Causal hierarchy

The causal hierarchy orders restrictions on the causal structure of a spacetime, from exclusion of closed timelike curves through stable causality and global hyperbolicity.

An observer's future event horizon bounds the events that can ever signal to its future-inextendible worldline. Its past event horizon bounds the events reachable by signals from its past-inextendible worldline. A complete inertial observer in Minkowski spacetime has neither. The uniformly accelerated worldline $t=a^{-1}\sinh (a\tau )$, $x=a^{-1}\cosh (a\tau )$ has $I^{-}(\gamma )=\{t-x<0\}$ and $I^{+}(\gamma )=\{t+x>0\}$. Their boundaries are its two Rindler horizons. This observer dependence is distinct from a black-hole horizon defined using future null infinity.

For a comoving observer in a spatially flat Friedmann-Lemaître-Robertson-Walker metric, a finite future conformal time interval gives an event horizon at the displayed comoving radius. Its physical radius at time $t$ is $a(t){\chi }_{\mathrm{eh}}(t)$.

A light signal sent from a comoving observer to a comoving reflector at separation $\chi$, and immediately returned, consumes conformal time $2\chi$. The supremum of the initial physical reflector distances allowing a return is therefore half the cosmological event horizon radius. Equality generally requires infinite return time; finite-time reception requires a strictly smaller distance. For a cosmological domain wall network with $a\propto t^2$, the supremum is $t/2=1/H$.

For two comoving observers in an Einstein-de Sitter universe, light observed at redshift $z$ today fixes their separation through the past null geodesic. A newly sent signal needs the same comoving distance for every one-way leg. Since $a=(t/t_0{)}^{2/3}$, each leg adds a fixed conformal time interval, producing the displayed reception time after $n$ legs when replies are immediate. At $z=1$ the first round-trip wait is $[(3-\sqrt{2}{)}^3-1]t_0$. There is no finite count limit because future conformal time diverges, though infinitely many legs require infinite cosmic time.

A time orientation is a continuous choice of future cone throughout a time-orientable spacetime. A continuous nonzero timelike vector field supplies such a choice. For metric signature $(-,+,+,+)$, a nonzero causal vector $Z$ is future-directed relative to a future timelike $W$ exactly when $g(W,Z)<0$.

A causal curve has an everywhere nonspacelike tangent of consistent time orientation; its smooth segments are timelike or null. More generally, curves with local Lipschitz continuity and future causal tangent almost everywhere are allowed. The causal future and causal past are defined by reachability along these curves.

The causal past is the time-reversed causal future. Relative to a specified component ${\mathcal{I}}^{+}$ of future null infinity, the black hole region is the complement of its causal past: $\mathcal{B}=\mathcal{M}\setminus J^{-}({\mathcal{I}}^{+})$.

The causal future of $S$ consists of points reached from $S$ by future-directed causal curves, including points of $S$ itself. In a globally hyperbolic spacetime, the causal future of a compact set is closed.

The future horismos is the part of the causal future outside the chronological future. For a compact set in a globally hyperbolic spacetime, it equals the boundary of the chronological future. Its normal null geodesic generators cannot have passed a conjugate point to a spacelike surface when the initial set is that surface.

The chronological future $I^{+}(S)$ of a subset $S$ of a spacetime is the set of points reachable from $S$ by future-directed timelike curves. The chronological future is an open set.

A subset $U$ is chronologically saturated in both directions when $I^{+}(U)=U=I^{-}(U)$. If the ambient spacetime is connected and $U$ is nonempty, then $U$ is the whole spacetime.

The chronological past $I^{-}(S)$ is the set of points from which a future-directed timelike curve reaches $S$. Equivalently, it is the time reverse of the chronological future.

A closed timelike curve is a closed curve whose tangent is everywhere timelike, so an observer following it returns to the same spacetime event.

In the flat quotient $t\equiv t+P_t$ with noncompact space, choose any sufficiently large positive or negative temporal winding to make a lifted displacement timelike. Every event is in both the causal future and causal past of every other event. Constant-position curves are closed timelike curves. The quotient is a valid Lorentzian manifold, but it violates chronology and cannot support arbitrary globally consistent initial-value evolution.

On the equatorial plane of the Kerr metric, the displayed component is negative for sufficiently small negative $r$. A curve of constant $t,r,\theta =\pi /2$, with $\varphi$ increasing through $2\pi$, is therefore a regular closed timelike curve. It avoids the ring at $r=0$. Such circles occur on the negative-radius sheet of the exact analytic extension, not in the exterior region accessible from positive-radius infinity.

A spacetime is stably causal when its light cones can be widened slightly without creating a closed timelike curve. Equivalently, it admits a continuous time function strictly increasing along every future-directed causal curve.

Strong causality means that every event has arbitrarily small neighborhoods which no causal curve can leave and then re-enter. Stable causality implies strong causality.

A spacetime is future-distinguishing when $I^{+}(p)=I^{+}(q)$ implies $p=q$. Every strongly causal spacetime is future-distinguishing.

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.