In
general relativity
General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
, an apparent horizon is a surface that is the
boundary
Boundary or Boundaries may refer to:
* Border, in political geography
Entertainment
* ''Boundaries'' (2016 film), a 2016 Canadian film
* ''Boundaries'' (2018 film), a 2018 American-Canadian road trip film
*Boundary (cricket), the edge of the pla ...
between
light ray
In optics a ray is an idealized geometrical model of light, obtained by choosing a curve that is perpendicular to the ''wavefronts'' of the actual light, and that points in the direction of energy flow. Rays are used to model the propagation o ...
s that are directed outwards and moving outwards and those directed outward but moving inward.
Apparent horizons are not invariant properties of
spacetime
In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differ ...
, and in particular, they are distinct from
event horizon
In astrophysics, an event horizon is a boundary beyond which events cannot affect an observer. Wolfgang Rindler coined the term in the 1950s.
In 1784, John Michell proposed that gravity can be strong enough in the vicinity of massive compact ob ...
s. Within an apparent horizon, light does not move outward; this is in contrast with the event horizon. In a dynamical spacetime, there can be outgoing light rays exterior to an apparent horizon (but still interior to the event horizon). An apparent horizon is a ''local'' notion of the boundary of a black hole, whereas an event horizon is a global notion.
The notion of a horizon in general relativity is subtle and depends on fine distinctions.
Definition
The notion of an "apparent horizon" begins with the notion of a
trapped null surface. A (
compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact
* Blood compact, an ancient ritual of the Philippines
* Compact government, a type of colonial rule utilized in British ...
,
orientable
In mathematics, orientability is a property of some topological spaces such as real vector spaces, Euclidean spaces, surfaces, and more generally manifolds that allows a consistent definition of "clockwise" and "counterclockwise". A space i ...
,
spacelike
In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differen ...
) surface always has two independent forward-in-time pointing,
lightlike
In mathematical physics, Minkowski space (or Minkowski spacetime) () is a combination of three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two events is independent of the inert ...
,
normal Normal(s) or The Normal(s) may refer to:
Film and television
* ''Normal'' (2003 film), starring Jessica Lange and Tom Wilkinson
* ''Normal'' (2007 film), starring Carrie-Anne Moss, Kevin Zegers, Callum Keith Rennie, and Andrew Airlie
* ''Norma ...
directions. For example, a (spacelike)
sphere
A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is th ...
in
Minkowski space
In mathematical physics, Minkowski space (or Minkowski spacetime) () is a combination of three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two events is independent of the iner ...
has lightlike vectors pointing inward and outward along the radial direction. In
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean ...
(i.e. flat and unaffected by gravitational effects), the inward-pointing, lightlike normal vectors converge, while the outward-pointing, lightlike normal vectors diverge. It can, however, happen that both inward-pointing and outward-pointing lightlike normal vectors converge. In such a case, the surface is called ''trapped''. The apparent horizon is the outermost of all trapped surfaces, also called the "marginally outer trapped surface" (MOTS).
Differences from the (absolute) event horizon
In the context of black holes, the term event horizon refers almost exclusively to the notion of the "
absolute horizon
In general relativity, an absolute horizon is a boundary in spacetime, defined with respect to the external universe, inside which events cannot affect an external observer. Light emitted inside the horizon can never reach the observer, and anythi ...
". Much confusion seems to arise concerning the differences between an apparent horizon (AH) and an event horizon (EH). In general, the two need not be the same. For example, in the case of a perturbed black hole, the EH and the AH generally do not coincide as long as either horizon fluctuates.
Event horizons can, in principle, arise and evolve in exactly flat regions of spacetime, having no black hole inside, if a hollow spherically symmetric thin shell of matter is collapsing in a vacuum spacetime. The exterior of the shell is a portion of Schwarzschild space and the interior of the hollow shell is exactly flat Minkowski space.
Bob Geroch has pointed out that if all the stars in the Milky Way gradually aggregate towards the galactic center while keeping their proportionate distances from each other, they will all fall within their joint Schwarzschild radius long before they are forced to collide.
In the simple picture of stellar collapse leading to formation of a black hole, an event horizon forms before an apparent horizon.
As the black hole settles down, the two horizons approach each other, and asymptotically become the same surface. If the AH exists, it is necessarily inside of the EH.
Apparent horizons depend on the "
slicing" of a spacetime. That is, the location and even existence of an apparent horizon depends on the way spacetime is divided into space and time. For example, it is possible to slice the Schwarzschild geometry in such a way that there is ''no'' apparent horizon, ever, despite the fact that there is certainly an event horizon.
See also
*
Absolute horizon
In general relativity, an absolute horizon is a boundary in spacetime, defined with respect to the external universe, inside which events cannot affect an external observer. Light emitted inside the horizon can never reach the observer, and anythi ...
*
Black hole
*
Cosmological horizon
A cosmological horizon is a measure of the distance from which one could possibly retrieve information. This observable constraint is due to various properties of general relativity, the expanding universe, and the physics of Big Bang cosmology. Co ...
*
Event horizon
In astrophysics, an event horizon is a boundary beyond which events cannot affect an observer. Wolfgang Rindler coined the term in the 1950s.
In 1784, John Michell proposed that gravity can be strong enough in the vicinity of massive compact ob ...
*
Particle horizon The particle horizon (also called the cosmological horizon, the comoving horizon (in Dodelson's text), or the cosmic light horizon) is the maximum distance from which light from particles could have traveled to the observer in the age of the univers ...
*
Trapped null surface
*
Ergosphere
file:Ergosphere_and_event_horizon_of_a_rotating_black_hole_(no_animation).gif, 300px, In the ergosphere (shown here in light gray), the component ''gtt'' is negative, i.e., acts like a purely spatial metric component. Consequently, timelike or ligh ...
*
Cauchy horizon
In physics, a Cauchy horizon is a light-like boundary of the domain of validity of a Cauchy problem (a particular boundary value problem of the theory of partial differential equations). One side of the horizon contains closed space-like geod ...
*
Reissner–Nordström solution
*
Photon sphere
A photon sphere or photon circle is an area or region of space where gravity is so strong that photons are forced to travel in orbits, which is also sometimes called the last photon orbit. The radius of the photon sphere, which is also the lower ...
*
Killing horizon
In physics, a Killing horizon is a geometrical construct used in general relativity and its generalizations to delineate spacetime boundaries without reference to the dynamic Einstein field equations. Mathematically a Killing horizon is a null hyp ...
*
Naked singularity
In general relativity, a naked singularity is a hypothetical gravitational singularity without an event horizon. In a black hole, the singularity is completely enclosed by a boundary known as the event horizon, inside which the curvature of space ...
References
External links
{{Spoken Wikipedia, Wikipedia Audio - Apparent Horizon.wav, date=2018-11-10
Mathematical methods in general relativity
Black holes