We present a short and direct derivation of Hawking radiation as a tunneling process, based on particles in a dynamical geometry. The imaginary part of the action for the classically forbidden process is related to the Boltzmann factor for emission at the Hawking temperature. Because the derivation respects conservation laws, the exact spectrum is no longer precisely thermal.

Hawking radiation is often intuitively visualized as particles that have tunneled across the horizon. Yet, at first sight, it is not apparent where the barrier is.Here I show that the barrier depends on the tunneling particle itself. The key is to implement energy conservation, so that the black hole contracts during the process of radiation. A direct consequence is that the radiation spectrum cannot be strictly thermal. The correction to the thermal spectrum is of precisely the form that one would expect from an underlying unitary quantum theory. This may have profound implications for the black hole information puzzle.Classically, a black hole is the ultimate prison: anything that enters is doomed;there is no escape. Moreover, since nothing can ever come out, a classical black hole can only grow bigger with time. Thus it came as a huge shock to physicists when Stephen Hawking demonstrated that, quantum mechanically, black holes could actually radiate particles. With the emission of Hawking radiation, black holes could lose energy, shrink, and eventually evaporate completely.How does this happen? When an object that is classically stable becomes quantum-mechanically unstable, it is natural to suspect tunneling. Indeed, whenHawking first proved the existence of black hole radiation [1], he described it as tunneling triggered by vacuum fluctations near the horizon. The idea is that when a virtual particle pair is created just inside the horizon, the positive energy virtual particle can tunnel out -no classical escape route exists -where it materializes as a real particle. Alternatively, for a pair created just outside the horizon, the negative energy virtual particle, which is forbidden outside, can tunnel inwards. In either case, the negative energy particle is absorbed by the black hole, resulting in a decrease in the mass of the black hole, while the positive energy particle escapes to infinity, appearing as Hawking radiation.This heuristic picture has obvious visual and intuitive appeal. But, oddly, actual derivations of Hawking radiation did not proceed in this way at all [1, 2].There were two apparent hurdles. The first was technical: in order to do a tunneling computation one needed to have a coordinate system that was wellbehaved at the horizon; none of the well-known coordinate systems were suitable. The second hurdle was conceptual: there didn't seem to be any barrier!

The membrane paradigm is the remarkable view that, to an external observer, a black hole appears to behave exactly like a dynamical fluid membrane, obeying such pre-relativistic equations as Ohm's law and the Navier-Stokes equation. It has traditionally been derived by manipulating the equations of motion. Here we provide an action formulation of this picture, clarifying what underlies the paradigm, and simplifying the derivations. Within this framework, we derive previous membrane results, and extend them to dyonic black hole solutions. We discuss how it is that an action can produce dissipative equations. Using a Euclidean path integral, we show that familiar semi-classical thermodynamic properties of black holes also emerge from the membrane action. Finally, in a Hamiltonian description, we establish the validity of a minimum entropy production principle for black holes.

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.

We propose that for every event in de Sitter space, there is a CPT-conjugate event at its antipode. Such an "elliptic" Z 2 -identification of de Sitter space provides a concrete realization of observer complementarity: every observer has complete information. It is possible to define the analog of an S-matrix for quantum gravity in elliptic de Sitter space that is measurable by all observers. In a holographic description, S-matrix elements may be represented by correlation functions of a dual (conformal field) theory that lives on the single boundary sphere. S-matrix elements are de Sitter-invariant, but have different interpretations for different observers. We argue that Hilbert states do not necessarily form representations of the full de Sitter group, but just of the subgroup of rotations. As a result, the Hilbert space can be finite-dimensional and still have positive norm. We also discuss the elliptic interpretation of de Sitter space in the context of type IIB* string theory.