Eduard Herlt

Static and stationary axially symmetric gravitational fields of bounded sources. I. Solutions obtainable from the van Stockum metric

This paper shows that a new class of axially symmetric static electrovacuum solutions and the Kerr solution are obtainable from the van Stockum metric. The new class contains an infinite set of asymptotically flat solutions (in closed form), each of which involves an arbitrary set of parameters. The parameters have to be interpreted as functions of massm, chargee, and higher electric multipole momentsαi of the particle. The casee=αi=0 leads to the Darmois metric. Well-known and new examples are given.

Stationary axisymmetric perfect fluid solutions in general relativity

A new class of exact solutions of Einsteins field equations with the energy-momentum tensor of a perfect fluid is given. The class of solutions is invariantly characterized by means of the following properties: (i) The energy-momentum tensor describes a perfect fluid. (ii) There are two commuting Killing vectors ξ andη which form an abelian groupG2 of motion. (iii) There is a timelike Killing vector parallel to the four-velocity of the fluid (rigid rotation of the fluid). (iv) The four-vector of the angular velocity of the fluid is a gradientΩi=−(1/4c)ɛirklUl (Ur:k−Uk:r)=χ′i. The last assumption is the reason that all solutions of this class can be found by solving an ordinary differential equation of the second order.

Kerr-Schild-Vaidya fields with axial symmetry

This paper contains the Kerr-Schild-Vaidya fields with axial symmetry (all metric functions independent of Vaidyas coordinateβ) in closed form. The general problem of Kerr-Schild pure radiation fields without any symmetry can be reduced to a single partial differential equation by means of Kerrs theorem.

Reduction of Einstein's field equations for twisting typeN-vacuum fields with anH2

The problem of vacuum typeN solutions of Einsteins field equations with Killing vector and homothetic group can be reduced to a single ordinary differential equation of the third order for a single real function. Hausers solution [4] is an exceptional case. If the homothetic parameterN takes the valueN=2 the third order differential equation becomes surprisingly simple. This caseN=2 is therefore the most promising one for the search of exact solutions.

A new solution of Schmutzer's projective relativistic cosmology

Schmutzers cosmological model on the basis of his projective unified field theory leads to a system of differential equations equivalent to a set of highly complicated Abel differential equations [1, 2]. This paper is concerned with the properties of this coupled system of differential equations. The main results of this paper are as follows, (i) We have found the explicit general solution of the cosmological equations for the caseγ=±5/2,ɛ= ±1. (ii) The procedure of solving the general caseγ arbitraryɛ=0 is easily within this formalism, (iii) After having found the solution withγ arbitrary,ɛ=±1, one gets the corresponding solutionγ= -γ, ɛ=±1. (iv) In general the coupled system of differential equations describing Schmutzers cosmological model is equivalent to a single generalized Emden differential equation and it is identical to Emdens differential equation for special values of the parameterγ.

Static and stationary axially symmetric gravitational fields of bounded sources II. solutions obtainable from Weyl's class

This paper shows that a new class of axially symmetric static electrovacuum/magnetovacuum solutions is obtainable from Weyls class of static vacuum solutions. The new class contains an infinite set of asymptotically flat solutions (in closed form) each of which involves an arbitrary set (d, αi) of parameters. These parameters have to be interpreted as functions of massm, chargee, and higher electric/magnetic multipole momentsαi of the particle. The cased = 0,αi=0 leads to the Darmois solution and the cased = 0,αi≠0 leads to the results of [1]. The case d≠=0, e=αi=0 leads to the Schwarzschild solution, the cased≠ 0,αi=0,e≠ 0 leads to the Reissner-Nordström solution. To get more general examples is a lengthy but straightforward exercise.

Exact Solutions of Einstein's Field Equations: Hypersurface-homogeneous space-times

The possible metrics This chapter is concerned with metrics admitting a group of motionstransitive on S 3 or T 3 . Some solutions, such as the well-known Taub– NUT (Newman, Unti, Tamburino) metrics (13.49), cover regions of both types, joined across a null hypersurface which is a special group orbit (metrics admitting a G r whose general orbits are N 3 are considered in Chapter 24). As in the case of the homogeneous space-times (Chapter 12) we first consider the cases with multiply-transitive groups. From Theorems 8.10 and 8.17 we see that only G 6 and G 4 are possible. Metrics with a G 6 on V 3 From §12.1, the space-times with a G 6 on S 3 have the metric (12.9); this always admits G 3 transitive on hypersurfaces t = const and the various cases are thus included in (13.1)–(13.3) and (13.20) below. The relevant G 3 types are V and VIIh if k = -1, I and VII 0 if k = 0, and IX if k = 1. Of the energy-momentum tensors considered in this book, the spacetimes with a G 6 on T 3 permit only vacuum and Λ-term Ricci tensors (see Chapter 5). Thus they will give only the spaces of constant curvature, with a complete G 10 , which also arise with G 6 on S 3 and those energymomentum types. Metrics with maximal G 6 on S 3 are non-empty and have an energy-momentum of perfect fluid type: see §14.2.

Exact Solutions of Einstein's Field Equations: List of tables

Exact solutions of Einsteins field equations. – 2nd ed. / H. Stephani. .. [et al.]. p. cm. – (Cambridge monographs on mathematical physics) Includes bibliographical references and index. Contents Preface xix List of tables xxiii Notation xxvii 1 Introduction 1 1.1 What are exact solutions, and why study them? 1 1.2 The development of the subject 3 1.3 The contents and arrangement of this book 4 1.4 Using this book as a catalogue 7

Exact Solutions of Einstein's Field Equations by Hans Stephani

Exact solutions of Einsteins field equations. – 2nd ed. / H. Stephani. .. [et al.]. p. cm. – (Cambridge monographs on mathematical physics) Includes bibliographical references and index. Contents Preface xix List of tables xxiii Notation xxvii 1 Introduction 1 1.1 What are exact solutions, and why study them? 1 1.2 The development of the subject 3 1.3 The contents and arrangement of this book 4 1.4 Using this book as a catalogue 7

Exact Solutions of Einstein's Field Equations: Frontmatter

Exact solutions of Einsteins field equations. – 2nd ed. / H. Stephani. .. [et al.]. p. cm. – (Cambridge monographs on mathematical physics) Includes bibliographical references and index. Contents Preface xix List of tables xxiii Notation xxvii 1 Introduction 1 1.1 What are exact solutions, and why study them? 1 1.2 The development of the subject 3 1.3 The contents and arrangement of this book 4 1.4 Using this book as a catalogue 7