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Tim-Torben Paetz

Publications and source records attributed to Tim-Torben Paetz.

At least 19 recordsLinked to original sources
arXiv2019

Multiple Killing Horizons: the initial value formulation for $\Lambda$-vacuum

In Class. Quantum Grav. 35 (2018) 155015 we have introduced the notion of "Multiple Killing Horizon" and analyzed some of its general properties. Multiple Killing Horizons are Killing horizons for two or more linearly independent Killing vectors simultaneously. In this paper we focus on the vacuum case, possibly with cosmological constant, and study the emergence of Multiple Killing Horizons in terms of characteristic initial value problems for two transversally intersecting null hypersurfaces. As a relevant outcome, a more general definition of Near Horizon Geometry is put forward. This new definition avoids the use of Gaussian null coordinates associated to the corresponding degenerate Killing vector and thereby allows for inclusion of its fixed points.

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arXiv2019

On the choice of a conformal Gauss gauge near the cylinder representing spatial infinity

A convenient approach to analyze spatial infinity is to use a cylinder representation $I$ and impose a gauge based on a congruence of conformal geodesics. This so-called conformal Gauss gauge comes along with the freedom to specify initial data for the conformal geodesics. Such a gauge has been constructed from an ordinary Cauchy surface and from past null infinity $\mathcal{J}^-$, respectively. The purpose of this note is to compare these gauges near the critical set $I^-$, where $I$ "touches" $\mathcal{J}^-$, as it turns out that they are related in a somewhat unexpected intricate way.

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arXiv2018

Multiple Killing Horizons and Near Horizon Geometries

Near Horizon Geometries with multiply degenerate Killing horizons $\mathcal{H}$ are considered, and their degenerate Killing vector fields identified. We prove that they all arise from hypersurface-orthogonal Killing vectors of any cut of $\mathcal{H}$ with the inherited metric -- cuts are spacelike co-dimension two submanifolds contained in $\mathcal{H}$. For each of these Killing vectors on a given cut, there are three different possibilities for the Near Horizon metric which are presented explicitly. The structure of the metric for Near Horizon Geometries with multiple Killing horizons of order $m\geq 3$ is thereby completely determined, and in particular we prove that the cuts on $\mathcal{H}$ must be warped products with maximally symmetric fibers (ergo of constant curvature). The question whether multiple degenerate Killing horizons may lead to inequivalent Near Horizon Geometries by using different degenerate Killings is addressed, and answered on the negative: all Near Horizon geometries built from a given multiple degenerate Killing horizon (using different degenerate Killings) are isometric.

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arXiv2018

On the smoothness of the critical sets of the cylinder at spatial infinity in vacuum spacetimes

We analyze the appearance of logarithmic terms at the critical sets of Friedrich's cylinder representation of spatial infinity. It is shown that if the radiation field vanishes at all orders at the critical sets no logarithmic terms are produced in the formal expansions. Conversely, it is proved that, under the additional hypothesis that the spacetime has constant (ADM) mass aspect and vanishing dual (ADM) mass aspect, this condition is also necessary for a spacetime to admit a smooth representation at the critical sets.

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arXiv2018

Multiple Killing Horizons

Killing horizons which can be such for two or more linearly independent Killing vectors are studied. We provide a rigorous definition and then show that the set of Killing vectors sharing a Killing horizon is a Lie algebra $\mathcal{A}_{\mathcal{H}}$ of dimension at most the dimension of the spacetime. We prove that one cannot attach different surface gravities to such multiple Killing horizons, as they have an essentially unique non-zero surface gravity (or none). $\mathcal{A}_{\mathcal{H}}$ always contains an Abelian (sub)-algebra ---whose elements all have vanishing surface gravity--- of dimension equal to or one less than dim $\mathcal{A}_{\mathcal{H}}$. There arise only two inequivalent possibilities, depending on whether or not there exists the non-zero surface gravity. We show the connection with Near Horizon geometries, and also present a linear system of PDEs, the master equation, for the proportionality function on the horizon between two Killing vectors of a multiple Killing horizon, with its integrability conditions. We provide explicit examples of all possible types of multiple Killing horizons, as well as a full classification of them in maximally symmetric spacetimes.

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arXiv2017

The limit of Kerr-de Sitter spacetime with infinite angular-momentum parameter $a$

We consider the limit $a\rightarrow \infty$ of the Kerr-de Sitter spacetime. The spacetime is a Petrov type-D solution of the vacuum Einstein field equations with a positive cosmological constant $\Lambda$, vanishing Mars-Simon tensor and conformally flat scri. It possesses an Abelian 2-dimensional group of symmetries whose orbits are spacelike or timelike in different regions, and it includes, as a particular case, de Sitter spacetime. The global structure of the solution is analyzed in detail, with particular attention to its Killing horizons: they are foliated by non-compact marginally trapped surfaces of finite area, and one of them `touches' the curvature singularity, which resembles a null 2-dimensional surface. Outside the region between these horizons there exist trapped surfaces that again are non-compact. The solution contains, apart from $\Lambda$, a unique free parameter which can be related to the angular momentum of the non-singular horizon in a precise way. A maximal extension of the (axis of the) spacetime is explicitly built. We also analyze the structure of scri, whose topology is $\mathbb{R}^3$.

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arXiv2017

Analysis of a Bianchi-like equation satisfied by the Mars-Simon tensor

The Mars-Simon tensor (MST), which e.g. plays a crucial role to provide gauge invariant characterizations of the Kerr-NUT-(A)(dS) family, satisfies a Bianchi-like equation. In this paper we analyze this equation in close analogy to the Bianchi equation, in particular it will be shown that the constraints are preserved supposing that a generalized Buchdahl condition holds. This permits the systematic construction of solutions to this equation in terms of a well-posed Cauchy problem. A particular emphasis lies on the asymptotic Cauchy problem, where data are prescribed on a spacelike scri (i.e. for $\Lambda >0$). In contrast to the Bianchi equation, the MST equation is of Fuchsian type at scri, for which existence and uniqueness results are derived.

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arXiv2016

Classification of Kerr-de Sitter-like spacetimes with conformally flat Scri

We provide a classification of $\Lambda>0$-vacuum spacetimes which admit a Killing vector field with respect to which the associated "Mars-Simon tensor" (MST) vanishes and having a conformally flat $\mathcal{J}^-$ (or $\mathcal{J}^+$). To that end, we also give a complete classification of conformal Killing vector fields on the $3$-sphere $\mathbb{S}^3$ up to M\"obius transformations shedding light on the two fundamental constants that characterize the family of Kerr-de Sitter-like spacetimes, which turn out to be well-defined geometrical invariants. The topology of $\mathcal{J}$ is determined in every case, and a characterization result at $\mathcal{J}$ of the Kerr-de Sitter family presented.

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arXiv2016

Characterization of (asymptotically) Kerr-de Sitter-like spacetimes at null infinity

We investigate solutions $(\mathcal{M}, g)$ to Einstein's vacuum field equations with positive cosmological constant $\Lambda$ which admit a smooth past null infinity $\mathcal{J}^-$ \`a la Penrose and a Killing vector field whose associated Mars-Simon tensor (MST) vanishes. The main purpose of this work is to provide a characterization of these spacetimes in terms of their Cauchy data on $\mathcal{J}^-$. Along the way, we also study spacetimes for which the MST does not vanish. In that case there is an ambiguity in its definition which is captured by a scalar function $Q$. We analyze properties of the MST for different choices of $Q$. In doing so, we are led to a definition of "asymptotically Kerr-de Sitter-like spacetimes", which we also characterize in terms of their asymptotic data on $\mathcal{J}^-$.

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arXiv2014

Conformally covariant systems of wave equations and their equivalence to Einstein's field equations

We derive, in 3+1 spacetime dimensions, two alternative systems of quasi-linear wave equations, based on Friedrich's conformal field equations. We analyse their equivalence to Einstein's vacuum field equations when appropriate constraint equations are satisfied by the initial data. As an application, the characteristic initial value problem for the Einstein equations with data on past null infinity is reduced to a characteristic initial value problem for wave equations with data on an ordinary light-cone.

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arXiv2014

Characteristic initial data and smoothness of Scri. I. Framework and results

We analyze the Cauchy problem for the vacuum Einstein equations with data on a complete light-cone in an asymptotically Minkowskian space-time. We provide conditions on the free initial data which guarantee existence of global solutions of the characteristic constraint equations. We present necessary-and-sufficient conditions on characteristic initial data in $3+1$ dimensions to have no logarithmic terms in an asymptotic expansion at null infinity.

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arXiv2014

Killing Initial Data on spacelike conformal boundaries

We analyze Killing Initial Data on Cauchy surfaces in conformally rescaled vacuum space-times satisfying Friedrich's conformal field equations. As an application, we derive the KID equations on a spacelike $\mathcal{J}^-$.

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arXiv2014

The mass of light-cones

We give an elementary proof of positivity of total gravitational energy in space-times containing complete smooth light-cones.

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