EinsteinsGRFieldEquations 2009-01-27 11:49:11 2009-01-27 11:51:12 Topic \section{Introduction} The following is a brief presentation cited from a \PMlinkexternal{PM entry reference on Einstein's Field Equations.}{http://planetmath.org/?op=getobj&from=objects&name=EinsteinFieldEquations} \subsection{Einstein's Field Equations in General Relativity} ``{\em Then the Einstein equations read as follows: \footnote{In the physics literature, the coefficient of $T_{\mu \nu}$ is written as $\frac{8\pi G}{c^4}$, where $G$ is the gravitational constant, $c$ is the light velocity constant but, since we are interested in the purely mathematical properties of these equations, we shall set $G = c = 1$ here, which may be accomplished by working in a suitable set of physical units. It might also be worth mentioning that, in physics, the tensor $T_{\mu \nu}$ is the stress-energy tensor, which encodes data pertaining to the mass, energy, and momentum densities of the surrounding space. The number $\Lambda$ is known as the cosmological constant because it determines large-scale properties of the universe, such as whether it collapses, remains stationary, or expands.} \[G_{\mu \nu} = \Lambda g_{\mu \nu} + 8 \pi T_{\mu \nu}\] Here, $G_{\mu\upsilon}=R_{\mu\upsilon}-\frac{1}{2}g_{\mu\upsilon}R$ is the \emph{Einstein Tensor}, $R_{\mu\upsilon}$ is the Ricci tensor, and $R=g^{\mu\nu}R_{\mu\nu}$ is the Ricci scalar, and $g^{\mu\nu}$ is the inverse metric tensor.} {\em One possibility is that the tensor field $T_{\mu \nu}$ is specified and that these equations are then solved to obtain $g_{\mu \nu}$. A noteworthy case of this is the \emph{vacuum Einstein equations}, in which $$T_{\mu \nu} = 0$$.Another possibility is that $T_{\mu \nu}$ is given in terms of some other fields on the manifold and that the Einstein equations are augmented by differential equations which describe those fields. In that case, one speaks of Einstein-Maxwell equations, Einstein-Yang-Mills equations, and the like depending on what these other fields may happen to be. It should be noted that, on account of the Bianchi identity, there is an integrability condition $\nabla_\mu (g) T^{\mu \nu} = 0$. (Here, $\nabla (g)$ denotes covariant differentiation with respect to the Levi-Civita connection of the metric tensor $g_{\mu \nu}$.) When choosing $T_{\mu \nu}$, these conditions must be taken into account in order to guarantee that a solution is possible.}