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.

Mach’s principle and the structure of dynamical theories

Abstract: A structure of dynamical theories is proposed that implements Mach’s ideas by being relational in its treatment of both motion and time. The resulting general dynamics, which is called intrinsic dynamics and by construction treats the evolution of the entire Universe, is shown to admit as special cases Newtonian dynamics and Lorentz-invariant field theory provided the angular momentum of the Universe is zero in the frame in which its momentum is zero. The formal structure of Einstein’s general theory of relativity also fits the pattern of intrinsic dynamics and is Machian according to the criteria of this paper provided the so-called thin-sandwich conjecture is generically correct.

Spin and angular momentum in general relativity

Abstract: In the general theory of relativity, the group of coordinate transformations gives rise to four point-to-point conservation laws, which are usually identified with energy and linear momentum. In the presence of a semiclassical Dirac field, it is convenient to introduce at each point of space-time an arbitrary set of four orthonormal vectors (quadrupeds, "beine") and to consider the group of "bein" transformations, which then play the role of local, nonholonomic lorentz transformations. A search for the corresponding conservation laws leads to terms that have the form of a spin angular momentum and which, in order to be conserved, must be supplemented by terms representing the orbital angular momentum. The technique of the so-called superpotentials has enabled us to introduce, in addition to the canonical stress-energy, a "contravariant" stress-energy which contains the usual symmetric Dirac and Maxwell terms and also asymmetric, purely gravitational terms. It is this set of expressions which enters into the orbital angular momentum. The techniques presented here are applicable to more general covariant theories, provided the gravitational field is represented by a metric tensor.

General Relativity and Flat Space. II

Abstract: The possibility is considered of interpreting the formalism of the general theory of relativity in terms of flat space, the fundamental tensor ${g}_{\ensuremath{\mu}\ensuremath{ u}}$ being regarded as describing the gravitational field but having no direct connection with geometry. The resulting theory in general leads to the same predictions as the Einstein theory, but there are cases where the predictions differ. The present theory may explain the principal results obtained by D. C. Miller in his "ether-drift" experiments. The implications of the theory for cosmology are briefly touched upon.

The Einstein evolution equations as a first-order quasi-linear symmetric hyperbolic system. I

Abstract: A systematic presentation of the quasi-linear first order symmetric hyperbolic systems of Friedrichs is presented A number of sharp regularity and smoothness properties of the solutions are obtained The present paper is devoted to the case ofRn with suitable asymptotic conditions imposed As an example, we apply this theory to give new proofs of the existence and uniqueness theorems for the Einstein equations in general relativity, due to Choquet-Bruhat and Lichnerowicz These new proofs usingfirst order techniques are considerably simplier than the classical proofs based onsecond order techniques Our existence results are as sharp as had been previously known, and our uniqueness results improve by one degree of differentiability those previously existing in the literature