Tony Rothman

Holonomy in the Schwarzschild-Droste geometry

Parallel transport of vectors in curved spacetimes generally results in a deficit angle between the directions of the initial and final vectors. We examine such a holonomy in the Schwarzschild-Droste geometry and find a number of interesting features that are not widely known. For example, parallel transport around circular orbits results in a quantized band structure of holonomy invariance. We also examine radial holonomy and extend the analysis to spinors and to the Reissner-Nordstrom metric, where we find qualitatively different behaviour for the extremal (Q = M) case. Our calculations provide a toolbox that will hopefully be useful in the investigation of quantum parallel transport in Hilbert-fibred spacetimes.

Hidden variables or positive probabilities

Despite claims that Bells inequalities are based on the Einstein locality condition, or equivalent, all derivations make an identical mathematical assumption that local hidden-variable theories produce a set of positive-definite probabilities for detecting a particle with a given spin orientation. The standard argument is that because quantum mechanics assumes that particles are emitted in a superposition of states the theory cannot produce such a set of probabilites. We examine a paper by Eberhard, and several similar papers, which claim to show that a generalized Bell inequality, the CHSH inequality, can be derived solely on the basis of the locality condition, without recourse to hidden variables. We point out that these authors nonetheless assumes a set of positive-definite probabilities, which supports the claim that hidden variables or “locality” is not at issue here, positive-definite probabilities are. We demonstrate that quantum mechanics does predict a set of probabilities that violate the CHSH inequality; however these probabilities are not positive-definite. Nevertheless, they are physically meaningful in that they give the usual quantum-mechanical predictions in physical situations. We discuss in what sense our results are related to the Wigner distribution.

A New Interpretation of Bell's Inequalities

Bells inequalities are always derived assuming that local hidden-variable theories give a set of positive-definite probabilities for detecting a particle with a given spin orientation. The usual claim is that quantum mechanics, by its very nature, cannot produce a set of such probabilities. We show that this is not the case if one allows for generalized (nonpositive-definite) “master probability distributions.” The master distributions give the usual quantum mechanical violation of Bells inequalities. Consequences for the interpretation of quantum mechanics are discussed.

EXTREMAL BLACK HOLES AND THE LIMITS OF THE THIRD LAW

Recent results of quantum field theory on a curved spacetime suggest that extremal black holes are not thermal objects and that the notion of zero temperature is ill-defined for them. If this is correct, one may have to go to a full semiclassical theory of gravity, including backreaction, in order to make sense of the third law of black hole thermodynamics. Alternatively it is possible that we shall have to drastically revise the status of extremality in black hole thermodynamics.

Upper limits on micro-mini black holes

The equations determining cosmic nucleosynthesis up to4He are modified to take account of black-hole evaporation during nucleosynthesis. Numerical calculation shows that deuterium is enhanced by the presence of these micro-mini black holes (μmBH); but very high-density universes are difficult to reconcile with observations, even with a very large abundance of μmBH.

Cosmic nucleosynthesis and nonlinear inhomogeneities

Abstract We present a number of numerical simulations of the synthesis of the light elements in an inhomogeneous big-bang cosmology. A completely relativistic, nonlinear plane-symmetric hydrodynamics and general relativity code provides the geometrical evolution, within which the nucleosynthesis occurs. We argue that our plane symmetry provides realistic modelling of the evolution during this epoch, and that local enhancements or depressions of the element abundances may persist to the present. In any case, the average abundances we find are not identical to the predictions of homogeneous cosmologies. Local abundances fluctuating in the range 22%−22% are typically produced in our models. These models are consistent with all observational constraints, in particular with the observed isotropy of the microwave radiation background.

A Phase Space Approach to Gravitational Entropy

We examine the definition S=lnΩ as a candidate function for “gravitational entropy.” We calculate its behavior for gravitational and density perturbations in closed, open and flat cosmological models and find that in all cases it increases monotonically. We are also able to calculate the entropy density of gravitational radiation produced by inflation. We compare the results with the behavior of the Weyl-tensor squared. Applying the formalism to black holes has proved more problematical.