Related papers: The vacuum weighted Einstein field equations on pr…
We establish the existence of smooth vacuum Gowdy solutions, which are asymptotically velocity term dominated (AVTD) and have T3-spatial topology, in an infinite dimensional family of generalized wave gauges. These results show that the…
This is the second paper of a two part work that establishes a definitive quantitative nonlinear scattering theory for asymptotically de Sitter vacuum solutions $(M,g)$ in $(n+1)$ dimensions with $n\geq4$ even, which are determined by small…
We determine the class of $p$-forms $F$ which possess vanishing scalar invariants (VSI) at arbitrary order in a $n$-dimensional spacetime. Namely, we prove that $F$ is VSI if and only if it is of type N, its multiple null direction $l$ is…
In Einstein's general relativity, with its nonlinear field equations, the discoveries and analyzes of various specific explicit solutions made a great impact on understanding many of the unforeseen features of the theory. Some solutions…
We consider $d$-dimensional static spacetimes in Einstein gravity with a cosmological constant in the presence of a minimally coupled massless scalar field. The spacetimes have a $(d-2)$-dimensional base manifold given by an Einstein space…
We obtain an exact solution for the Einstein's equations with cosmological constant coupled to a scalar, static particle in static, "spherically" symmetric background in 2+1 dimensions.
A new semi-supervised machine learning package is introduced which successfully solves the Euclidean vacuum Einstein equations with a cosmological constant, without any symmetry assumptions. The model architecture contains subnetworks for…
1- It is shown that the upper bound for $\alpha$ in the general solutions of spherically symmetric vacuum field equations(gr-qc/9812081,$\Lambda$=0) is nearly 10^3.This has been obtained by comparing the theoretical prediction for bending…
The stationary, axisymmetric reduction of the vacuum Einstein equations, the so-called Ernst equation, is an integrable nonlinear PDE in two dimensions. There now exists a general method for analyzing boundary value problems for integrable…
We study the geometric structure of weighted Einstein smooth metric measure spaces with weighted harmonic Weyl tensor. A complete local classification is provided, showing that either the underlying manifold is Einstein, or decomposes as a…
This paper presents solutions to Einstein's equation -- and the numerical methods used to construct them -- that describe simple cosmological models on manifolds with compact non-orientable spatial slices. These solutions have been…
We review the problem of describing the gravitational field of compact stars in general relativity. We focus on the deviations from spherical symmetry which are expected to be due to rotation and to the natural deformations of mass…
The Newtonian theory of gravitation and electrostatics admit equilibrium configurations of charged fluids where the charge density can be equal to the mass density, in appropriate units. The general relativistic analog for charged dust…
We demonstrate how the Einstein's equations for the $D$-dimensional spherical gravity can be written in the covariant vector-like form. These equations reveal easily the causal structure of curved spherically symmetric manifolds and may…
In this paper, we consider the weighted $p$-Laplacian equation $$ \Delta_{p,f}u+au^{\sigma}\ln u=0$$ defined on a complete smooth metric measure space under the conditon that the $m$-Bakry-\'{E}mery Ricci curvature has a lower bound, where…
A $p$-form $F$ is VSI (i.e., all its scalar invariants of arbitrary order vanish) in a $n$-dimensional spacetime if and only if it is of type N, its multiple null direction $l$ is "degenerate Kundt", and $\pounds_{l}F=0$. This recent result…
The incompressible Euler equations on a compact Riemannian manifold $(M,g)$ take the form \begin{align*} \partial_t u + \nabla_u u &= - \mathrm{grad}_g p \\ \mathrm{div}_g u &= 0, \end{align*} where $u: [0,T] \to \Gamma(T M)$ is the…
We present the solution space of the field equations in the Einstein-aether theory for the case of a vacuum Bianchi Type V space-time. We also find that there are portions of the initial parameters space for which no solution is admitted by…
It is shown that four dimensional vacuum Einstein solutions simply embedded in five dimensions obey the Gauss-Bonnet-Einstein field equations: $G_{ab}+\alpha GB_{ab}+\delta^{55}_{ab}\alpha\exp(-2\chi/\sqrt{\alpha})GB_4=0$ and the…
A Riemannian manifold $(M,g)$ is called \emph{weakly Einstein} if the tensor $R_{iabc}R_{j}^{~~abc}$ is a scalar multiple of the metric tensor $g_{ij}$. We give a complete classification of weakly Einstein hypersurfaces in the spaces of…