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For axially symmetric solutions of Einstein equations there exists a gauge which has the remarkable property that the total mass can be written as a conserved, positive definite, integral on the spacelike slices. The mass integral provides…
The energy density of the gravitational field is a full-fledged source of the gravitational field. This principle of Einstein was not implemented by him in the Einstein equation. Not long ago it was possible to find an energy-momentum…
Basic principles of the Hamilton approach developed for the metric General Relativity (Einstein`s GR) are discussed. In particular, we derive the Hamiltonian of the metric GR in the explicit form. This Hamiltonian is a quadratic function of…
A general covariant extension of Einstein's field equations is considered with a view to Numerical Relativity applications. The basic variables are taken to be the metric tensor and an additional four-vector. The extended field equations,…
A recent proposal equates the circuit complexity of a quantum gravity state with the gravitational action of a certain patch of spacetime. Since Einstein's equations follow from varying the action, it should be possible to derive them by…
The generalized harmonic representation of Einstein's equation is manifestly hyperbolic for a large class of gauge conditions. Unfortunately most of the useful gauges developed over the past several decades by the numerical relativity…
We present a general method which can be used for geometrical and physical interpretation of an arbitrary spacetime in four or any higher number of dimensions. It is based on the systematic analysis of relative motion of free test…
Numerical schemes for Einstein's vacuum equation are developed. Einstein's equation in harmonic gauge is second order symmetric hyperbolic. It is discretized in four-dimensional spacetime by Finite Differences, Finite Elements, and Interior…
Gravitational waves allow us to test general relativity in the highly dynamical regime. While current observations have been consistent with waves emitted by quasi-circular binaries, eccentric binaries may also produce detectable signals in…
Adaptive techniques are crucial for successful numerical modeling of gravitational waves from astrophysical sources such as coalescing compact binaries, since the radiation typically has wavelengths much larger than the scale of the…
We derive the general solution to the coupled Einstein and Dirac field equations in static and hyperplane-symmetric spacetime of arbitrary dimension including a cosmological constant of either sign. As a result, only a massful Dirac field…
I give a compact, pedagogical review of our present understanding of the universe as based on general relativity. This includes the uniform models, with special reference to the cosmological 'constant'; and the equations for…
Recent results on solutions of the Einstein equations with matter are surveyed and a number of open questions are stated. The first group of results presented concern asymptotically flat spacetimes, both stationary and dynamical. Then there…
Modifications to gravity that add additional functions of the Ricci curvature to the Einstein-Hilbert action -- collectively known as $f(R)$ theories -- have been studied in great detail. When considered as complete theories of gravity they…
We prove the global stability of the Minkowski space viewed as the trivial solution of the Einstein-Vlasov system. To estimate the Vlasov field, we use the vector field and modified vector field techniques developed in [FJS15; FJS17]. In…
Certain features associated with the symmetry reduction of the vacuum Einstein equations by two commuting, space-like Killing vector fields are studied. In particular, the discussion encompasses the equations for the Gowdy $T^3$ cosmology…
The linear stability of warped product Einstein metrics as fixed points of the Ricci flow is investigated. We generalise the results of Gibbons, Hartnoll and Pope and show that in sufficiently low dimensions, all warped product Einstein…
In contrast to electrodynamics, Einstein's gravitation equations are not invariant with respect to a wide class of the mapping of field variables which leave equations of motion of test particles in a given coordinate system invariant. It…
We study a general class of gravitational theories formulated in the Palatini approach and derive the equations governing the evolution of tensor perturbations. In the absence of torsion, the connection can be solved as the Christoffel…
The study of the properties and dynamics of self-gravitating bosonic objects in Einstein gravity was conducted. We studied self-coupled boson stars and determined the quasinormal mode (QNM) frequencies of stable boson stars in spherical…