Abstract
We shall not significantly manipulate tensors in our account of GR, but even just in order to exhibit the field equations, we need the notation of general tensors. Moreover, no idea of the flavor of actual work in GR can be conveyed without at least some examples of tensors in action. If he wishes, however, the reader may skim lightly over the present section and only refer back to it as need arises.
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References
See, for example, A. S. Eddington, “The Mathematical Theory of Relativity,” p. 85, Cambridge University Press,.1924. For a more modern computational technique see Appendix A of C. W. Misner, J. Math. Phys., 4, 924 (1963).
For a rigorous proof, see W. B. Bonnor’s article in “Recent Developments in General Relativity,” p. 167, New York, Pergamon Press, Inc., 1962.
Phys. Rev. 119, 1743 (1960). Kruskal mistakenly seems to credit E. Kasner with the original discovery in 1921; Kasner’s work is discussed and modified by C. Fronsdal in Phys. Rev. 116, 778 (1959). Actually, it was A. S. Eddington (Nature 113, 192 (1924)) who first transformed Schwarzschild’s metric into a form not singular at r = 2m, but he seems not to have noticed this. (His paper, incidentally, contains a misleading misprint: eq. (2) should have r - 2m instead of r - m.) Eddington’s transformation was rediscovered by D. Finkelstein Phys. Rev. 110, 965 (1958). G. Lemaître’s paper is in Ann. Soc. Sci. Bruxelles A53, 51 (1933).
For further details of the analogy between Kruskal space and the hyperbolic field, see W. Rindler, Am. J. Phys. 34, 1174 (1966). † Theoretically, intelligent beings could stop each star in a galaxy in its motion by ejecting a (small) part of its substance. (See exercise 5–16.) The residual stars would then fall towards the galactic mass center under their mutual gravitation, and eventually disappear through the galactic Schwarzschild radius.
J. L. Anderson, “Principles of Relativity Physics,” New York, Academic Press, 1967; see also his Chapter 9 in “Gravitation and Relativity,” Chiu and Hoffmann, editors, New York, W. A. Benjamin, 1964.
See, for example, J. Weber, “Gravitational Waves,” Phys. Today 21, 34 (April 1968).
Cf. Eddington, loc. cit., Section 45.
See A. S. Eddington, “Space, Time, and Gravitation,” Chapter X, Cambridge University Press, 1920 (and New York, Harper Torchbooks, 1959). Also Eddington, loc. cit. (Mathematical Theory) p. 166.
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Rindler, W. (1969). Formal Development of General Relativity. In: Essential Relativity. Springer, New York, NY. https://doi.org/10.1007/978-1-4757-1135-6_8
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