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.

Some solutions of complex Einstein equations

Abstract: Complex V4’s are investigated where ΓȦḂ =0 and therefore a fortiori equations Gab=0 are fulfilled. A general theory of spaces of this type is outlined and examples of nontrivial solutions of all degenerate algebraic types are provided.

Gravity: An Introduction to Einstein's General Relativity

Abstract: Einstein's theory of general relativity is a cornerstone of modern physics. It also touches upon a wealth of topics that students find fascinating – black holes, warped spacetime, gravitational waves, and cosmology. Now reissued by Cambridge University Press, this ground-breaking text helped to bring general relativity into the undergraduate curriculum, making it accessible to virtually all physics majors. One of the pioneers of the 'physics-first' approach to the subject, renowned relativist James B. Hartle, recognized that there is typically not enough time in a short introductory course for the traditional, mathematics-first, approach. In this text, he provides a fluent and accessible physics-first introduction to general relativity that begins with the essential physical applications and uses a minimum of new mathematics. This market-leading text is ideal for a one-semester course for undergraduates, with only introductory mechanics as a prerequisite.

Space-time structure

Abstract: Introduction Part I. The Unconnected Manifold: 1. Invariance 2. Integrals Part II. Affinely Connected Manifold: 3. Invariant derivatives 4. Some relations between ordinary and invariant derivatives 5. The notion of parallel transfer 6. The curvature tensor 7. The geodesics of an affine connexion 8. The general geometrical hypothesis about gravitation Part III. Metrically Connected Manifold: 9. Metrical affinities 10. The meaning of the metric according to the special theory of relativity 11. Conservation laws and variational principles 12. Generalizations of Einstein's theory.

The teleparallel equivalent of general relativity

Abstract: A review of the teleparallel equivalent of general relativity is presented. It is emphasized that general relativity may be formulated in terms of the tetrad fields and of the torsion tensor, and that this geometrical formulation leads to alternative insights into the theory. The equivalence with the standard formulation in terms of the metric and curvature tensors takes place at the level of field equations. The review starts with a brief account of the history of teleparallel theories of gravity. Then the ordinary interpretation of the tetrad fields as reference frames adapted to arbitrary observers in space–time is discussed, and the tensor of inertial accelerations on frames is obtained. It is shown that the Lagrangian and Hamiltonian field equations allow us to define the energy, momentum and angular momentum of the gravitational field, as surface integrals of the field quantities. In the phase space of the theory, these quantities satisfy the algebra of the Poincare group.

Physical equivalence between nonlinear gravity theories and a general-relativistic self-gravitating scalar field.

Abstract: We argue that in a nonlinear gravity theory (the Lagrangian being an arbitrary function of the curvature scalar R), which according to well-known results is dynamically equivalent to a self-gravitating scalar field in general relativity, the true physical variables are exactly those which describe the equivalent general-relativistic model (these variables are known as the Einstein frame). Whenever such variables cannot be defined, there are strong indications that the original theory is unphysical, in the sense that Minkowski space is unstable due to the existence of negative-energy solutions close to it. To this aim we first clarify the global net of relationships between the nonlinear gravity theories, scalar-tensor theories, and general relativity, showing that in a sense these are ``canonically conjugated'' to each other. We stress that the isomorphisms are in most cases local; in the regions where these are defined, we explicitly show how to map, in the presence of matter, the Jordan frame to the Einstein one and vice versa. We study energetics for asymptotically flat solutions for those Lagrangians which admit conformal rescaling to the Einstein frame in the vicinity of flat space. This is based on the second-order dynamics obtained, without changing the metric, by the use of a Helmholtz Lagrangian. We prove for a large class of these Lagrangians that the ADM energy is positive for solutions close to flat space, and this is determined by the lowest-order terms R+${\mathit{aR}}^{2}$ in the Lagrangian. The proof of this positive-energy theorem relies on the existence of the Einstein frame, since in the (Helmholtz-)Jordan frame the dominant energy condition does not hold and the field variables are unrelated to the total energy of the system. This is why we regard the Jordan frame as unphysical, while the Einstein frame is physical and reveals the physical contents of the theory. The latter should hence be viewed as physically equivalent to a self-interacting general-relativistic scalar field.