Mathematics of general relativity

The mathematical framework of general relativity uses differential geometry and tensor calculus to describe spacetime curvature and gravitation.

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Differential geometry: The study of geometric structures that are locally modeled on linear spaces and can vary smoothly. It provides the broader geometric framework in which curved spacetime is modeled.
Einstein field equations: Equations relating spacetime curvature to the distribution of energy, momentum, and stress. They are the central equations connecting geometry with gravitating matter.
Schwarzschild metric: The spherically symmetric vacuum solution of Einstein's field equations outside a nonrotating mass. It models the exterior gravitational field of an idealized spherical body.
Newtonian gravity: A theory in which masses exert an instantaneous gravitational force described by a scalar potential. It is the weak-field, slow-motion limit that general relativity must recover.
Singularity theorem: A theorem establishing conditions under which spacetime contains incomplete causal geodesics. These results identify breakdowns of classical spacetime without specifying what replaces them.
Tensor calculus: A coordinate-independent calculus for quantities that transform consistently under changes of coordinates. Its tensors express physical laws without tying them to one coordinate system.
Einstein tensor: A divergence-free tensor built from the Ricci curvature and scalar curvature of a metric. It forms the geometric side of the field equations.
Kerr metric: The stationary, axisymmetric vacuum solution describing the exterior spacetime of a rotating uncharged mass. It supplies the standard geometry for rotating black holes.
Teleparallel gravity: A formulation of gravity using spacetime torsion rather than curvature as its principal geometric description. It can reproduce general relativity's field equations with a different geometric emphasis.
Cosmic censorship hypothesis: The conjecture that singularities formed by gravitational collapse are generally hidden behind event horizons. Its unresolved status concerns the predictability of solutions to Einstein's equations.