In mathematical

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 ...

, the Penrose inequality, first conjectured by Sir

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 ...

, estimates the mass of a

spacetime 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 ...

in terms of the total area of its

black holes A black hole is a region of spacetime where gravity is so strong that nothing, including light or other electromagnetic waves, has enough energy to escape it. The theory of general relativity predicts that a sufficiently compact mass can def ...

and is a generalization of the

positive mass theorem The positive energy theorem (also known as the positive mass theorem) refers to a collection of foundational results in general relativity and differential geometry. Its standard form, broadly speaking, asserts that the gravitational energy of an ...

. The Riemannian Penrose inequality is an important special case. Specifically, if (''M'', ''g'') is an

asymptotically flat An asymptotically flat spacetime is a Lorentzian manifold in which, roughly speaking, the curvature vanishes at large distances from some region, so that at large distances, the geometry becomes indistinguishable from that of Minkowski spacetime. ...

Riemannian

3-manifold In mathematics, a 3-manifold is a space that locally looks like Euclidean 3-dimensional space. A 3-manifold can be thought of as a possible shape of the universe. Just as a sphere looks like a plane to a small enough observer, all 3-manifolds lo ...

with nonnegative

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 ...

and

ADM mass The ADM formalism (named for its authors Richard Arnowitt, Stanley Deser and Charles W. Misner) is a Hamiltonian formulation of general relativity that plays an important role in canonical quantum gravity and numerical relativity. It was first ...

''m'', and ''A'' is the area of the outermost

minimal surface In mathematics, a minimal surface is a surface that locally minimizes its area. This is equivalent to having zero mean curvature (see definitions below). The term "minimal surface" is used because these surfaces originally arose as surfaces that ...

(possibly with multiple connected components), then the Riemannian Penrose inequality asserts :

m \geq \sqrt.

This is purely a geometrical fact, and it corresponds to the case of a complete three-dimensional,

space-like 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 ...

totally geodesic This is a glossary of some terms used in Riemannian geometry and metric geometry — it doesn't cover the terminology of differential topology. The following articles may also be useful; they either contain specialised vocabulary or provi ...

submanifold In mathematics, a submanifold of a manifold ''M'' is a subset ''S'' which itself has the structure of a manifold, and for which the inclusion map satisfies certain properties. There are different types of submanifolds depending on exactly which p ...

of a (3 + 1)-dimensional spacetime. Such a submanifold is often called a time-symmetric initial data set for a spacetime. The condition of (''M'', ''g'') having nonnegative scalar curvature is equivalent to the spacetime obeying the dominant energy condition. This inequality was first proved by

Gerhard Huisken Gerhard Huisken (born 20 May 1958) is a German mathematician whose research concerns differential geometry and partial differential equations. He is known for foundational contributions to the theory of the mean curvature flow, including Huiske ...

and

Tom Ilmanen Tom or TOM may refer to: * Tom (given name), a diminutive of Thomas or Tomás or an independent Aramaic given name (and a list of people with the name) Characters * Tom Anderson, a character in ''List of Beavis and Butt-Head characters#Local r ...

in 1997 in the case where ''A'' is the area of the largest component of the outermost minimal surface. Their proof relied on the machinery of weakly defined

inverse mean curvature flow In the mathematical fields of differential geometry and geometric analysis, inverse mean curvature flow (IMCF) is a geometric flow of submanifolds of a Riemannian or pseudo-Riemannian manifold. It has been used to prove a certain case of the R ...

, which they developed. In 1999,

Hubert Bray Hubert Lewis Bray is a mathematician and differential geometry, differential geometer. He is known for having proved the Riemannian Penrose inequality. He works as professor of mathematics and physics at Duke University. Early life and education ...

gave the first complete proof of the above inequality using a conformal

flow Flow may refer to: Science and technology * Fluid flow, the motion of a gas or liquid * Flow (geomorphology), a type of mass wasting or slope movement in geomorphology * Flow (mathematics), a group action of the real numbers on a set * Flow (psych ...

of metrics. Both of the papers were published in 2001.

Physical motivation

The original physical argument that led Penrose to conjecture such an inequality invoked the Hawking area theorem and the

cosmic censorship hypothesis The weak and the strong cosmic censorship hypotheses are two mathematical conjectures about the structure of gravitational singularities arising in general relativity. Singularities that arise in the solutions of Einstein's equations are typically ...

Case of equality

Both the Bray and Huisken–Ilmanen proofs of the Riemannian Penrose inequality state that under the hypotheses, if :

m = \sqrt,

then the manifold in question is isometric to a slice of the Schwarzschild spacetime outside of the outermost minimal surface.

Penrose conjecture

More generally, Penrose conjectured that an inequality as above should hold for spacelike submanifolds of spacetimes that are not necessarily time-symmetric. In this case, nonnegative scalar curvature is replaced with the dominant energy condition, and one possibility is to replace the minimal surface condition with 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 ...

condition. Proving such an inequality remains an open problem in general relativity, called the Penrose conjecture.

In popular culture

*In episode 6 of season 8 of the television sitcom ''

The Big Bang Theory ''The Big Bang Theory'' is an American television sitcom created by Chuck Lorre and Bill Prady, both of whom served as executive producers on the series, along with Steven Molaro, all of whom also served as head writers. It premiered on CBS ...

'', Dr. Sheldon Cooper claims to be in the process of solving the Penrose Conjecture while at the same time composing his Nobel Prize acceptance speech.

References

* * * * Riemannian geometry Geometric inequalities General relativity Theorems in geometry {{relativity-stub