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Carla Cederbaum

Publications and source records attributed to Carla Cederbaum.

30 records · Page 2Linked to original sources
arXiv2017

Asymptotically flat extensions of CMC Bartnik data

Let $g$ be a metric on the $2$-sphere $\mathbb{S}^2$ with positive Gaussian curvature and $H$ be a positive constant. Under suitable conditions on $(g, H)$, we construct smooth, asymptotically flat $3$-manifolds $M$ with non-negative scalar curvature, with outer-minimizing boundary isometric to $(\mathbb{S}^2, g)$ and having mean curvature $H$, such that near infinity $M$ is isometric to a spatial Schwarzschild manifold whose mass $m$ can be made arbitrarily close to a constant multiple of the Hawking mass of $(\mathbb{S}^2,g,H)$. Moreover, this constant multiplicative factor depends only on $(g, H)$ and tends to $1$ as $H$ tends to $0$. The result provides a new upper bound of the Bartnik mass associated to such boundary data.

math.DG↗
arXiv2015

Uniqueness of photon spheres via positive mass rigidity

In a recent paper the first author established the uniqueness of photon spheres, suitably defined, in static vacuum asymptotically flat spacetimes by adapting Israel's proof of static black hole uniqueness. In this note we establish uniqueness of photon spheres by adapting the argument of Bunting and Masood-ul-Alam, which then allows certain assumptions to be relaxed. In particular, multiple photon spheres are allowed a priori. As a consequence of our result, we can rule out the existence of static configurations involving multiple "very compact" bodies and black holes.

math.DG↗
arXiv2015

Uniqueness of photon spheres in electro-vacuum spacetimes

In a recent paper, the authors established the uniqueness of photon spheres in static vacuum asymptotically flat spacetimes by adapting Bunting and Masood-ul-Alam's proof of static vacuum black hole uniqueness. Here, we establish uniqueness of suitably defined sub-extremal photon spheres in static electro-vacuum asymptotically flat spacetimes by adapting the argument of Masood-ul-Alam. As a consequence of our result, we can rule out the existence of electrostatic configurations involving multiple "very compact" electrically charged bodies and sub-extremal black holes.

math.DG↗
arXiv2015

Level sets of the lapse function in static GR

We present a novel physical interpretation of the level sets of the (canonical) lapse function in static isolated general relativistic space-times. Our interpretation uses a notion of 'constrained test particles'. It leads to a definition of gravitational force on test particles and to a previously unknown uniqueness result for the lapse function. In Section 5, we discuss 'photon spheres' in static isolated relativistic space-times and relate them to the level sets of the lapse function.

gr-qc↗
arXiv2015

On the center of mass of asymptotically hyperbolic initial data sets

We define the (total) center of mass for suitably asymptotically hyperbolic time-slices of asymptotically anti-de Sitter spacetimes in general relativity. We do so in analogy to the picture that has been consolidated for the (total) center of mass of suitably asymptotically Euclidean time-slices of asymptotically Minkowskian spacetimes (isolated systems). In particular, we unite -- an altered version of -- the approach based on Hamiltonian charges with an approach based on CMC-foliations near infinity. The newly defined center of mass transforms appropriately under changes of the asymptotic coordinates and evolves in the direction of an appropriately defined linear momentum under the Einstein evolution equations.

math.DG↗
arXiv2013

Explicit Riemannian manifolds with unexpectedly behaving center of mass

The (relativistic) center of mass of an asymptotically flat Riemannian manifold is often defined by certain surface integral expressions evaluated along a foliation of the manifold near infinity, e. g. by Arnowitt, Deser, and Misner (ADM). There are also what we call 'abstract' definitions of the center of mass in terms of a foliation near infinity itself, going back to the constant mean curvature (CMC-) foliation studied by Huisken and Yau; these give rise to surface integral expressions when equipped with suitable systems of coordinates. We discuss subtle asymptotic convergence issues regarding the ADM- and the coordinate expressions related to the CMC-center of mass. In particular, we give explicit examples demonstrating that both can diverge -- in a setting where Einstein's equation is satisfied. We also give explicit examples of the same asymptotic order of decay with prescribed mass and center of mass. We illustrate both phenomena by providing analogous examples in Newtonian gravity. Our examples conflict with some results in the literature.

math.AP↗
arXiv2012

Geometrostatics: the geometry of static space-times

We present a new geometric approach to the study of static isolated general relativistic systems for which we suggest the name geometrostatics. After describing the setup, we introduce localized formulas for the ADM-mass and ADM/CMC-center of mass of geometrostatic systems. We then explain the pseudo-Newtonian character of these formulas and show that they converge to Newtonian mass and center of mass in the Newtonian limit, respectively, using Ehlers' frame theory. Moreover, we present a novel physical interpretation of the level sets of the canonical lapse function and apply it to prove uniqueness results. Finally, we suggest a notion of force on test particles in geometrostatic space-times.

math.DG↗
arXiv2012

The Newtonian Limit of Geometrostatics

This thesis discusses the Newtonian limit of General Relativity for static isolated systems with compactly supported matter. We call these systems "geometrostatic" to underline their geometric nature. We introduce new quasi-local notions of mass and center of mass that can be read off locally in the vicinity of the matter/black holes. We prove that these notions asymptotically coincide with ADM-mass and the Huisken-Yau CMC-center of mass. Moreover, we prove that they converge to Newtonian mass and center of mass in the Newtonian limit. The Newtonian limit is discussed in the language of Ehlers' frame theory. Furthermore, we prove several uniqueness claims in geometrostatics as well as other geometric and physical properties of these systems. We analyze equipotential sets, provide a pseudo-Newtonian reformulation of geometrostatics, and prove uniqueness of static photon spheres.

gr-qc↗
arXiv2011

Universal properties of distorted Kerr-Newman black holes

We discuss universal properties of axisymmetric and stationary configurations consisting of a central black hole and surrounding matter in Einstein-Maxwell theory. In particular, we find that certain physical equations and inequalities (involving angular momentum, electric charge and horizon area) are not restricted to the Kerr-Newman solution but can be generalized to the situation where the black hole is distorted by an arbitrary axisymmetric and stationary surrounding matter distribution.

gr-qc↗