Stress-energy tensor

The stress-energy tensor (sometimes stress-energy-momentum tensor) is a tensor quantity in physics. It describes the density and flux of energy and momentum in space, generalizing the stress tensor of Newtonian physics. It is the source of the gravitational field in general relativity, just as mass is the source of such a field in Newtonian gravity. One very important use of the stress-energy tensor is in the Einstein field equations.

Definition

Please note that throughout we will assume the use of the Einstein summation notation. When using coordinates, x⁰ will represent time, while the other coordinates x¹, x² and x³ will be the remaining spatial components.

The Stress-energy tensor is defined as the tensor $T^{ab}$ of rank two that gives the flux of the a ^th component of the momentum vector across a surface with constant x^b coordinate. (In the theory of relativity this momentum vector is taken as the four-momentum). It is also important to note that the stress-energy tensor is symmetric (when the spin tensor is zero), as in

T^{ab} = T^{ba}

If the spin tensor S is nonzero, then

\partial_{\alpha}S^{\mu\nu\alpha}=T^{\mu\nu}-T^{\nu\mu}

Examples

Here we will present some specific cases:

T^{00}

This represents the energy density.

T^{0i}

This represents the flux of energy across the xⁱ surface, which is equivalent to

T^{i0},

the density of the i^th momentum.

The components

T^{ij}

represent flux of i momentum across the x^j surface. In particular,

T^{ii}

represents a pressure-like quantity, normal stress, whereas

T^{ij}, \quad i \ne j

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.

As a Noether current

The stress-energy tensor satisfies the continuity equation

\nabla_b T^{ab}=T^{ab}{}_{;b}=0

The quantity

\int d^3x T^{a0}

over a spacelike slice gives the energy-momentum vector. The components

T^{a0}

can therefore be interpreted as the local density of (non-gravitational) energy and momentum, and the first component of the continuity equation

\nabla_b T^{0b} = \nabla \cdot \mathbf{p} - \frac{\partial E}{\partial t} = 0

is simply a statement of energy conservation. The spatial components $T^{ij}$ (i, j = 1, 2, 3) correspond to components of local non-gravitational stresses, including pressure. This tensor is the conserved Noether current associated with spacetime translations.

In general relativity

The relations given above do not uniquely define the tensor. In general relativity, the symmetric form additionally satisfying

T^{ab} = T^{ba}

acts as the source of spacetime curvature, and is the current density associated with gauge transformations (in this case 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 given above are actually covariant derivatives. What this means is that the continuity equation no longer implies that the energy and momentum expressed by the tensor are absolutely conserved. 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 no way to define physical quantities corresponding to 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 the general case, we must remain satisfied with a partial "covariant conservation" of the stress-energy tensor.

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.

Relativistic stress tensor for an idealized fluid

For an idealized fluid, with no viscosity and no heat conduction, the stress tensor takes on a particularly simple form:

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

where $\rho$ is the mass-energy density (mass per unit 3-volume), $p$ is the hydrostatic pressure, $u^{\alpha}$ is the fluid's 4-velocity, and $g^{\alpha \beta}$ is the inverse metric of the manifold.

Furthermore, if the tensor components are being measured in a local inertial frame comoving with the fluid, then the metric tensor is simply Minkowski's metric

g^{\alpha \beta} \, = \eta^{\alpha \beta} = \mathrm{diag}(-1,1,1,1)

and the squared magnitude of the 4-velocity

u^{\alpha}u^{\beta} \, = \mathrm{diag}(c^2,0,0,0)

The stress tensor is then a diagonal matrix:

T^{\alpha \beta} = \left( \begin{matrix} \rho c^2 & 0 & 0 & 0 \\ 0 & p & 0 & 0 \\ 0 & 0 & p & 0 \\ 0 & 0 & 0 & p \end{matrix} \right)

The various stress-energy tensors

There are a number of inequivalent stress-energy tensors out there.

Canonical stress-energy tensor

This is the Noether current associated with spacetime translations. In flat spacetime, this isn't symmetric in general and if we have some gauge theory, it won't be gauge invariant because space-dependent gauge transformations obviously don't commute with spatial translations. In general relativity, the translations are with respect to the coordinate system and as such, don't transform covariantly. This is called a pseudostress-energy tensor.

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^{\mu\nu}(x)=\frac{1}{\sqrt{-|g|}}\frac{\delta S_{matter}}{\delta g_{\mu\nu}(x)}

where S_matter is the nongravitational part of the action. This is symmetric and gauge-invariant.

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.

Pseudotensors

Both the Einstein pseudotensor and the Landau-Lifschitz pseudotensor are pseudotensors.

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

Tensors in general relativity | Variational formalism of general relativity | Tensors

Gourt Home::Gourt :: Stress-energy tensor