6.2: Field Equations of the General Theory of Relativity

The general theory of relativity is a classical field theory of gravitation in which all variables are assumed to be continuous and are uniquely specified. Thus, the Heisenberg uncertainty principle and quantum mechanics play no direct role in the theory. Although it is traditional to present general relativity in a system of units where c = h = G = 1, we adopt the nontraditional notion of generally maintaining the physical constants in the expressions in the hopes that the physical interpretation of the various terms may be clearer to the readers. However, we adopt the Einstein summation convention where repeated indices are summation indices for this section, to avoid the host of summation signs that would otherwise accompany the tensor calculus.

The basic philosophy of general relativity is to relate the geometry of space-time, which determines the motion of matter, to the density of matter-energy, known as the stress energy tensor. This relation is accomplished through the Einstein field equations. The geometry of space-time is dictated by the metric tensor which defines the properties of that geometry and basically describes how travel in one coordinate involves another coordinate, so that \[d s^2=\mathbf{g}_{\mu \nu} d x^\mu d x^\nu\label{6.1.1}\]

The elements of the metric tensor are dimensionless; for ordinary Euclidean space they are all unity if \(\mu=\nu\) and zero otherwise. If one were doing geometry on a deformed rubber sheet, this would not necessarily be true. In general, the distance traveled, expressed in terms of any set of local coordinates, will depend on the orientation of those coordinates on the rubber sheet. The coefficients that "weight" the role played by each coordinate in determining the distance according to equation \ref{6.1.1}, for all directions traveled, are the elements of the metric tensor. Now the field equations relate second derivatives of the metric tensor to the properties of the local matter-energy density expressed in terms of the stress-energy tensor. Specifically the Einstein field equations are \[\mathbf{G}_{\mu \nu}=\frac{8 \pi G}{c^2} \mathbf{T}_{\mu \nu}\label{6.1.2}\]

Here \(\mathbf{G}_{\mu\nu}\) is known as the Einstein tensor and \(\mathbf{T}_{\mu\nu}\) is the stress energy tensor in physical units (say grams per cubic centimeter). The quantity \(\mathrm{G/c^2}\) is a very small number in any common system of units, which shows that the departure from Euclidean space is small unless the stress-energy is exceptionally large. The specific relation of the metric tensor to the Einstein tensor is extremely complicated and for completeness is given below.

Define \[\Gamma_{\beta \mu \nu} \equiv \frac{1}{2}\left(\frac{\partial \mathbf{g}_{\beta \nu}}{\partial x^\mu}+\frac{\partial \mathbf{g}_{\beta \mu}}{\partial x^\nu}+\frac{\partial \mathbf{g}_{\mu \nu}}{\partial x^\beta}\right)\label{6.1.3}\]

and \[\Gamma_{\mu \nu}^\alpha \equiv \mathbf{g}^{\alpha \beta} \Gamma_{\beta \mu \nu}\label{6.1.4}\]

where \(\mathbf{g}^{\alpha\beta}\) is the matrix inverse of \(\mathbf{g}_{\alpha\beta}\). The symbol \(\Gamma_{\beta \mu \nu}\) is known as the Christoffel symbol. The Christoffel symbols and their derivatives can be combined to produce the Riemann curvature tensor \[\mathbf{R}_{\beta \gamma \delta}^\alpha=\frac{\partial \Gamma_{\beta \delta}^\alpha}{\partial x^\gamma}-\frac{\partial \Gamma_{\beta \gamma}^\alpha}{\partial x^\delta}+\Gamma_{\mu \gamma}^\alpha \Gamma_{\beta \delta}^\mu-\Gamma_{\mu \delta}^\alpha \Gamma_{\beta \gamma}^\mu\label{6.1.5}\]

which when summed over two of its indices produces the Ricci tensor \[\mathbf{R}_{\mu \nu}=R^\alpha{ }_{\mu \alpha \nu}\label{6.1.6}\]

This can be further summed (contracted) over the remaining two indices to yield a quantity known as the scalar curvature \[R=\mathbf{R}_\mu^{\mu}\label{6.1.7}\]

Finally, the Einstein tensor can be expressed in terms of the Ricci tensor, the scalar curvature, and the metric tensor itself as \[\mathbf{G}_{\mu \nu}=\mathbf{R}_{\mu \nu}-\mathbf{g}_{\mu \nu} R\label{6.1.8}\]

For a given arbitrary metric, the calculations implied by equations \ref{6.1.4} through \ref{6.1.8} are extremely tedious, but conceptually simple. Since the metric tensor depends on only the geometry, and since the operations described in forming the Riemann and Ricci tensors, and scalar curvature are essentially geometric, nothing but geometry appears in the Einstein tensor. Hence the saying, "the left-hand side of the Einstein field equations is geometry, while the right-hand side is physics".