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 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 ...

. 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.

Exact solutions

The higher-dimensional generalization of the

Kerr metric The Kerr metric or Kerr geometry describes the geometry of empty spacetime around a rotating uncharged axially symmetric black hole with a quasispherical event horizon. The Kerr metric is an exact solution of the Einstein field equations of gen ...

was discovered by

Myers Myers as a surname has several possible origins, e.g. Old French ("physician"), Old English ("mayor"), and Old Norse ("marsh"). People * Abram F. Myers (born 1889), chair of the Federal Trade Commission and later general counsel and board ch ...

and

Perry Perry, also known as pear cider, is an alcoholic beverage made from fermented pears, traditionally the perry pear. It has been common for centuries in England, particularly in Gloucestershire, Herefordshire, and Worcestershire. It is also mad ...

. Like the Kerr metric, the Myers-Perry metric has spherical horizon topology. The construction involves making a Kerr-Schild

ansatz In physics and mathematics, an ansatz (; , meaning: "initial placement of a tool at a work piece", plural Ansätze ; ) is an educated guess or an additional assumption made to help solve a problem, and which may later be verified to be part of the ...

; by a similar method, the solution has been generalized to include a

cosmological constant In cosmology, the cosmological constant (usually denoted by the Greek capital letter lambda: ), alternatively called Einstein's cosmological constant, is the constant coefficient of a term that Albert Einstein temporarily added to his field eq ...

. 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. In 2014, Hari Kunduri and James Lucietti proved the existence of a black hole with

Lens space A lens space is an example of a topological space, considered in mathematics. The term often refers to a specific class of 3-manifolds, but in general can be defined for higher dimensions. In the 3-manifold case, a lens space can be visualized ...

topology of the ''L''(2, 1) type in five dimensions, this was next extended to all ''L''(p, 1) with positive integers p by Shinya Tomizawa and Masato Nozawa in 2016 and finally in a preprint to all ''L''(p, q) and any dimension by Marcus Khuri and Jordan Rainone in 2022, a black lens doesn't necessarily need to rotate as a black ring but all examples so far need a matter field coming from the extra dimensions to remain stable.

Black hole uniqueness

In four dimensions,

Hawking Hawking may refer to: People * Stephen Hawking (1942–2018), English theoretical physicist and cosmologist * Hawking (surname), a family name (including a list of other persons with the name) Film * ''Hawking'' (2004 film), about Stephen Ha ...

proved that the topology of the

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 ...

of a non-rotating black hole must be spherical. Because the proof uses the

Gauss–Bonnet theorem In the mathematical field of differential geometry, the Gauss–Bonnet theorem (or Gauss–Bonnet formula) is a fundamental formula which links the curvature of a surface to its underlying topology. In the simplest application, the case of a t ...

, 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 In the mathematical field of Riemannian geometry, the scalar curvature (or the Ricci scalar) is a measure of the curvature of a Riemannian manifold. To each point on a Riemannian manifold, it assigns a single real number determined by the geometr ...

References

{{DEFAULTSORT:Higher-Dimensional Einstein Gravity Theories of gravity Albert Einstein Exact solutions in general relativity

Exact solutions

Black hole uniqueness

See also

References