In the case when the energy-momentum tensor
is present (supposing however that it describes the influence of matter weak enough in order to keep the basic solution unchanged), one must use the full Einstein's tensor on the right-hand side.
Its 4-curvature is determined from the various contributions to its total energy-momentum tensor
density, namely in the form of matter energy density, radiation pressure and cosmological constant.
mu]v] is the energy-momentum tensor
of matter, c is the light speed in free space, and G is the gravitational constant.
In general relativity the matter content of the spacetime is described by the energy-momentum tensor
T which is to be determined from the physical considerations dealing with the distribution of matter and energy.
Ueno begins by describing Riemann spaces and stable curves, including compact Riemann surfaces and pointed curves, then moves to affine Lie algebras and integrable highest weight representations, with an explanation of the energy-momentum tensor
, Uechi then moves to conformal blocks and correlation functions, the sheaf of conformal blocks associated with a family of pointed Reimann surfaces with coordinates, the sheaf's support of projectively flat connections, one of the most important facts of conformal field theory.
We write the conservation law for the Einstein tensor density derived from the Bianchi identities, which cannot apply to the energy-momentum tensor
density as a source;
The energy-momentum tensor
of the electromagnetic field at macroscopic scales [section]4.
Using this notation and taking into account that in the surrounding spacetime of the material particle there is an electric current j, as given by equation (242), the energy-momentum tensor
[19-21] of the electromagnetic field is given by the tensor
This, of course, is part of the the emergence of a purely geometric energy-momentum tensor
alpha][beta]] is the energy-momentum tensor
, and [alpha], [beta] = 0, 1, 2, 3 are the space-time indices.
Thus, the energy-momentum tensor
of the system will thus be chosen to be
Montesinos and Flores  deduce the electromagnetic energy-momentum tensor
via Noether's theorem .