The stress-energy tensor (sometimes stress-energy-momentum tensor) is a tensor quantity in physics that describes the density and flux of energy and momentum in spacetime, generalizing the stress tensor of Newtonian physics. It is an attribute of matter, radiation, and non-gravitational force fields. The stress-energy tensor is the source of the gravitational field in the Einstein field equations of general relativity, just as mass is the source of such a field in Newtonian gravity.

Definition

In the following, the Einstein summation notation is used. The components of the position 4-vector are given by: x⁰ = t (time in seconds), x¹ = x (in meters), x² = y (in meters), and x³ = z (in meters).

The Stress-energy tensor is defined as the tensor $T^\{alpha\; beta\}$ of rank two that gives the flux of the α^th component of the momentum vector across a surface with constant x^β coordinate. In the theory of relativity this momentum vector is taken as the four-momentum. The stress-energy tensor is symmetric,

$T^\{alpha\; beta\}\; =\; T^\{beta\; alpha\}\; !.$

Some people have speculated that it could be non-symmetric. In those hypotheses, when the spin tensor S is nonzero,

$partial\_\{alpha\}S^\{munualpha\}\; =\; T^\{munu\}\; -\; T^\{numu\}\; !.$

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,

$T^\{00\}\; =\; rho.\; !$

The flux of relativistic mass across the xⁱ 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 called pressure when it is independent of direction. Whereas

$T^\{ik\},\; quad\; i\; ne\; k$

represents shear stress (compare with the stress tensor).

Warning: In solid state physics and fluid mechanics, the stress tensor is defined to be the spatial components of the stress-energy tensor in the comoving frame of reference. In other words, the stress energy tensor in engineering 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, $T^\{mu\; nu\}!$ of the stress-energy tensor. However, it is often necessary to work with the covariant form

$T\_\{mu\; nu\}\; =\; g\_\{mu\; alpha\}\; g\_\{nu\; beta\}\; T^\{alpha\; beta\}!$

or the mixed form

$T\_\{mu\}^\{nu\}\; =\; g\_\{mu\; alpha\}\; T^\{alpha\; nu\}.$

Indeed, one could argue that the most correct form is the mixed density

$mathfrak\{T\}\_\{mu\}^\{nu\}\; =\; T\_\{mu\}^\{nu\}\; sqrt\{-g\}.$

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,

$0\; =\; T^\{mu\; nu\}\{\}\_\{,nu\}\; =\; partial\_\{nu\}\; T^\{mu\; nu\}.\; !$

The integral form of this is

$0\; =\; int\_\{partial\; N\}\; T^\{mu\; nu\}\; mathrm\{d\}^3\; s\_\{nu\}\; !$

where N is any compact four-dimensional region of spacetime; $partial\; N$ is its boundary, a three dimensional hypersurface; and $mathrm\{d\}^3\; s\_\{nu\}$ 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,

$0\; =\; (x^\{alpha\}\; T^\{mu\; nu\}\; -\; x^\{mu\}\; T^\{alpha\; nu\})\_\{,nu\}\; .\; !$

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 the covariant derivative

$0\; =\; T^\{mu\; nu\}\{\}\_\{;nu\}\; =\; nabla\_\{nu\}\; T^\{mu\; nu\}\; =\; T^\{mu\; nu\}\{\}\_\{,nu\}\; +\; T^\{sigma\; nu\}\; Gamma^\{mu\}\{\}\_\{sigma\; nu\}\; +\; T^\{mu\; sigma\}\; Gamma^\{nu\}\{\}\_\{sigma\; nu\}$

where $Gamma^\{mu\}\{\}\_\{sigma\; nu\}$ 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

$0\; =\; (xi^\{mu\}\; mathfrak\{T\}\_\{mu\}^\{nu\})\_\{,nu\}\; .$

The integral form of this is

$0\; =\; int\_\{partial\; N\}\; xi^\{mu\}\; mathfrak\{T\}\_\{mu\}^\{nu\}\; mathrm\{d\}^3\; s\_\{nu\}\; .$

In general relativity

In general relativity, the symmetric stress-energy tensor acts as the source of spacetime curvature, and is the current density associated with gauge transformations of gravity which are general curvilinear coordinate transformations. (If there is torsion, then the tensor is no longer symmetric. This corresponds to the case with a nonzero spin tensor. See Einstein-Cartan gravity.)

In general relativity, the partial derivatives 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: 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. 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

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

where $R\_\{alpha\; beta\}$ is the Ricci tensor, $R$ is the Ricci scalar (the tensor contraction of the Ricci tensor), 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 mass m is

$T^\{alpha\; beta\}[t,x,y,z]\; =\; frac\{m\; ,\; v^\{alpha\}[t]\; v^\{beta\}[t]\}\{sqrt\{1\; -\; (v/c)^2\}\}\; delta(x\; -\; x[t])\; delta(y\; -\; y[t])\; delta(z\; -\; z[t])$

where δ is the Dirac delta function and $v^\{alpha\}\; !$ is the velocity vector

$$

begin{pmatrix} v^0 [t] v^1 [t] v^2 [t] v^3 [t] end{pmatrix} = begin{pmatrix} 1 {d x [t] over d t} {d y [t] over d t} {d z [t] over d t} end{pmatrix} .

Stress-energy of a fluid in equilibrium

For a fluid in thermodynamic equilibrium, the stress-energy tensor takes on a particularly simple form

$T^\{alpha\; beta\}\; ,\; =\; (rho\; +\; \{p\; over\; c^2\})u^\{alpha\}u^\{beta\}\; +\; p\; g^\{alpha\; beta\}$

where $rho$ 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 $g^\{alpha\; beta\}$ is the reciprocal of the metric tensor.

The four velocity satisfies

$u^\{alpha\}\; u^\{beta\}\; g\_\{alpha\; beta\}\; =\; -\; c^2\; ,.$

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 beta} , = left(begin{matrix} - c^{-2} & 0 & 0 & 0

                  0 & 1 & 0 & 0 

                  0 & 0 & 1 & 0 

                  0 & 0 & 0 & 1

end{matrix} right) ,,

and the stress-energy tensor is a diagonal matrix

$$

T^{alpha beta} = left(begin{matrix}

                  rho & 0 & 0 & 0 

                  0 & p & 0 & 0 

                  0 & 0 & p & 0 

                  0 & 0 & 0 & p

end{matrix} right).

Electromagnetic stress-energy tensor

The stress-energy tensor of a source-free electromagnetic field is

$T^\{mu\; nu\}\; (x)\; =\; frac\{1\}\{mu\_0\}\; left(F^\{mu\; alpha\}\; g\_\{alpha\; beta\}\; F^\{nu\; beta\}\; -\; frac\{1\}\{4\}\; g^\{mu\; nu\}\; F\_\{delta\; gamma\}\; F^\{delta\; gamma\}\; right)$

where $F\_\{mu\; nu\}$ is the electromagnetic field tensor.

Scalar Field

The stress-energy tensor for a scalar field $phi$ which satisfies the Klein–Gordon equation is

$T^\{munu\}\; =\; frac\{hbar^2\}\{m\}\; (g^\{mu\; alpha\}\; g^\{nu\; beta\}\; +\; g^\{mu\; beta\}\; g^\{nu\; alpha\}\; -\; g^\{munu\}\; g^\{alpha\; beta\})\; partial\_\{alpha\}barphi\; partial\_\{beta\}phi\; -\; g^\{munu\}\; m\; c^2\; barphi\; phi\; .$

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 a functional derivative

$T^\{munu\}\; =\; frac\{2\}\{sqrt\{-g\}\}frac\{delta\; (mathcal\{L\}\_\{mathrm\{matter\}\}\; sqrt\{-g\})\; \}\{delta\; g\_\{munu\}\}\; =\; 2\; frac\{delta\; mathcal\{L\}\_mathrm\{matter\}\}\{delta\; g\_\{munu\}\}\; +\; g^\{munu\}\; mathcal\{L\}\_mathrm\{matter\}.$

where L_matter is the nongravitational part of the Lagrangian density of the action. 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 transformations 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 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-Lifschitz pseudotensor. The Landau-Lifschitz pseudotensor can be reduced to zero at any event in spacetime by choosing an appropriate coordinate system.

See also

• Energy condition
• Maxwell stress tensor
• Poynting vector
• Energy density of electric and magnetic fields
• Electromagnetic stress-energy tensor
• Segre classification

External links

• Lecture, Stephan Waner
• Caltech Tutorial on Relativity — A simple discussion of the relation between the Stress-Energy tensor of General Relativity and the metric