Editorial note to Wolfgang Kundt and Manfred Trüm

Authors: Malcolm A H MacCallum

Publish Date: 2016/03/18

Volume: 48, Issue: 4, Pages: 43-

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Abstract

This is the fifth and last of the famous ‘Hamburg bible’ series to appear as a Golden Oldie for the first four see 1 2 3 4 The series as a whole was described in Ellis’s editorial note to the translation of paper I 1 and its several authors are named on the title page of the present paper This fifth contribution 5 summarizes key points of the earlier papers and applies them in particular results from papers II and IV in the context of the propagation of gravitational radiation when matter is present As Ellis noted the paper “is noteworthy for the systematic use of the covariant full Bianchi identities to determine exact properties of such solutions with null type N Weyl tensors proving interesting nonexistence theorems for the matter flows in these cases and examining properties of pure radiation fields” The study of algebraically special solutions with perfect fluids which this paper began now fills a chapter Chapter 33 of 6 and many algebraically special solutions with electromagnetic or pure radiation fields appear elsewhere in Part III of 6The paper opens by giving the field equations and Bianchi identities In the latter the decomposition of curvature 113 is used leading to the equivalent set of equations 116 and 117 and the identity 1110 As the authors remark “119 are the equations of motion for the sources 1110 are differential equations for the free part of the field and 118 give the interaction between the sources and the free part of the field”The next section gives some of the formulae for matter discussed by Ehlers 4 in the special case of a perfect fluid and substitutes these into the equations of the first section slightly generalizing Ehlers’ version of 128 by allowing nonbarotropic fluids Section 13 then opens the issue of finding solutions containing both perfect fluid and a gravitational null field a field of Petrov type N Type N fields had been identified in earlier work as the “far fields of spatially bounded sources of radiation” and the objective here was to begin to study how such radiation linked to the fields’ sources The formalism and equations used based on the tangent vector ka to the null rays largely follow paper II of the series 2Theorem 131 summarizes the resulting constraints on the properties of the fluid and in particular shows that the only conformally flat perfect fluid spacetimes with an equation of state mu = mu p t are the Friedman universes a converse appears as Theorem 141 In 7 Ellis ascribes this result to Trümper and notes that if the equation of state assumption is dropped inhomogeneous models are allowed 8 In fact all conformally flat perfect fluids were found in the same year as 8 by Stephani 9 see also 6 Theorem 3717 They are either generalized Schwarzschild or generalized Friedman solutionsSection 15 ends Part 1 with a discussion of Petrov N fields with an irrotational fluid Evaluating the previous equations under this restriction leads to the Theorem 151 that there are no such solutions if the fluid is ‘dust’ ie pressureless or if the fluid flow is geodesic It was later shown that even if rotation is allowed there are still no solutions with geodesic flow 10 Kundt and Trümper end by inferring that there could only be irrotational solutions if mu = p + At which they discard on physical grounds these solutions were later determined by Oleson 10 11 cf 6 section 334 See also 12As the authors state the notation and results of paper II in the series 2 are used heavily the main change is that the ka of paper II becomes ell a here Section 22 reviews the properties of geodesic null congruences the correspondence with the near contemporary and now widelyused Newman–Penrose notation 13 as used in 6 is that z sigma beta varOmega and zeta are respectively the NP quantities rho barsigma gamma + bargamma tau and baralpha +beta Theorem1 21 states an easy consequence of the NP equation for the derivative of rho along the rays which complements a result in paper II There it was shown that when R abell aell b=0 with the present ell a sigma =theta =0 Rightarrow omega =0 here it is shown that similarly omega =theta =0 Rightarrow sigma =0

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