Globally hyperbolic spacetime

A globally hyperbolic spacetime has no causal pathologies and admits a Cauchy hypersurface intersected exactly once by every inextendible causal curve.

Gravitational initial data on a spatial manifold consist of a Riemannian metric $h$ and an extrinsic curvature $K$ satisfying the Hamiltonian constraint and momentum constraint. Matter theories require their own data and constraints as well. The data determine a maximal Cauchy development.

A maximal Cauchy development is the largest globally hyperbolic solution, unique up to an appropriate isometry, determined by admissible initial data, with the initial hypersurface a Cauchy hypersurface. It can sometimes be extended as a spacetime across a Cauchy horizon even though that extension is no longer globally determined by the initial hypersurface.

The strong cosmic censorship conjecture asserts that the maximal Cauchy development of generic admissible initial data is inextendible in a specified regularity class. The matter model, topology on initial data, and extension regularity are part of the assertion. An exceptional exact solution with a smooth Cauchy horizon need not violate this generic claim.

An achronal set contains no pair of points connected by a timelike curve.

An achronal boundary is the boundary of a chronological future or past. Its null generators cease to belong to the boundary after developing a conjugate point or intersecting another generator in the relevant way.

A Cauchy hypersurface is an achronal hypersurface met exactly once by every inextendible timelike curve. Its topology records the global spatial topology of a globally hyperbolic spacetime.

The set $D^{+}(\Sigma )$ of events for which every past-inextendible causal curve meets $\Sigma$. This causal characterization permits an initial sign condition on a null geodesic congruence to propagate throughout the regular region by integrating a focusing inequality from $\Sigma$.

The future or past Cauchy horizon of a partial Cauchy hypersurface is the boundary of the region whose physical evolution is determined by data on that hypersurface. Crossing it can destroy uniqueness of the maximal globally hyperbolic development.

A complete Cauchy hypersurface in Minkowski spacetime, such as $t=0$, has full domain of dependence and empty Cauchy horizons. This does not forbid horizons for a non-Cauchy initial surface in the same spacetime. The smooth edgeless spacelike hyperboloid $t=\sqrt{a^2+r^2}$, $a>0$, has domain of dependence $t>r$ and a past Cauchy horizon on the future light cone of the origin. Every causal curve starting inside that cone crosses the hyperboloid in the direction needed for dependence: $t^2-r^2$ passes through $a^2$, while a null generator of the cone never meets it. The time-reversed hyperboloid gives the analogous future horizon.

Near a nondegenerate outgoing inner Cauchy horizon, a regular null coordinate behaves as $V_{-}\propto -e^{-{\kappa }_{-}v}$, where ${\kappa }_{-}>0$ is the magnitude of the inner-horizon surface gravity. Transforming stress components gives $T_{V_{-}{V}_{-}}\propto e^{2{\kappa }_{-}v}{T}_{vv}$. Even a small power-law tail can therefore cause an unbounded locally measured flux, motivating mass inflation.

Mass inflation is the growth of the interior effective mass and curvature near an inner Cauchy horizon under suitably perturbed black-hole data. It can obstruct regular spacetime extensions that exist for an exact Reissner-Nordstrom spacetime. The extension regularity matters: curvature blowup alone does not rule out every continuous-metric extension.