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Embeddings and time evolution of the Schwarzschild wormhole
1. K. Schwarzschild, “Über das Gravitationsfeld eines Massenpunktes nach der Einsteinschen Theorie,” Sitzungsber. Preuss. Akad. Wiss., Phys. Math. Kl., 189–196 (1916).
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5. M. Visser, Lorentzian Wormholes: From Einstein To Hawking (American Institute of Physics Press, New York, 1996)
8. G. Szekeres, “On the singularities of a Riemannian manifold,” Publ. Math. Debrecen 7, 285–301 (1960) and
8. G. Szekeres, Reproduced in Gen. Relativ. Gravit. 34, 2001–2016 (2002).
9. We refer to this coordinate system as Kruskal coordinates for simplicity.
11. Several general relativity textbooks include visual representations of the Schwarzschild wormhole but calculate at most only the elementary embedding at Kruskal time = 0.12–14 Other texts show profile curves of the dynamical evolution of the wormhole but without the required embedding calculations.15–18 Reference 19 includes only a sketch of the = 0 embedding, but without embedding calculations. References 20 and 21 also provide insights into Schwarzschild wormhole embeddings, but the mathematical arguments and associated numerical calculations for the dynamics of the wormhole are not presented.
12. J. Plebanski and A. Krasiński, An Introduction to General Relativity and Cosmology (Cambridge U.P., Cambridge, 2006).
13. Ø. Grøn and S. Hervik, Einstein’s General Theory of Relativity (Springer-Verlag, London, 2007).
14. N. M. J. Woodhouse, General Relativity (Springer-Verlag, London, 2007).
15. C. W. Misner, K. S. Thorne, and J. A. Wheeler, Gravitation (W. H. Freeman, San Francisco, 1973).
16. M. P. Hobson, G. P. Efstathiou, and A. N. Lasenby, General Relativity, An Introduction for Physicists (Cambridge U.P., Cambridge, 2006).
17. L. Ryder, Introduction to General Relativity (Cambridge U.P., Cambridge, 2009).
18. S. Carroll, Spacetime and Geometry, An Introduction to General Relativity (Addison-Wesley, San Francisco, 2004).
19. R. M. Wald, General Relativity (University of Chicago Press, Chicago, 1984).
20. A. P. Lightman, W. H. Press, R. H. Price, and S. A. Teukolsky, Problem Book in Relativity and Gravitation (Princeton U.P., Princeton, 1975), pp. 90–and.
21. B. K. Harrison, K. S. Thorne, M. Wakano, and J. A. Wheeler, Gravitation Theory and Gravitational Collapse, (University of Chicago Press, Chicago, 1965).
22. R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey, and D. E. Knuth, “On the Lambert W function,” Adv. Comput. Math. 5, 329–359 (1996) and
23. B. O’Neill, Elementary Differential Geometry, rev. 2nd ed. (Elsevier Academic Press, San Diego, 2006).
24. This restriction was not observed in Ref. 18. We explain in Secs. V and VI that slices must meet the singularity hyperbola tangentially (see also Ref. 25), so slices A and E in Fig. 5.14 of Ref. 18 are inappropriate because they do not meet the singularity hyperbola tangentially, and the parts “inside” r = 0 have no meaning. Thus they do not result in the corresponding embeddings in Fig. 5.15. Similar errors occur in Ref. 10 in their Figs. 3 (1) and (5). These errors were corrected in Fig. 31.6, p. 839 of Ref. 15.
25. More generally we can show, using Eq. (4.22) of Ref. 23, that if a spacelike slice (u) has a Taylor expansion about a point u1 given by (u) = s(u1) + [u1/ s(u1)](u − u1) + [ ′′(u1)/2](u − u1)2 + O((u − u1)3), and thus meets the singularity hyperbola s(u) tangentially at u1, then in an open interval containing u1, where C is a positive constant, provided that .
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