Closed trapped surfaces are a concept used in black hole solutions of
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 ...
which describe the inner region of an
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 obj ...
.
Roger Penrose
Sir Roger Penrose (born 8 August 1931) is an English mathematician, mathematical physicist, philosopher of science and Nobel Laureate in Physics. He is Emeritus Rouse Ball Professor of Mathematics in the University of Oxford, an emeritus fello ...
defined the notion of closed trapped surfaces in 1965. A trapped surface is one where light is not moving away from the black hole. The boundary of the union of all trapped surfaces around a black hole is called an
apparent horizon
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 outward but moving inward.
Apparent horizons are not invariant properties of spacetim ...
.
A related term
trapped null surface
Closed trapped surfaces are a concept used in black hole solutions of general relativity which describe the inner region of an event horizon. Roger Penrose defined the notion of closed trapped surfaces in 1965. A trapped surface is one where light ...
is often used interchangeably. However, when discussing
causal horizons, trapped null surfaces are defined as only null vector fields giving rise to null surfaces. But marginally trapped surfaces may be spacelike, timelike or null.
Definition
They are
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 ...
surfaces (topological spheres, tubes, etc.) with restricted bounds, their area tending to decrease locally along any possible future direction and with a dual definition with respect to the past. The trapped surface is a spacelike surface of co-dimension 2, in a
Lorentzian spacetime. It follows
that any
normal vector
In geometry, a normal is an object such as a line, ray, or vector that is perpendicular to a given object. For example, the normal line to a plane curve at a given point is the (infinite) line perpendicular to the tangent line to the curve at ...
can be expressed as a linear combination of two future directed null vectors, normalised by:
k
+ · k
− = −2
The k
+ vector is directed “outwards” and k
− “inwards”. The set of all such vectors engenders one outgoing and one ingoing null congruence. The surface is designated trapped if the cross sections of both congruences decrease in area as they exit the surface; and this is apparent in the mean curvature vector, which is:
H
ɑ= −θ
+k
−ɑ − θ
−k
+ɑ
The surface is trapped if both the null expansions θ
± are negative, signifying that the mean curvature vector is timelike and future directed. The surface is marginally trapped if the outer expansion θ
+ = 0 and the inner expansion θ
− ≤ 0.
Trapped null surface
A trapped null surface is a
set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
of points defined in the context of
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 ...
as a closed surface on which outward-pointing
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 are actually converging (moving inwards).
Trapped null surfaces are used in the definition of the
apparent horizon
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 outward but moving inward.
Apparent horizons are not invariant properties of spacetim ...
which typically surrounds a
black hole
A black hole is a region of spacetime where gravitation, gravity is so strong that nothing, including light or other Electromagnetic radiation, electromagnetic waves, has enough energy to escape it. The theory of general relativity predicts t ...
.
Definition
We take 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 is ...
,
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, and find its outward pointing
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 ...
vectors. The basic picture to think of here is a ball with pins sticking out of it; the pins are the normal vectors.
Now we look at light rays that are directed outward, along these normal vectors. The rays will either be diverging (the usual case one would expect) or converging. Intuitively, if the light rays are converging, this means that the light is moving backwards inside of the ball. If all the rays around the entire surface are converging, we say that there is a trapped null surface.
More formally, if every null congruence orthogonal to a spacelike two-surface has negative expansion, then such surface is said to be trapped.
See also
*
Null hypersurface In relativity and in pseudo-Riemannian geometry, a null hypersurface is a hypersurface whose normal vector at every point is a null vector (has zero length with respect to the local metric tensor). A light cone is an example.
An alternative ch ...
*
Raychaudhuri equation
In general relativity, the Raychaudhuri equation, or Landau–Raychaudhuri equation, is a fundamental result describing the motion of nearby bits of matter.
The equation is important as a fundamental lemma for the Penrose–Hawking singularity the ...
References
* This is the gold standard in black holes because of its place in history. It is also quite thorough.
* This book is somewhat more up-to-date.
{{Roger Penrose
Mathematical methods in general relativity
Black holes