Higher-dimensional Einstein gravity
Encyclopedia
Higher-dimensional Einstein gravity is any of various physical theories that attempt to generalise to higher dimensions various results of the well established theory of standard (four-dimensional) Einstein gravity, that is, general relativity
. This attempt at generalisation has been strongly influenced in recent decades by string theory
.
At present, this work can probably be most fairly described as extended theoretical speculation. Currently, it has no direct observational and experimental support, in contrast to four-dimensional general relativity. However, this theoretical work has led to the possibility of proving the existence of extra dimensions. This is best demonstrated by the proof of Harvey Reall and Roberto Emparan that there is a 'black ring' solution in 5 dimensions. If such a 'black ring' could be produced in a particle accelerator such as the Large Hadron Collider
, this would provide the evidence that higher dimensions exist.
was discovered by Myers and Perry
. Like the Kerr metric, the Myers-Perry metric has spherical horizon topology. The construction involves making a Kerr-Schild ansatz
; by a similar method, the solution has been generalized to include a cosmological constant
. The black ring is a solution of five-dimensional general relativity. It inherits its name from the fact that its event horizon is topologically S1 × S2. This is in contrast to other known black hole solutions in five dimensions which have horizon topology S3.
proved that the topology of the event horizon
of a non-rotating black hole
must be spherical. Because the proof uses the Gauss-Bonnet theorem, it does not generalize to higher dimensions. The discovery of black ring solutions in five dimensions shows that other topologies are allowed in higher dimensions, but it is unclear precisely which topologies are allowed. It has been shown that the horizon must be of positive Yamabe type, meaning that it must admit a metric of positive scalar curvature
.
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...
. This attempt at generalisation has been strongly influenced in recent decades by string theory
String theory
String theory is an active research framework in particle physics that attempts to reconcile quantum mechanics and general relativity. It is a contender for a theory of everything , a manner of describing the known fundamental forces and matter in a mathematically complete system...
.
At present, this work can probably be most fairly described as extended theoretical speculation. Currently, it has no direct observational and experimental support, in contrast to four-dimensional general relativity. However, this theoretical work has led to the possibility of proving the existence of extra dimensions. This is best demonstrated by the proof of Harvey Reall and Roberto Emparan that there is a 'black ring' solution in 5 dimensions. If such a 'black ring' could be produced in a particle accelerator such as the Large Hadron Collider
Large Hadron Collider
The Large Hadron Collider is the world's largest and highest-energy particle accelerator. It is expected to address some of the most fundamental questions of physics, advancing the understanding of the deepest laws of nature....
, this would provide the evidence that higher dimensions exist.
Exact solutions
The higher-dimensional generalization of the Kerr metricKerr metric
The Kerr metric describes the geometry of empty spacetime around an uncharged axially-symmetric black-hole with an event horizon which is topologically a sphere. The Kerr metric is an exact solution of the Einstein field equations of general relativity; these equations are highly non-linear, which...
was discovered by Myers and Perry
Malcolm Perry (physicist)
Malcolm John Perry, is a theoretical physicist. Perry is a professor of theoretical physics at the University of Cambridge. His research mainly concerns general relativity, supergravity and string theory....
. Like the Kerr metric, the Myers-Perry metric has spherical horizon topology. The construction involves making a Kerr-Schild ansatz
Ansatz
Ansatz is a German noun with several meanings in the English language.It is widely encountered in physics and mathematics literature.Since ansatz is a noun, in German texts the initial a of this word is always capitalised.-Definition:...
; by a similar method, the solution has been generalized to include a cosmological constant
Cosmological constant
In physical cosmology, the cosmological constant was proposed by Albert Einstein as a modification of his original theory of general relativity to achieve a stationary universe...
. The black ring is a solution of five-dimensional general relativity. It inherits its name from the fact that its event horizon is topologically S1 × S2. This is in contrast to other known black hole solutions in five dimensions which have horizon topology S3.
Black hole uniqueness
In four dimensions, HawkingStephen Hawking
Stephen William Hawking, CH, CBE, FRS, FRSA is an English theoretical physicist and cosmologist, whose scientific books and public appearances have made him an academic celebrity...
proved that the topology of the event horizon
Event horizon
In 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...
of a non-rotating 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...
must be spherical. Because the proof uses the Gauss-Bonnet theorem, it does not generalize to higher dimensions. The discovery of black ring solutions in five dimensions shows that other topologies are allowed in higher dimensions, but it is unclear precisely which topologies are allowed. It has been shown that the horizon must be of positive Yamabe type, meaning that it must admit a metric of positive scalar curvature
Scalar curvature
In Riemannian geometry, the scalar curvature is the simplest curvature invariant of a Riemannian manifold. To each point on a Riemannian manifold, it assigns a single real number determined by the intrinsic geometry of the manifold near that point...
.