In general relativity

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...

, Birkhoff's theorem states that any spherically symmetric solution

Spherically symmetric spacetime

A spherically symmetric spacetime is a spacetime whose isometry group contains a subgroup which is isomorphic to the group SO and the orbits of this group are 2-dimensional spheres . The isometries are then interpreted as rotations and a spherically symmetric spacetime is often described as one...

 of the vacuum field equations must be static

Static spacetime

In general relativity, a spacetime is said to be static if it admits a global, non-vanishing, timelike Killing vector field K which is irrotational, i.e., whose orthogonal distribution is involutive...

 and asymptotically flat. This means that the exterior solution must be given by the Schwarzschild metric

Schwarzschild metric

In Einstein's theory of general relativity, the Schwarzschild solution describes the gravitational field outside a spherical, uncharged, non-rotating mass such as a star, planet, or black hole. It is also a good approximation to the gravitational field of a slowly rotating body like the Earth or...

.

The theorem was proven in 1923 by G. D. Birkhoff (author of an even more famous Birkhoff theorem, the pointwise ergodic theorem which lies at the foundation of ergodic theory

Ergodic theory

Ergodic theory is a branch of mathematics that studies dynamical systems with an invariant measure and related problems. Its initial development was motivated by problems of statistical physics....

). However, Stanley Deser

Stanley Deser

Stanley Deser is an American physicist known for his contributions to general relativity. Currently, he is the Ancell Professor of Physics at Brandeis University in Waltham, Massachusetts....

 recently pointed out that it was published two years earlier by a little-known Norwegian physicist, Jørg Tofte Jebsen

Jørg Tofte Jebsen

Jørg Tofte Jebsen was a Norwegian physicist who discovered and published an important theorem concerning general relativity which is now known as Birkhoff's theorem. This result was rediscovered by George David Birkhoff a few years later....

.

Intuitive rationale

The intuitive idea of Birkhoff's theorem is that a spherically symmetric gravitational field should be produced by some massive object at the origin; if there were another concentration of mass-energy somewhere else, this would disturb the spherical symmetry, so we can expect the solution to represent an isolated object. That is, the field should vanish at large distances, which is (partly) what we mean by saying the solution is asymptotically flat. Thus, this part of the theorem is just what we would expect from the fact that general relativity reduces to Newtonian

Newtonian

Newtonian refers to the work of Isaac Newton, in particular:* Newtonian mechanics, also known as classical mechanics* Newtonian telescope, a type of reflecting telescope* Newtonian cosmology* Newtonian dynamics...

 gravitation

Gravitation

Gravitation, or gravity, is a natural phenomenon by which physical bodies attract with a force proportional to their mass. Gravitation is most familiar as the agent that gives weight to objects with mass and causes them to fall to the ground when dropped...

 in the Newtonian limit

Newtonian limit

In physics, the Newtonian limit refers to physical systems without significantly intense gravitation, in the sense that Newton's law of universal gravitation may be used to obtain values that are correct to a high order...

.

Implications

The conclusion that the exterior field must also be stationary is more surprising, and has an interesting consequence. Suppose we have a spherically symmetric star of fixed mass which is experiencing spherical pulsations. Then Birkhoff's theorem says that the exterior geometry must be Schwarzschild; the only effect of the pulsation is to change the location of the stellar surface. This means that a spherically pulsating star cannot emit gravitational waves.

Another interesting consequence of Birkhoff's theorem is that for a spherically symmetric thin shell, the interior solution must be given by the Minkowski metric; in other words, the gravitational field must vanish inside a spherically symmetric shell. This agrees with what happens in Newtonian gravitation.

Generalizations

Birkhoff's theorem can be generalized: any spherically symmetric solution of the Einstein/Maxwell field equations must be stationary and asymptotically flat, so the exterior geometry of a spherically symmetric charged star must be given by the Reissner-Nordström electrovacuum

Reissner-Nordström metric

In physics and astronomy, the Reissner–Nordström metric is a static solution to the Einstein-Maxwell field equations, which corresponds to the gravitational field of a charged, non-rotating, spherically symmetric body of mass M.-The metric:...

.

External links

• Birkhoff's Theorem on ScienceWorld
  ScienceWorld
  ScienceWorld, also known as Eric Weisstein's World of Science, is a Web site that opened to the general public in January 2002. As of November 2007, ScienceWorld includes more than 4,000 entries in fields of science including astronomy, chemistry, physics, as well as biographies of many scientists...