arXiv research

The study finds regular null hypersurfaces in a perturbed Schwarzschild black hole exterior.

Revises Schwarzschild manifold rigidity proof for spin manifolds.

Study on curvature blow-up rates in black hole interiors from gravitational collapse.

Warped-product black hole spacetimes are $C^0$-inextendible.

Proves uniqueness of black holes in Lovelock gravity, a generalization of Einstein's theory.

Stability of Schwarzschild singularity in near-Schwarzschild black holes under perturbations.

Study linear perturbations in Schwarzschild black hole spacetime.

Proves stability of Schwarzschild black holes without symmetry assumptions.

The celebrated uniqueness's theorem of the Schwarzschild solution by Israel, Robinson et al, and Bunting/Masood-ul-Alam, asserts that the only asymptotically flat static solution of the vacuum Einstein equations with compact but non-necessarily connected horizon is Schwarzschild. Between this article and its sequel we …

Scattering theory developed for linearised gravity near Schwarzschild black hole.

Study gluing event horizons of Minkowski and Schwarzschild spacetimes.

Scattering theory for linearised gravity on Schwarzschild black hole exterior.

We study a multiply warped products manifold associated with the Reissner-Nordstrom metric to investigate the physical properties inside the black hole event horizons. It is shown that, different from the uncharged Schwarzschild metric, the Ricci curvature components inside the Reissner-Nordstrom black hole horizons ar…

This is the second in a series of papers in which we take a systematic study of gauge field theories such as the Maxwell equations and the Yang-Mills equations, on curved space-times. In this paper, we study the Maxwell equations in the domain of outer-communication of the Schwarzschild black hole. We show that if we a…

New static black hole uniqueness theorems for negative cosmological constant.

This is the second article of a series or two, proving a generalisation of the uniqueness theorem of the Schwarzschild solution. The theorem to be shown classifies all (metrically complete) solutions of the static vacuum Einstein equations with compact but non-necessarily connected horizon without any further assumptio…

New concept of Lorentzian-Euclidean black holes and metric transitions explored.

This paper explores charged black holes in 3+1 dimensions, finding limitations on their existence.

In a paper \cite{P} in 1973, R. Penrose made a physical argument that the total mass of a spacetime which contains black holes with event horizons of total area $A$ should be at least $\sqrt{A/16π}$. An important special case of this physical statement translates into a very beautiful mathematical inequality in Riemann…

Study of marginally trapped surfaces in a perturbed Schwarzschild spacetime.

New black hole solutions cannot be rotated without breaking their structure.

New method solves 'googly problem' for Schwarzschild black holes.

These lecture notes, based on a course given at the Zurich Clay Summer School (June 23-July 18, 2008), review our current mathematical understanding of the global behaviour of waves on black hole exterior backgrounds. Interest in this problem stems from its relationship to the non-linear stability of the black hole spa…

Researchers create black holes isometric to extremal Reissner-Nordström.

We construct dynamical many-black-hole spacetimes with well-controlled asymptotic behavior as solutions of the Einstein vacuum equation with positive cosmological constant. We accomplish this by gluing Schwarzschild-de Sitter or Kerr-de Sitter black hole metrics into neighborhoods of points on the future conformal boun…

Novel Hodge theory developed for Carrollian bundles, addressing black hole event horizons.

Adapting Israel's proof of static black hole uniqueness, we show that the Schwarzschild spacetime is the only static vacuum asymptotically flat spacetime that possesses a suitably defined photon sphere.

We prove the Riemannian Penrose conjecture, an important case of a conjecture made by Roger Penrose in 1973, by defining a new flow of metrics. This flow of metrics stays inside the class of asymptotically flat Riemannian 3-manifolds with nonnegative scalar curvature which contain minimal spheres. In particular, if we …

We model pseudo-Finsler geometries, with pseudo-Euclidean signatures of metrics, for two classes of four dimensional nonholonomic manifolds: a) tangent bundles with two dimensional base manifolds and b) pseudo-Riemannian/ Einstein manifolds. Such spacetimes are enabled with nonholonomic distributions and associated non…

In this work, we consider solutions of the Maxwell equations on the Schwarzschild-de Sitter family of black hole spacetimes. We prove that, in the static region bounded by black hole and cosmological horizons, solutions of the Maxwell equations decay to stationary Coulomb solutions at a super-polynomial rate, with deca…

We study Finsler black holes induced from Einstein gravity as possible effects of quantum spacetime noncommutativity. Such Finsler models are defined by nonholonomic frames not on tangent bundles but on (pseudo) Riemannian manifolds being compatible with standard theories of physics. We focus on noncommutative deformat…

The Positive Mass Theorem states that a complete asymptotically flat manifold of nonnegative scalar curvature has nonnegative mass. The Riemannian Penrose inequality provides a sharp lower bound for the mass when black holes are present. More precisely, this lower bound is given in terms of the area of an outermost min…

Proves existence of solutions to Einstein constraints with specific boundary conditions and verifies Penrose inequality.

In a seminal paper of 1917, H. Weyl presented a remarkable reduction of the static axisymmetric vacuum Einstein equations, serving as a relatively straightforward technique to generate and explore new solutions. Weyl's reduction was used by Myers in 1987, and independently by Korotkin-Nicolai in 1994, to construct a ne…

We prove the linear stability of slowly rotating Kerr black holes as solutions of the Einstein vacuum equation: linearized perturbations of a Kerr metric decay at an inverse polynomial rate to a linearized Kerr metric plus a pure gauge term. We work in a natural wave map/DeTurck gauge and show that the pure gauge term …

We show that the maximal future development of asymptotically flat spherically symmetric black hole initial data for a self-gravitating nonlinear scalar field, also called a Higgs field, contains a connected, achronal marginally trapped tube which is asymptotic to the event horizon of the black hole, provided the initi…

We prove in this paper the linear stability of the celebrated Schwarzschild family of black holes in general relativity: Solutions to the linearisation of the Einstein vacuum equations around a Schwarzschild metric arising from regular initial data remain globally bounded on the black hole exterior and in fact decay to…

In this paper we use the anholonomic frames method to construct exact solutions for vacuum 5D gravity with metrics having off-diagonal components. The solutions are in general anisotropic and possess interesting features such as an anisotropic warp factor with respect to the extra dimension, or a gravitational scaling/…

We study the perturbations of two classes of static black ellipsoid solutions of four dimensional vacuum Einstein equations. Such solutions are described by generic off--diagonal metrics which are generated by anholonomic transforms of diagonal metrics. The analysis is performed in the approximation of small eccentrici…

Global existence and decay for quasilinear wave equations on various spacetimes, including Kerr black holes.

The horizon and geodesic structure of static configurations generated by anisotropic conformal transforms of the Schwarzschild metric is analyzed. We construct the maximal analytic extension of such off--diagonal vacuum metrics and conclude that for small deformations there are different classes of vacuum solutions of …

We consider solutions to the linear wave equation $\Box_gφ=0$ on a non-extremal maximally extended Schwarzschild-de Sitter spacetime arising from arbitrary smooth initial data prescribed on an arbitrary Cauchy hypersurface. (In particular, no symmetry is assumed on initial data, and the support of the solutions may con…

We prove the linear stability of Schwarzschild-Tangherlini spacetimes and their Anti-de Sitter counterparts under Ricci flow for a special class of perturbations. This is useful in the choice of suitable initial conditions in numerical Ricci-flow-based algorithms for obtaining new solutions to the Einstein equation whe…

The paper investigates various generalizations of semisymmetric and pseudosymmetric manifolds.

Study on spacelike submanifolds in generalized Schwarzschild spacetimes with lightlike foliations.

This paper addresses strong cosmic censorship for spacetimes with self-gravitating collisionless matter, evolving from surface-symmetric compact initial data. The global dynamics exhibit qualitatively different features according to the sign of the curvature $k$ of the symmetric surfaces and the cosmological constant $…

We study noncommutative bundles and Riemannian geometry at the semiclassical level of first order in a deformation parameter $λ$, using a functorial approach. The data for quantisation of the cotangent bundle is known to be a Poisson structure and Poisson preconnection and we now show that this data defines to a functo…

We construct a large class of dynamical vacuum black hole spacetimes whose exterior geometry asymptotically settles down to a fixed Schwarzschild or Kerr metric. The construction proceeds by solving a backwards scattering problem for the Einstein vacuum equations with characteristic data prescribed on the event horizon…