arXiv subjects

Walter Simon

Publications and source records attributed to Walter Simon.

26 records · Page 2Linked to original sources

arXiv2002

On the Penrose Inequality

We summarize results on the Penrose inequality bounding the ADM-mass or the Bondi mass in terms of the area of an outermost apparent horizon for asymptotically flat initial data of Einstein's equations. We first recall the proof, due to Geroch and to Jang and Wald, of monotonicity of the Geroch-Hawking mass under a smooth inverse mean curvature flow for data with non-negative Ricci scalar, which leads to a Penrose inequality if the apparent horizon is a minimal surface.We then sketch a proof of the Penrose inequality of Malec, Mars and Simon which holds for general horizons and for data satisfying the dominant energy condition, but imposes (in addition to smooth inverse mean curvature flow) suitable restrictions on the data on a spacelike surface. These conditions can, however, at least locally be fulfilled by a suitable choice of the initial surface in a given spacetime. Remarkably, they are also (formally) identical to ones employed earlier by Hayward in order to define a 2+1 foliation on null surfaces, with respect to which the Hawking mass is again monotonic.

gr-qc↗

arXiv2002

On uniqueness of static Einstein-Maxwell-dilaton black holes

We prove uniqueness of static, asymptotically flat spacetimes with non-degenerate black holes for three special cases of Einstein-Maxwell-dilaton theory: For the coupling ``$α= 1$'' (which is the low energy limit of string theory) on the one hand, and for vanishing magnetic or vanishing electric field (but arbitrary coupling) on the other hand. Our work generalizes in a natural, but non-trivial way the uniqueness result obtained by Masood-ul-Alam who requires both $α= 1$ and absence of magnetic fields, as well as relations between the mass and the charges. Moreover, we simplify Masood-ul-Alam's proof as we do not require any non-trivial extensions of Witten's positive mass theorem. We also obtain partial results on the uniqueness problem for general harmonic maps.

gr-qc↗

arXiv2002

Criteria for (in)finite extent of static perfect fluids

In Newton's and in Einstein's 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 extent 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 asumptions. 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 results in a unified way.

gr-qc↗

arXiv2002

On the Penrose Inequality for general horizons

For asymptotically flat initial data of Einstein's equations satisfying an energy condition, we show that the Penrose inequality holds between the ADM mass and the area of an outermost apparent horizon, if the data are restricted suitably. We prove this by generalizing Geroch's 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. Modulo smoothness issues we also show that our restrictions on the data can locally be fulfilled by a suitable choice of the initial surface in a given spacetime.

gr-qc↗

arXiv2000

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 space-times. 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.

gr-qc↗

arXiv2000

Conformal positive mass theorems

We show the following two extensions of the standard positive mass theorem (one for either sign): Let (N,g) and (N,g') be asymptotically flat Riemannian 3-manifolds with compact interior and finite mass, such that g and g' are twice Hoelder differentiable and related via the conformal rescaling g' = (phi^4).g, with a twice Hoelder differentiable function phi>0. Assume further that the corresponding Ricci scalars satisfy either R + (phi^4).R' >= 0 or R - (phi^4).R' >= 0. Then the corresponding masses satisfy m + m' >= 0 or m - m' >= 0, respectively. Moreover, in the case of the minus signs, equality holds iff g and g' are isometric, whereas for the plus signs equality holds iff both (N,g) and (N,g') are flat Euclidean spaces. While the proof of the case with the minus signs is rather obvious, the case with the plus signs requires a subtle extension of Witten's proof of the standard positive mass theorem. The idea for this extension is due to Masood-ul-Alam who, in the course of an application, proved the rigidity part m + m' = 0 of this theorem, for a special conformal factor. We observe that Masood-ul-Alam's method extends to the general situation.

gr-qc↗

arXiv2000

A Proof of Uniqueness of the Taub-bolt Instanton

We show that the Riemannian Schwarzschild and the ``Taub-bolt'' instanton solutions are the only spaces (M,g) such that 1) M is a 4-dimensional, simply connected manifold with a Riemannian, Ricci-flat C^2-metric g which admits (at least) a 1-parameter group of isometries H without isolated fixed points on M. 2) The quotient (M L)/H (where L is the set of fixed points of H) is an asymptotically flat manifold, and the length of the Killing field corresponding to H tends to a constant at infinity.

gr-qc↗

arXiv1995

Nuts have no hair

We show that the Riemannian Kerr solutions are the only Riemannian, Ricci-flat and asymptotically flat ${\rm C}^{2}$-metrics $g_{μν}$ on a 4-dimensional complete manifold ${\cal M}$ of topology ${\rm R}^{2} \times {\rm S}^{2}$ which have (at least) a 1-parameter group of periodic isometries with only isolated fixed points ("nuts") and with orbits of bounded length at infinity.

gr-qc↗