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Deichmann Edwards, Miguel A. De La Torre, George E. Demacopoulos, Thomas de Mayo, Leah DeVun, Beatriz de Vasconcellos Dias, Dennis C. Dickerson, John M. Dillon, Luis Miguel Donatello, Igor Dorfmann-Lazarev, Susanna Drake, Jonathan A. Draper, N. Dreher Martin, Otto Dreydoppel, Angelyn Dries, A. J. Droge, Francis X. D'Sa, Marilyn Dunn, Nicole Wilkinson Duran, Rifaat Ebied, Mark J. Edwards, William H. Edwards, Leonard H. Ehrlich, Nancy L. Eiesland, Martin Elbel, J. Harold Ellens, Stephen Ellingson, Marvin M. Ellison, Robert Ellsberg, Jean Bethke Elshtain, Eldon Jay Epp, Peter C. Erb, Tassilo Erhardt, Maria Erling, Noel Leo Erskine, Gillian R. Evans, Virginia Fabella, Michael A. Fahey, Edward Farley, Margaret A. Farley, Wendy Farley, Robert Fastiggi, Seena Fazel, Duncan S. Ferguson, Helwar Figueroa, Paul Corby Finney, Kyriaki Karidoyanes FitzGerald, Thomas E. FitzGerald, John R. Fitzmier, Marie Therese Flanagan, Sabina Flanagan, Claude Flipo, Ronald B. 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Phan, Isabel Apawo Phiri, William S. F. Pickering, Derrick G. Pitard, William Elvis Plata, Zlatko Plese, John Plummer, James Newton Poling, Ronald Popivchak, Andrew Porter, Ute Possekel, James M. Powell, Enos Das Pradhan, Devadasan Premnath, Jaime Adrían Prieto Valladares, Anne Primavesi, Randall Prior, María Alicia Puente Lutteroth, Eduardo Guzmão Quadros, Albert Rabil, Laurent William Ramambason, Apolonio M. Ranche, Vololona Randriamanantena Andriamitandrina, Lawrence R. Rast, Paul L. Redditt, Adele Reinhartz, Rolf Rendtorff, Pål Repstad, James N. Rhodes, John K. Riches, Joerg Rieger, Sharon H. Ringe, Sandra Rios, Tyler Roberts, David M. Robinson, James M. Robinson, Joanne Maguire Robinson, Richard A. H. Robinson, Roy R. Robson, Jack B. Rogers, Maria Roginska, Sidney Rooy, Rev. Garnett Roper, Maria José Fontelas Rosado-Nunes, Andrew C. Ross, Stefan Rossbach, François Rossier, John D. Roth, John K. Roth, Phillip Rothwell, Richard E. Rubenstein, Rosemary Radford Ruether, Markku Ruotsila, John E. Rybolt, Risto Saarinen, John Saillant, Juan Sanchez, Wagner Lopes Sanchez, Hugo N. Santos, Gerhard Sauter, Gloria L. Schaab, Sandra M. Schneiders, Quentin J. Schultze, Fernando F. Segovia, Turid Karlsen Seim, Carsten Selch Jensen, Alan P. F. Sell, Frank C. Senn, Kent Davis Sensenig, Damían Setton, Bal Krishna Sharma, Carolyn J. Sharp, Thomas Sheehan, N. Gerald Shenk, Christian Sheppard, Charles Sherlock, Tabona Shoko, Walter B. Shurden, Marguerite Shuster, B. Mark Sietsema, Batara Sihombing, Neil Silberman, Clodomiro Siller, Samuel Silva-Gotay, Heikki Silvet, John K. Simmons, Hagith Sivan, James C. Skedros, Abraham Smith, Ashley A. Smith, Ted A. Smith, Daud Soesilo, Pia Søltoft, Choan-Seng (C. S.) Song, Kathryn Spink, Bryan Spinks, Eric O. Springsted, Nicolas Standaert, Brian Stanley, Glen H. Stassen, Karel Steenbrink, Stephen J. Stein, Andrea Sterk, Gregory E. Sterling, Columba Stewart, Jacques Stewart, Robert B. 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Yee, Viktor Yelensky, Yeo Khiok-Khng, Gustav K. K. Yeung, Angela Yiu, Amos Yong, Yong Ting Jin, You Bin, Youhanna Nessim Youssef, Eliana Yunes, Robert Michael Zaller, Valarie H. Ziegler, Barbara Brown Zikmund, Joyce Ann Zimmerman, Aurora Zlotnik, Zhuo Xinping
- Edited by Daniel Patte, Vanderbilt University, Tennessee
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18 - Kundt solutions
- Jerry B. Griffiths, Loughborough University, Jiří Podolský, Charles University, Prague
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Summary
Solutions are said to belong to Kundt's class if they admit a null geodesic congruence, generated by a vector field k, which is shear-free, twist-free and expansion-free. They include the pp-waves and plane wave space-times that have been described in the previous chapter. The whole class, initially investigated by Kundt (1961, 1962) in the case of vacuum or with an aligned pure radiation field, is however much wider. Because the null vector field k is not in general covariantly constant, the rays of the corresponding non-expanding waves are not necessarily parallel, as in the case of pp-waves, and the wave surfaces need not be planar. This greater freedom permits, for example, the presence of a cosmological constant Λ or aligned electromagnetic fields, both null and non-null.
For vacuum or some specific matter content, generalised Goldberg—Sachs theorems imply that the Kundt space-times must be algebraically special, that is of Petrov type II, D, III or N (or conformally flat), with k being a repeated principal null direction of the Weyl tensor. Any Einstein—Maxwell and pure radiation fields must also be aligned: that is, k is the common eigendirection of the Weyl and Ricci tensor.
A number of physically interesting space-times of this class are explicitly known. These will be briefly described in this chapter, with emphasis given to those that describe exact non-expanding gravitational waves of various kinds.
Appendix B - 3-spaces of constant curvature
- Jerry B. Griffiths, Loughborough University, Jiří Podolský, Charles University, Prague
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1 - Introduction
- Jerry B. Griffiths, Loughborough University, Jiří Podolský, Charles University, Prague
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Summary
After Einstein first presented his theory of general relativity in 1915, a few exact solutions of his field equations were found very quickly. All of these assumed a high degree of symmetry. Some could be interpreted as representing physically significant situations such as the exterior field of a spherical star, or a homogeneous and isotropic universe, or plane or cylindrical gravitational waves. Yet it took a long time before some of the more subtle properties of these solutions were widely understood.
In their seminal review of “exact solutions of the gravitational field equations”, Ehlers and Kundt (1962) included the following statement. “At present the main problem concerning solutions, in our opinion, is not to construct more but rather to understand more completely the known solutions with respect to their local geometry, symmetries, singularities, sources, extensions, completeness, topology, and stability.” Since this was written, considerable progress has been made in the understanding of many exact solutions. However, this development has been very restricted compared to the enormous effort that has been put into the derivation of further “new” solutions. Although significant advance has been achieved in the interpretation of many solutions, it is a fact that some aspects of even the most frequently quoted exact solutions still remain poorly understood. The opinion of Ehlers and Kundt thus still indicates an even more urgent task.
In this work, the very traditional approach will be adopted that an exact solution of Einstein's equations is expressed in terms of a metric in particular coordinates. Specifically, it will be represented in the form of a 3+1-dimensional line element in which the coordinates have certain ranges.
2 - Basic tools and concepts
- Jerry B. Griffiths, Loughborough University, Jiří Podolský, Charles University, Prague
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9 - Space-times related to Schwarzschild
- Jerry B. Griffiths, Loughborough University, Jiří Podolský, Charles University, Prague
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Contents
- Jerry B. Griffiths, Loughborough University, Jiří Podolský, Charles University, Prague
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14 - Accelerating black holes
- Jerry B. Griffiths, Loughborough University, Jiří Podolský, Charles University, Prague
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This chapter considers the vacuum solution that was referred to as the C-metric in the classic review of Ehlers and Kundt (1962) – a label that has generally been used ever since. In fact, the static form of this solution was originally found by Levi-Civita (1918) and Weyl (1919a), and has subsequently been rediscovered many times. Its basic properties were first interpreted by Kinnersley and Walker (1970) and Bonnor (1983). Specifi- cally, it was shown that, with its analytic extension, this solution describes a pair of causally separated black holes which accelerate away from each other due to the presence of strings or struts that are represented by conical singularities.
The C-metric is a generalisation of the Schwarzschild solution which includes an additional parameter that is related to the acceleration of the black holes. In fact, generalisations to “accelerating” versions of all three A-metrics have been described by Ishikawa and Miyashita (1983). These include what may be called the CI, CII and CIII-metrics. However, it is only the CI-metric, which describes accelerating black holes, that will be considered in the present chapter.
General properties of space-times such as this, which admit boost and rotationsymmetries, were described by Bičák (1968). Asymptotic and other properties of the C-metric were further investigated by Farhoosh and Zimmerman (1980a), Ashtekar and Dray (1981), Dray (1982), Bičák (1985) and Cornish and Uttley (1995a). For more recent work see e.g. Pravda and Pravdová (2000) and Griffiths, Krtouš and Podolsky (2006), on which the present chapter is based and from where the figures are taken.
13 - Stationary, axially symmetric space-times
- Jerry B. Griffiths, Loughborough University, Jiří Podolský, Charles University, Prague
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10 - Static axially symmetric space-times
- Jerry B. Griffiths, Loughborough University, Jiří Podolský, Charles University, Prague
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Preface
- Jerry B. Griffiths, Loughborough University, Jiří Podolský, Charles University, Prague
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Summary
In the now extensive literature on general relativity and its related subjects, references abound to “known solutions” or even “well-known solutions” of Einstein's field equations. Yet, apart from a few familiar space-times, such as those of Schwarzschild, Kerr and Friedmann, often little more is widely known about such solutions than that they exist and can be expressed in terms of a particular line element using some standard coordinate system.
With the most welcome publication of the second edition of the “exact solutions” book of Stephani et al. (2003), an amazing number of solutions, and even families of solutions, have been identified and classified. This is of enormous benefit. However, when it comes to understanding the physical meaning of these solutions, the situation is much less satisfactory – even for some of the most fundamental ones.
Of course, there are now many excellent textbooks on general relativity which present the subject in a coherent way to students with a variety of primary interests. These always describe the basic properties of the Schwarzschild solution and usually a few others as well. Yet, beyond these, when trying to find out what is known about any particular exact solution, there is normally still no alternative to searching through original papers dating back many years and published in journals that are often not available locally or freely available on the internet. Proceeding in this way, it is possible to miss significant contributions or to repeat errors or unhelpful emphases.
Frontmatter
- Jerry B. Griffiths, Loughborough University, Jiří Podolský, Charles University, Prague
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12 - Taub–NUT space-time
- Jerry B. Griffiths, Loughborough University, Jiří Podolský, Charles University, Prague
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In this chapter, we describe what is widely known as the Taub–NUT solution. This was first discovered by Taub (1951), but expressed in a coordinate system which only covers the time-dependent part of what is now considered as the complete space-time. It was initially constructed on the assumption of the existence of a four-dimensional group of isometries so that it could be interpreted as a possible vacuum homogeneous cosmological model.
This solution was subsequently rediscovered by Newman, Tamburino and Unti (1963) as a simple generalisation of the Schwarzschild space-time. And, although they presented it with an emphasis on the exterior stationary region, they expressed it in terms of coordinates which cover both stationary and time-dependent regions. In addition to a Schwarzschild-like parameter m which is interpreted as the mass of the source, it contained two additional parameters – a continuous parameter l which is now known as the NUT parameter, and the discrete 2-space curvature parameter which is denoted here by ∈. It is only the case in which ∈ = +1, which includes the Schwarzschild solution, that was obtained by Taub. The cases with other values of ∈ are generalisations of the other A-metrics.
We will follow the usual convention of referring to the case in which ∈ = +1 as the Taub–NUT solution. However, there are two very different interpretations of this particular case. Both of these have unsatisfactory aspects in terms of their global physical properties. In one interpretation, the space time contains a semi-infinite line singularity, part of which is surrounded by a region that contains closed time like curves.
21 - Colliding plane waves
- Jerry B. Griffiths, Loughborough University, Jiří Podolský, Charles University, Prague
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Summary
Since Einstein's equations are essentially nonlinear, gravitational fields and waves cannot be simply superposed. In general relativity, even electromagnetic waves experience a nonlinear interaction through the gravitational equations, in spite of the fact that Maxwell's equations remain linear. The physical phenomena that arise as a result of this nonlinearity need to be understood. And the simplest situation for which this can be modelled exactly is in the collision and subsequent interaction of plane waves in a flat Minkowski background. In fact, many explicit solutions are now known which describe situations of this type. Thorough reviews of early work on this topic can be found in the book by Griffiths (1991) and also in Chapter 25 of Stephani et al. (2003). The purpose of the present chapter is to use an up-to-date approach to review the basic results that have been found, with a particular emphasis on the physically significant features that arise. It will also be shown how certain solutions that have been studied in previous chapters reappear in this context.
Clearly, the collision of plane waves, is a highly idealised situation. Realistic waves have convex wavefronts, and only become approximately planar at a large distance from their source. Moreover, exact plane waves are infinite in transverse spatial directions and therefore have unbounded energy. These features may have unfortunate consequences in exact colliding plane wave space-times. Thus, when seeking to interpret these solutions, care has to be taken to distinguish properties that apply to general wave interactions from those that arise as a consequence of the idealised assumptions.
8 - Schwarzschild space–time
- Jerry B. Griffiths, Loughborough University, Jiří Podolský, Charles University, Prague
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- Exact Space-Times in Einstein's General Relativity
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- 04 February 2010
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- 15 October 2009, pp 106-126
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Summary
The Schwarzschild solution is undoubtedly the best known nontrivial exact solution of Einstein's equations. It was found only a few months after Einstein published his field equations. And, not only is it one of the simplest exact vacuum solutions, but it is also the most physically significant. It is widely applied both in astrophysics and in considerations of orbital motions about the Sun or the Earth. Until recently, it was only on the assumption of the applicability of this space-time that general relativity had been demonstrated to be a superior theory to the classical gravitational theory of Newton, in a quantitatively precise manner. It predicts the tiny departures from Newtonian theory that are observed in orbital motions in the solar system, in the deflection of light by the Sun, in the gravitational redshift of light and in time-delay effects. In addition, it provides a model for a theory of strong gravitational fields that is widely applied in astrophysics in the final stages of stellar evolution and the formation of black holes.
For all these reasons, the properties of the Schwarzschild solution are explained even in the most introductory texts on general relativity. Nevertheless, it is still useful to describe this space-time here as some important concepts, such as black hole horizons and analytic extensions, are best introduced in this context. These concepts and some associated techniques, which arise naturally in the Schwarzschild space-time, will be developed further and applied in the more complicated solutions that will be described in following chapters.
In this chapter, we present the familiar interpretation of the Schwarzschild space-time that is based on the assumption of global spherical symmetry.
References
- Jerry B. Griffiths, Loughborough University, Jiří Podolský, Charles University, Prague
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- Book:
- Exact Space-Times in Einstein's General Relativity
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- 04 February 2010
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- 15 October 2009, pp 477-519
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6 - Friedmann–Lemaître–Robertson–Walker space-times
- Jerry B. Griffiths, Loughborough University, Jiří Podolský, Charles University, Prague
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- Exact Space-Times in Einstein's General Relativity
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- 04 February 2010
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- 15 October 2009, pp 67-94
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Summary
On a sufficiently large scale, the universe we live in appears to be both spatially homogeneous and isotropic (that is, on an appropriate spatial section its matter content is uniformly distributed on average, and it looks qualitatively the same in all directions). Space-times with these properties were systematically investigated from different points of view in the pioneering work particularly of Friedmann, Lemaître, Robertson and Walker. The solutions they developed underlie the foundation of modern cosmology. They provide a wide range of possible dynamical models of the universe, among which cosmologists can identify that which most closely resembles our own on appropriately large scales. In particular, they have lead to the prediction of an initial cosmological singularity known as the big bang.
In this chapter, we will describe such a family of spatially homogeneous and isotropic space-times. These are considered as idealised cosmological models containing a perfect fluid satisfying some equation of state. As such, they represent various possible types of uniformly distributed matter, including the most important special cases of dust and radiation. They also admit a non-trivial cosmological constant. Like the vacuum space-times described in the previous three chapters, they are also conformally flat. The geometrical reason for this is that their natural three-dimensional spatial subspaces have constant curvature. This curvature can be positive, zero or negative, giving rise to different models of closed or open universes whose dynamics are uniquely determined by the specific matter content.
19 - Robinson–Trautman solutions
- Jerry B. Griffiths, Loughborough University, Jiří Podolský, Charles University, Prague
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- Book:
- Exact Space-Times in Einstein's General Relativity
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- 04 February 2010
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- 15 October 2009, pp 361-391
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Exact Space-Times in Einstein's General Relativity
- Jerry B. Griffiths, Jiří Podolský
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- Published online:
- 04 February 2010
- Print publication:
- 15 October 2009
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Einstein's theory of general relativity is a theory of gravity and, as in the earlier Newtonian theory, much can be learnt about the character of gravitation and its effects by investigating particular idealised examples. This book describes the basic solutions of Einstein's equations with a particular emphasis on what they mean, both geometrically and physically. Concepts such as big bang and big crunch-types of singularities, different kinds of horizons and gravitational waves, are described in the context of the particular space-times in which they naturally arise. These notions are initially introduced using the most simple and symmetric cases. Various important coordinate forms of each solution are presented, thus enabling the global structure of the corresponding space-time and its other properties to be analysed. The book is an invaluable resource both for graduate students and academic researchers working in gravitational physics.
Appendix A - 2-spaces of constant curvature
- Jerry B. Griffiths, Loughborough University, Jiří Podolský, Charles University, Prague
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- Book:
- Exact Space-Times in Einstein's General Relativity
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- 04 February 2010
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- 15 October 2009, pp 462-469
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