On the Physics in Fundamental General Relativity Theory

2008

2010

The General Theory of Relativity (GTR) is essentially a theory of gravitation. It is built on the Principle of Relativity. It is bonafide knowledge, known even to Einstein the founder, that the GTR violates the very principle upon which it is founded i.e., it violates the Principle of Relativity; because a central equation i.e., the geodesic law which emerges from the GTR, is well known to be in conflict with the Principle of Relativity because the geodesic law, must in complete violation of the Principle of Relativity, be formulated in special (or privileged) coordinate systems i.e., Gaussian coordinate systems. The Principle of Relativity clearly and strictly forbids the existence/use of special (or privileged) coordinate systems in the same way the Special Theory of Relativity forbids the existence of privileged and or special reference systems. In the pursuit of a more Generalized Theory of Relativity i.e., an all-encampusing unified field theory to include the Electromagnetic, Weak & the Strong force, Einstein and many other researchers, have successfully failed to resolve this problem. In this reading, we propose a solution to this dilemma faced by Einstein and many other researchers i.e., the dilemma of obtaining a more Generalized Theory of Relativity. Our solution brings together the Gravitational, Electromagnetic, Weak & the Strong force under a single roof via an extension of Riemann geometry to a new hybrid geometry that we have coined the Riemann-Hilbert Space (RHS). This geometry is a fusion of Riemann geometry and the Hilbert space. Unlike Riemann geometry, the RHS preserves both the length and the angle of a vector under parallel transport because the affine connection of this new geometry, is a tensor. This tensorial affine leads us to a geodesic law that truly upholds the Principle of Relativity. It is seen that the unified field equations derived herein are seen to reduce to the well known Maxwell-Procca equation, the non-Abelian nuclear force field equations, the Lorentz equation of motion for charged particles and the Dirac equation.

Reviews of Modern Physics, 1948

' 'N the following we shall give a new presentation of the generalized theory of gravitation, which constitutes a certain progress in clarity as compared to the previous presentations. * It is our aim to achieve a theory of the total field by a generalization of the concepts and methods of the relativistic theory of gravitation. i. THE FIELD STRUCTURE The theory of gravitation represents the field by a symmetric tensor g;~, i.e. , g;q=gq;(i, k=&,~, 4), where the g,t ar e real functions of Xgp ' ' ' X4 In the generalized theory the total field is represented by a Hermitian tensor. The symmetry property of the (complex) g;& is gik gkiĨ f we decompose g,& into its real and imaginary components, then the former is a symmetric tensor (g;&), the latter an antisymmetric tensor (gp). The g;s are still functions of the real variables x~, .~, x4.

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

Filomat, 2015

This paper gives a brief survey of the development of general relativity theory starting from Newtonian theory and Euclidean geometry and proceeding through to special relativity and finally to general relativity and relativistic cosmology.

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.

International Journal of Theoretical and Applied Physics (IJTAP), Vol. 4, No. I , pp. 9-26, 2014

The axiomatization of general theory of relativity (GR) is done. The axioms of GR are compared with the axioms of the metric theory of relativity and the covariant theory of gravitation. The need to use the covariant form of the total derivative with respect to the proper time of the invariant quantities, the 4-vectors and tensors is indicated. The definition of the 4-vector of force density in Riemannian spacetime is deduced.

General Relativity and Gravitation, 2011

We prove that some basic aspects of gravity commonly attributed to the modern connection-based approaches, can be reached naturally within the usual Riemannian geometry-based approach, by assuming the independence between the metric and the connection of the background manifold. These aspects are: 1) the BFlike field theory structure of the Einstein-Hilbert action, of the cosmological term, and of the corresponding equations of motion; 2) the formulation of Maxwellian field theories using only the Riemannian connection and its corresponding curvature tensor, and the subsequent unification of gravity and gauge interactions in a four dimensional field theory; 3) the construction of four and three dimensional geometrical invariants in terms of the Riemann tensor and its traces, particularly the formulation of an anomalous Chern-Simons topological model where the action of diffeomorphisms is identified with the action of a gauge symmetry, close to Witten's formulation of threedimensional gravity as a Chern-Simon gauge theory. 4) Tordions as propagating and non-propagating fields are also formulated in this approach. This new formulation collapses to the usual one when the metric connection is invoked, and certain geometrical structures very known in the traditional literature can be identified as remanent structures in this collapse.