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Timelike curves of limited acceleration in general relativity
1.See, for example, S. W. Hawking and G. F. R. Ellis, The Large Scale Structure of Space‐Time (Cambridge U.P., Cambridge, 1973), p. 159.
2.K. Gödel, Rev. Mod. Phys. 21, 447 (1949);
2.S. Chandrasekhar and J. P. Wright, Proc. Natl. Acad. Sci. 47, 341 (1961).
3.The same conclusion holds for charged‐particle trajectories with finite integrated nonelectromagnetic acceleration, provided where κ is the charge/mass ratio of the charged particle, and q is the Reissner‐Nordström charge. To see this, modify (3) for this case to read where φ, the electric potential, is determined by For Reissner‐Nordström, and So, under the above condition on the charge, the aE on the right is bounded by a multiple of a The argument now goes through as before.
4.See, for example, J. L. Synge, Relativity: The General Theory (North‐Holland, Amsterdam, 1964), p. 309.
5.See, for example, A. Das, J. Math. Phys. 12, 1136 (1971).
6.These objects may be constructed as follows. Let V be a four‐dimensional vector space with metric of signature (−,−,+,+) Consider the locus of vectors in V with Then S is the covering space of this three‐submanifold of V. while is the metric induced from The Killing fields of are the generators of G‐preserving linear maps on V, and so are described by antisymmetric tensors over V. In particular, is described by with antisymmetric, and (by norm‐condition on ) and (Such F’s exist in this signature.) So, F defines a complex structure on V, and so we may regard V as a complex two‐dimensional vector space, on which becomes a Hermitian metric of signature (−,+). The Killing fields of are the generators of linear maps on Kwhich preserve G and F; i.e., the generators of complex‐linear maps on the complex vector space which preserve its Hermitian metric. So, for example, the Killing fields of the full Godel space‐time form the Lie algebra of
7.R. Penrose, in Relativity, Groups and Topology, edited by C. DeWitt and B. DeWitt (Gordon and Breach, New York, 1964);
7.R. Geroch, in Asymptotic Structure of Space‐Time, edited by F. P. Esposito and L. Witten (Plenum, New York, 1977).
8.H. Bondi, M. G. J. Van der Burg, and A. W. K. Metzner, Proc. R. Soc. London, Ser. A 270, 103 (1962);
8.R. Sachs, Phys. Rev. 128, 2851 (1962).
9.R. Geroch and J. Winicour, J. Math. Phys. 22, 803 (1981).
10.To see this, let be a smooth curve reaching p. Then the line integral of on γ equals the line integral on (which is finite), plus the integral of the curl of on a two‐surface connecting γ and (which is also finite).
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