Maulik K. Parikh

A SECRET TUNNEL THROUGH THE HORIZON

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.

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.

A First-Quantized Formalism for Cosmological Particle Production

Given suitable boundary conditions, we show that the initial state and the amount of particle production in a cosmological spacetime are encoded in the Feynman propagator. The propagator can be represented in terms of a particle path integral in an auxiliary spacetime, and particle production can be extracted from the auxiliary propagator. This provides a first-quantized formalism for computing cosmological particle production which, unlike conventional Bogolubov transformations, may be amenable to a string-theoretic generalization.

A Secret tunnel through the horizon

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.

Energy Conservation and Hawking Radiation

The conservation of energy implies that an isolated radiating black hole cannot have an emission spectrum that is precisely thermal. Moreover, the no-hair theorem is only approximately applicable. We consider the implications for the black hole information puzzle.

Elliptic de Sitter space: dS/Z(2)

We propose that for every event in de Sitter space, there is a CPT-conjugate event at its antipode. Such an “elliptic” Z2-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.

Volume of black holes

We propose a simple definition of volume for stationary spacetimes. The proposed volume is constant in time, independent of the choice of stationary time slicing, and applies even in the absence of a globally timelike Killing vector. We then consider whether it is possible to construct spacetimes that have finite horizon area but infinite volume, by letting the radius go to infinity while making discrete identifications to preserve the horizon area. We show that, in three or four dimensions, no such solutions exist that are not inconsistent in some way. This may constrain the statistical interpretation of the Bekenstein-Hawking entropy.

Universal correction to the inflationary vacuum

The Bunch-Davies state appears precisely thermal to a free-falling observer in de Sitter space. However, precise thermality is unphysical because it violates energy conservation. Instead, the true spectrum must take a certain different form, with the Boltzmann factor exp (−βωk) replaced by exp (ΔS), where S is the entropy of the de Sitter horizon. The deviation from precise thermality can be regarded as an explicitly calculable correction to the Bunch-Davies state. This correction is mandatory in that it relies only on energy conservation. The modified Bunch-Davies state leads, in turn, to an (H/Mp)2 modification of the primordial power spectrum of inflationary perturbations, which we determine.

De sitter space with finitely many states: A Toy story

The finite entropy of de Sitter space suggests that in a theory of quantum gravity there are only finitely many states. It has been argued that in this case there is no action of the de Sitter group consistent with unitarity. In this note we propose a way out of this if we give up the requirement of having a hermitian Hamiltonian. We argue that some of the generators of the de Sitter group act in a novel way, namely by mixing inand out-states. In this way it is possible to have a unitary S-matrix that is finite-dimensional and, moreover, de Sitter-invariant. Using Dirac spinors, we construct a simple toy model that exhibits these features.

Two roads to the null energy condition

The null energy condition has sweeping consequences in general relativity. I argue here that it has been misunderstood as a property exclusively of matter, when in fact it arises only in a theory of both matter and gravity. I then derive an equivalent geometric formulation of the null energy condition from worldsheet string theory, where it arises beautifully as simply Einsteins equations in two dimensions. But further, I show that this condition also has a thermodynamic origin, following from a local version of the second law of thermodynamics, applied to gravitational entropy. Thus, far from being an incidental property of matter, the validity of the null energy condition hints at the deep dual origins of gravity.