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.

Of pots and holes: Einstein's bumpy road to general relativity

Abstract: Readers of this volume will notice that it contains only a few papers on general relativity. This is because most papers documenting the genesis and early development of general relativity were not published in Annalen der Physik.After Einstein took up his new prestigious position at the PrussianAcademy of Sciences in the spring of 1914, the Sitzungsberichte of the Berlin academy almost by default became the main outlet for his scientific production. Two of the more important papers on general relativity, however, did find their way into the pages of the Annalen [35,41].Although I shall discuss both papers in this essay, the main focus will be on [35], the first systematic exposition of general relativity, submitted in March 1916 and published in May of that year. Einstein’s first paper on a metric theory of gravity, co-authored with his mathematician friend Marcel Grossmann, was published as a separatum in early 1913 and was reprinted the following year in Zeitschrift fur Mathematik und Physik [50,51]. Their second (and last) joint paper on the theory also appeared in this journal [52]. Most of the formalism of general relativity as we know it today was already in place in this Einstein-Grossmann theory. Still missing were the generally-covariant Einstein field equations. As is clear from research notes on gravitation from the winter of 1912–1913 preserved in the so-called “Zurich Notebook,” Einstein had considered candidate field equations of broad if not general covariance, but had found all such candidates wanting on physical grounds. In the end he had settled on equations constructed specifically to be compatible with energy-momentum conservation and with Newtonian theory in the limit of weak static fields, even though it remained unclear whether these equations would be invariant under any non-linear transformations. In view of this uncertainty, Einstein and Grossmann chose a fairly modest title for their paper: “Outline (“Entwurf”) of a Generalized Theory of Relativity and of a Theory of Gravitation.” The Einstein-Grossmann theory and its fields equations are therefore also known as the “Entwurf” theory and the “Entwurf” field equations. Much of Einstein’s subsequent work on the “Entwurf” theory went into clarifying the covariance properties of its field equations. By the following year he had convinced himself of three things. First, generallycovariant field equations are physically inadmissible since they cannot determine the metric field uniquely. This was the upshot of the so-called “hole argument” (“Lochbetrachtung”) first published in an appendix to [51]. Second, the class of transformations leaving the “Entwurf” field equations invariant was as broad ∗ E-mail: janss011@tc.umn.edu 1 An annotated transcription of the gravitational portion of the “Zurich Notebook” is published as Doc.10 in [11]. For facsimile reproductions of these pages, a new transcription, and a running commentary, see [89]. 2 See Sect. 2 for further discussion of the hole argument.

Hyperbolicity of the 3 + 1 system of einstein equations

Abstract: By a suitable choice of the lapse, which in a natural way is connected to the space metric, we obtain a hyperbolic system from the 3+1 system of Einstein equations with zero shift; this is accomplished by combining the evolution equations with the constraints.

General Relativity for the Experimentalist

Abstract: Einstein's general theory of relativity is broken down and simplified under limitations usually satisfied in experimentally realizable situations. Following the work of Moller,1 an analogy between electromagnetism and gravitation is presented which allows calculation of various gravitational forces by considering the equivalent electromagnetic problem. A number of examples are included. Tensor formulation is not used except in the Appendix, where justification for the analogies is given.

Invariants of General Relativity and the Classification of Spaces

Abstract: A unique equivalence is established between the Riemann curvature tensor and two spinors. The fourteen invariants which can be constructed from the curvature tensor are listed in terms of the spinors. The vanishing of the invariants for several different types of spaces is described. A classification of Einstein spaces is made together with a few additional remarks concerning classification of spaces.