Contributions to the relativistic mechanics of continuous media

Abstract

This is a translation from German of an article originally published inProceedings of the Mathematical-Natural Science Section of the Mainz Academy of Science and Literature, Nr. 11, 1961 (pp. 792–837) (printed by Franz Steiner and Co, Wiesbaden), which is Paper IV in the series ldquoExact Solutions of the Field Equations of General Relativity Theoryrdquo by Pascual Jordan, Jürgen Ehlers, Wolfgang Kundt and Rainer K. Sachs. The translation has been carried out by G. F. R. Ellis (Department of Applied Mathematics, University of Cape Town), assisted by P. K. S. Dunsby, so that this outstanding review paper can be readily accessible to workers in the field today. As far as possible, the translation has preserved both the spirit and the form of the original paper. Despite its age, it remains one of the best reviews available in this area

Cosmological dynamics on the brane

“…Rg ν]α ) produce the fundamental evolution and constraint equations governing the above covariant quantities [11]. Einstein's equations are incorporated via the algebraic replacement of the Ricci tensor R µν by the effective total energy-momentum tensor, according to Eq.…”

“…We find the bulk corrections to the propagation and constraint equations, using the covariant Lagrangian approach [11,12]. This approach is well-suited to identifying the geometric and physical quantities that determine inhomogeneity and anisotropy on the brane, and it is also the basis for a gauge-invariant perturbation theory [13].…”

Editorial note to: Pascual Jordan, Jürgen Ehlers and Wolfgang Kundt, Exact solutions of the field equations of the general theory of relativity

“…The fourth paper [8], by Ehlers, dealt with the general-relativistic treatment of the mechanics of continuous media. It particularly developed the use of the kinematic quantities for timelike curves to characterise fluid solutions, and gives a particularly clear covariant derivation of the fundamentally important Raychaudhuri equation [15], 123 2182 G. Ellis generalized to non-geodesic curves and arbitrary matter.…”

“…The fourth one (by Ehlers alone) has already been published as a Golden Oldie, translated into English [8]; the others in the series will now be so published, beginning with the classic paper by Ehlers and Kundt [5].…”

“…5 More than that, the focus was on exact solutions (Strenge Lösungen) of the Einstein Field Equations of General Relativity Theory, seeing what could be achieved without approximations: a crucial effort in the case of a theory as non-linear as General Relativity. It resulted in a major series of papers, the so-called Hamburg Bible [5][6][7][8][9], setting out the results of this work. 6 Jordan's name was on the first three of the papers, although he in fact did not take part in writing them: his name was there simply because he was the head of the group from which they came.…”