Tensor Calculus in General Relativity: A Bridge Between Mathematics and Theoretical Physics

Science sets itself apart from other paths to truth by recognizing that even its greate practitioners sometimes err.

Einstein's field equations are based on Riemannian geometry. One of the great important in Riemannian geometry is the curves that minimize the distance between two given points. The Ricci curvature tensors are also broadly applicable to modern Riemannian geometry and general theory of relativity (GTR). The solution of Einstein's field equations, need Ricci tensors. The line elements based on the ellipsoidal coordinates systems usually are difficult to work but are more perfect than spherical coordinate systems. Here we are trying to get all non-zero Ricci tensors coefficients of a line element in an oblate spheroidal coordinate system for an elliptical shaped object. These objects can be as galaxies, stars or planets and other celestial objects that we know today.

Seven new solutions to the interior static and spherically symmetric Einstein's field equations (EFE) are found and investigated. These new solutions are a generalisation of the quadratic density fall-off profile of the Tolman VII solution. The generalisation involves the addition of anisotropic pressures and electric charge to the density profile. Of these new solutions three are found to obey all the necessary conditions of physical acceptability, including linear stability under radial perturbations, and causality of the speed of pressure waves inside the object. Additionally an equation of state can be found for all the physically viable solutions. The generalised pulsation equation for interior solutions to the EFE that include both electric charge and pressure anisotropy is derived and used to determine the stability of the solutions. However the pulsation equation found is general and can be used for all new solutions that contain these ingredients.

A paperback edition of a classic text, this book gives a unique survey of the known solutions of Einstein's field equations for vacuum, Einstein-Maxwell, pure radiation and perfect fluid sources. It introduces the foundations of differential geometry and Riemannian geometry and the methods used to characterize, find or construct solutions. The solutions are then considered, ordered by their symmetry group, their algebraic structure (Petrov type) or other invariant properties such as special subspaces or tensor fields and embedding properties. Includes all the developments in the field since the first edition and contains six completely new chapters, covering topics including generation methods and their application, colliding waves, classification of metrics by invariants and treatments of homothetic motions. This book is an important resource for graduates and researchers in relativity, theoretical physics, astrophysics and mathematics. It can also be used as an introductory text on some mathematical aspects of general relativity.

In this paper, we develop a new method to find the exact solutions of the Einstein's field equations by using which we construct time-periodic solutions. The singularities of the time-periodic solutions are investigated and some new physical phenomena, such as the time-periodic event horizon, are found. The applications of these solutions in modern cosmology and general relativity are expected.

After a brief introduction to classical relativity, we describe how to solve the Cauchy problem in general relativity. In particular, we introduce the notion of gauge source functions and explain how they can be used in order to reduce the problem to that of solving a system of hyperbolic partial differential equations. We then go on to explain how the initial value problem is formulated for the so-called Einstein-Vlasov system and describe a recent future global non-linear stability result in this setting. In particular, this result applies to models of the universe which are consistent with observations.

In recent years there had been a growing interest in analog models of general relativity, with certain superfluid solutions simulating black hole solutions of Einstein's gravitational field equation. The quantization of a superfluid, composed of discrete particles (helium atoms), treated as a nonrelativistic many body problem does not lead to divergencies as the quantization of Einstein's field equations. Quantization of gravity is possible in string theory, but only if one introduces the daring hypothesis of higher dimensions. But if the gravitational field is made up of discrete elements as superfluid helium is made up of helium atoms, then gravity can be quantized without difficulty in three space and one time dimension. Such a hypothesis, of course, implies that Lorentz invariance is a dynamic symmetry caused by real rod and clock deformations, as it was assumed in the pre-Einstein theory of relativity by Lorentz and Poincare, which required the existence of an aether. Making the hypothesis that this aether is a kind of superfluid plasma made up of positive and negative Planck mass particles interacting with the Planck force over a Planck length, one obtains an analog of the standard model, including gravity, which can be quantized as a nonrelativistic many body problem. In this model nonrelativistic vortex rings in three space dimensions and one time dimension simulate the relativistic theory of closed strings in ten space-time dimensions. But because in the vortex lattice, one obtains a large dimensionless number conceivably advancing our understanding of the finestructure constant.

In recent years there had been a growing interest in analog models of general relativity, with certain superfluid solutions simulating black hole solutions of Einstein's gravitational field equation. The quantization of a superfluid, composed of discrete particles (helium atoms), treated as a nonrelativistic many body problem does not lead to divergencies as the quantization of Einstein's field equations. Quantization of gravity is possible in string theory, but only if one introduces the daring hypothesis of higher dimensions. But if the gravitational field is made up of discrete elements as superfluid helium is made up of helium atoms, then gravity can be quantized without difficulty in three space and one time dimension. Such a hypothesis, of course, implies that Lorentz invariance is a dynamic symmetry caused by real rod and clock deformations, as it was assumed in the pre-Einstein theory of relativity by Lorentz and Poincaré, which required the existence of an aether. Making the hypothesis that this aether is a kind of superfluid plasma made up of positive and negative Planck mass particles interacting with the Planck force over a Planck length, one obtains an analog of the standard model, including gravity, which can be quantized as a nonrelativistic many body problem. In this model nonrelativistic vortex rings in three space dimensions and one time dimension simulate the relativistic theory of closed strings in ten space-time dimensions. But because in the vortex lattice, one obtains a large dimensionless number conceivably advancing our understanding of the finestructure constant.

Given a truncated incoming null cone and a truncated outgoing null cone intersecting at a two sphere $S$ with smooth characteristic initial data, a theorem of Rendall shows that the vacuum Einstein equations can be solved in a small neighborhood of $S$ in the future of $S$. We show that in fact the vacuum Einstein equations can be solved in a neighborhood in the future of the cones, as long as the constraint equations are initially satisfied on the null cones. The proof is based on energy type estimates and relies heavily on the null structure of the Einstein equations in the double null foliation.

The global existence problem in general relativity does not reduce toa global existence theorem for a solution of the Einstein equations withsome choice of time coordinate. The physical problem is the existenceof spacetimes for an infinite proper time. But proper time depends on theobserver, i.e. on the timelike line along which it is observed. The singularitytheorems of general relativity consider as a singularity the timelike or nullgeodesic incompleteness of a spacetime and are established under variousgeometric hypotheses.On the other hand, existence theorems for a solution of generic Cauchyproblems for the Einstein equations with an infinite proper time of existencefor a family of observers have been obtained but their timelike or null

Starting from Newton's gravitational theory, we give a general introduction into the spherically symmetric solution of Einstein's vacuum field equation, the Schwarzschild(–Droste) solution, and into one specific stationary axially symmetric solution, the Kerr solution. The Schwarzschild solution is unique and its metric can be interpreted as the exterior gravitational field of a spherically symmetric mass. The Kerr solution is only unique if the multipole moments of its mass and its angular momentum take on prescribed values. Its metric can be interpreted as the exterior gravitational field of a suitably rotating mass distribution. Both solutions describe objects exhibiting an event horizon, a frontier of no return. The corresponding notion of a black hole is explained to some extent. Eventually, we present some generalizations of the Kerr solution.

We analyze Einstein's vacuum field equations in generalized harmonic coordinates on a compact spatial domain with boundaries. We specify a class of boundary conditions, which is constraint-preserving and sufficiently general to include recent proposals for reducing the amount of spurious reflections of gravitational radiation. In particular, our class comprises the boundary conditions recently proposed by Kreiss and Winicour, a geometric modification thereof, the freezing-Ψ0 boundary condition and the hierarchy of absorbing boundary conditions introduced by Buchman and Sarbach. Using the recent technique developed by Kreiss and Winicour based on an appropriate reduction to a pseudo-differential first-order system, we prove well posedness of the resulting initial-boundary value problem in the frozen coefficient approximation. In view of the theory of pseudo-differential operators, it is expected that the full nonlinear problem is also well posed. Furthermore, we implement some of our boundary conditions numerically and study their effectiveness in a test problem consisting of a perturbed Schwarzschild black hole.

After reviewing the known stability results for vacuum, U(1) symmetric solutions to Einstein’s field equations, we shall describe some mathematical ideas and techniques that were unavailable during the earlier research in this area and which might conceivably be exploited to shed new light on the large data global Cauchy problem for this interesting class of spacetimes.

Late-time acceleration with a scalar field source: Observational constraints and statefinder diagnostics

The different kinds of self-similarity in general relativity are discussed, with special emphasis on similarity of the `first' kind, corresponding to spacetimes admitting a homothetic vector. We then survey the various classes of self-similar solutions to Einstein's field equations and the different mathematical approaches used in studying them. We focus mainly on spatially homogenous and spherically symmetric self-similar solutions, emphasizing their possible roles as asymptotic states for more general models. Perfect fluid spherically symmetric similarity solutions have recently been completely classified, and we discuss various astrophysical and cosmological applications of such solutions. Finally, we consider more general types of self-similar models.