No abstract

The authors prove boundedness on an exterior Schwarzschild wedge for Cinfinity solutions of the covariant Klein-Gordon equation which have compact support on Cauchy surfaces in Kruskal spacetime. Previously used methods enable such boundedness to be proven only for solutions whose initial data satisfy the additional restriction of vanishing at the bifurcation 2-sphere of the horizon. By employing a rarely considered discrete isometry of Kruskal spacetime and the causal propagation property of the equation, they remove this restriction. This also enables them to prove boundedness exterior to the horizon of a spacetime representing the collapse to a black hole of a spherically symmetric compact star for solutions of the same equation having Cinfinity initial data on a Cauchy surface drawn prior to the collapse.

We give mathematically rigorous results on the quantization of the covariant Klein Gordon field with an external stationary scalar interaction in a stationary curved space-time.We show how, following Segal, Weinless etc., the problem reduces to finding a "one particle structure" for the corresponding classical system.Our main result is an existence theorem for such a one-particle structure for a precisely specified class of stationary space-times. Byproducts of our approach are : 1) A discussion of when a given "equal-time" surface in a given stationary space-time is Cauchy.2) A modification and extension of the methods of Chernoff [3] for proving the essential self-adjointness of certain partial differential operators.

A theory recently proposed by the author aims to explain decoherence and the thermodynamical behaviour of closed systems within a conservative, unitary, framework for quantum gravity by assuming that the operators tied to the gravitational degrees of freedom are unobservable and equating physical entropy with matter-gravity entanglement entropy. Here we obtain preliminary results on the extent of decoherence this theory predicts. We treat first a static state which, if one were to ignore quantum gravitational effects, would be a quantum superposition of two spatially displaced states of a single classically well describable ball of uniform mass density in empty space. Estimating the quantum gravitational effects on this system within a simple Newtonian approximation, we obtain formulae which predict e.g. that as long as the mass of the ball is considerably larger than the Planck mass, such a would-be-coherent static superposition will actually be decohered whenever the separation of the centres of mass of the two ball-states excedes a small fraction (which decreases as the mass of the ball increases) of the ball radius. We then obtain a formula for the quantum gravitational correction to the would-be-pure density matrix of a nonrelativistic many-body Schrödinger wave function and argue that this formula predicts decoherence between configurations which differ (at least) in the 'relocation' of a cluster of particles of Planck mass. We estimate the entropy of some simple model closed systems, finding a tendency for it to increase with 'matter-clumping' suggestive of a link with existing phenomenological discussions of cosmological entropy increase.In [1] a theory was proposed for the origin both of decoherence and of thermodynamics in which quantum gravity plays a fundamental role. The starting point is the following, conservative, set of assumptions: Any closed quantum gravitational system is described by a total Hilbert space H total which takes the form H total = H matter ⊗ H gravity and any full description of the system is given, at all times, by a pure density operator