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];
> 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 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, ∇_{a}(*X*_{m}* X*^{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.

@ __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.

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