In general relativity, the de Sitter–Schwarzschild solution describes a black hole in a causal patch of de Sitter space. 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 be mapped to each other by a discrete reflection symmetry in any causal patch. Introduction In general relativity, space-times can have black hole event horizons and also cosmological horizons. The de Sitter–Schwarzschild solution is the simplest solution which has both. Metric The metric of any spherically symmetric solution in Schwarzschild form is: The vacuum Einstein equations give a linear equation for ƒ(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 magnitude 3b. Superposing the two solutions gives the de Sitter–Schwarzschild solution: The two parameters a and b give the black hole mass and the cosmological constant respectively. In d + 1 dimensions, the inverse power law falloff in the black hole part is d − 2. In 2 + 1 dimensions, where the exponent is zero, the analogous solution starts with 2 + 1 de Sitter space, cuts out a wedge, and pastes the two sides of the wedge together to make a conical space. The geodesic equation gives for the radial, and for the time component. Horizon properties The de Sitter space is the simplest solution of Einstein's equation with a positive cosmological constant. It is spherically symmetric and it has a cosmological horizon surrounding any observer, and describes an inflating universe. The Schwarzschild solution is the simplest spherically symmetric solution of the Einstein equations with zero cosmological constant, and it describes a black hole event horizon in otherwise empty space. The de Sitter–Schwarzschild space-time is a combination of the two, and describes a black hole horizon spherically centered in an otherwise de Sitter universe. An observer which hasn't fallen into the black hole, and which can still see the black hole 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 fundamentally the same. Classically the two types of horizon look different. A