Stress–energy tensor

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 four-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 $T^\{\backslash alpha\; \backslash beta\}$ 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. In general relativity, the stress-energy tensor is symmetric, :$T^\{\backslash alpha\; \backslash beta\}\; =\; T^\{\backslash beta\; \backslash alpha\}\; \backslash !.$

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

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

The flux of relativistic mass across the xⁱ surface is equivalent to the density of the i^th component of linear momentum, :$T^\{0i\}\; =\; T^\{i0\}.\; \backslash !$

The components :$T^\{ik\}\; \backslash !$ represent flux of i momentum across the x^k surface. In particular, :$T^\{ii\}\; \backslash !$ (not summed) represents normal stress which is called pressure when it is independent of direction. Whereas :$T^\{ik\},\; \backslash quad\; i\; \backslash ne\; k$ 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, $T^\{\backslash mu\; \backslash nu\}\backslash !$ of the stress–energy tensor. However, it is often necessary to work with the covariant form :$T\_\{\backslash mu\; \backslash nu\}\; =\; g\_\{\backslash mu\; \backslash alpha\}\; g\_\{\backslash nu\; \backslash beta\}\; T^\{\backslash alpha\; \backslash beta\}\backslash !$

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

Or as a mixed tensor density :$\backslash mathfrak\{T\}\_\backslash mu^\backslash nu\; =\; T\_\backslash mu^\backslash nu\; \backslash sqrt\{-g\}.$

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

The integral form of this is :$0\; =\; \backslash int\_\{\backslash partial\; N\}\; T^\{\backslash mu\; \backslash nu\}\; \backslash mathrm\{d\}^3\; s\_\{\backslash nu\}\; \backslash !$

where N is any compact four-dimensional region of spacetime; $\backslash partial\; N$ is its boundary, a three dimensional hypersurface; and $\backslash mathrm\{d\}^3\; s\_\{\backslash nu\}$ 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, :$0\; =\; (x^\{\backslash alpha\}\; T^\{\backslash mu\; \backslash nu\}\; -\; x^\{\backslash mu\}\; T^\{\backslash alpha\; \backslash nu\})\_\{,\backslash nu\}\; .\; \backslash !$

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

where $\backslash Gamma^\{\backslash mu\}\{\}\_\{\backslash sigma\; \backslash nu\}$ is the Christoffel symbol which is the gravitational force field.

Consequently, if $\backslash xi^\{\backslash mu\}$ is any Killing vector field, then the conservation law associated with the symmetry generated by the Killing vector field may be expressed as :$0\; =\; (\backslash xi^\{\backslash mu\}\; \backslash mathfrak\{T\}\_\{\backslash mu\}^\{\backslash nu\})\_\{,\backslash nu\}\; .$

The integral form of this is :$0\; =\; \backslash int\_\{\backslash partial\; N\}\; \backslash xi^\{\backslash mu\}\; \backslash mathfrak\{T\}\_\{\backslash mu\}^\{\backslash nu\}\; \backslash mathrm\{d\}^3\; s\_\{\backslash nu\}\; .$

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. 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 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

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

where $R\_\{\backslash mu\; \backslash nu\}$ 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 and trajectory $\backslash mathbf\{x\}\_\backslash text\{p\}(t)$ is: :$T^\{\backslash alpha\; \backslash beta\}(\backslash mathbf\{x\},t)\; =\; \backslash frac\{m\; \backslash ,\; v^\{\backslash alpha\}(t)\; v^\{\backslash beta\}(t)\}\{\backslash sqrt\{1\; -\; (v/c)^2\}\}\backslash ;\backslash ,\; \backslash delta(\backslash mathbf\{x\}\; -\; \backslash mathbf\{x\}\_\backslash text\{p\}(t))\; =\; E\; \backslash frac\{v^\{\backslash alpha\}(t)\; v^\{\backslash beta\}(t)\}\{c^2\}\backslash ;\backslash ,\; \backslash delta(\backslash mathbf\{x\}\; -\; \backslash mathbf\{x\}\_\backslash text\{p\}(t))$

where $v^\{\backslash alpha\}\; \backslash !$ is the velocity vector (which should not be confused with four-velocity) :$v^\{\backslash alpha\}\; =\; \backslash left(1,\; \backslash frac\{d\; \backslash mathbf\{x\}\_\backslash text\{p\}\}\{dt\}(t)\; \backslash right)\; \backslash ,,$ δ is the Dirac delta function and $E\; =\; \backslash sqrt\{p^2\; c^2\; +\; m^2\; c^4\}$ is the energy of the particle.

Stress-energy of a fluid in equilibrium

For a fluid in thermodynamic equilibrium, the stress-energy tensor takes on a particularly simple form :$T^\{\backslash alpha\; \backslash beta\}\; \backslash ,\; =\; \backslash left(\backslash rho\; +\; \{p\; \backslash over\; c^2\}\backslash right)u^\{\backslash alpha\}u^\{\backslash beta\}\; +\; p\; g^\{\backslash alpha\; \backslash beta\}$

where $\backslash rho$ is the mass-energy density (kilograms per cubic meter), $p$ is the hydrostatic pressure (pascals), $u^\{\backslash alpha\}$ is the fluid's four velocity, and $g^\{\backslash alpha\; \backslash beta\}$ is the reciprocal of the metric tensor.

The four velocity satisfies :$u^\{\backslash alpha\}\; u^\{\backslash beta\}\; g\_\{\backslash alpha\; \backslash beta\}\; =\; -\; c^2\; \backslash ,.$

In an inertial frame of reference comoving with the fluid, the four velocity is :$u^\{\backslash alpha\}\; =\; (1,\; 0,\; 0,\; 0)\; \backslash ,,$

the reciprocal of the metric tensor is simply :

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

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

Scalar field

The stress-energy tensor for a scalar field $\backslash phi$ which satisfies the Klein–Gordon equation is :$T^\{\backslash mu\backslash nu\}\; =\; \backslash frac\{\backslash hbar^2\}\{m\}\; (g^\{\backslash mu\; \backslash alpha\}\; g^\{\backslash nu\; \backslash beta\}\; +\; g^\{\backslash mu\; \backslash beta\}\; g^\{\backslash nu\; \backslash alpha\}\; -\; g^\{\backslash mu\backslash nu\}\; g^\{\backslash alpha\; \backslash beta\})\; \backslash partial\_\{\backslash alpha\}\backslash bar\backslash phi\; \backslash partial\_\{\backslash beta\}\backslash phi\; -\; g^\{\backslash mu\backslash nu\}\; m\; c^2\; \backslash bar\backslash phi\; \backslash phi\; .$

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 :$T^\{\backslash mu\backslash nu\}\; =\; \backslash frac\{2\}\{\backslash sqrt\{-g\}\}\backslash frac\{\backslash delta\; (\backslash mathcal\{L\}\_\{\backslash mathrm\{matter\}\}\; \backslash sqrt\{-g\})\; \}\{\backslash delta\; g\_\{\backslash mu\backslash nu\}\}\; =\; 2\; \backslash frac\{\backslash delta\; \backslash mathcal\{L\}\_\backslash mathrm\{matter\}\}\{\backslash delta\; g\_\{\backslash mu\backslash nu\}\}\; +\; g^\{\backslash mu\backslash nu\}\; \backslash mathcal\{L\}\_\backslash mathrm\{matter\}.$

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

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. 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–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

Poynting vector

Segre classification

Notes and references

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

Category:Fundamental physics concepts Category:Tensors in general relativity Category:Variational formalism of general relativity Category:Tensors