Einstein's General Theory Of Relativity

Contents 1. Special Relativity 2. Oblique Axes 3. Curvilinear Coordinates 4. Nontensors 5. Curved Space 6. Parallel Displacement 7. Christoffel Symbols 8. Geodesics 9. The Stationary Property of Geodesics 10. Covariant Differentiation 11. The Curvature Tensor 12. The Condition for Flat Space 13. The Bianci Relations 14. The Ricci Tensor 15. Einstein's Law of Gravitation 16. The Newtonian Approximation 17. The Gravitational Red Shift 18. The Schwarzchild Solution 19. Black Holes 20. Tensor Densities 21. Gauss and Stokes Theorems 22. Harmonic Coordinates 23. The Electromagnetic Field 24. Modification of the Einstein Equations by the Presence of Matter 25. The Material Energy Tensor 26. The Gravitational Action Principle 27. The Action for a Continuous Distribution of Matter 28. The Action for the Electromagnetic Field

2018

A general theory of relativity is formulated without Einstein's equation. Einstein's tensor ties the space metric to the stress-energy tensor of a gravitational field. A homogeneous isotropic field metric is under consideration. In particular, the metric for a homogeneous isotropic universe possesses the anticipated density of our own universe. The speed of time progression at a distance from the observer slows according to an acceleration law asymptotically equal to Hubble's Law. A continuous field is homogeneous within the limits of small domains. This makes it possible to write a metric for a general continuous field and the single parameter given by the wave equation. The Schwarzschild problem has a continuous solution for r > 0. This solution approaches the Schwarzschild solution beyond the Schwarzschild radius and a second solution is obtained from the Schwarzschild problem-a stationary, expulsive, self-consistent field.

2011

Hawking ironically once said, at the beginning of the twentieth century, the only cosmological observation was the statement that "the night sky is dark." The night-sky-paradox remained unsolved in the system that it arose from, collections of ideas and associated mathematical formalism, called today "classical physics". To understand and describe the existence of isotropic and homogeneous universe it is necessary to use the concepts and the language of relativity theory. In the next few chapters we will introduce the concepts of relativity, which allow formulating a model of the Universe.

2006

General relativity is a cornerstone of modern physics, and is of major importance in its applications to cosmology. Plebanski and Krasinski are experts in the field and in this 2006 book they provide a thorough introduction to general relativity, guiding the reader through complete derivations of the most important results. Providing coverage from a unique viewpoint, geometrical, physical and astrophysical properties of inhomogeneous cosmological models are all systematically and clearly presented, allowing the reader to follow and verify all derivations. For advanced undergraduates and graduates in physics and astronomy, this textbook will enable students to develop expertise in the mathematical techniques necessary to study general relativity.

These lecture notes have been prepared as a rapid introduction to Einstein's General Theory of Relativity. Consequently, I have restricted to the standard four dimensional, metric theory of gravity with no torsion. A basic exposure to geometrical notions of tensors, their algebra and calculus, Riemann-Christoffel connection, curvature tensors, etc has been presupposed being covered by other lecturers. Given the time constraint, the emphasis is on explaining the concepts and the physical ideas. Calculational details and techniques have largely been given reference to. The First two lectures discuss the arguments leading to the beautiful synthesis of the idea of space-time geometry, the relativity of observers and the phenomenon of gravity. Heuristic 'derivations' of the Einstein Field equations are presented and some of their mathematical properties are discussed. The (simplest) Schwarzschild solution is presented. The next lecture discusses the standard solar system tests of Einstein's theory. The fourth lecture returns to static, spherically symmetric solutions namely the interiors of stars. This topic is discussed both to illustrate how non-vacuum solutions are constructed, how the Einstein's gravity affects stellar equilibria and hold out the possibility of complete, un-stoppable gravitational collapse. The concept of a black hole is introduced via the example of the Schwarzschild solution with the possibility of a physical realization justified by the interior solution. The fifth lecture describes the Kerr-Newman family of black holes. More general (nonstationary) black holes are defined and the laws of black hole mechanics are introduced. Their analogy with the laws of thermodynamics is discussed. This topic is of importance because it provides an arena from where the glimpses of interaction of GR and quantum theory can be hoped for. The cosmos is too large and too real to be ignored. So the last lecture is devoted to a view of the standard cosmology. Some additional material is included in an appendix. A collection of exercises meant for practice are also included.

Classical and Quantum Gravity, 1999

This paper considers the evolution of the relation between gravitational theory and cosmology from the development of the first simple quantitative cosmological models in 1917 to the sophistication of our cosmological models at the turn of the millenium. It is structured around a series of major ideas that have been fundamental in developing today's models, namely: 1, the idea of a cosmological model; 2, the idea of an evolving universe; 3, the idea of astronomical observational tests; 4, the idea of physical structure development; 5, the idea of causal and visual horizons; 6, the idea of an explanation of spacetime geometry; and 7, the idea of a beginning to the universe. A final section considers relating our simplified models to the real universe, and a series of related unresolved issues that need investigation.

, except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights.

Researchgate , 2019

We consider a (4+D)-dimensional Friedmann-Robertson-Walker type universe having complex scale factor R + iR I , where R is the scale factor corresponding to the usual 4-dimensional Universe while iR I is that of D-dimensional space. It is then compared with (4+D)-dimensional Kaluza-Klein Cosmology having two scale factors R and a(= iR I). It is shown that the rate of compactification of higher dimension depends on extra dimension 'D'. The Wheeler-DeWitt equation is constructed and general solution is obtained. It is found that for D = 6 (i.e. in 10 dimension), the Wheeler-DeWitt equation is symmetric under the exchange of R I R. I. Introduction: In 1915 Einstein published the general theory of relativity. He expected the universe to be 'closed' and to be filled with matter. Now, if we go outside the gravitating sphere, we see the gravitation would be weaker and weaker. According to Einstein's theory of general relativity, the matter-space-time cannot be separated by any cost. Thus, outside the Einstein's universe, where real time cannot be defined, the corresponding space (although, the matter belongs to another phase) must be measured as imaginary. Thus the space-time of the universe is actually a complex space-time. Here we consider the real space-time (unfolded space-time) for Einstein and imaginary space-time (folded space-time) for us. We found a relation between folded and unfolded space-time of the universe by using Wheeler De-Witt equation. The generalized solution for the Einstein field equations for a homogeneous universe was first presented by Alexander Friedmann. The Friedmann equation for the evolution of the cosmic scale factor R(t) which represents the size of the universe, is Differentiating the above equation with respect to time t and since the total matter in a given expanding volume is unchanged, that means = constant. We have, i.e

Springer Proceedings in Mathematics & Statistics, 2014

The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. While the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein.

General Relativity, 2020

Notes on general relativity. The topics that are covered -Fast introduction and recap of special relativity -Gravity and metrics (Rindler spacetime) -Basics in differential geometry -Free Point particles dynamics -Covariant derivatives -Newtonian limit -Curvature and Gravity: Eintein equation (+perfect fluid) -Gravitational waves -Vielbein formalism ( Schwarzschild) -Symmetries and study of Schawazschild -Cosmological space-time (De Sitter spacetime, Godesic and Perfect Fluid) -BH Introduction (Approaching a horizon, Carter-Penrose diagram , hyperbolic spacetime, Surface gravity, charged BH, rotating BH, Intro to Thermodynamics) -Exercises