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

A remark on kinks and time machines

We describe an elementary proof that a manifold with the topology of the Politzer time machine does not admit a nonsingular, asymptotically flat Lorentz metric.Comment: 4 page

Branes as BIons

A BIon may be defined as a finite energy solution of a non-linear field theory with distributional sources. By contrast a soliton is usually defined to have no sources. I show how harmonic coordinates map the exteriors of the topologically and causally non-trivial spacetimes of extreme p-branes to BIonic solutions of the Einstein equations in a topologically trivial spacetime in which the combined gravitational and matter energy momentum is located on distributional sources. As a consequence the tension of BPS p-branes is classically unrenormalized. The result holds equally for spacetimes with singularities and for those, like the M-5-brane, which are everywhere singularity free.Comment: Latex, 9 pages, no figure

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

Eisenhart lifts and symmetries of time-dependent systems

Certain dissipative systems, such as Caldirola and Kannai's damped simple harmonic oscillator, may be modelled by time-dependent Lagrangian and hence time dependent Hamiltonian systems with $n$ degrees of freedom. In this paper we treat these systems, their projective and conformal symmetries as well as their quantisation from the point of view of the Eisenhart lift to a Bargmann spacetime in $n+2$ dimensions, equipped with its covariantly constant null Killing vector field. Reparametrization of the time variable corresponds to conformal rescalings of the Bargmann metric. We show how the Arnold map lifts to Bargmann spacetime. We contrast the greater generality of the Caldirola-Kannai approach with that of Arnold and Bateman. At the level of quantum mechanics, we are able to show how the relevant Schr\"odinger equation emerges naturally using the techniques of quantum field theory in curved spacetimes, since a covariantly constant null Killing vector field gives rise to well defined one particle Hilbert space. Time-dependent Lagrangians arise naturally also in cosmology and give rise to the phenomenon of Hubble friction. We provide an account of this for Friedmann-Lemaitre and Bianchi cosmologies and how it fits in with our previous discussion in the non-relativistic limit.Comment: 34 pages, no figures. Minor corrections, some references adde

The light curve of the companion to PSR B1957+20

We present a new analysis of the light curve for the secondary star in the eclipsing binary millisecond pulsar system PSR B1957+20. Combining previous data and new data points at minimum from the Hubble Space Telescope, we have 100% coverage in the R-band. We also have a number of new K_s-band data points, which we use to constrain the infrared magnitude of the system. We model this with the Eclipsing Light Curve code (ELC). From the modelling with the ELC code we obtain colour information about the secondary at minimum light in BVRI and K. For our best fit model we are able to constrain the system inclination to 65 +/- 2 degrees for pulsar masses ranging from 1.3 -- 1.9 M_sun. The pulsar mass is unconstrained. We also find that the secondary star is not filling its Roche lobe. The temperature of the un-irradiated side of the companion is in agreement with previous estimates and we find that the observed temperature gradient across the secondary star is physically sustainable.Comment: 6 pages, 4 figures & 3tables. Accepted for publication in MNRA