The stress–energy tensor, sometimes called the stress–energy–momentum tensor or the energy–momentum tensor, is a tensor field quantity that describes the density and flux of energy and momentum at each point 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 electromagnetic stress–energy tensor was introduced by Hermann Minkowski in 1907, and later generalized by Max von Laue in 1911.
Definition The stress–energy tensor involves the use of superscripted variables (not exponents; see Tensor index notation and Einstein summation notation). The four coordinates of an event of spacetime x are given by x0, x1, x2, x3. These are customarily set as t, x, y, z, where t is the time coordinate, and x, y, and z are spatial coordinates. 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 coordinate xβ. In the theory of relativity, this momentum vector is taken as the four-momentum. In general relativity, the stress–energy tensor is symmetric,
T α β = T β α . {\displaystyle T^{\alpha \beta }=T^{\beta \alpha }.}
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.
Components Because the stress–energy tensor is of order 2, its components can be displayed in 4 × 4 matrix form:
T μ ν = ( T 00 T 01 T 02 T 03 T 10 T 11 T 12 T 13 T 20 T 21 T 22 T 23 T 30 T 31 T 32 T 33 ) , {\displaystyle T^{\mu \nu }={\begin{pmatrix}T^{00}&T^{01}&T^{02}&T^{03}\\T^{10}&T^{11}&T^{12}&T^{13}\\T^{20}&T^{21}&T^{22}&T^{23}\\T^{30}&T^{31}&T^{32}&T^{33}\end{pmatrix}}\,,}
where the indices μ and ν take on the values 0, 1, 2, 3. Each component of the stress–energy tensor has a direct physical interpretation. 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.
Covariant and mixed forms Most of this article works with the contravariant form, Tμν of the stress–energy tensor. However, it is often convenient to work with the covariant form,
T μ ν = T α β g α μ g β ν , {\displaystyle T_{\mu \nu }=T^{\alpha \beta }g_{\alpha \mu }g_{\beta \nu },}
or the mixed form,
T μ
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