Relativity and gravitational theory

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• ConceptEinstein–Hilbert action

  The Einstein–Hilbert action is the functional on Lorentzian metrics of a four-dimensional spacetime given by the integral of the scalar curvature against the volume form. Its critical points are exactly the solutions of the vacuum Einstein field equations. It casts general relativity as a variational principle, like the other field theories of physics.

• ExampleSchwarzschild metric

  Fix a mass parameter $m > 0$, in units where the gravitational constant and the speed of light are $1$. On $M = \R \times (2m, \infty) \times S^2$ with coordinates $(t, r, \theta, \phi)$, the Schwarzschild metric is $$g = -\left(1 - \frac{2m}{r}\right) dt^2 + \left(1 - \frac{2m}{r}\right)^{-1} dr^2 + r^2 \left(d\theta^2 + \sin^2\theta \, d\phi^2\right).$$ Check that $(M, g)$ is a Lorentzian manifold: for $r > 2m$ the number $1 - 2m/r$ is positive. So the coefficient of $dt^2$ is negative and the coefficient of $dr^2$ is positive. The last term is $r^2$ times the round metric of $S^2$, which is positive definite. Hence $g$ has signature $(-, +, +, +)$ at every point. A direct computation of the Christoffel symbols shows that the Ricci curvature of $g$ is zero, so $g$ solves the vacuum Einstein equations. It is static and spherically symmetric, and $g$ tends to the Minkowski metric as $r \to \infty$.

• TheoremPenrose–Hawking singularity theorems

  A spacetime here is a connected, time-oriented, four-dimensional Lorentzian manifold $(M, g)$ of signature $(-, +, +, +)$. It is timelike (resp. null) geodesically complete if every inextendible timelike (resp. null) geodesic is defined for all values of its affine parameter. A closed trapped surface is a compact spacelike surface without boundary in $M$. On it, both families of future-directed null geodesics orthogonal to it must have negative expansion. Let $\operatorname{Ric}$ be the Ricci curvature of $g$.