Introduction to the mathematics of general relativity

About: Introduction to the mathematics of general relativity is a research topic. Over the lifetime, 2583 publications have been published within this topic receiving 73295 citations.

On Gauss' theorem and the concept of mass in general relativity

Abstract: The present communication is concerned with the extension to general relativity theory of the well-known theorem of Gauss on the Newtonian potential, viz., that the total flux of gravitational force through a simple closed surface is equal to (-4π) x the total gravitating mass contained within the surface: and to various questions which arise in connection with this. In the extended theorem, which is found in 2, the Newtonian concept of "gravitating mass" is naturally replaced by that of the energy-tensor, which does not in general consist solely of the "material" energy-tensor, and need not involve any "matter" at all. This new feature is illustrated in 3 by an example in which the "gravitating mass" is simply an electrostatic field. In 4 a theorem of "energy" is obtained which is required later, and which enables us to make precise the concept of the "potential energy" of an infinitesimal particle in a statical field in general relativity; this "potential energy" is shown to be the product of two factors, one depending on the particle alone (which may be called its "potential mass") and the other depending solely on its position. It is shown in 5 that the definition of "potential mass" introduced in 4 enables us to express the generalized Gauss' theorem of 2, in the case when the energy-tensor is due to actual masses, by a simple statement practically identical with the original Gauss' theorem of Newtonian theory. Finally in 6 it is shown that the electrostatical form of Gauss' theorem in Newtonian physics, viz., that the total strength of the tubes of force issuing from a closed surface is equal to the total electric charge within the surface, can also be extended to General Relativity, but that this extension is different in character from the gravitational theorem of 2.

The definition of multipole moments for extended bodies

Abstract: There is considerable freedom in the definition of multipole moments of the energy-momentum tensor of an extended body in general relativity. By studying the corresponding Newtonian theory we obtain guidelines which enable us to choose the most suitable definitions in the relativistic theory. In this way we find two sets, the complete moments and the reduced moments, of which the latter are the most natural choice for studying the dynamics of extended bodies. Expressions as explicit integrals are give for both sets, and the multipole equations of motion of the body are given in a form exact to all orders. Proofs of the relativistic results will appear elsewhere.

100 years of relativity: Space-time structure: Einstein and beyond

Abstract: From Newton to Einstein: Development of the Concepts of Space, Time and Space-Time from Newton to Einstein (J Stachel) Einstein's Universe: Gravitational Billiards, Dualities and Hidden Symmetries (H Nicolai) The Nature of Spacetime Singularities (A D Rendall) Black Holes -- An Introduction (P T Chru ciel) The Physical Basis of Black Hole Astrophysics (R H Price) Probing Space-Time Through Numerical Simulations (P Laguna) Understanding Our Universe: Current Status and Open Issues (T Padmanabhan) Was Einstein Right? Testing Relativity at the Centenary (C M Will) Receiving Gravitational Waves (P R Saulson) Relativity in the Global Positioning System (N Ashby) Beyond Einstein: Spacetime in Semiclassical Gravity (L H Ford) Space Time in String Theory (T Banks) Quantum Geometry and Its Ramifications (A Ashtekar) Loop Quantum Cosmology (M Bojowald) Consistent Discrete Space-Time (R Gambini & J Pullin) Causal Sets and the Deep Structure of Spacetime (F Dowker) The Twistor Approach to Space-Time Structures (R Penrose).

Quadratic lagrangians and general relativity

Abstract: Quadratic invariants of the Riemann-Christoffel curvature tensor and its contractions in a four-dimensional Riemann space are used as the Lagrangians in three variational principles. The field equations are derived by treating the metric tensor and the arbitrary symmetric affine connection as independent variables (following the method of Palatini), and specializing to the Christoffel connection after the variation. It is shown that the field equations derived from two of these variational principles in this way have as a class of solutions all solutions of Einstein’s equations with cosmological term, whilst all three sets of field equations are satisfied by the Schwarzschild metric and have vanishing divergence. This suggests alternative forms of the field equations for gravitation, quadratic in the Riemann-Christoffel tensor and with zero trace, which give the same results for the three «crucial tests» of general relativity as Einstein’s equationsR ik =0.