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It is therefore convenient to collect many of these terms in the Maxwell stress tensor, and to use tensor arithmetic to find the answer to the problem at hand. In the relativistic formulation of electromagnetism, the nine components of the Maxwell stress tensor appear, negated, as components of the electromagnetic stress–energy tensor , which ...
For example, a linear operator is represented in a basis as a two-dimensional square n × n array. The numbers in the multidimensional array are known as the components of the tensor. They are denoted by indices giving their position in the array, as subscripts and superscripts, following the symbolic name of the tensor.
where n is the dimension of the manifold, g is the metric, R is the Riemann tensor, Ric is the Ricci tensor, s is the scalar curvature, and denotes the Kulkarni–Nomizu product of two symmetric (0,2) tensors:
There are a variety of notations for the d'Alembertian. The most common are the box symbol (Unicode: U+2610 ☐ BALLOT BOX) whose four sides represent the four dimensions of space-time and the box-squared symbol which emphasizes the scalar property through the squared term (much like the Laplacian).
or using the index notation with square brackets for the antisymmetric part of the tensor: ∂ [ α F β γ ] = 0 {\displaystyle \partial _{[\alpha }F_{\beta \gamma ]}=0} Using the expression relating the Faraday tensor to the four-potential, one can prove that the above antisymmetric quantity turns to zero identically ( ≡ 0 {\displaystyle ...
Xerus [52] is a C++ tensor algebra library for tensors of arbitrary dimensions and tensor decomposition into general tensor networks (focusing on matrix product states). It offers Einstein notation like syntax and optimizes the contraction order of any network of tensors at runtime so that dimensions need not be fixed at compile-time.
A common solution is to model these terms by simple ad hoc prescriptions. The theory of the Reynolds stress is quite analogous to the kinetic theory of gases, and indeed the stress tensor in a fluid at a point may be seen to be the ensemble average of the stress due to the thermal velocities of molecules at a given point in a fluid. Thus, by ...
But if one requires an exact solution or a solution describing strong fields, the evolution of both the metric and the stress–energy tensor must be solved for at once. To obtain solutions, the relevant equations are the above quoted EFE (in either form) plus the continuity equation (to determine the evolution of the stress–energy tensor):