Horizons

In General > s.a. black holes; black-hole thermodynamics; event horizon; isolated and dynamical horizons; Rigidity; rindler space.
* Remark: As originally defined, the event horizon is a highly non-local object, defined in terms of the causal past of future null infinity; Alternative, (quasi)local definitions of horizons are often used, which include apparent, trapping, isolated and dynamical horizons, and are closely associated with 2-surfaces of zero outward null expansion.
* Universal horizons: A special spacelike hypersurface in certain solutions of Einstein-Aether and Hořava gravity theories which acts as a one-way membrane; Signals with arbitrarily high velocities from beyond this hypersurface cannot escape to infinity and are destined to hit the singularity, so the hypersurface acts like a black-hole horizon.
@ General references: Mäkelä gq/01 [microscopic properties]; Chruściel et al JGP(02) [differentiability]; Hubeny & Rangamani JHEP(02)ht [stability]; Aros JHEP(03)gq/02 [diffeomorphism algebra]; Booth & Fairhurst CQG(05)gq [energy and angular momentum]; Korzyński PRD(06)gq; 't Hooft IJMPD(06)gq-in [black hole, as a dynamical system]; Faraoni a1309-in [evolving black holes, apparent and trapping horizons]; Ellis & Uzan CRP(15)-a1612 [in cosmology, rev]; Grumiller et al a2005-GRF [horizon degrees of freedom].
@ Evolving: Nielsen & Visser CQG(06)gq/05 [spherically symmetric, relation to various approaches]; Kavanagh & Booth PRD(06)gq [slowly].
@ Entropy: Wang PRD(05)gq, Mitra et al PLB(14) [in modified gravity]; Oriti et al PRL(16)-a1510 [from quantum gravity condensates].
@ Other thermodynamics: Padmanabhan MPLA(02)gq, gq/02, CQG(02)gq, GRG(02)gq, PRP(05)gq/03 [intro]; Jacobson & Parentani FP(03)gq; Padmanabhan MPLA(04)gq [observer-dependent, and tunneling]; Malafarina & Joshi a1106 [and gravitational collapse]; Majhi & Padmanabhan PRD(12)-a1204, PRD(12) [entropy from Virasoro algebra of Noether currents from diffeomorphisms]; Mohd & Sarkar PRD(13)-a1304 [local causal horizons]; Skákala & Shankaranarayanan IJMPD(16)-a1406 [relativistic Bose gas from projected gravity equations]; Faraoni & Vitagliano PRD(14) [for evolving horizons]; Hansen et al PLB(17)-a1610 [from Einstein's equation of state].
@ Trapping horizons: Nielsen JKPS(09)-a0802-conf; Hayward et al CQG(09)-a0806 [spherical, local temperature]; Nielsen IJMPD(08)-a0809, GRG(09)-a0809 [and black holes]; Hayward a0810-ASL [rev]; Nielsen CQG(10)-a1006 [relationship with event horizon, and thermodynamics].
@ Universal horizons: Michel & Parentani PRD(15)-a1504 [black-hole radiation]; Maciel PRD(16)-a1511 [quasilocal approach]; Lin et al PRD(16)-a1603 [static and rotating]; > s.a. black-hole thermodynamics.
@ Related topics: Hosler et al PRA(12)-a1111 [effective horizons for quantum communication]; Coley & McNutt CQG(18)-a1710 [geometric, using scalar curvature invariants]; > s.a. black-hole laws; semiclassical general relativity [reliability horizon]; entropy.

Apparent Horizon > s.a. black-hole geometry [vs firewall]; models in numerical relativity; Penrose Inequality.
$ Def: Given a spacelike hypersurface Σ, the apparent horizon is the boundary of the region of Σ containing trapped surfaces lying in Σ; In terms of the advanced time v, it is defined by dr/dv = 0.
* Relationships: A marginally-trapped surface; It need not coincide with the event horizon, since some light rays may start out diverging and reconverge again later, but it either coincides or is contained in the event horizon; (Not true? York in Corvallis: light rays may start out converging but not be trapped, e.g., for an evaporating hole).
@ General references: Gundlach PRD(98)gq/97, Alcubierre et al CQG(00) [algorithms]; Dasgupta ht/03 [entropy]; Zhou et al PLB(07) [thermodynamics]; Andersson & Metzger CMP(09)-a0708 [area estimate]; Faraoni 15.
@ Specific spacetimes: Wang GRG(05)gq/03 [no outer ones with 2 Killing vector fields]; Goncalves PRD(03)gq, CQG(03)gq [none with isometries]; Biswas et al a1106 [FLRW spacetime, Hawking-like radiation]; Faraoni PRD(11)-a1106 [FLRW spacetime, particle, event, and apparent horizons]; Viaggiu GRG(15)-a1506 [dynamical apparent horizons, first law of thermodynamics].

Cauchy Horizon > s.a. censorship; gowdy spacetime; Rigidity.
$ Def: The future and past Cauchy horizons are given, respectively, by \({\cal H}^\pm(S):= \partial\{D^\pm(S)\} - I^\pm(D^\pm(S))\), the boudary of the region where evolution can be predicted from data on S.
* Existence of isometry groups: & Isenberg and Moncrief.
@ Differentiability: Królak & Beem gq/97; Chruściel CQG(98)gq; Budzyński et al JMP(99)gq.
@ Other properties: Borde PLA(84) [compact]; Beem GRG(95) [stability]; Burko gq/97-MG8, PRL(97)gq [singularity, Klein-Gordon-perturbed Reissner-Nordström]; Beem & Królak JMP(98)gq/97 [endpoints]; Budzyński et al gq/00-conf [properties]; Rácz gq/01-MG9 [spacetime rigidity]; Minguzzi JMP(14)-a1406 [past completeness of generators], CMP(15)-a1406 [area theorem and smoothness]; Krasnikov CQG(14)-a1407 [simpler version of past completeness proof]; Juárez-Aubry IJMPD(15)-a1502-proc [particle detector crossing a Cauchy horizon].

Particle (Cosmological) Horizon > s.a. general relativity [problems]; Horizon Problem.
* Idea: The particle horizon of a point p in spacetime, given a congruence of timelike lines (particles), is the boundary of the set of world-lines that p can see; One does not expect an entropy associated to this type of horizon to be meaningful.
* Example: If the particles are geodesics, there are particle horizons iff past null infinity is spacelike (e.g., de Sitter spacetime).
@ References: in Hawking & Ellis 73, p128; Ellis & Rothman AJP(93)oct; Boya et al PRD(02)gq [graphical approach]; Melia MNRAS(07)-a0711; Oirschot et al MNRAS(10)-a1001; Hu PLB(11)-a1007 [and Hawking radiation]; Margalef-Bentabol et al JCAP(12)-a1302 [evolution, for different equations of state]; Bolotin & Tanatarov a1310 [problems]; Fermi et al Univ(19)-a1901 [and phantom fields].

Killing Horizon > s.a. quantum field theory in curved spacetime [vacua].
* Idea: The locus of points in spacetime where a Killing vector field X^a is null.
* Remark: It is often a source of coordinate singularities (similarly to the case when the Killing vector field vanishes), if one uses coordinates adapted to the action of the isometry group generated by the Killing vector field.
* Special cases: It is called degenerate when the surface gravity vanishes, \(\nabla_{\!a}\,(X_mX^m) = 0\); If the Killing horizon is non-degenerate, the Killing vector field has to change character from timelike to spacelike across the Killing horizon; In general, non-degenerate Killing horizons cross each other–they are bifurcate; The Killing horizon is called bifurcate if it is the union of two null surfaces which intersect in a codimension-2 spacelike surface (e.g., Rindler space, Schwarzschild, de Sitter spacetime).
* Examples: Spacetimes that have Killing horizons are some black holes, Rindler, de Sitter, Taub-NUT and Taub-Bolt spaces.
@ General references: Griffiths GRG(05)gq [Killing-Cauchy horizons for colliding plane waves, instability]; Jacobson & Parentani CQG(08)-a0806 [surface gravity, as expansion rate]; Smolić CQG(12)-a1205 [proof that Killing horizons are equipotential hypersurfaces for the electric and magnetic scalar potentials]; da Cunha & de Queiroz PRD(14)-a1312 [near-horizon geometry]; > s.a. Kundt Spacetimes.
@ Thermodynamics: Cvetič et al PRD(18)-a1806 [negative temperatures and entropy super-additivity].
@ Special types: Mars et al CQG(18)-a1803 [for two or more linearly independent Killing vectors].

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