A result, first conjectured by Geroch, is proved to the extent, that the multipole moments of a s... more A result, first conjectured by Geroch, is proved to the extent, that the multipole moments of a static space-time characterize this space-time uniquely. As an offshoot of the proof one obtains an essentially coordinate-free algorithm for explicitly writing down a geometry in terms of it's moments in a purely algebraic manner. This algorithm seems suited for symbolic manipulation on a computer.
For asymptotically flat initial data of Einstein's equations satisfying an energy condition, we s... more 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.
We present a theorem which establishes uniqueness, in particular spherical symmetry, of a wide cl... more We present a theorem which establishes uniqueness, in particular spherical symmetry, of a wide class of general relativistic, static perfect-fluid models provided there exists a spherically symmetric model with the same equation of state and surface potential. The method of proof, which is inspired by recent work of Masood-ul-Alam, is illustrated by demonstrating uniqueness of a class of solutions due to Buchdahl which correspond to an extreme case of the inequality on the equation of state required by our theorem.
Motivated by the conjectured Penrose inequality and by the work of Hawking, Geroch, Huisken and I... more Motivated by the conjectured Penrose inequality and by the work of Hawking, Geroch, Huisken and Ilmanen in the null and the Riemannian case, we examine necessary conditions on flows of two-surfaces in spacetime under which the Hawking quasilocal mass is monotone. We focus on a subclass of such flows which we call uniformly expanding, which can be considered for null as well as for spacelike directions. In the null case, local existence of the flow is guaranteed. In the spacelike case, the uniformly expanding condition leaves a 1-parameter freedom, but for the whole family, the embedding functions satisfy a forward-backward parabolic system for which local existence does not hold in general. Nevertheless, we have obtained a generalization of the weak (distributional) formulation of this class of flows, generalizing the corresponding step of Huisken and Ilmanen’s proof of the Riemannian Penrose inequality.
The present work extends our short communication Phys. Rev. Lett. 95, 111102 (2005). For smooth m... more The present work extends our short communication Phys. Rev. Lett. 95, 111102 (2005). For smooth marginally outer trapped surfaces (MOTS) in a smooth spacetime we define stability with respect to variations along arbitrary vectors v normal to the MOTS. After giving some introductory material about linear non self-adjoint elliptic operators, we introduce the stability operator L_v and we characterize stable MOTS in terms of sign conditions on the principal eigenvalue of L_v. The main result shows that given a strictly stable MOTS S contained in one leaf of a given reference foliation in a spacetime, there is an open marginally outer trapped tube (MOTT), adapted to the reference foliation, which contains S. We give conditions under which the MOTT can be completed. Finally, we show that under standard energy conditions on the spacetime, the MOTT must be either locally achronal, spacelike or null.
Given a spacelike foliation of a spacetime and a marginally outer trapped surface S on some initi... more Given a spacelike foliation of a spacetime and a marginally outer trapped surface S on some initial leaf, we prove that under a suitable stability condition S is contained in a ``horizon'', i.e. a smooth 3-surface foliated by marginally outer trapped slices which lie in the leaves of the given foliation. We also show that under rather weak energy conditions this horizon must be either achronal or spacelike everywhere. Furthermore, we discuss the relation between ``bounding'' and ``stability'' properties of marginally outer trapped surfaces.
Any stationary, asymptotically flat solution to Einstein's equation is shown to asymptotically ap... more Any stationary, asymptotically flat solution to Einstein's 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.
We prove uniqueness of static, asymptotically flat spacetimes with non-degenerate black holes for... more 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 ``$\alpha = 1$'' (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 $\alpha = 1$ 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.
Following earlier work of Masood-ul-Alam, we consider a uniqueness problem for non-rotating stell... more Following earlier work of Masood-ul-Alam, we consider a uniqueness problem for non-rotating stellar models. Given a static, asymptotically flat perfectfluid spacetime with barotropic equation of state θ(p), and given another such spacetime which is spherically symmetric and has the same θ(p) and the same surface potential: we prove that both are identical provided θ(p) satisfies a certain differential inequality. This inequality is more natural and less restrictive than the conditions required by Masood-ul-Alam.
The stationary Einstein-Maxwell equations are rewritten in a form which permits the introduction ... more 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.
In Newton’s and in Einstein’s theory we give criteria on the equation of state of a barotropic pe... more 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 oneparameter 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 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 results in a unified way.
A mis padres, que me apoyaron siempre a ser la mejor en todo lo que me proponga, a mi mamá que me... more A mis padres, que me apoyaron siempre a ser la mejor en todo lo que me proponga, a mi mamá que me brindó palabras de aliento desde el inicio de la maestría hasta el final.
A result, first conjectured by Geroch, is proved to the extent, that the multipole moments of a s... more A result, first conjectured by Geroch, is proved to the extent, that the multipole moments of a static space-time characterize this space-time uniquely. As an offshoot of the proof one obtains an essentially coordinate-free algorithm for explicitly writing down a geometry in terms of it's moments in a purely algebraic manner. This algorithm seems suited for symbolic manipulation on a computer.
For asymptotically flat initial data of Einstein's equations satisfying an energy condition, we s... more 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.
We present a theorem which establishes uniqueness, in particular spherical symmetry, of a wide cl... more We present a theorem which establishes uniqueness, in particular spherical symmetry, of a wide class of general relativistic, static perfect-fluid models provided there exists a spherically symmetric model with the same equation of state and surface potential. The method of proof, which is inspired by recent work of Masood-ul-Alam, is illustrated by demonstrating uniqueness of a class of solutions due to Buchdahl which correspond to an extreme case of the inequality on the equation of state required by our theorem.
Motivated by the conjectured Penrose inequality and by the work of Hawking, Geroch, Huisken and I... more Motivated by the conjectured Penrose inequality and by the work of Hawking, Geroch, Huisken and Ilmanen in the null and the Riemannian case, we examine necessary conditions on flows of two-surfaces in spacetime under which the Hawking quasilocal mass is monotone. We focus on a subclass of such flows which we call uniformly expanding, which can be considered for null as well as for spacelike directions. In the null case, local existence of the flow is guaranteed. In the spacelike case, the uniformly expanding condition leaves a 1-parameter freedom, but for the whole family, the embedding functions satisfy a forward-backward parabolic system for which local existence does not hold in general. Nevertheless, we have obtained a generalization of the weak (distributional) formulation of this class of flows, generalizing the corresponding step of Huisken and Ilmanen’s proof of the Riemannian Penrose inequality.
The present work extends our short communication Phys. Rev. Lett. 95, 111102 (2005). For smooth m... more The present work extends our short communication Phys. Rev. Lett. 95, 111102 (2005). For smooth marginally outer trapped surfaces (MOTS) in a smooth spacetime we define stability with respect to variations along arbitrary vectors v normal to the MOTS. After giving some introductory material about linear non self-adjoint elliptic operators, we introduce the stability operator L_v and we characterize stable MOTS in terms of sign conditions on the principal eigenvalue of L_v. The main result shows that given a strictly stable MOTS S contained in one leaf of a given reference foliation in a spacetime, there is an open marginally outer trapped tube (MOTT), adapted to the reference foliation, which contains S. We give conditions under which the MOTT can be completed. Finally, we show that under standard energy conditions on the spacetime, the MOTT must be either locally achronal, spacelike or null.
Given a spacelike foliation of a spacetime and a marginally outer trapped surface S on some initi... more Given a spacelike foliation of a spacetime and a marginally outer trapped surface S on some initial leaf, we prove that under a suitable stability condition S is contained in a ``horizon'', i.e. a smooth 3-surface foliated by marginally outer trapped slices which lie in the leaves of the given foliation. We also show that under rather weak energy conditions this horizon must be either achronal or spacelike everywhere. Furthermore, we discuss the relation between ``bounding'' and ``stability'' properties of marginally outer trapped surfaces.
Any stationary, asymptotically flat solution to Einstein's equation is shown to asymptotically ap... more Any stationary, asymptotically flat solution to Einstein's 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.
We prove uniqueness of static, asymptotically flat spacetimes with non-degenerate black holes for... more 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 ``$\alpha = 1$'' (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 $\alpha = 1$ 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.
Following earlier work of Masood-ul-Alam, we consider a uniqueness problem for non-rotating stell... more Following earlier work of Masood-ul-Alam, we consider a uniqueness problem for non-rotating stellar models. Given a static, asymptotically flat perfectfluid spacetime with barotropic equation of state θ(p), and given another such spacetime which is spherically symmetric and has the same θ(p) and the same surface potential: we prove that both are identical provided θ(p) satisfies a certain differential inequality. This inequality is more natural and less restrictive than the conditions required by Masood-ul-Alam.
The stationary Einstein-Maxwell equations are rewritten in a form which permits the introduction ... more 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.
In Newton’s and in Einstein’s theory we give criteria on the equation of state of a barotropic pe... more 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 oneparameter 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 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 results in a unified way.
A mis padres, que me apoyaron siempre a ser la mejor en todo lo que me proponga, a mi mamá que me... more A mis padres, que me apoyaron siempre a ser la mejor en todo lo que me proponga, a mi mamá que me brindó palabras de aliento desde el inicio de la maestría hasta el final.
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Papers by Walter Simon