Apparent horizon
Encyclopedia
In general relativity
, an apparent horizon is a surface that is the boundary
between light rays that are directed outwards and moving outwards, and those directed outwards but moving inwards.
Apparent horizons are not invariant properties of a spacetime. They are observer-dependent, and in particular they are distinct from absolute horizon
s.
See, however, the articles on ergosphere
, Cauchy horizon
, the Reissner-Nordström solution, photon sphere, Killing horizon
and naked singularity
; the notion of a horizon in general relativity is subtle, and depends on fine distinctions.
. A (compact
, orientable
, spacelike) surface always has 2 independent forward-in-time pointing, lightlike, normal
directions. For example, a (spacelike) sphere
in Minkowski space
has lightlike vectors pointing inward and outward along the radial direction. 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.
We can take the set of all such trapped surfaces. In terms of a simple Schwarzschild
black hole
, these surfaces fill up the black hole. The apparent horizon is then defined as the boundary of these surfaces — essentially, it is the outermost surface of the black hole, in this sense. Note, however, that a black hole is defined with respect to the event horizon, which is not always the same as the apparent horizon.
Any apparent horizon is observer dependent.
. 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.
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.
General relativity
General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...
, an apparent horizon is a surface that is the boundary
Boundary (topology)
In topology and mathematics in general, the boundary of a subset S of a topological space X is the set of points which can be approached both from S and from the outside of S. More precisely, it is the set of points in the closure of S, not belonging to the interior of S. An element of the boundary...
between light rays that are directed outwards and moving outwards, and those directed outwards but moving inwards.
Apparent horizons are not invariant properties of a spacetime. They are observer-dependent, and in particular they are distinct from absolute horizon
Absolute horizon
In general relativity, an absolute horizon is a boundary in spacetime, defined with respect to the external universe, inside of which events cannot affect an external observer. Light emitted inside the horizon can never reach the observer, and anything that passes through the horizon from the...
s.
See, however, the articles on ergosphere
Ergosphere
The ergosphere is a region located outside a rotating black hole. Its name is derived from the Greek word ergon, which means “work”. It received this name because it is theoretically possible to extract energy and mass from the black hole in this region...
, Cauchy horizon
Cauchy horizon
In physics, a Cauchy horizon is a light-like boundary of the domain of validity of a Cauchy problem...
, the Reissner-Nordström solution, photon sphere, Killing horizon
Killing horizon
A Killing horizon is a null hypersurface on which there is a null Killing vector field .Associated to a Killing horizon is a geometrical quantity known as surface gravity, \kappa...
and naked singularity
Naked singularity
In general relativity, a naked singularity is a gravitational singularity, without an event horizon. In a black hole, there is a region around the singularity, the event horizon, where the gravitational force of the singularity is strong enough so that light cannot escape. Hence, the singularity...
; 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 surfaceTrapped null surface
A trapped null surface is a set of points defined in the context of general relativity as a closed surface on which outward-pointing light rays are actually converging ....
. A (compact
Compact space
In mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...
, orientable
Orientability
In mathematics, orientability is a property of surfaces in Euclidean space measuring whether or not it is possible to make a consistent choice of surface normal vector at every point. A choice of surface normal allows one to use the right-hand rule to define a "clockwise" direction of loops in the...
, spacelike) surface always has 2 independent forward-in-time pointing, lightlike, normal
Surface normal
A surface normal, or simply normal, to a flat surface is a vector that is perpendicular to that surface. A normal to a non-flat surface at a point P on the surface is a vector perpendicular to the tangent plane to that surface at P. The word "normal" is also used as an adjective: a line normal to a...
directions. For example, a (spacelike) sphere
Sphere
A sphere is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball. Like a circle in two dimensions, a perfect sphere is completely symmetrical around its center, with all points on the surface lying the same distance r from the center point...
in Minkowski space
Minkowski space
In physics and mathematics, Minkowski space or Minkowski spacetime is the mathematical setting in which Einstein's theory of special relativity is most conveniently formulated...
has lightlike vectors pointing inward and outward along the radial direction. 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.
We can take the set of all such trapped surfaces. In terms of a simple Schwarzschild
Schwarzschild metric
In Einstein's theory of general relativity, the Schwarzschild solution describes the gravitational field outside a spherical, uncharged, non-rotating mass such as a star, planet, or black hole. It is also a good approximation to the gravitational field of a slowly rotating body like the Earth or...
black hole
Black hole
A black hole is a region of spacetime from which nothing, not even light, can escape. The theory of general relativity predicts that a sufficiently compact mass will deform spacetime to form a black hole. Around a black hole there is a mathematically defined surface called an event horizon that...
, these surfaces fill up the black hole. The apparent horizon is then defined as the boundary of these surfaces — essentially, it is the outermost surface of the black hole, in this sense. Note, however, that a black hole is defined with respect to the event horizon, which is not always the same as the apparent horizon.
Any apparent horizon is observer dependent.
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 horizonAbsolute horizon
In general relativity, an absolute horizon is a boundary in spacetime, defined with respect to the external universe, inside of which events cannot affect an external observer. Light emitted inside the horizon can never reach the observer, and anything that passes through the horizon from the...
. 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.
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 horizonAbsolute horizonIn general relativity, an absolute horizon is a boundary in spacetime, defined with respect to the external universe, inside of which events cannot affect an external observer. Light emitted inside the horizon can never reach the observer, and anything that passes through the horizon from the...
- Black holeBlack holeA black hole is a region of spacetime from which nothing, not even light, can escape. The theory of general relativity predicts that a sufficiently compact mass will deform spacetime to form a black hole. Around a black hole there is a mathematically defined surface called an event horizon that...
- Cosmological horizon
- Event horizonEvent horizonIn general relativity, an event horizon is a boundary in spacetime beyond which events cannot affect an outside observer. In layman's terms it is defined as "the point of no return" i.e. the point at which the gravitational pull becomes so great as to make escape impossible. The most common case...
- Particle horizon
- Trapped null surfaceTrapped null surfaceA trapped null surface is a set of points defined in the context of general relativity as a closed surface on which outward-pointing light rays are actually converging ....