Proves stability of spacetime Penrose inequality for spherical symmetric initial data.
problem Stability of the Penrose inequality for spherical symmetric spacetimes.
method Formulated and proved stability statement using spherical symmetry and asymptotically flat initial data.
result Initial data must arise from an isometric embedding into a static spacetime close to Schwarzschild spacetime.
Proves a new Penrose inequality for static spacetimes.
problem Establishing a lower bound on mass using area and static references.
method Develops a new quasi-local Penrose inequality for spherically symmetric static spacetimes.
result Proves a quasi-local Penrose inequality for any spherically symmetric static spacetime.
Weak Bianchi identity found for spacetimes with timelike singularities.
problem Weak Bianchi identity for spacetimes with curvature singularities.
method Found sufficient conditions for a weak version of the second Bianchi identity.
result Identified a class of spherically symmetric static spacetimes with timelike singularities for which the identity holds.
The paper characterizes photon surfaces in static spacetimes and proves their uniqueness.
problem Characterizing and proving uniqueness of photon surfaces in static spacetimes.
method Local characterization and proof of uniqueness using specific spacetime properties.
result Static, vacuum, asymptotically isotropic spacetimes with equipotential photon surfaces are isometric to Schwarzschild spacetime.
The central object of study of this thesis is inverse mean curvature vector flow of two-dimensional surfaces in four-dimensional spacetimes. Being a system of forward-backward parabolic PDEs, inverse mean curvature vector flow equation lacks a general existence theory. Our main contribution is proving that there exist …
Study of caustics in Einstein-dust system, showing spacetime singularities and diverging curvature.
problem Understanding caustics and singularities in the Einstein-dust system.
method Established local existence result for spherically symmetric spacetimes containing caustics, constructed from solutions to a PDE problem.
result Obtained spherically symmetric spacetimes with diverging curvature and singular boundary.
Characterizes photon surfaces in static spacetimes, proving uniqueness.
problem Understanding photon surfaces in static spacetimes of arbitrary dimension.
method Complete characterization and new insights into spacetime geometry.
result Proves uniqueness of certain electrostatic spacetimes.
We survey many of the important properties of spherically symmetric spacetimes as follows. We present several different ways of describing a spherically symmetric spacetime and the resulting metrics. We then focus our discussion on an especially useful form of the metric of a spherically symmetric spacetime in polar-ar…
Static spherically symmetric solutions to the Einstein-Euler equations with prescribed central densities are known to exist, be unique and smooth for reasonable equations of state. Some criteria are also available to decide whether solutions have finite extent (stars with a vacuum exterior) or infinite extent. In the l…
The paper explores how a geometric flow can turn a black hole into a traversable wormhole.
problem The study investigates how a static, spherically symmetric black hole can be transformed into a traversable wormhole.
method The approach involves analyzing almost η-Ricci-Yamabe solitons and their geometric coupling with the Hawking temperature. result The geometric flow successfully transforms the black hole into a traversable wormhole, opening the throat and preserving the exact cosmological spacetime.
New exact spherically symmetric vacuum solutions found in Finsler gravity.
problem Finding exact vacuum solutions in Finsler gravity.
method Spherically symmetric, asymptotically flat Berwald spacetimes solved for Finsler gravity vacuum equation.
result Only one class of spherically symmetric Berwald spacetimes is compatible with asymptotic flatness and a well-defined causal structure.
Study of spacelike singularities in spherical spacetimes with scalar matter.
problem Characterize spacelike singularities in spherically symmetric spacetimes with scalar matter.
method Analyzes the properties of spacelike singularities in spherically symmetric spacetimes with scalar matter, proving inverse polynomial blow-up rates and providing a BKL-type expansion.
result Provides a rigorous description of Kasner-like singularities in spherically symmetric gravitational collapse.
The paper investigates geometrical aspects of static spacetime with almost gradient Ricci solitons.
problem Geometrical properties of static spacetime with almost gradient Ricci solitons.
method Analyzing conditions and properties of static spacetime with almost gradient Ricci solitons.
result Conditions and properties of static spacetime with almost gradient Ricci solitons are determined.
We formulate certain inequalities for the geometric quantities characterizing causal diamonds in curved and Minkowski spacetimes. These inequalities involve the red-shift factor which, as we show explicitly in the spherically symmetric case, is monotonic in the radial direction and it takes its maximal value at the cen…
Establishes a Penrose-type inequality for static spacetimes.
problem Finding a lower bound on the total mass of static spacetimes.
method Analyzes (n+1)-dimensional asymptotically flat standard static spacetimes under timelike convergence condition.
result Extends Penrose-type inequalities to all dimensions and characterizes equality conditions.
We solve spacelike spherically symmetric constant mean curvature (SS-CMC) hypersurfaces in Schwarzschild spacetimes and analyze their asymptotic behavior near the coordinate singularity r = 2M. Furthermore, we join SS-CMC hypersurfaces in the Kruskal extension to obtain complete ones and discuss the smooth properties.
The classical Minkowski formula is extended to spacelike codimension-two submanifolds in spacetimes which admit "hidden symmetry" from conformal Killing-Yano two-forms. As an application, we obtain an Alexandrov type theorem for spacelike codimension-two submanifolds in a static spherically symmetric spacetime: a codim…
The paper proves stability of the positive mass theorem in spherical symmetry.
problem The stability of the positive mass theorem in spherical symmetry when mass is small.
method Formulated a conjecture and proved it under spherical symmetry assumption.
result A sequence of asymptotically flat initial data converging to Minkowski space.
The study characterizes spacetimes with quasi-constant sectional curvature and explores their properties in F(R)-gravity.
problem Characterizing spacetimes with quasi-constant sectional curvature.
method Investigation through examples, proofs, and analysis of energy conditions.
result A spacetime of quasi-constant sectional curvature can represent a Robertson Walker spacetime or a static spacetime.
We first summarize the characterization of smooth spacelike spherically symmetric constant mean curvature (SS-CMC) hypersurfaces in the Schwarzschild spacetime and Kruskal extension. Then use the characterization to prove special SS-CMC foliation property, and verify part of the conjecture by Malec and Ó Murchadha in t…
In this paper, we present two observations about static spherically symmetric solutions of the Einstein-Klein-Gordon equations. The first is a comment extending the well-known result of the existence of static states (i.e. standing wave solutions) of the Einstein-Klein-Gordon equations. The second more important observ…
The paper examines geometric properties of a unique spacetime model.
problem Investigating the geometric properties of a point-like global monopole spacetime.
method Analyzing the spacetime's pseudosymmetry structures, energy-momentum tensor, and curvature properties.
result The point-like global monopole spacetime exhibits various pseudosymmetry structures and properties.
Study spherical doubly warped spacetimes for stellar collapse and cosmology.
problem Analyzing spherically symmetric spacetimes for stellar collapse and cosmology.
method Obtained results for Weyl and Ricci tensors on general doubly warped spacetimes.
result Friedmann equations deviate from standard FRW cosmology due to electric tensor terms.
The paper examines gravitational singularities in spacetimes and proves inextendibility.
problem Investigating gravitational singularities in spacetimes.
method Analyzing local holonomy and using it to prove inextendibility.
result Proves the Cloc0,1-inextendibility of certain spacetimes. Jebsen-Birkhoff theorem extended to Berwald spacetimes.
problem Generalizing Birkhoff theorem to Berwald spacetimes.
method Proving Ricci-flat, spatially spherically symmetric Berwald spacetimes are pseudo-Riemannian or flat.
result Jebsen-Birkhoff theorem extended to Berwald spacetimes.
It is shown that the initial singularities in spatially compact spacetimes with spherical, plane or hyperbolic symmetry admitting a compact constant mean curvature hypersurface are crushing singularities when the matter content of spacetime is described by the Vlasov equation (collisionless matter) or the wave equation…
New proof shows FLRW spacetimes can't be extended smoothly in certain axisymmetric cases.
problem Proving smooth extension of FLRW spacetimes in specific spacetime classes.
method Extending previous work on spherically symmetric spacetimes to axisymmetric spacetimes.
result Demonstrates C0-inextendibility for FLRW spacetimes in a subclass of axisymmetric spacetimes. New symmetries found in Riemann-Cartan geometries.
problem Investigating symmetries in geometries with curvature and torsion.
method Mathematical tools to determine symmetries and subclasses of geometries.
result Determined all static and stationary spherically symmetric Riemann-Cartan geometries and subclasses with specific symmetries.
The study proves strong cosmic censorship violation for spherically symmetric dust clouds.
problem Violation of strong cosmic censorship for spherically symmetric dust clouds.
method Derived an ordinary differential equation for light rays and used it to prove strong cosmic censorship violation.
result Generic violation of strong cosmic censorship for spherically symmetric dust clouds.
We prove the existence and uniqueness of the Dirichlet problem for spacelike, spherically symmetric, constant mean curvature equation with symmetric boundary data in the extended Schwarzschild spacetime. As an application, we completely solve the CMC foliation conjecture which is posted by Malec and O Murchadha in 2003…
We give conditions on a general stress-energy tensor T_{αβ} in a spherically symmetric black hole spacetime which are sufficient to guarantee that the black hole will contain a (spherically symmetric) marginally trapped tube which is eventually achronal, connected, and asymptotic to the event horizon. Price law decay p…
We solve the spacelike, spherically symmetric, constant mean curvature hypersurfaces in the maximally extended Reissner-Nordstrom spacetime with the charge smaller than the mass. Based on these results, we construct constant mean curvature foliations with fixed or varied mean curvature in each slice in this spacetime.
Study wormholes in Einstein-Yang-Mills theory with a phantom field.
problem Existence of wormholes in Einstein-Yang-Mills theory with a phantom scalar field.
method Rigorous mathematical proof and numerical analysis of wormhole solutions.
result Existence of an infinite sequence of symmetric wormhole solutions.
The main aim of this thesis is to study the properties of trapped surfaces in spacetimes with symmetries and their possible relation with the theory of black holes. We will concetrate specially on one aspect of this possible equivalence, namely whether the static black hole uniqueness theorems extend to static spacetim…
This paper tackles gauge fixing and regularity for perturbations around spherical backgrounds.
problem Understanding gauge freedom and regularity in perturbation theory for symmetric tensors.
method Analyzing Hodge-type decomposition for axially symmetric and axistationary tensors, showing existence and uniqueness of gauge tensors.
result Stationary and axially symmetric second order perturbations can be rendered in a canonical form with only one degree of differentiability loss near the origin.
We prove that the maximal development of any spherically symmetric spacetime with collisionless matter (obeying the Vlasov equation) or a massless scalar field (obeying the massless wave equation) and possessing a constant mean curvature S1×S2 Cauchy surface also contains a maximal Cauchy surface. Combining …
New black hole models with both null and spacelike singularities.
problem Understanding singularities in black hole spacetimes.
method Developed a new spacelike-characteristic gluing method to construct black hole spacetimes.
result First examples of black holes with coexisting null and spacelike singularities.
The study finds regular null hypersurfaces in a perturbed Schwarzschild black hole exterior.
problem Existence of regular null hypersurfaces in a perturbed Schwarzschild black hole.
method Proof of existence for null hypersurfaces in a perturbed Schwarzschild spacetime.
result Existence of many foliations by regular null hypersurfaces in the exterior region of a perturbed Schwarzschild black hole.
Global existence and geometry of constant mass aspect function foliation in perturbed Schwarzschild spacetime studied.
problem Null Penrose inequality on a null hypersurface.
method Global existence of constant mass aspect function foliation on a nearly spherically symmetric incoming null hypersurface in a vacuum perturbed Schwarzschild spacetime.
result Geometry of the constant mass aspect function foliation compared to the spherically symmetric foliation in the Schwarzschild spacetime.
The paper solves the isoperimetric problem in Riemannian optical geometry, proving circles minimize lengths with area constraints.
problem Optical geometry of static spherically symmetric spacetimes.
method Applying isoperimetric problem results to curves in Riemannian optical geometry.
result Length-minimizing curves with area constraints are circles, with implications for photon spheres.
Paper proves a rigidity result for static perfect fluids.
problem Proving a rigidity result for static perfect fluids.
method Robinson's divergence formula and boundary conditions.
result Rigidity result for static perfect fluids.
We solve Bartnik's stationary extension problem near Schwarzschild spheres.
problem Existence and uniqueness of asymptotically flat stationary vacuum spacetimes.
method Developed a double geodesic gauge, reducing equations to elliptic and transport-type problems.
result Local well-posedness for Bartnik stationary metric extension problem near Schwarzschild spheres.
Warped-product black hole spacetimes are C0-inextendible.
problem Future C0-inextendibility of warped-product black hole spacetimes method Adapting Sbierski's proof for a broad class of warped-product black hole spacetimes
result Future C0-inextendibility established for spacetimes with a static exterior region A multi-neck spacetime wormhole is constructed with a simple metric tensor.
problem Existence of multi-neck spacetime wormholes.
method Spherical inversion of a 3-torus to create a 3-neck spacetime wormhole.
result Exact solution of Einstein's field equations for a multi-neck spacetime wormhole.
This paper describes the black hole threshold in a moduli space of spherically symmetric spacetimes.
problem Understanding the black hole threshold in a moduli space of spherically symmetric spacetimes.
method Complete description and analysis of the black hole threshold in the moduli space M. result The black hole threshold is the extremal leaf of a C1 foliation of the moduli space, separating black hole solutions from non-collapsing solutions. The study explores spacetimes with changing spatial curvature, leading to topological transitions.
problem The need for a model that avoids infinite matter and energy after the Big Bang.
method Investigates spacetimes with time-dependent spatial curvature, allowing it to change sign.
result Topological transitions are possible in spacetimes with time-dependent spatial curvature.
Proves uniqueness of certain spacetime solutions with extremal horizons.
problem Proving uniqueness of extremal Schwarzschild de Sitter spacetime solutions.
method Analytic proof in four and higher dimensions, spectral problem for hyperbolic surfaces.
result Proves extremal Schwarzschild de Sitter solutions are unique up to identifications.
In this paper we provide a method capable of producing an infinite number of solutions for Einstein's equation on static spacetimes with perfect fluid as a matter field. All spacetimes of this type which are symmetric with respect to a given group of translations and whose spatial factor is conformally flat, are charac…