K-causality and degenerate spacetimes

The causal relation $K^+$ was introduced by Sorkin and Woolgar to extend the standard causal analysis of $C^2$ spacetimes to those that are only $C^0$. Most of their results also hold true in the case of spacetimes with degeneracies. In this paper we seek to examine $K^+$ explicitly in the case of Lorentzian topology changing Morse spacetimes containing isolated degeneracies. We first demonstrate some interesting features of this relation in globally Lorentzian spacetimes. In particular, we show that $K^+$ is robust and that it coincides with the Seifert relation when the spacetime is stably causal. Moreover, the Hawking and Sachs characterisation of causal continuity translates into a natural expression in terms of $K^+$ for general spacetimes. We then examine $K^+$ in topology changing Morse spacetimes both with and without the degeneracies and find further characterisations of causal continuity.Comment: Latex, 23 pages, 4 figure

Causality in Time-Neutral Cosmologies

Gell-Mann and Hartle (GMH) have recently considered time-neutral cosmological models in which the initial and final conditions are independently specified, and several authors have investigated experimental tests of such models. We point out here that GMH time-neutral models can allow superluminal signalling, in the sense that it can be possible for observers in those cosmologies, by detecting and exploiting regularities in the final state, to construct devices which send and receive signals between space-like separated points. In suitable cosmologies, any single superluminal message can be transmitted with probability arbitrarily close to one by the use of redundant signals. However, the outcome probabilities of quantum measurements generally depend on precisely which past {\it and future} measurements take place. As the transmission of any signal relies on quantum measurements, its transmission probability is similarly context-dependent. As a result, the standard superluminal signalling paradoxes do not apply. Despite their unusual features, the models are internally consistent. These results illustrate an interesting conceptual point. The standard view of Minkowski causality is not an absolutely indispensable part of the mathematical formalism of relativistic quantum theory. It is contingent on the empirical observation that naturally occurring ensembles can be naturally pre-selected but not post-selected.Comment: 5 pages, RevTeX. Published version -- minor typos correcte

Determinants on lens spaces and cyclotomic units

The Laplacian functional determinants for conformal scalars and coexact one-forms are evaluated in closed form on inhomogeneous lens spaces of certain orders, including all odd primes when the essential part of the expression is given, formally as a cyclotomic unitComment: 18 pages, 1 figur

Morse index and causal continuity. A criterion for topology change in quantum gravity

Studies in 1+1 dimensions suggest that causally discontinuous topology changing spacetimes are suppressed in quantum gravity. Borde and Sorkin have conjectured that causal discontinuities are associated precisely with index 1 or n-1 Morse points in topology changing spacetimes built from Morse functions. We establish a weaker form of this conjecture. Namely, if a Morse function f on a compact cobordism has critical points of index 1 or n-1, then all the Morse geometries associated with f are causally discontinuous, while if f has no critical points of index 1 or n-1, then there exist associated Morse geometries which are causally continuous.Comment: Latex, 20 pages, 3 figure

Causal continuity in degenerate spacetimes

A change of spatial topology in a causal, compact spacetime cannot occur when the metric is globally Lorentzian. One can however construct a causal metric from a Riemannian metric and a Morse function on the background cobordism manifold, which is Lorentzian almost everywhere except that it is degenerate at each critical point of the function. We investigate causal structure in the neighbourhood of such a degeneracy, when the auxiliary Riemannian metric is taken to be Cartesian flat in appropriate coordinates. For these geometries, we verify Borde and Sorkin's conjecture that causal discontinuity occurs if and only if the Morse index is 1 or n-1.Comment: 34 pages, 11 figures, Latex2e, important references added, introduction and discussions sections reworded slightl

Quantum Dynamics without the Wave Function

When suitably generalized and interpreted, the path-integral offers an alternative to the more familiar quantal formalism based on state-vectors, selfadjoint operators, and external observers. Mathematically one generalizes the path-integral-as-propagator to a {\it quantal measure} $\mu$ on the space $\Omega$ of all ``conceivable worlds'', and this generalized measure expresses the dynamics or law of motion of the theory, much as Wiener measure expresses the dynamics of Brownian motion. Within such ``histories-based'' schemes new, and more ``realistic'' possibilities open up for resolving the philosophical problems of the state-vector formalism. In particular, one can dispense with the need for external agents by locating the predictive content of $\mu$ in its sets of measure zero: such sets are to be ``precluded''. But unrestricted application of this rule engenders contradictions. One possible response would remove the contradictions by circumscribing the application of the preclusion concept. Another response, more in the tradition of ``quantum logic'', would accommodate the contradictions by dualizing $\Omega$ to a space of ``co-events'' and effectively identifying reality with an element of this dual space.Comment: plainTeX, 24 pages, no figures. To appear in a special volume of {\it Journal of Physics A: Mathematical and General} entitled ``The Quantum Universe'' and dedicated to Giancarlo Ghirardi on the occasion of his 70th birthday. Most current version is available at http://www.physics.syr.edu/~sorkin/some.papers/ (or wherever my home-page may be

Topology Change and Causal Continuity

The result that, for a scalar quantum field propagating on a ``trousers'' topology in 1+1 dimensions, the crotch singularity is a source for an infinite burst of energy has been used to argue against the occurrence of topology change in quantum gravity. We draw attention to a conjecture due to Sorkin that it may be the particular type of topology change involved in the trousers transition that is problematic and that other topology changes may not cause the same difficulties. The conjecture links the singular behaviour to the existence of ``causal discontinuities'' in the spacetime and relies on a classification of topology changes using Morse theory. We investigate various topology changing transitions, including the pair production of black holes and of topological geons, in the light of these ideas.Comment: Latex, 28 pages, 10 figures, small changes in text (one figure removed), conclusions remain unchanged. Accepted for publication in Physical Review

Dualities, Twists, and Gauge Theories with Non-Constant Non-Commutativity

We study the world volume theory of D3-branes wrapping the Melvin universe supported by background NSNS B-field. In the appropriate decoupling limit, the open string dynamics is that of non-commutative guage field theory with non-constant non-commutativity. We identify this model as a simple Melvin twist of flat D3 branes. Along similar lines, one recognizes the model of Hashimoto and Sethi as being the Melvin null twist, and the model of Dolan and Nappi as being the null Melvin twist, of the flat D3-brane. This construction therefore offers a unified perspective on most of the known explicit constructions of non-commutative gauge theories as a decoupled theory of D-branes in a B-field background. We also describe the world volume theory on the D3-brane in Melvin universe which is decaying via the nucleation of monopole anti-monopole pair.Comment: 18 pages, 1 figure, References added, typo correcte

Many accelerating black holes

We show how the Weyl formalism allows metrics to be written down which correspond to arbitrary numbers of collinear accelerating neutral black holes in 3+1 dimensions. The black holes have arbitrary masses and different accelerations and share a common acceleration horizon. In the general case, the black holes are joined by cosmic strings or struts that provide the necessary forces that, together with the inter black hole gravitational attractions, produce the acceleration. In the cases of two and three black holes, the parameters may be chosen so that the outermost black hole is pulled along by a cosmic string and the inner black holes follow behind accelerated purely by gravitational forces. We conjecture that similar solutions exist for any number of black holes.Comment: 12 pages, LaTe