James W. York

Conformally invariant orthogonal decomposition of symmetric tensors on Riemannian manifolds and the initial‐value problem of general relativity

It is shown that an arbitrary symmetric tensor ψab (or ψab) of any weight can be covariantly decomposed on a Riemannian manifold (M,g) into a unique sum of transverse‐traceless, longitudinal, and pure trace parts. The summands involve only linear operators and are mutually orthogonal in the global scalar product on (M,g). Each summand transforms separately into itself if the decomposition is carried out properly in a conformally related space (M,g). The decomposition is therefore determined by a conformal equivalence class of Riemannian manifolds. This property makes the decomposition ideally suited to the initial‐value problem of general relativity, which becomes, as a result, a well‐defined system of elliptic equations. Three of the four initial‐value equations are linear and determine the decomposition of a symmetric tensor. The fourth equation is quasilinear and determines the conformal factor. The decomposition applied to the space of symmetric tensors on (M,g) can be written in terms of a direct sum...

Path integral over black hole fluctuations

Evaluating a functional integral exactly over a subset of metrics that represent the quantum fluctuations of the horizon of a black hole, we obtain a Schroedinger equation in null coordinate time for the key component of the metric. The equation yields a current that preserves probability if we use the most natural choice of functional measure. This establishes the existence of blurred horizons and a thermal atmosphere. It has been argued previously that the existence of a thermal atmosphere is a direct concomitant of the thermal radiation of black holes when the temperature of the hole is greater than that of its larger environment, which we take as zero.

Unified Form of the Initial Value Conditions

In this paper both the initial value problem and the conformal thin sandwich problem are written in a unified way.