arXiv subjects
Publications and source records attributed to Piotr T. Chruściel.
We derive a formula for the energy of asymptotically locally hyperbolic (ALH) manifolds obtained by a gluing at infinity of two ALH manifolds. As an application we show that there exist three-dimensional conformally compact ALH manifolds either without boundary or with toroidal black hole boundary, with connected conformal infinity of higher genus, with constant negative scalar curvature, and with negative mass.
We review the theory of stationary black hole solutions of vacuum Einstein equations.
We establish several results on gluing/embedding/extending geometric structures in vacuum spacetimes with a cosmological constant in any spacetime dimensions $d\ge 4$, with emphasis on characteristic data. A useful tool is provided by the notion of submanifold-data of order $k$. As an application of our methods we prove that vacuum Cauchy data on a spacelike Cauchy surface with boundary can always be extended to vacuum data defined beyond the boundary.
We analyse the Noether charges for scalar and Maxwell fields on light cones on a de Sitter, Minkowski, and anti-de Sitter backgrounds. Somewhat surprisingly, under natural asymptotic conditions all charges for the Maxwell fields on both the de Sitter and anti-de Sitter backgrounds are finite. On the other hand, one needs to renormalise the charges for the conformally-covariant scalar field when the cosmological constant does not vanish. In both cases well-defined renormalised charges, with well-defined fluxes, are obtained. Again surprisingly, a Hamiltonian analysis of a suitably rescaled scalar field leads to finite charges, without the need to renormalise. Last but not least, we indicate natural phase spaces where the Poisson algebra of charges is well defined.
We finish the proof of the no-hair theorem for stationary, analytic, connected, suitably regular, four dimensional vacuum black holes. We show how to define the surface gravity and the angular velocity of horizons without assuming analyticity. We point out that, under the usual regularity conditions, vacuum near-horizon geometries are Kerrian without assuming analyticity.
We analyze the Cauchy problem for the Proca equation in gravitating dielectric media.
Original abstract: "We construct periodic solutions of nonlinear wave equations using analytic continuation. The construction applies in particular to Einstein equations, leading to infinite-dimensional families of time-periodic solutions of the vacuum, or of the Einstein-Maxwell-dilaton-scalar fields-Yang-Mills-Higgs-Chern-Simons-$f(R)$ equations, with a negative cosmological constant." However, there is a gap in the proof, and it is unlikely that the strategy presented can be upgraded to a full proof.
We show that the Proca equation in vacuum, as well as its plausible modifications in dielectric media, is incompatible with experimental evidence, no matter how small the Proca mass is.
We construct families of asymptotically locally hyperbolic Riemannian metrics with constant scalar curvature (i.e., time symmetric vacuum general relativistic initial data sets with negative cosmological constant), with prescribed topology of apparent horizons and of the conformal boundary at infinity, and with controlled mass. In particular we obtain new classes of solutions with negative mass.
The standard method of proving analyticity of stationary vacuum metrics invokes the quotient-space version of Einstein equations. We verify that the same conclusion can be obtained using the KID equations on maximal surfaces.
We prove uniqueness, existence, and regularity results for maximal hypersurfaces in spacetimes with a conformal completion at timelike infinity and asymptotically constant scalar curvature, as relevant for asymptotically AdS spacetimes. This work is dedicated to Yvonne Choquet-Bruhat on the occasion of her upcoming 99th birthday.
We carry out a Carlotto-Schoen-type gluing with interpolating scalar curvature on cone-like sets, or deformations thereof, in the category of smooth Riemannian asymptotically Euclidean metrics.
We analyse the response of laser interferometric gravitational wave detectors using the full Maxwell equations in curved spacetime in the presence of weak gravitational waves. Existence and uniqueness of solutions is ensured by setting up a suitable boundary value problem. This puts on solid ground previous approximate calculations. We find consistency with previous results obtained from eikonal expansions at the level of accuracy accessible to current gravitational wave detectors.
We prove positive mass theorems for asymptotically hyperbolic and asymptotically locally hyperbolic Riemannian manifolds with black-hole-type boundaries.
The Kerr geometry is believed to represent the exterior spacetime of astrophysical black holes. We here re-analyze the geometry of Kerr-like metrics (Kerr, Kerr-Newman, Kerr-de Sitter, and Kerr-anti de Sitter), paying particular attention to the region near the singular set. We find that, although the Kretschmann scalar vanishes at the singular set along a given direction, a certain combination of curvature invariants diverges regardless of the direction of approach. We also find that the two-dimensional geometry induced by the spacetime metric on the orbits of the isometry group also possesses a singularity regardless of the direction of approach. Likewise, the two-dimensional geometry in the directions orthogonal to the isometry orbits is $C^{2}$-divergent, but extends continuously at the singular set as a cone with opening angle $4π$. We conclude by showing that tidal forces lead to infinite stresses on neighboring geodesics that approach the singular set, destroying any such observers in finite proper time. Those geodesics that come in from infinity and do not hit the singular set but approach it are found to need tremendous energy to get close to the singular set, experiencing an acceleration transversal to the equatorial plane which grows without bound when the minimal distance of approach goes to zero. While establishing these results, we also present an alternative description of some other known properties, as well as introducing toroidal coordinates that provide a hands-on description of the double-covering for the geometry near the singular set.
We show how to parameterise solutions of the general relativistic vector constraint equation on Einstein manifolds by unconstrained potentials. We provide a similar construction for the trace-free part of tensors satisfying the linearised scalar constraint. Previous work of ours has provided similar different constructions for solutions of the linearized constraints in the case where the cosmological constant $Λ$ is zero. We use our new potentials to show that one can shield linearised gravitational fields using linearised gravitational fields without imposing the TT gauge (as done in previous work), for any value of $Λ\in \mathbb{R}$.
We present results supporting the Horowitz-Myers conjecture, that the Horowitz-Myers metrics minimise energy in the relevant classes of metrics.
Generalising a proof by Bartnik in the asymptotically Euclidean case, we give an elementary proof of positivity of the hyperbolic mass near the hyperbolic space. It is a pleasure to dedicate this work to Robert Bartnik on the occasion of his 60th birthday.