Superstring theoryをめぐって(放談室)

Gravitational Field of a Spinning Mass as an Example of Algebraically Special Metrics

We review black-hole solutions of higher-dimensional vacuum gravity and higher-dimensional supergravity theories. The discussion of vacuum gravity is pedagogical, with detailed reviews of Myers-Perry solutions, black rings, and solution-generating techniques. We discuss black-hole solutions of maximal supergravity theories, including black holes in anti-de Sitter space. General results and open problems are discussed throughout.

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Black Holes in Higher Dimensions

The title immediately brings to mind a standard reference of almost the same title [1]. The authors are quick to point out the relationship between these two works: they are complementary. The purpose of this work is to explain what is known about a selection of exact solutions. As the authors state, it is often much easier to find a new solution of Einstein's equations than it is to understand it. Even at first glance it is very clear that great effort went into the production of this reference. The book is replete with beautifully detailed diagrams that reflect deep geometric intuition. In many parts of the text there are detailed calculations that are not readily available elsewhere. The book begins with a review of basic tools that allows the authors to set the notation. Then follows a discussion of Minkowski space with an emphasis on the conformal structure and applications such as simple cosmic strings. The next two chapters give an in-depth review of de Sitter space and then anti-de Sitter space. Both chapters contain a remarkable collection of useful diagrams. The standard model in cosmology these days is the ICDM model and whereas the chapter on the Friedmann-Lema?tre?Robertson?Walker space-times contains much useful information, I found the discussion of the currently popular a representation rather too brief. After a brief but interesting excursion into electrovacuum, the authors consider the Schwarzschild space-time. This chapter does mention the Swiss cheese model but the discussion is too brief and certainly dated. Space-times related to Schwarzschild are covered in some detail and include not only the addition of charge and the cosmological constant but also the addition of radiation (the Vaidya solution). Just prior to a discussion of the Kerr space-time, static axially symmetric space-times are reviewed. Here one can find a very interesting discussion of the Curzon?Chazy space-time. The chapter on rotating black holes is rather brief and, for example, does not contain reference to the insights found by Pretorius and Israel [2]. This is perhaps justifiable in view of the many specialized texts devoted to the Kerr space-time (e.g. [3]). The large clear diagrams that one becomes accustomed to in this book show off the Taub-NUT (and related) space-times in the next chapter. After perhaps a somewhat standard discussion of stationary axially symmetric space-times, there is a very informative discussion of accelerating black holes. For example, the global structure of the C-metric is considered in detail. This is followed by a brief discussion of solutions for uniformly accelerating particles. The discussion of the Pleba?ski-Demia?ski solutions contains two very useful flow charts that help to systematize two rather complex families of solutions. After a somewhat brief discussion of plane and pp-waves, the authors give an extensive discussion of the Kunt solutions. I note here that after this text was in production the importance of the Kunt space-times as regards the characterization of space-times by scalar curvature invariants was made clear [4]. The discussion of the Robinson-Trautman solutions that follows is extensive, containing, for example, details of the singularity structure and of the global structure. The final formal chapter in this text covers colliding plane waves. This contains, for example, discussions of the Khan?Penrose, Ferrari?Iba?ez and Chandrasekhar?Xanthopoulos solutions. The text ends with a `final miscellany'. This covers a number of interesting topics, but I found the discussion of the Lema?tre?Tolman solutions rather weak (compare e.g. [5]). The book has two quite useful appendices covering 2-spaces and 3-spaces of constant curvature. To conclude, I will quote from the dust jacket: `The book is an invaluable resource for both graduate students and academic researchers working in gravitational physics'. I highly recommend it. References [1] Stephani H, Kramer D, MacCallum M, Hoenselaers C and Herlt E 2003 Exact Solutions of Einstein's Field Equations (Second Edition) (Cambridge: Cambridge University Press) [2] Pretorius F and Israel W 1998 Class. Quantum Grav.15 2289 [3] Wiltshire D, Visser M and Scott S (ed) 2008 The Kerr Spacetime: Rotating Black Holes in General Relativity (Cambridge: Cambridge University Press) [4] Coley A, Hervik S and Pelavas N 2009 Class. Quantum Grav. 26 025013 [5] Pleba?ski J and Krasi?ski A 2006 An Introduction to General Relativity and Cosmology (Cambridge: Cambridge University Press)

Exact Space-Times in Einstein's General Relativity

: The final report covering the research performed under the Air Force Office of Scientific Research by the Dallas relativity group, summarizes information on personnel involved, on the scientific work done, and on the results obtained in the four years of the grant's duration. Topics investigated were in areas of General relativity, cosmology, and electrodynamics.

Exact Solutions of Einstein's Field Equations.

We discuss the algebraic classification of the Weyl tensor in higher-dimensional Lorentzian manifolds. This is done by characterizing algebraically special Weyl tensors by means of the existence of aligned null vectors of various orders of alignment. Further classification is obtained by specifying the alignment type and utilizing the notion of reducibility. For a complete classification it is then necessary to count aligned directions, the dimension of the alignment variety and the multiplicity of principal directions. The present classification reduces to the classical Petrov classification in four dimensions. Some applications are briefly discussed.

Classification of the Weyl tensor in higher dimensions

We develop a dimension-independent theory of alignment in Lorentzian geometry, and apply it to the tensor classification problem for the Weyl and Ricci tensors. First, we show that the alignment condition is equivalent to the principal null direction equation. In 4 dimensions this recovers the usual Petrov types are recovered. For higher dimensions we prove that, in general, a Weyl tensor does not possess any aligned directions. We then go on to describe a number of additional algebraic types for the various alignment configurations. For the case of second-order symmetric (Ricci) tensors, we perform the classification by considering the geometric properties of the corresponding alignment variety.

Alignment and algebraically special tensors in lorentzian geometry

We study manifolds with Lorentzian signature and prove that all scalar curvature invariants of all orders vanish in a higher dimensional Lorentzian spacetime if and only if there exists an aligned non-expanding, non-twisting, geodesic null direction along which the Riemann tensor has negative boost order.

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Vanishing scalar invariant spacetimes in higher dimensions

All Lorentzian spacetimes with vanishing invariants constructed from the Riemann tensor and its covariant derivatives are determined. A subclass of the Kundt spacetimes results and we display the corresponding metrics in local coordinates. Some potential applications of these spacetimes are discussed.

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All spacetimes with vanishing curvature invariants

A higher dimensional frame formalism is developed in order to study implications of the Bianchi identities for the Weyl tensor in vacuum spacetimes of the algebraic types III and N in arbitrary dimension n. It follows that the principal null congruence is geodesic and expands isotropically in two dimensions and does not expand in n − 4 spacelike dimensions or does not expand at all. It is shown that the existence of such principal geodesic null congruence in vacuum (together with an additional condition on twist) implies an algebraically special spacetime. We also use the Myers–Perry metric as an explicit example of a vacuum type D spacetime to show that principal geodesic null congruences in vacuum type D spacetimes do not share this property.

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Vojtech Pravda

Papers

Classification of the Weyl tensor in higher dimensions

Alignment and algebraically special tensors in lorentzian geometry

Vanishing scalar invariant spacetimes in higher dimensions

All spacetimes with vanishing curvature invariants

Bianchi identities in higher dimensions