New Coordinates for de Sitter Space and de Sitter Radiation

We introduce a simple coordinate system covering half of de Sitter space. The new coordinates have several attractive properties: the time direction is a Killing vector, the metric is smooth at the horizon, and constant-time slices are just flat Euclidean space. We demonstrate the usefulness of the coordinates by calculating the rate at which particles tunnel across the horizon. When self-gravitation is taken into account, the resulting tunneling rate is only approximately thermal. The effective temperature decreases through the emission of radiation. 1 e-mail: [email protected] 1 Motivation At the heart of Einstein’s theory of gravity is local diffeomorphism invariance: coordinate systems are unimportant, only diffeomorphism invariants matter. However, a poor choice of local coordinates can sometimes obscure the nature of the global aspects of spacetime, such as horizons or causal boundaries. Indeed, the true nature of the “coordinate singularity” at the Schwarzschild radius eluded Einstein himself, and was only fully illuminated with the discovery of coordinate systems that were regular at the horizon. Coordinate systems that cover larger patches of spacetime are especially useful to have in dealing with physical phenomena that are in some sense nonlocalized. In this note, we present a simple new coordinate system for de Sitter space, covering the causal future/past of an observer. The new coordinates, which we shall call Painlevéde Sitter coordinates, are a cross between static coordinates and planar coordinates, and inherit the strengths of each of these. For example, like static coordinates but unlike planar coordinates, the new coordinates have a direction of time that is a Killing vector, making them well-adapted to thermodynamics. On the other hand, like planar coordinates, but unlike static coordinates, Painlevé-de Sitter coordinates continue smoothly through the horizon, and constant time slices are just flat Euclidean space. Painlevé-de Sitter coordinates differ also from Eddington-Finkelstein type coordinates in that the coordinates are all either timelike or spacelike, rather than null. The combination of a Killing time direction and regularity at the horizon is particularly powerful as it allows one to study across-horizon physics as seen by an observer. A natural application is de Sitter radiation. Heuristically, one envisions de Sitter radiance as arising in much the same way that Hawking radiation does. That is to say, a particle-pair forms just inside the horizon, one member of the pair tunnels across the horizon, and the virtual pair becomes real. To show that this is actually what happens, one would like to compute the amplitude for traversing the horizon; because of their regularity, Painlevé-de Sitter coordinates make the calculation feasible. By directly evaluating the imaginary part of the action, we obtain the emission amplitude associated with a tunneling particle. In the s-wave limit, it is in fact possible to extend the computation to include the effects of self-gravitation. As a result, the de Sitter spectrum turns out to be only approximately thermal. In particular it has a cutoff at high energies. The back-reaction is such that, unlike Schwarzschild black holes, de Sitter space lowers its temperature as it radiates. 2 Painlevé-de Sitter Coordinates Many different coordinate systems are known for de Sitter space (see, e.g., [1, 2]). One commonly used metric is the static metric, analogous to the familiar Schwarzschild metric for an uncharged black hole. This metric covers a static patch, that part of de Sitter space that an observer at the origin can interact with. The time coordinate, ts, corresponds to a timelike Killing vector, which makes it suitable for thermodynamics; thermal equilibrium requires among other things that the spatial metric be at equilibrium. However, the static metric has the limitation that it is only valid upto the horizon; it therefore covers only a very small region of the full space. Another drawback is that in a static background one cannot