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].

@ __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 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 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 d*r*/d*v* = 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 \(\cal H\)^{±}(*S*):=
∂{*D*^{ ±}(*S*)}
– *I*^{ ±}(*D*^{ ±}(*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].

**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 a1806 [negative temperatures and entropy super-additivity].

@ __Special types__: Mars et al a1803 [for two or more linearly independent Killing vectors].

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