First I'd like to apply Hamilton's Principle to Einstein's theory of General Relativity. We need to find a Lagrangian that incorporates the effect of mass on the manifold's metric. It must, as usual, be a scalar, and we suppose it is a functional of the metric and the matter fields. Hence, we suppose for free space

\[A[g]=\int\frac{1}{2\kappa}R\sqrt{-g}\,d^4x\]

where \(\sqrt{-g}\,d^4x\) is the invariant 4-volume element of our Riemannian manifold (and \(g=\det(g_{\alpha\beta})/c^2\) is the determinant of the metric), \(\kappa=8\pi G/c^4\) is a universal constant, and \(R\) is the Ricci scalar (the simplest curvature invariant of a Riemannian manifold). If we were deriving the Einstein field equations for the first time, we would not know what these constants and scalars were, but we would still be able to write it in this form based on our assumptions of locality, isotropy of free space, etc.

Let us now suppose that the full action is this free-space action (the Einstein-Hilbert action) plus a term that describes matter fields:

\[A[g]=\int\left[\frac{1}{2\kappa}R+\mathcal{L}_M\right]\sqrt{-g}\,d^4x\]

When we apply the Euler-Lagrange equations from last time, we find that

\[\frac{\delta R}{\delta g^{\mu\nu}}+\frac{R}{\sqrt{-g}}\frac{\delta\sqrt{-g}}{\delta g^{\mu\nu}}=-2\kappa\frac{1}{\sqrt{-g}}\frac{\delta(\sqrt{-g}\mathcal{L}_M)}{\delta g^{\mu\nu}}\]

We define the right hand side as (\(\kappa\) times) the stress-energy tensor \(T_{\mu\nu}\) responsible for the curvature of spacetime due to the presence of matter. The left hand side is completely geometric (no physics is required), and from the theory of differential geometry, can be shown to equal \(G_{\mu\nu}=R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R\), the Einstein tensor, in terms of the Ricci tensor, the Ricci scalar, and the metric. The Einstein field equations, then, are

\[G_{\mu\nu}=\frac{8\pi G}{c^4}T_{\mu\nu}\]

This rather beautiful relation between geometry (left hand side) and physics (right hand side) can be viewed in either direction: a curving geometry tells matter and energy how to move; or equivalently, the presence of matter and energy curves spacetime. The theory does not, however, give us either one a priori, and the constant must be determined by experimental results. We can place restrictions on the stress-energy tensor, however: as the source of the gravitational field, we expect a particular simplicity in the non-relativistic limit (Newton's law of gravitation). Because of the symmetries of spacetime, we can apply Noether's theorem to obtain the conservation law \(\partial_{\nu}T^{\mu\nu}=0\). In many cases (where the spin tensor is zero), the tensor is symmetric and angular momentum is conserved as well. The radiation portions of the tensor can be gotten from Maxwell's theory, which I'll discuss next time. From the field equations, one can extract information about the nature of black holes, galactic rotation, etc., and as such these equations encapsulate most, if not all, of mechanics in the non-quantum regime.