- Stress-energy tensor
The

**stress-energy tensor**(sometimes**stress-energy-momentum**tensor) is atensor quantity inphysics that describes thedensity andflux ofenergy andmomentum inspacetime , generalizing the stress tensor of Newtonian physics. It is an attribute ofmatter ,radiation , and non-gravitational force fields. The stress-energy tensor is the source of thegravitational field in theEinstein field equations ofgeneral relativity , just as mass is the source of such a field inNewtonian gravity .**Definition**In the following, the Einstein summation notation is used. The components of the position

4-vector are given by: "x"^{0}= "t" (time in seconds), "x"^{1}= "x" (in meters), "x"^{2}= "y" (in meters), and "x"^{3}= "z" (in meters).The Stress-energy tensor is defined as the

tensor $T^\{alpha\; eta\}$ of rank two that gives theflux of the α^{th}component of themomentum vector across a surface with constant "x"^{β}coordinate . In the theory of relativity this momentum vector is taken as thefour-momentum . The stress-energy tensor is symmetric,:$T^\{alpha\; eta\}\; =\; T^\{eta\; alpha\}\; !.$Some people have speculated that it could be non-symmetric. In those hypotheses, when the

spin tensor S is nonzero,:$partial\_\{alpha\}S^\{mu\; ualpha\}\; =\; T^\{mu\; u\}\; -\; T^\{\; umu\}\; !.$**Identifying the components of the contravariant tensor**The time-time component is the density of relativistic mass, i.e. the

energy density divided by the speed of light squared,:$T^\{00\}\; =\; ho.\; !$The flux of relativistic mass across the "x"

^{"i"}surface is equivalent to the density of the "i"^{th}component of linear momentum, :$T^\{0i\}\; =\; T^\{i0\}.\; !$The components:$T^\{ik\}\; !$represent flux of "i" momentum across the "x"

^{"k"}surface. In particular,:$T^\{ii\}\; !$(not summed) represents normal stress which is calledpressure when it is independent of direction. Whereas:$T^\{ik\},\; quad\; i\; e\; k$representsshear stress (compare with the stress tensor).**Warning**: Insolid state physics andfluid mechanics , the stress tensor is defined to be the spatial components ofthe stress-energy tensor in thecomoving frame of reference. In other words, the stress energy tensor inengineering differs from the stress energy tensor here by a momentum convective term.**Covariant and mixed forms**In most of this article we work with the contravariant form, $T^\{mu\; u\}!$ of the stress-energy tensor. However, it is often necessary to work with the covariant form:$T\_\{mu\; u\}\; =\; g\_\{mu\; alpha\}\; g\_\{\; u\; eta\}\; T^\{alpha\; eta\}!$

or the mixed form:$T\_\{mu\}^\{\; u\}\; =\; g\_\{mu\; alpha\}\; T^\{alpha\; u\}.$

Indeed, one could argue that the most correct form is the mixed density:$mathfrak\{T\}\_\{mu\}^\{\; u\}\; =\; T\_\{mu\}^\{\; u\}\; sqrt\{-g\}.$

**Conservation law****In special relativity**The stress-energy tensor is the conserved Noether current associated with

spacetime translations.When gravity is negligible and using a

Cartesian coordinate system for spacetime, the divergence of the non-gravitational stress-energy will be zero. In other words, non-gravitational energy and momentum are conserved,:$0\; =\; T^\{mu\; u\}\{\}\_\{,\; u\}\; =\; partial\_\{\; u\}\; T^\{mu\; u\}.\; !$The integral form of this is:$0\; =\; int\_\{partial\; N\}\; T^\{mu\; u\}\; mathrm\{d\}^3\; s\_\{\; u\}\; !$

where "N" is any compact four-dimensional region of spacetime; $partial\; N$ is its boundary, a three dimensional hypersurface; and $mathrm\{d\}^3\; s\_\{\; u\}$ is an element of the boundary regarded as the outward pointing normal.

If one combines this with the symmetry of the stress-energy tensor, one can show that

angular momentum is also conserved,:$0\; =\; (x^\{alpha\}\; T^\{mu\; u\}\; -\; x^\{mu\}\; T^\{alpha\; u\})\_\{,\; u\}\; .\; !$**In general relativity**However, when gravity is non-negligible or when using arbitrary coordinate systems, the divergence of the non-gravitational stress-energy may fail to be zero. In this case, we have to use a more general

continuity equation which incorporates thecovariant derivative :$0\; =\; T^\{mu\; u\}\{\}\_\{;\; u\}\; =\; abla\_\{\; u\}\; T^\{mu\; u\}\; =\; T^\{mu\; u\}\{\}\_\{,\; u\}\; +\; T^\{sigma\; u\}\; Gamma^\{mu\}\{\}\_\{sigma\; u\}\; +\; T^\{mu\; sigma\}\; Gamma^\{\; u\}\{\}\_\{sigma\; u\}$where $Gamma^\{mu\}\{\}\_\{sigma\; u\}$ is the

Christoffel symbol which is the gravitational force field.Consequently, if $xi^\{mu\}$ is any

Killing vector field , then the conservation law associated with the symmetry generated by the Killing vector field may be expressed as:$0\; =\; (xi^\{mu\}\; mathfrak\{T\}\_\{mu\}^\{\; u\})\_\{,\; u\}\; .$The integral form of this is:$0\; =\; int\_\{partial\; N\}\; xi^\{mu\}\; mathfrak\{T\}\_\{mu\}^\{\; u\}\; mathrm\{d\}^3\; s\_\{\; u\}\; .$

**In general relativity**In

general relativity , thesymmetric stress-energy tensor acts as the source of spacetime curvature, and is the current density associated withgauge transformation s of gravity which are general curvilinearcoordinate transformation s. (If there istorsion , then the tensor is no longer symmetric. This corresponds to the case with a nonzerospin tensor . SeeEinstein-Cartan gravity .)In general relativity, the

partial derivatives used in special relativity are replaced bycovariant 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 ofNewtonian gravity , this has a simple interpretation: energy is being exchanged with gravitationalpotential energy , which is not included in the tensor, and momentum is being transferred through the field to other bodies. However, in general relativity there is not a unique way to define densities of "gravitational" field energy and field momentum. Any "pseudo-tensor" purporting to define them 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**In general relativity, the stress tensor is studied in the context of the Einstein field equations which are often written as

:$R\_\{alpha\; eta\}\; -\; \{1\; over\; 2\}R,g\_\{alpha\; eta\}\; =\; \{8\; pi\; G\; over\; c^4\}\; T\_\{alpha\; eta\},$

where $R\_\{alpha\; eta\}$ is the

Ricci tensor , $R$ is the Ricci scalar (thetensor contraction of the Ricci tensor), and $G$ is theuniversal gravitational constant .**tress-energy in special situations****Isolated particle**In special relativity, the stress-energy of a non-interacting particle with mass "m" is:$T^\{alpha\; eta\}\; [t,x,y,z]\; =\; frac\{m\; ,\; v^\{alpha\}\; [t]\; v^\{eta\}\; [t]\; \}\{sqrt\{1\; -\; (v/c)^2\; delta(x\; -\; x\; [t]\; )\; delta(y\; -\; y\; [t]\; )\; delta(z\; -\; z\; [t]\; )$

where δ is the

Dirac delta function and $v^\{alpha\}\; !$ is the velocity vector:$egin\{pmatrix\}v^0\; [t]\; \backslash \; v^1\; [t]\; \backslash \; v^2\; [t]\; \backslash \; v^3\; [t]\; end\{pmatrix\}\; =\; egin\{pmatrix\}1\; \backslash \; \{d\; x\; [t]\; over\; d\; t\}\; \backslash \; \{d\; y\; [t]\; over\; d\; t\}\; \backslash \; \{d\; z\; [t]\; over\; d\; t\}end\{pmatrix\}.$**tress-energy of a fluid in equilibrium**For a fluid in

thermodynamic equilibrium , the stress-energy tensor takes on a particularly simple form:$T^\{alpha\; eta\}\; ,\; =\; (\; ho\; +\; \{p\; over\; c^2\})u^\{alpha\}u^\{eta\}\; +\; p\; g^\{alpha\; eta\}$where $ho$ is the mass-energy density (kilograms per cubic meter), $p$ is the hydrostatic pressure (Newtons per square meter), $u^\{alpha\}$ is the fluid's

four velocity , and $g^\{alpha\; eta\}$ is the reciprocal of the metric tensor.The four velocity satisfies:$u^\{alpha\}\; u^\{eta\}\; g\_\{alpha\; eta\}\; =\; -\; c^2\; ,.$

In an

inertial frame of reference comoving with the fluid, the four velocity is :$u^\{alpha\}\; =\; (1,\; 0,\; 0,\; 0)\; ,,$the reciprocal of the metric tensor is simply:$g^\{alpha\; eta\}\; ,\; =\; left(\; egin\{matrix\}\; -\; c^\{-2\}\; 0\; 0\; 0\; \backslash \; 0\; 1\; 0\; 0\; \backslash \; 0\; 0\; 1\; 0\; \backslash \; 0\; 0\; 0\; 1\; end\{matrix\}\; ight),,$

and the stress-energy tensor is a diagonal matrix:$$

T^{alpha eta} = left( egin{matrix} ho & 0 & 0 & 0 \ 0 & p & 0 & 0 \ 0 & 0 & p & 0 \ 0 & 0 & 0 & p end{matrix} ight).

**Electromagnetic stress-energy tensor**The stress-energy tensor of a source-free electromagnetic field is:$T^\{mu\; u\}\; (x)\; =\; frac\{1\}\{mu\_0\}\; left(\; F^\{mu\; alpha\}\; g\_\{alpha\; eta\}\; F^\{\; u\; eta\}\; -\; frac\{1\}\{4\}\; g^\{mu\; u\}\; F\_\{delta\; gamma\}\; F^\{delta\; gamma\}\; ight)$

where $F\_\{mu\; u\}$ is the

electromagnetic field tensor .**calar Field**The stress-energy tensor for a scalar field $phi$ which satisfies the Klein–Gordon equation is:$T^\{mu\; u\}\; =\; frac\{hbar^2\}\{m\}\; (g^\{mu\; alpha\}\; g^\{\; u\; eta\}\; +\; g^\{mu\; eta\}\; g^\{\; u\; alpha\}\; -\; g^\{mu\; u\}\; g^\{alpha\; eta\})\; partial\_\{alpha\}arphi\; partial\_\{eta\}phi\; -\; g^\{mu\; u\}\; m\; c^2\; arphi\; phi\; .$

**Variant definitions of stress-energy**There are a number of inequivalent definitions of non-gravitational stress-energy.

**Hilbert stress-energy tensor**This stress-energy tensor can only be defined in

general relativity with a dynamical metric. It is defined as afunctional derivative :$T^\{mu\; u\}\; =\; frac\{2\}\{sqrt\{-gfrac\{delta\; (mathcal\{L\}\_\{mathrm\{matter\; sqrt\{-g\})\; \}\{delta\; g\_\{mu\; u\; =\; 2\; frac\{delta\; mathcal\{L\}\_mathrm\{matter\{delta\; g\_\{mu\; u\; +\; g^\{mu\; u\}\; mathcal\{L\}\_mathrm\{matter\}.$where "L"

_{matter}is the nongravitational part of theLagrangian density of the action. This is symmetric and gauge-invariant. SeeEinstein–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 begauge invariant because space-dependentgauge 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**This is a symmetric and gauge-invariant stress energy tensor defined over flat spacetimes. There is a construction to get the Belinfante-Rosenfeld tensor from the canonical stress-energy tensor. In GR, this tensor agrees with the Hilbert stress-energy tensor. See the article

Belinfante-Rosenfeld stress-energy tensor for more details.**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 apseudotensor .In general relativity, there are many possible distinct definitions of the gravitational stress-energy-momentum

pseudotensor . These include the Einstein pseudotensor and the Landau-Lifschitz pseudotensor. The Landau-Lifschitz pseudotensor can be reduced to zero at any event in spacetime by choosing an appropriate coordinate system.**ee also***

Energy condition

*Maxwell stress tensor

*Poynting vector

*Energy density of electric and magnetic fields

*Electromagnetic stress-energy tensor

*Segre classification **External links*** [

*http://people.hofstra.edu/faculty/Stefan_Waner/diff_geom/Sec12.html Lecture, Stephan Waner*]

* [*http://www.black-holes.org/numrel1.html Caltech Tutorial on Relativity*] — A simple discussion of the relation between the Stress-Energy tensor of General Relativity and the metric

*Wikimedia Foundation.
2010.*

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