De SitterSchwarzschild metric 1

De SitterSchwarzschild metricIn general relativity, the de SitterSchwarzschild solution describes a black hole in a causal patch of de Sitterspace. Unlike a flat-space black hole, there is a largest possible de Sitter black hole, which is the Nariai spacetime.The Nariai limit has no singularities, the cosmological and black hole horizons have the same area, and they can bemapped to each other by a discrete reflection symmetry in any causal patch.[1]

IntroductionIn general relativity, spacetimes can have black hole event horizons and also cosmological horizons. The deSitterSchwarzschild solution is the simplest solution which has both.

MetricThe metric of any spherically symmetric solution in Schwarzschild form is:

The vacuum Einstein equations give a linear equation for (r), which has as solutions:

The first is a zero stress energy solution describing a black hole in empty space time, the second (with b positive)describes de Sitter space with a stress-energy of a positive cosmological constant of magnitude3b. Superposing thetwo solutions gives the de SitterSchwarzschild solution:

The two parameters a and b give the black hole mass and the cosmological constant respectively. In d+1dimensions, the inverse power law falloff in the black hole part is d2. In 2+1 dimensions, where the exponent iszero, the analogous solution starts with 2+1 de Sitter space, cuts out a wedge, and pastes the two sides of the wedgetogether to make a conical space.

Horizon propertiesDe Sitter space is the simplest solution of Einstein's equation with a positive cosmological constant. It is sphericallysymmetric and it has a cosmological horizon surrounding any observer, and describes an inflating universe. TheSchwarzschild solution is the simplest spherically symmetric solution of the Einstein equations with zerocosmological constant, and it describes a black hole event horizon in otherwise empty space. deSitterSchwarzschild is a combination of the two, and describes a black hole horizon spherically centered in anotherwise de Sitter universe. An observer which hasn't fallen into the black hole, and which can still see the blackhole despite the inflation, is sandwiched between the two horizons.One natural question to ask is whether the two horizons are different kinds of objects or whether they are theyfundamentally the same. Classically the two types of horizon look different. A black hole horizon is a future horizon,things can go in, but don't come out. The cosmological horizon in a big bang type cosmology is a past horizon,things come out, but nothing goes in.But in a semiclassical treatment, the de Sitter cosmological horizon can be thought of as absorbing or emitting depending on the point of view. Similarly, for a black hole that has been around for a long time, the horizon can be thought of as emitting or absorbing depending on whether you take the point of view of infalling matter or outgoing

De SitterSchwarzschild metric 2

Hawking radiation. Hawking argued based on thermodynamics that the past horizon of a white hole is in factphysically the same as the future horizon of a black hole, so that past and future horizons are physically identical.This was elaborated by Susskind into black hole complementarity, which states that any interior parts of a black holesolution, in either the past and future horizon interpretation, can be holographically related by a unitary change ofbasis to the quantum mechanical description of the horizon itself.The Nariai solution is the limit of the largest is a black hole in a space which is de Sitter at large distances, it has twohorizons, the cosmological de Sitter horizon and a Schwarzschild black hole horizon. For small mass black holes, thetwo are very different--- there is a singularity at the center of the black hole, and there is no singularity past thecosmological horizon. But the Nariai limit considers making the black hole bigger and bigger, until its event horizonhas the same area as the cosmological de Sitter horizon. At this point, the spacetime becomes regular, the black holesingularity runs off to infinity, and the two horizons are related by a space-time symmetry.In the Nariai limit, the black hole and de Sitter horizon can be interchanged just by changing the sign of thecoordinate z. When there is additional matter density, the solution can be thought of as an Einstein spherical universewith two antipodal black holes. Whichever black hole becomes larger becomes the cosmological horizon.

Nariai solutionStarting with de SitterSchwarzschild:

with

The two parameters a and b give the black hole mass and the cosmological constant respectively. In higherdimensions, the power law for the black hole part is faster.When a is small, (r) has two zeros at positive values of r, which are the location of the black hole and cosmologicalhorizon respectively. As the parameter a increases, keeping the cosmological constant fixed, the two positive zeroscome closer. At some value of a, they collide.Approaching this value of a, the black hole and cosmological horizons are at nearly the same value of r. But thedistance between them doesn't go to zero, because (r) is very small between the two zeros, and the square root of itsreciprocal integrates to a finite value. If the two zeros of are at R+ and R taking the small limit whilerescaling r to remove the dependence gives the Nariai solution.The form of near the almost-double-zero in terms of the new coordinate u given by r=R+u is:

The metric on the causal patch between the two horizons reduces to

which is the metric of . This form is local for an observer sandwiched between the black hole and thecosmological horizon, which reveal their presence as the two horizons at z=R and z=R respectively.The coordinate z can be replaced by a global coordinate for the 1+1-dimensional de Sitter space part, and then themetric can be written as:

In these global coordinates, the isotropy of de Sitter space makes shifts of the coordinate x isometries, so that it is possible to identify x with x+A, and make the space dimension into a circle. The constant-time radius of the circle

De SitterSchwarzschild metric 3

expands exponentially into the future and the past, and this is Nariai's original form.Rotating one of the horizons in Nariai space makes the other horizon rotate in the opposite sense. This is amanifestation of Mach's principle in self-contained causal patches, if the cosmological horizon is included as"matter", like it's symmetric counterpart, the black hole.

Hawking temperatureThe temperature of the small and large horizon in the de SitterSchwarzschild can be calculated as the period inimaginary time of the solution, or equivalently as the surface gravity near the horizon. The temperature of thesmaller black hole is relatively larger, so there is heat flow from the smaller to the larger horizon. The quantity whichis the temperature of the black hole is hard to define, because there is no asymptotically flat space to measure itrelative to.

References[1] http:/ / arxiv. org/ abs/ hep-th/ 0205177v1

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De SitterSchwarzschild metricIntroduction Metric Horizon properties Nariai solution Hawking temperature References

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