Walter Simon

Towards the classification of static vacuum spacetimes with negative cosmological constant

We present a systematic study of static solutions of the vacuum Einstein equations with negative cosmological constant which asymptotically approach the generalized Kottler (“Schwarzschild–anti-de Sitter”) solution, within (mainly) a conformal framework. We show connectedness of conformal infinity for appropriately regular such spacetimes. We give an explicit expression for the Hamiltonian mass of the (not necessarily static) metrics within the class considered; in the static case we show that they have a finite and well-defined Hawking mass. We prove inequalities relating the mass and the horizon area of the (static) metrics considered to those of appropriate reference generalized Kottler metrics. Those inequalities yield an inequality which is opposite to the conjectured generalized Penrose inequality. They can thus be used to prove a uniqueness theorem for the generalized Kottler black holes if the generalized Penrose inequality can be established.

The multipole structure of stationary space‐times

A definition of multipole moments for stationary asymptotically flat solutions of Einstein’s equations is proposed. It is shown that these moments characterize a given space‐time uniquely. Conversely, they can be arbitrarily prescribed, i.e., they generate power series for the field variables which satisfy the field equations to all orders. Despite their apparently rather different origin, they are shown to be identical with the Geroch–Hansen ones.

The multipole expansion of stationary Einstein–Maxwell fields

The stationary Einstein–Maxwell equations are rewritten in a form which permits the introduction of (Geroch‐) multipole moments for asymptotically flat solutions. Some known theorems on the moments in the stationary case are generalized to include Einstein–Maxwell fields.

The stationary gravitational field near spatial infinity

Any stationary, asymptotically flat solution to Einsteins equation is shown to asymptotically approach the Kerr solution in a precise sense. As an application of this result we prove a technical lemma on the existence of harmonic coordinates near infinity.

On the Penrose inequality for general horizons.

For asymptotically flat initial data of Einsteins equations satisfying an energy condition, we show that the Penrose inequality holds between the Arnowitt-Deser-Misner mass and the area of an outermost apparent horizon, if the data are suitably restricted. We prove this by generalizing Gerochs proof of monotonicity of the Hawking mass under a smooth inverse mean curvature flow, for data with non-negative Ricci scalar. Unlike Geroch we need not confine ourselves to minimal surfaces as horizons. Leaving smoothness issues aside, we also show that our restrictions on the data can be locally fulfilled by a suitable choice of the initial surface in a given spacetime.

Bounds on area and charge for marginally trapped surfaces with a cosmological constant

We sharpen the known inequalities A? ? 4?(1 ? g) (Hayward et al 1994 Phys. Rev. D 49 5080, Woolgar 1999 Class. Quantum Grav. 16 3005) and A ? 4?Q2 (Dain et al 2012 Class. Quantum Grav. 29 035013) between the area A and the electric charge Q of a stable marginally outer-trapped surface (MOTS) of genus g in the presence of a cosmological constant ?. In particular, instead of requiring stability we include the principal eigenvalue ? of the stability operator. For ?* = ??+?? > 0, we obtain a lower and an upper bound for ?*A in terms of ?*Q2, as well as the upper bound for the charge, which reduces to in the stable case ? ? 0. For ?* < 0, there only remains a lower bound on A. In the spherically symmetric, static, stable case, one of our area inequalities is saturated iff the surface gravity vanishes. We also discuss implications of our inequalities for ?jumps? and mergers of charged MOTS.

Criteria for (In)finite Extent of Static Perfect Fluids

In Newtons and in Einsteins theory we give criteria on the equation of state of a barotropic perfect fluid which guarantee that the corresponding one- parameter family of static, spherically symmetric solutions has finite extent. These criteria are closely related to ones which are known to ensure finite or infinite ex- tent of the fluid region if the assumption of spherical symmetry is replaced by certain asymptotic falloff conditions on the solutions. We improve this result by relaxing the asymptotic assumptions. Our conditions on the equation of state are also related to (but less restrictive than) ones under which it has been shown in Relativity that static, asymptotically flat fluid solutions are spherically symmetric. We present all these re- sults in a unified way.

Gravitational field strength and generalized Komar integral

We define a “gravitational field strength” in theories of the Einstein-Cartan type admitting a Killing vector. This field strength is a second rank, antisymmetric, divergence-free tensor, whose (“Komar”) integral over a closed 2-surface gives a physically meaningful quantity. We find conditions on the Lagrange density of the theory which ensure the existence of such a tensor, and show that they are satisfied forN=2-supergravity and for a special case of the bosonic sector ofN=4-supergravity. We discuss a possible application of the generalized Komar integral in the theory of stationary black holes. We also consider the “field strength problem” in Kaluza-Klein theory, where the application to black holes is particularly interesting.

A Simple proof of the generalized electrostatic Israel theorem

A simple proof of a generalization of the theorem of Israel concerning the uniqueness of the Reissner-Nordström black holes is presented. The present method is also applied to show, in the bosonic sector ofN=4-supergravity, the uniqueness of a static black hole solution found by Gibbons.

A proof of uniqueness of the Taub-bolt instanton

Abstract We show that the Riemannian Schwarzschild and the “Taub-bolt” instanton solutions are the only spaces (M, gμv) such that: 1. ṡM is a four-dimensional, simply connected manifold with a Riemannian, Ricci-flat C2-metric gμv which admits (at least) a 1-parameter group μτ of isometries without isolated fixed points on M. 2. ṡ The quotient (M β L M ) μ τ (where LM is the set of fixed points of μτ) is an asymptotically flat manifold, and the length of the Killing field corresponding to μτ tends to a constant at infinity.