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Concept
Stress–energy tensor
Summary
The stress–energy tensor, sometimes called the stress–energy–momentum tensor or the energy–momentum tensor, is a tensor physical quantity 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. This density and flux of energy and momentum are the sources of the gravitational field in the Einstein field equations of general relativity, just as mass density is the source of such a field in Newtonian gravity.
The stress–energy tensor involves the use of superscripted variables (not exponents; see tensor index notation and Einstein summation notation). If Cartesian coordinates in SI units are used, then the components of the position four-vector x are given by: (x^0, x^1, x^2, x^3) = (t, x, y, z), where t is time in seconds, and x, y, and z are distances in meters.
The stress–energy tensor is defined as the tensor Tαβ of order two that gives the flux of the αth component of the momentum vector across a surface with constant xβ coordinate. In the theory of relativity, this momentum vector is taken as the four-momentum. In general relativity, the stress–energy tensor is symmetric,
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.
Because the stress–energy tensor is of order 2, its components can be displayed in 4 × 4 matrix form:
where the indices μ and ν take on the values 0, 1, 2, 3.
In the following, k and l range from 1 through 3:
In solid state physics and fluid mechanics, the stress tensor is defined to be the spatial components of the stress–energy tensor in the proper frame of reference. In other words, the stress–energy tensor in engineering differs from the relativistic stress–energy tensor by a momentum-convective term.
Most of this article works with the contravariant form, T^μν of the stress–energy tensor.
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