Paul Tod

Spherically-symmetric solutions of the Schrödinger-Newton equations

As part of a programme in which quantum state reduction is understood as a gravitational phenomenon, we consider the Schrodinger-Newton equations. For a single particle, this is a coupled system consisting of the Schrodinger equation for the particle moving in its own gravitational field, where this is generated by its own probability density via the Poisson equation. Restricting to the spherically-symmetric case, we find numerical evidence for a discrete family of solutions, everywhere regular, and with normalizable wavefunctions. The solutions are labelled by the non-negative integers, the nth solution having n zeros in the wavefunction. Furthermore, these are the only globally defined solutions. Analytical support is provided for some of the features found numerically.

Einstein–Weyl geometry, the dKP equation and twistor theory

Abstract It is shown that Einstein–Weyl (EW) equations in 2+1 dimensions contain the dispersionless Kadomtsev–Petviashvili (dKP) equation as a special case: if an EW structure admits a constant-weighted vector then it is locally given by h= d y 2 −4 d x d t−4u d t 2 , ν=−4u x d t , where u=u(x,y,t) satisfies the dKP equation (ut−uux)x=uyy. Linearised solutions to the dKP equation are shown to give rise to four-dimensional anti-self-dual conformal structures with symmetries. All four-dimensional hyper-Kahler metrics in signature (++−−) for which the self-dual part of the derivative of a Killing vector is null arise by this construction. Two new classes of examples of EW metrics which depend on one arbitrary function of one variable are given, and characterised. A Lax representation of the EW condition is found and used to show that all EW spaces arise as symmetry reductions of hyper-Hermitian metrics in four dimensions. The EW equations are reformulated in terms of a simple and closed two-form on the CP 1 -bundle over a Weyl space. It is proved that complex solutions to the dKP equations, modulo a certain coordinate freedom, are in a one-to-one correspondence with mini-twistor spaces (two-dimensional complex manifolds Z containing a rational curve with normal bundle O (2) ) that admit a section of κ−1/4, where κ is the canonical bundle of Z . Real solutions are obtained if the mini-twistor space also admits an anti-holomorphic involution with fixed points together with a rational curve and section of κ−1/4 that are invariant under the involution.

An analytical approach to the Schrödinger-Newton equations

In this paper we present the second part of a study of spherically-symmetric solutions of the Schrodinger-Newton equations for a single particle (Penrose R 1998 Quantum computation, entanglement and state reduction Phil. Trans. R. Soc. 356 1-13). We show that there exists an infinite family of normalizable, finite energy solutions which are characterized by being smooth and bounded for all values of the radial coordinate. We therefore provide analytical support for our earlier numerical integrations (Moroz I M et al 1998 Spherically-symmetric solutions of the Schrodinger-Newton equations Classical and Quantum Gravity 1998 15 2733-42).

Paraconformal geometry of n th-order ODEs, and exotic holonomy in dimension four

We characterise n th-order ODEs for which the space of solutions M is equipped with a particular paraconformal structure in the sense of Bailey and Eastwood [T.N. Bailey, M.G. Eastwood, Complex Paraconformal manifolds, their differential geometry and twistor theory, Forum Math. 3 (1991) 61–103], that is a splitting of the tangent bundle as a symmetric tensor product of rank-two vector bundles. This leads to the vanishing of (n−2) quantities constructed from of the ODE. If n=4 the paraconformal structure is shown to be equivalent to the exotic G3 holonomy of Bryant. If n=4, or n≥6 and M admits a torsion-free connection compatible with the paraconformal structure then the ODE is trivialisable by point or contact transformations, respectively. If n=2 or 3M admits an affine paraconformal connection with no torsion. In these cases additional constraints can be imposed on the ODE so that M admits a projective structure if n=2, or an Einstein–Weyl structure if n=3. The third-order ODE can in this case be reconstructed from the Einstein–Weyl data.

On non-existence of static vacuum black holes with degenerate components of the event horizon

We present a simple proof of the non-existence of degenerate components of the event horizon in static, vacuum, regular, four-dimensional black hole spacetimes. We discuss the generalization to higher dimensions and the inclusion of a cosmological constant.

The Classification of Static Electro–Vacuum Space–Times Containing an Asymptotically Flat Spacelike Hypersurface with Compact Interior

We show that static electro–vacuum black hole space–times containing an asymptotically flat spacelike hypersurface with compact interior and with both degenerate and non–degenerate components of the event horizon do not exist. This is done by a careful study of the near-horizon geometry of degenerate horizons, which allows us to eliminate the last restriction of the static electro-vacuum no-hair theory.

On Israel-Wilson-Perjés black holes

We show, under certain conditions, that regular Israel?Wilson?Perj?s black holes necessarily belong to the Majumdar?Papapetrou family.

A relation between local and total energy in general relativity

It is shown that, for an asymptotically flat space-time, there exists a collection of conserved vector fields which depend on the local stress energy of the matter and whose integrals over space-like hypersurfaces yield the total ADM or Bondi energy-momentum of the space-time. These vector fields can be used to prove the positivity of the ADM and Bondi energies.

Rigid upper bounds for the angular momentum and centre of mass of non-singular asymptotically anti-de Sitter space-times

We prove upper bounds on angular momentum and centre of mass in terms of the Hamiltonian mass and cosmological constant for non-singular asymptotically anti-de Sitter initial data sets on spin manifolds satisfying the dominant energy condition. We work in space-dimensions larger than or equal to three, and allow a large class of asymptotic backgrounds, with spherical and non-spherical conformal infinities; in the latter case, a spin-structure compatibility condition is imposed. We give classes of non-trivial examples saturating the inequality. We analyse the borderline case in space-time dimension four: for spherical cross-sections of Scri, equality together with completeness occurs only in anti-de Sitter space-time. On the other hand, in the toroidal case, regular non-trivial initial data sets saturating the bound exist.

Cosmological Einstein-Maxwell instantons and Euclidean supersymmetry: anti-self-dual solutions

We classify super-symmetric solutions of the minimal