Gravitational Instantons, Confocal Quadrics and Separability of the Schr\"odinger and Hamilton-Jacobi equations

A hyperk\"ahler 4-metric with a triholomorphic SU(2) action gives rise to a family of confocal quadrics in Euclidean 3-space when cast in the canonical form of a hyperk\"ahler 4-metric metric with a triholomorphic circle action. Moreover, at least in the case of geodesics orthogonal to the U(1) fibres, both the covariant Schr\"odinger and the Hamilton-Jacobi equation is separable and the system integrable.Comment: 10 pages Late

Bohm and Einstein-Sasaki Metrics, Black Holes and Cosmological Event Horizons

We study physical applications of the Bohm metrics, which are infinite sequences of inhomogeneous Einstein metrics on spheres and products of spheres of dimension 5 <= d <= 9. We prove that all the Bohm metrics on S^3 x S^2 and S^3 x S^3 have negative eigenvalue modes of the Lichnerowicz operator and by numerical methods we establish that Bohm metrics on S^5 have negative eigenvalues too. We argue that all the Bohm metrics will have negative modes. These results imply that higher-dimensional black-hole spacetimes where the Bohm metric replaces the usual round sphere metric are classically unstable. We also show that the stability criterion for Freund-Rubin solutions is the same as for black-hole stability, and hence such solutions using Bohm metrics will also be unstable. We consider possible endpoints of the instabilities, and show that all Einstein-Sasaki manifolds give stable solutions. We show how Wick rotation of Bohm metrics gives spacetimes that provide counterexamples to a strict form of the Cosmic Baldness conjecture, but they are still consistent with the intuition behind the cosmic No-Hair conjectures. We show how the Lorentzian metrics may be created ``from nothing'' in a no-boundary setting. We argue that Lorentzian Bohm metrics are unstable to decay to de Sitter spacetime. We also argue that noncompact versions of the Bohm metrics have infinitely many negative Lichernowicz modes, and we conjecture a general relation between Lichnerowicz eigenvalues and non-uniqueness of the Dirichlet problem for Einstein's equations.Comment: 53 pages, 11 figure

Generalized Taub-NUT metrics and Killing-Yano tensors

A necessary condition that a St\"ackel-Killing tensor of valence 2 be the contracted product of a Killing-Yano tensor of valence 2 with itself is re-derived for a Riemannian manifold. This condition is applied to the generalized Euclidean Taub-NUT metrics which admit a Kepler type symmetry. It is shown that in general the St\"ackel-Killing tensors involved in the Runge-Lenz vector cannot be expressed as a product of Killing-Yano tensors. The only exception is the original Taub-NUT metric.Comment: 14 pages, LaTeX. Final version to appear in J.Phys.A:Math.Ge

Non-Abelian pp-waves in D=4 supergravity theories

The non-Abelian plane waves, first found in flat spacetime by Coleman and subsequently generalized to give pp-waves in Einstein-Yang-Mills theory, are shown to be 1/2 supersymmetric solutions of a wide variety of N=1 supergravity theories coupled to scalar and vector multiplets, including the theory of SU(2) Yang-Mills coupled to an axion \sigma and dilaton \phi recently obtained as the reduction to four-dimensions of the six-dimensional Salam-Sezgin model. In this latter case they provide the most general supersymmetric solution. Passing to the Riemannian formulation of this theory we show that the most general supersymmetric solution may be constructed starting from a self-dual Yang-Mills connection on a self-dual metric and solving a Poisson equation for e^\phi. We also present the generalization of these solutions to non-Abelian AdS pp-waves which allow a negative cosmological constant and preserve 1/4 of supersymmetry.Comment: Latex, 1+12 page

Brane Worlds in Collision

We obtain an exact solution of the supergravity equations of motion in which the four-dimensional observed universe is one of a number of colliding D3-branes in a Calabi-Yau background. The collision results in the ten-dimensional spacetime splitting into disconnected regions, bounded by curvature singularities. However, near the D3-branes the metric remains static during and after the collision. We also obtain a general class of solutions representing $p$-brane collisions in arbitrary dimensions, including one in which the universe ends with the mutual annihilation of a positive-tension and negative-tension 3-brane.Comment: RevTex, 4 pages, 1 figure, typos and minor errors correcte

Statistical Mechanics of Charged Particles in Einstein-Maxwell-Scalar Theory

We consider an $N$-body system of charged particle coupled to gravitational, electromagnetic, and scalar fields. The metric on moduli space for the system can be considered if a relation among the charges and mass is satisfied, which includes the BPS relation for monopoles and the extreme condition for charged black holes. Using the metric on moduli space in the long distance approximation, we study the statistical mechanics of the charged particles at low velocities. The partition function is evaluated as the leading order of the large $d$ expansion, where $d$ is the spatial dimension of the system and will be substituted finally as $d=3$.Comment: 11 pages, RevTeX3.

Killing vectors in asymptotically flat space-times: I. Asymptotically translational Killing vectors and the rigid positive energy theorem

We study Killing vector fields in asymptotically flat space-times. We prove the following result, implicitly assumed in the uniqueness theory of stationary black holes. If the conditions of the rigidity part of the positive energy theorem are met, then in such space-times there are no asymptotically null Killing vector fields except if the initial data set can be embedded in Minkowski space-time. We also give a proof of the non-existence of non-singular (in an appropriate sense) asymptotically flat space-times which satisfy an energy condition and which have a null ADM four-momentum, under conditions weaker than previously considered.Comment: 30 page

Rotating Black Holes which Saturate a Bogomol'nyi Bound

We construct and study the electrically charged, rotating black hole solution in heterotic string theory compactified on a $(10-D)$ dimensional torus. This black hole is characterized by its mass, angular momentum, and a $(36-2D)$ dimensional electric charge vector. One of the novel features of this solution is that for $D >5$, its extremal limit saturates the Bogomol'nyi bound. This is in contrast with the $D=4$ case where the rotating black hole solution develops a naked singularity before the Bogomol'nyi bound is reached. The extremal black holes can be superposed, and by taking a periodic array in $D>5$, one obtains effectively four dimensional solutions without naked singularities.Comment: 13 pages, no figure

The Action of Instantons with Nut Charge

We examine the effect of a non-trivial nut charge on the action of non-compact four-dimensional instantons with a U(1) isometry. If the instanton action is calculated by dimensionally reducing along the isometry, then the nut charge is found to make an explicit non-zero contribution. For metrics satisfying AF, ALF or ALE boundary conditions, the action can be expressed entirely in terms of quantities (including the nut charge) defined on the fixed point set of the isometry. A source (or sink) of nut charge also implies the presence of a Misner string coordinate singularity, which will have an important effect on the Hamiltonian of the instanton.Comment: 25 page

General Very Special Relativity is Finsler Geometry

We ask whether Cohen and Glashow's Very Special Relativity model for Lorentz violation might be modified, perhaps by quantum corrections, possibly producing a curved spacetime with a cosmological constant. We show that its symmetry group ISIM(2) does admit a 2-parameter family of continuous deformations, but none of these give rise to non-commutative translations analogous to those of the de Sitter deformation of the Poincar\'e group: spacetime remains flat. Only a 1-parameter family DISIM_b(2) of deformations of SIM(2) is physically acceptable. Since this could arise through quantum corrections, its implications for tests of Lorentz violations via the Cohen-Glashow proposal should be taken into account. The Lorentz-violating point particle action invariant under DISIM_b(2) is of Finsler type, for which the line element is homogeneous of degree 1 in displacements, but anisotropic. We derive DISIM_b(2)-invariant wave equations for particles of spins 0, 1/2 and 1. The experimental bound, $|b|<10^{-26}$, raises the question ``Why is the dimensionless constant $b$ so small in Very Special Relativity?''Comment: 4 pages, minor corrections, references adde