The study of Killing tensor fields in Weyl's class of stationary and axially symmetric space-times.
problem Understanding the space of Killing tensor fields of valence 3 in Weyl's class.
method Analyzing the static vacuum sub-class of stationary and axially symmetric space-times, proving reducibility of valence-3 Killing tensors.
result The space of Killing tensor fields of valence 3 in Weyl's class is reducible to Killing vector fields and quadratic Killing tensor fields.
Study shows nonexistence of Killing tensors for specific axially symmetric vacuum space-times.
problem Existence of Killing tensors in stationary and axially symmetric space-times.
method Mathematically rigorous, computer algebra computations with complex equations.
result Nonexistence of nontrivial Killing tensors for certain metrics up to high valences.
Paper proposes NOSTILL-GP for accurate space-time modeling in environmental monitoring.
problem Accurate modeling of space-time dynamics in environmental phenomena.
method NOSTILL-GP - a non-stationary, spatio-temporal Gaussian Process model with efficient training strategies.
result Demonstrates the effectiveness and general applicability of NOSTILL-GP for environmental monitoring.
In this paper, based on an intrinsic definition of asymptotically AdS space-times, we show that the standard anti-de Sitter space-time is the unique strictly stationary asymptotically AdS solution to the vacuum Einstein equations with negative cosmological constant in dimension less than 7. Instead of using the positiv…
Method proves ellipticity of vacuum spacetime boundary problems.
problem Proving ellipticity of boundary value problems for stationary vacuum spacetimes.
method Developed a general method using the projection formalism.
result Proved manifold theorem for moduli space of stationary vacuum spacetimes.
Algebraic curvature tensors possess generators which can be formed from symmetric or alternating tensors S, A or tensors θwith an irreducible (2,1)-symmetry. In differential geometry examples of curvature formulas are known which contain generators on the basis of S or A realized by differentiable tensor fields in a na…
Generalizing Riemannian theorems of Anderson-Herzlich and Biquard, we show that two (n+1)-dimensional stationary vacuum space-times (possibly with cosmological constant Λ∈R) that coincide up to order one along a timelike hypersurface $\mycal T$ are isometric in a neighbourhood of $\mycal T$. We further prove th…
We show the existence of a Hawking vector field in a full neighborhood of a local, regular, bifurcate, non-expanding horizon embedded in a smooth Einstein-Maxwell space-time without assuming the underlying space-time is analytic. It extends one result of Friedrich, Rácz and Wald, which was limited to the interior of th…
New non-singular spacetimes found with negative cosmological constant.
problem Finding non-singular spacetimes with a negative cosmological constant.
method Constructing infinite-dimensional families of solutions to complex equations.
result Infinite-dimensional families of non-singular stationary space-times with negative cosmological constant.
Proves uniqueness of non-extremal Kerr-Newman black holes under small perturbations.
problem Uniqueness of non-extremal Kerr-Newman black holes under small perturbations.
method Perturbative analysis using Mars-Simon type tensors.
result Proves that a space-time close to the Kerr-Newman family must be one of the Kerr-Newman solutions.
A trivial projective change of a Finsler metric F is the Finsler metric F+df. I explain when it is possible to make a given Finsler metric both forward and backward complete by a trivial projective change. The problem actually came from lorentz geometry and mathematical relativity: it was observed that it is poss…
The correspondence of stationary, axisymmetric, asymptotically flat space-times and bundles over a reduced twistor space has been established in four dimensions. The main impediment for an application of this correspondence to examples in higher dimensions is the lack of a higher-dimensional equivalent of the Ernst pot…
We study focal points and Maslov index of a horizontal geodesic γ:I→M in the total space of a semi-Riemannian submersion π:M→B by determining an explicit relation with the corresponding objects along the projected geodesic π∘γ:I→B in the base space. We use this result to calculate the focal Maslov in…
New findings on GRW space-times with constant scalar curvature.
problem Understanding GRW space-times in different subspaces.
method Analyzing orthogonal subspaces of Gray's decomposition.
result Generalized quasi-Einstein GRW space-times reduce to known types of space-times.
The paper examines properties of W-curvature tensor in relativistic space-times.
problem Investigating the properties and implications of the W-curvature tensor in relativistic space-times. method Analyzing the semi-symmetry and divergence properties of the energy-momentum tensor in relation to the W-curvature tensor. result Space-times with specific properties of the W-curvature tensor are classified as Einstein or Codazzi type. We investigate refocusing and strong refocusing of light rays in a space-time. A strongly refocusing space-time is refocusing. The converse is unknown. We construct examples of space-times which are refocusing, but not strongly so, at a particular point. These space-times are strongly refocusing at other points. The ge…
The paper examines isotropic cosmological space-times with changing sectional curvature.
problem Cosmological space-times with changing sectional curvature.
method Analysis of a family of geometrically well-behaved cosmological space-times foliated by isotropic hypersurfaces.
result Only space-time isometries ensure the rigidity properties of isotropic cosmological space-times.
New distances defined between space-times, proving some definite.
problem Defining distances between space-times.
method Introducing causal-null-compactifiable space-times and using cosmological time and null distance.
result Various definite distances defined, proving convergence of space-times.
We uniquely and explicitly reconstruct the instantaneous intrinsic metric of the Kerr-Newman Event Horizon from the spectrum of its Laplacian. In the process we find that the angular momentum parameter, radius, area; and in the uncharged case, mass, can be written in terms of these eigenvalues. In the uncharged case th…
We consider Killing vector fields on standard static space-times and obtain equations for a vector field on a standard static space-time to be Killing. We also provide a characterization of Killing vector fields on standard static space-times with compact Riemannian parts.
A large class of vacuum space-times is constructed in dimension 4+1 from hyperboloidal initial data sets which are not small perturbations of empty space data. These space-times are future geodesically complete, smooth up to their future null infinity, and extend as vacuum space-times through their Cauchy horizon. Dime…
New conditions for GRW space-times to be perfect-fluid space-times.
problem Conditions for GRW space-times to be perfect-fluid.
method Gray's decomposition of the gradient of the Ricci tensor, determining Ricci tensor forms in invariant subspaces.
result For most GRW space-times, the Ricci tensor is Einstein or perfect fluid.
Study Codazzi tensors in space-times, linking to Cotton gravity.
problem Understanding Codazzi tensors and their role in space-times.
method Analyzing geometric properties and proving conditions for Codazzi tensors.
result Codazzi tensors restrict space-times, influencing energy-momentum tensors in Cotton gravity.
Mathematical framework for caustics of world hyper-sheets in Minkowski space-time.
problem Defining holographic domains in space-time.
method Developed a mathematical framework to describe caustics of world hyper-sheets.
result Investigated singularities of caustics and their geometrical meanings.
The paper analyzes tensors in generalized Robertson-Walker space-times.
problem Analyzing tensors in generalized Robertson-Walker space-times.
method Proving theorems about Ricci and Weyl tensors, decomposing Ricci tensor, showing conditions for harmonic Weyl tensor, and generalizing Riemann tensor structure.
result Conditions for a GRW space-time to be a quasi-Einstein manifold and the structure of Riemann tensor.
New geometric flow equations describe how space-time dimensions change.
problem Understanding how the number of space-time dimensions affects geometry.
method Developed D-flow equations to model the variation of space-time geometries.
result Solutions for D-flow equations on D-dimensional spheres and Freund-Rubin Compactification.
The notion of maximal extension of a globally hyperbolic space-time arises from the notion of maximal solutions of the Cauchy problem associated to the Einstein's equations of general relativity. In 1969 Choquet-Bruhat and Geroch proved that if the Cauchy problem has a local solution, this solution has a unique maximal…
We investigate a generalization of the so-called metric splitting of globally hyperbolic space-times to non-smooth Lorentzian manifolds and show the existence of this metric splitting for a class of wave-type space-times. Our approach is based on smooth approximations of non-smooth space-times by families (or sequences…
Perfect fluids in higher dimensions become generalized Robertson-Walker spaces under specific conditions.
problem Characterizing perfect-fluid space-times as generalized Robertson-Walker spaces.
method Analyzing conditions for a perfect-fluid space-time to be a generalized Robertson-Walker space-time.
result Conditions for a perfect-fluid space-time to be a generalized Robertson-Walker space-time are verified.
New measure defined for Brakke flow, linking classical and new definitions.
problem Defining and characterizing the Brakke flow.
method Introduced a space-time-Grassmann measure to characterize the flow.
result Equivalence between classical and new definitions of the Brakke flow.
Study on static perfect fluid space-time geometry and boundary estimates.
problem Investigate the geometry and boundary properties of static perfect fluid space-time.
method Used generalized Reilly's formula to establish geometric inequalities and boundary estimates.
result Obtained new boundary estimates involving the Brown-York mass and first eigenvalue of the Jacobi operator.
The paper proves geodesic connectedness for convex functions in space-times.
problem Geodesic connectedness of space-times and semi-Riemannian manifolds.
method Geometric-topological proofs for specific classes of space-times.
result Geodesic connectedness for null-disprisoning space-times and timelike strictly convex hypersurfaces.
Causal fermion systems explore geometric space-time structures.
problem Understanding geometric structures in space-time.
method Causal fermion systems approach.
result Exploration of geometric space-time structures.
Study null sectional curvatures in warped product spaces.
problem Investigate null sectional curvatures in warped product spaces.
method Derive formulas for null sectional curvatures of specific warped product models.
result Formulas for null sectional curvatures of various space-time models.
Lorentzian LCS spaces are equivalent to Generalized Robertson-Walker spaces.
problem Identifying equivalent space-time structures.
method Direct demonstration of equivalence between LCS and GRW spaces.
result Lorentzian LCS spaces are equivalent to Generalized Robertson-Walker spaces.
M-CaStLe discovers causal structures in multivariate space-time data.
problem Challenges in causal graph discovery for high-dimensional gridded data.
method Generalizes CaStLe to multivariate analyses, using local embeddings and pooling spatial replicates.
result More accurately recovers multivariate causal structure and identifies physical dynamics.
Essentially, some conditions for the Riemannian factor and the warping function of a standard static space-time are obtained in order to guarantee that no nontrivial warping function on the Riemannian factor can make the standard static space-time Einstein.
In this paper, we study Ricci-flat and Einstein Lorentzian multiply warped products. We also consider the case of having constant scalar curvatures for this class of warped products. Finally, after we introduce a new class of spacetimes called as generalized Kasner space-times, we apply our results to this kind of spac…
Solves PDE system for minimal space-like surfaces in Minkowski space-time.
problem Solving the system of natural PDE's for minimal space-like surfaces.
method Using canonical Weierstrass representations, solves the system explicitly.
result Expresses solutions by means of two holomorphic functions.
Simple conformally recurrent spaces are identified as pp-waves.
problem Characterizing conformally recurrent space-times.
method Analyzing dimension n>3 space-times.
result Simple conformally recurrent space-times are conformally recurrent pp-waves.
Proves stability of Minkowski space-time for Einstein-Yang-Mills equations.
problem Stability of Minkowski space-time for perturbations governed by Einstein-Yang-Mills equations.
method Proves exterior energy estimates for tensorial non-linear wave equations in Minkowski space-time.
result Proves exterior stability of Minkowski space-time for Einstein-Yang-Mills equations.
Survey on Margulis space-times and anti-de Sitter manifolds.
problem Understanding Margulis space-times and their properties.
method Analyzing anti-de Sitter manifolds and their deformations.
result Margulis space-times are homeomorphic to the interior of a handlebody and admit a fundamental domain bounded by crooked planes.
Study of Moncrief lines' behavior in curved space-times.
problem Understanding the asymptotic behavior of Moncrief lines in curved space-times.
method Analysis of geodesic laminations and convergence to Thurston boundary.
result Moncrief lines converge to a unique point in the Thurston boundary.
A study of proper affine vector fields in plane symmetric static space-times by using the rank of the Rieman matrix and holonomy. Studying proper affine vector fields in each case, It is shown that the special class of the above space-times admit proper affine vector fields.
In this paper we describe the stable and unstable leaves for the geodesic flow on the space of non-wandering spacelike geodesics of a Margulis Space Time and prove contraction properties of the leaves under the flow. We also show that monodromy of Margulis Space Times are "Anosov representations in non semi-simple Lie …
Paper develops fast, flexible Hawkes process inference for space-time data.
problem Capturing self-exciting, clustering spatio-temporal data.
method Finite support kernels, discretization, precomputations, ℓ2 gradient-based solver. result Statistically accurate and fast inference for space-time Hawkes processes.
Study on stochastic covariant derivatives in curved space-time.
problem Analyzing covariant derivatives in curved space-time under stochastic processes.
method Using Itô-Wiener processes and stochastic calculus, including Besov spaces, Schrödinger operators, and white noise.
result Developed a framework for stochastic geodesics and white noise in fractoid spaces.
Brakke flow support is parabolically rectifiable
problem Support of Brakke flow is parabolically rectifiable
method Developed approach to Brakke flow as space-time-Grassmann measure
result Standard convergence of Brakke flows is equivalent to space-time-Grassmann Radon measures