}. \!</math>
When gravity is negligible and using a [[Cartesian coordinate system]] for spacetime, this may be expressed in terms of partial derivatives as 
:<math>0 = T^{\mu \nu}{}_{,\nu} = \partial_{\nu} T^{\mu \nu}. \!</math>

The integral form of this is
:<math>0 = \int_{\partial N} T^{\mu \nu} \mathrm{d}^3 s_{\nu} \!</math>

where ''N'' is any compact four-dimensional region of spacetime; <math>\partial N</math> is its boundary, a three-dimensional hypersurface; and <math>\mathrm{d}^3 s_{\nu}</math> 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]] is also conserved:
:<math>0 = (x^{\alpha} T^{\mu \nu} - x^{\mu} T^{\alpha \nu})_{,\nu} . \!</math>

===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 [[Divergence#Generalizations|coordinate-free definition of the divergence]] is used which incorporates the [[covariant derivative]]
:<math>0 = \operatorname{div} T = T^{\mu \nu}{}_{;\nu} = \nabla_{\nu} T^{\mu \nu} = T^{\mu \nu}{}_{,\nu} +  \Gamma^{\mu}{}_{\sigma \nu}T^{\sigma \nu} + \Gamma^{\nu}{}_{\sigma \nu} T^{\mu \sigma}</math>

where <math>\Gamma^{\mu}{}_{\sigma \nu} </math> is the [[Christoffel symbol]] which is the gravitational [[force field (physics)|force field]].

Consequently, if <math>\xi^{\mu}</math> is any [[Killing vector field]], then the conservation law associated with the symmetry generated by the Killing vector field may be expressed as
:<math>0 = \nabla_\nu (\xi^{\mu} T_{\mu}^{\nu}) = \frac{1}{\sqrt{-g}} \partial_\nu( \sqrt{-g} \ \xi^{\mu} T_{\mu}^{\nu}) </math>

The integral form of this is
:<math>0 = \int_{\partial N} \sqrt{-g} \ \xi^{\mu} T_{\mu}^{\nu} \ \mathrm{d}^3 s_{\nu} = \int_{\partial N}  \xi^{\mu} \mathfrak{T}_{\mu}^{\nu} \ \mathrm{d}^3 s_{\nu}</math>

==In general relativity==
In [[general relativity]], the [[symmetric]] stress–energy tensor acts as the source of spacetime [[Riemann curvature tensor|curvature]], and is the current density associated with [[gauge transformation]]s of gravity which are general curvilinear [[coordinate transformation]]s. (If there is [[Torsion tensor|torsion]], then the tensor is no longer symmetric. This corresponds to the case with a nonzero [[spin tensor]] in [[Einstein–Cartan theory|Einstein–Cartan gravity theory]].)

In general relativity, the [[partial derivatives]] used in special relativity are replaced by [[covariant derivative]]s. 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]], which is not included in the tensor, and momentum is being transferred through the field to other bodies. In general relativity the [[Stress–energy–momentum pseudotensor#Landau–Lifshitz pseudotensor|Landau–Lifshitz pseudotensor]] is a unique way to define the ''gravitational'' field energy and momentum densities. Any such [[Stress–energy–momentum pseudotensor|stress–energy pseudotensor]] can be made to vanish locally by a coordinate transformation.

In curved spacetime, the spacelike [[integral]] 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.

===The Einstein field equations===
{{Main|Einstein field equations}}

In general relativity, the stress tensor is studied in the context of the Einstein field equations which are often written as

:<math>R_{\mu \nu} - \tfrac{1}{2} R\,g_{\mu \nu} + \Lambda g_{\mu \nu} = {8 \pi G \over c^4} T_{\mu \nu},</math>

where <math>R_{\mu \nu}</math> is the [[Ricci tensor]], <math>R</math> is the Ricci scalar (the [[tensor contraction]] of the Ricci tensor), <math>g_{\mu \nu}\,</math> is the [[metric tensor (general relativity)|metric tensor]], {{mvar|Λ}} is the [[cosmological constant]] (negligible at the scale of a galaxy or smaller), and <math>G</math> is the [[universal gravitational constant]].

==Stress–energy in special situations==

===Isolated particle===
In special relativity, the stress–energy of a non-interacting particle with mass ''m'' and trajectory <math> \mathbf{x}_\text{p}(t) </math> is:
:<math>T^{\alpha \beta}(\mathbf{x},t) = \frac{m \, v^{\alpha}(t) v^{\beta}(t)}{\sqrt{1 - (v/c)^2}}\;\, \delta(\mathbf{x} - \mathbf{x}_\text{p}(t)) = \frac{E}{c^2}\;
 v^{\alpha}(t) v^{\beta}(t)\;\, \delta(\mathbf{x} - \mathbf{x}_\text{p}(t)) </math>

where <math>(v^{\alpha})_{\alpha=0,1,2,3} \!</math> is the velocity vector (which should not be confused with [[four-velocity]], since it is missing a <math>\gamma</math>)
:<math> (v^{\alpha})_{\alpha=0,1,2,3} = \left(1, \frac{d \mathbf{x}_\text{p}}{dt}(t) \right) \,,</math>
δ is the [[Dirac delta function]] and <math> E = \sqrt{p^2 c^2 + m^2 c^4} </math> is the [[energy]] of the particle.

===Stress–energy of a fluid in equilibrium===
For a [[perfect fluid]] in [[thermodynamic equilibrium]], the stress–energy tensor takes on a particularly simple form
:<math>T^{\alpha \beta} \, = \left(\rho + {p \over c^2}\right)u^{\alpha}u^{\beta} + p g^{\alpha \beta}</math>

where <math>\rho</math> is the mass–energy density (kilograms per cubic meter), <math>p</math> is the hydrostatic pressure ([[pascal (unit)|pascals]]), <math>u^{\alpha}</math> is the fluid's [[four velocity]], and <math>g^{\alpha \beta}</math> is the reciprocal of the [[metric tensor (general relativity)|metric tensor]]. Therefore, the trace is given by
:<math>T  = 3p - \rho c^2 \,.</math>

The four velocity satisfies
:<math>u^{\alpha} u^{\beta} g_{\alpha \beta} = - c^2 \,.</math>

In an [[inertial frame of reference]] comoving with the fluid, better known as the fluid's [[proper frame]] of reference, the four velocity is 
:<math>(u^{\alpha})_{\alpha=0,1,2,3} = (1, 0, 0, 0) \,,</math>

the reciprocal of the metric tensor is simply
:<math>
(g^{\alpha \beta})_{\alpha,\beta=0,1,2,3} \, = \left( \begin{matrix}
                   - c^{-2} & 0 & 0 & 0 \\
                   0 & 1 & 0 & 0 \\
                   0 & 0 & 1 & 0 \\
                   0 & 0 & 0 & 1    
      \end{matrix} \right)
\, </math>

and the stress–energy tensor is a diagonal matrix
:<math>
(T^{\alpha \beta})_{\alpha,\beta=0,1,2,3} = \left( \begin{matrix}
                   \rho & 0 & 0 & 0 \\
                   0 & p & 0 & 0 \\
                   0 & 0 & p & 0 \\
                   0 & 0 & 0 & p    
      \end{matrix} \right).
</math>

===Electromagnetic stress–energy tensor===
{{Main|Electromagnetic stress–energy tensor}}
The Hilbert stress–energy tensor of a source-free electromagnetic field is
:<math> T^{\mu \nu} = \frac{1}{\mu_0} \left( F^{\mu \alpha} g_{\alpha \beta} F^{\nu \beta} - \frac{1}{4} g^{\mu \nu} F_{\delta \gamma} F^{\delta \gamma} \right) </math>

where <math> F_{\mu \nu} </math> is the [[electromagnetic field tensor]].

===Scalar field===
{{Main|Klein–Gordon equation}}
The stress–energy tensor for a complex scalar field <math>\phi </math> which satisfies the Klein–Gordon equation is
:<math>T^{\mu\nu} = \frac{\hbar^2}{m} (g^{\mu \alpha} g^{\nu \beta} + g^{\mu \beta} g^{\nu \alpha} - g^{\mu\nu} g^{\alpha \beta}) \partial_{\alpha}\bar\phi \partial_{\beta}\phi - g^{\mu\nu} m c^2 \bar\phi \phi ,</math>
and when the metric is flat (Minkowski) its components work out to be:
:<math>\begin{align} T^{00} & =  \frac{\hbar^2}{m c^4} \left(\partial_0 \bar{\phi} \partial_0 \phi + c^2 \partial_k \bar{\phi} \partial_k \phi \right) + m \bar{\phi} \phi, \\
T^{0i} = T^{i0} & = - \frac{\hbar^2}{m c^2} \left(\partial_0 \bar{\phi} \partial_i \phi + \partial_i \bar{\phi} \partial_0 \phi \right),\ \mathrm{and} \\
T^{ij} & = \frac{\hbar^2}{m} \left(\partial_i \bar{\phi} \partial_j \phi + \partial_j \bar{\phi} \partial_i \phi \right) - \delta_{ij} \left(\frac{\hbar^2}{m} \eta^{\alpha\beta} \partial_\alpha \bar{\phi} \partial_\beta \phi + m c^2 \bar{\phi} \phi\right). 
\end{align}</math>

==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]]
:<math>T_{\mu\nu} = \frac{-2}{\sqrt{-g}}\frac{\delta S_{\mathrm{matter}} }{\delta g^{\mu\nu}} =\frac{-2}{\sqrt{-g}}\frac{\partial(\sqrt{-g}\mathcal{L}_{\mathrm{matter}})}{\partial g^{\mu\nu}}= - 2 \frac{\partial \mathcal{L}_\mathrm{matter}}{\partial g^{\mu\nu}} + g_{\mu\nu} \mathcal{L}_\mathrm{matter} ,</math>

where <math>S_{\mathrm{matter}}</math> is the nongravitational part of the [[action (physics)|action]], <math>\mathcal{L}_{\mathrm{matter}}</math> is the nongravitational part of the [[Lagrangian (field theory)|Lagrangian]] 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]] 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]] because space-dependent [[gauge transformation]]s do not commute with spatial translations.

In [[general relativity]], 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===
{{Main|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==
{{Main|Stress–energy–momentum pseudotensor}}
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==
*[[Cooperstock's energy-localization hypothesis]]
*[[Electromagnetic stress–energy tensor]]
*[[Energy condition]]
*[[Energy density#Energy density of electric and magnetic fields|Energy density of electric and magnetic fields]]
*[[Maxwell stress tensor]]
*[[Poynting vector]]
*[[Ricci calculus]]
*[[Segre classification]]

==Notes and references==
<references/>
*{{cite article|title= The Energy-Momentum Tensor in Classical Field Theory |author=W. Wyss|location=Colorado, USA|year=2005|url=http://merlin.fic.uni.lodz.pl/concepts/2005_3_4/2005_3_4_295.pdf}}

==External links==
* [https://web.archive.org/web/20060430094645/http://people.hofstra.edu/faculty/Stefan_Waner/diff_geom/Sec12.html Lecture, Stephan Waner]
* [https://web.archive.org/web/20140530175713/http://www.black-holes.org/numrel1.html Caltech Tutorial on Relativity] &mdash; A simple discussion of the relation between the Stress–Energy tensor of General Relativity and the metric

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