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)

Albert Einstein Albert Einstein (14 March 187918 April 1955) was a German-born theoretical physicist who is best known for developing the theory of relativity. Einstein also made important contributions to quantum mechanics. His mass–energy equivalence f ...

's gravitational theory, that is,

general relativity General relativity, also known as the general theory of relativity, and as Einstein's theory of gravity, is the differential geometry, geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of grav ...

. This attempt at generalisation has been strongly influenced in recent decades by

string theory In physics, string theory is a theoretical framework in which the point-like particles of particle physics are replaced by one-dimensional objects called strings. String theory describes how these strings propagate through space and intera ...

. 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 The Large Hadron Collider (LHC) is the world's largest and highest-energy particle accelerator. It was built by the CERN, European Organization for Nuclear Research (CERN) between 1998 and 2008, in collaboration with over 10,000 scientists, ...

, this would provide the evidence that higher dimensions exist.

Exact solutions

The higher-dimensional generalization of the Kerr metric was discovered by Robert Myers and Malcolm Perry. 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 ansatzes or, from German, ansätze ; ) is an educated guess or an additional assumption made to help solve a problem, and which may later be ...

; 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 a coefficient that Albert Einstein initially added to his field equations of general rel ...

. 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 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 proved that the topology of the

event horizon In astrophysics, an event horizon is a boundary beyond which events cannot affect an outside 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 c ...

of a non-rotating

black hole A black hole is a massive, compact astronomical object so dense that its gravity prevents anything from escaping, even light. Albert Einstein's theory of general relativity predicts that a sufficiently compact mass will form a black hole. Th ...

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 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 geometry ...

References

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

Exact solutions

Black hole uniqueness

See also

References