Stress–energy tensor

If Cartesian coordinates in SI units are used, then the components of the position four-vector x are given by: [ x0, x1, x2, x3 ].

Because the stress–energy tensor is of order 2, its components can be displayed in 4 × 4 matrix form:

is the relativistic mass per unit volume, and for an electromagnetic field in otherwise empty space this component is

Most of this article works with the contravariant form, Tμν of the stress–energy tensor.

This article uses the spacelike sign convention (− + + +) for the metric signature.

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, this may be expressed in terms of partial derivatives as

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:

When gravity is non-negligible or when using arbitrary coordinate systems, the divergence of the stress–energy still vanishes.

But in this case, a coordinate-free definition of the divergence is used which incorporates the covariant derivative

Note that this divergenceless property of this tensor is equivalent to four continuity equations.

That is, fields have at least four sets of quantities that obey the continuity equation.

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.

This corresponds to the case with a nonzero spin tensor in Einstein–Cartan gravity theory.)

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 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.

There is in fact no way to define a global energy–momentum vector in a general curved spacetime.

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

is the metric tensor, Λ is the cosmological constant (negligible at the scale of a galaxy or smaller), and

In special relativity, the stress–energy of a non-interacting particle with rest mass m and trajectory

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

and when the metric is flat (Minkowski in Cartesian coordinates) its components work out to be:

There are a number of inequivalent definitions[5] of non-gravitational stress–energy: The Hilbert stress–energy tensor is defined as the functional derivative

Noether's theorem implies that there is a conserved current associated with translations through space and time; for details see the section above on the stress–energy tensor in special relativity.

In general relativity, the translations are with respect to the coordinate system and as such, do not transform covariantly.

In the presence of spin or other intrinsic angular momentum, the canonical Noether stress–energy tensor fails to be symmetric.

The Landau–Lifshitz pseudotensor can be reduced to zero at any event in spacetime by choosing an appropriate coordinate system.