John Stachel

Einstein and the History of General Relativity

This book is a collection of essays by the authors and other people that deal with scientific opinions that led Einstein and his contemporaries to their views of general relativity. Some of the essays explore Einsteins passage from the special theory through a sequence of gravitational theories to the discovery of the field equations of the grand theory in November 1915. Two other essays discuss Einsteins public and private exchanges with Max Abraham and Tullio Levi-Civita in 1913 and 1914. A sympathetic picture of H.A. Lorentzs reaction to the general theory of relativity is included, and a careful and insightful essay on the early understanding of the Schwarzschild-Droste solution to the field equations of general relativity is presented. One paper presents a discussion on the state of the enterprise of general relativity between 1925 and 1928, and a short essay details the history of steps toward quantum gravitational through canonical quantization. A discussion of the history of derivations of the geodesic equation of motion from the field equation and conservation laws of the general theory is presented. The early history of geometrical unified field theories is included.

Conformal two‐structure as the gravitational degrees of freedom in general relativity

In this paper, we suggest that what we shall call the conformal 2‐structure may, in an appropriate coordinate system, serve to embody the two gravitational degrees of freedom of the Einstein (vacuum) field equations. The conformal 2‐structure essentially gives information concerning the manner in which a family of 2‐surfaces is embedded in a 3‐surface. We show that, formally at least, this prescription works for the exact plane and cylindrical gravitational wave solutions, for the double‐null and null‐timelike characteristic initial value problems, and for the usual Cauchy spacelike initial value problem. We conclude with a preliminary consideration of a two‐plus‐two breakup of the field equations aimed at unifying these and other initial value problems; and a discussion of some aspirations and remaining problems of this approach.

Cylindrical Gravitational News

The concepts of news function and mass aspect are generalized to a class of cylindrically symmetric metrics containing both degrees of freedom of the gravitational field. It is proved that the mass/unit length always decreases if there is any cylindrical news. The asymptotic behavior of the Riemann tensor in the cylindrical case is analyzed and a peeling theorem proved for this case. An example is given to show that asymptotic conditions on the metric or the Riemann tensor which are analogous to the conditions used in the asymptotically spherical case do not exclude certain infinite incoming radiation trains in the cylindrical case. Pure incoming and outgoing solutions are defined for the cylindrical case, and their generalization to the asymptotically spherical case is suggested. An exactly conserved quantity is shown to exist which may be the cylindrical analog of the ten exactly conserved quantities recently discovered by Newman and Penrose.

Did Malament prove the non-conventionality of simultaneity in the special theory of relativity?

David Malaments (1977) well-known result, which is often taken to show the uniqueness of the Poincare-Einstein convention for defining simultaneity, involves an unwarranted physical assumption: that any simultaneity relation must remain invariant under temporal reflections. Once that assumption is removed, his other criteria for defining simultaneity are also satisfied by membership in the same backward (forward) null cone of the family of such cones with vertices on an inertial path. What is then unique about the Poincare-Einstein convention is that it is independent of the choice of inertial path in a given inertial frame, confirming a remark in Einstein 1905. Similarly, what is unique about the backward (forward) null cone definition is that it is independent of the state of motion of an observer at a point on the inertial path.

Hilbert's foundation of physics : from a theory of everything to a constituent of general relativity

Remarkably, Einstein was not the first to discover the correct form of the law of warpage [of space-time, i.e. the gravitational field equations], the form that obeys his relativity principle. Recognition for the first discovery must go to Hilbert. In autumn 1915, even as Einstein was struggling toward the right law, making mathematical mistake after mistake, Hilbert was mulling over the things he had learned from Einstein’s summer visit to Göttingen. While he was on an autumn vacation on the island of Rugen in the Baltic the key idea came to him, and within a few weeks he had the right law–derived not by the arduous trial-and-error path of Einstein, but by an elegant, succinct mathematical route. Hilbert presented his derivation and the resulting law at a meeting of the Royal Academy of Sciences in Göttingen on 20 November 1915, just five days before Einstein’s presentation of the same law at the Prussian Academy meeting in Berlin. 2

Einstein Tensor and Spherical Symmetry

The classification of symmetric second‐rank tensors in Minkowski space and its application to the Einstein tensor is reviewed. It is shown that, for spherically symmetric metrics, the Einstein tensor always has a spacelike double eigenvector; and the possible types of Einstein tensor that this degeneracy allows are discussed. A complete classification of all spherically symmetric metrics with two double eigenvalues is given. A study of the timelike eigencongruence, in the case when one timelike and two spacelike eigenvectors exist, is carried out. Canonical forms for the metric, the Einstein tensor, and the Weyl tensor (which is always of type D) are given for each of the various possible types.

Einstein Tensor and 3‐Parameter Groups of Isometries with 2‐Dimensional Orbits

The algebraic classification of the Weyl and Ricci tensors and the relation between them in a Riemann space with an isometry group possessing a nontrivial isotropy group are reviewed. All metrics with Minkowski signature, invariant under a 3‐parameter isometry group with 2‐dimensional orbits having nondegenerate metrics, are constructed from the group properties and are shown to have Ricci tensors with a double eigenvalue, and the orbits are shown to be surfaces of constant curvature. The null orbits are shown to have a triply degenerate eigenvalue of the Ricci tensor. The various additionally degenerate metrics are classified in further detail, extending the work of Plebanski and Stachel.

A theory of everything

In his later years, Einstein sought a unified theory that would extend general relativity and provide an alternative to quantum theory. There is now talk of a ‘theory of everything’ (although Einstein himself never used the phrase). Fifty years after his death, how close are we to such a theory?

The genesis of general relativity

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The hole argument for covariant theories

The hole argument was developed by Einstein in 1913 while he was searching for a relativistic theory of gravitation. Einstein used the language of coordinate systems and coordinate invariance, rather than the language of manifolds and diffeomorphism invariance. He formulated the hole argument against covariant field equations and later found a way to avoid it using coordinate language. In this paper we shall use the invariant language of categories, manifolds and natural objects to give a coordinate-free description of the hole argument and a way of avoiding it. Finally we shall point out some important implications of further extensions of the hole argument to sets and relations for the problem of quantum gravity.