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Visualizing curved spacetime
1.R. D’Inverno, Introducing Einstein’s Relativity (Oxford U.P., Oxford, 1998), pp. 99–101.
2.Equation (1) defines a so-called distance function, or a metric. It can also be used considering events where Then is negative which simply means that it is related to spatial distance rather than temporal distance. A distance function like Eq. (1) corresponds to a flat spacetime, but see Eq. (10) for an example of a distance function corresponding to a curved spacetime.
3.Steven Weinberg, Gravitation and Cosmology (Wiley, New York, 1972), pp. 337–338.
4.Assuming that there is no black hole inside the crust.
5.As was pointed out to me by Ingemar Bengtsson, this relation is also used by Hawking and Ellis (Ref. 18), although for completely different purposes than those of this article.
6.If we had instead considered a metric of the form where α is some general number, the inverse would have been It is only in the case that we can simply raise the indices of the absolute metric with the original metric to get the inverse of the absolute metric.
7.The gamma factor is defined as where is the relative velocity. With this definition, it follows from Eq. (1) that
8.A geometry has a Killing symmetry if there exists a vector field (called a Killing field such that when we shift our coordinates –the metric has the same form. As an example we can consider a geometry that can be embedded as a surface of revolution. Then there exists a Killing field directed around the surface (in the azimuthal direction) with a length proportional to the embedding radius. Also, if there are coordinates where the metric is independent of one coordinate, then there is a Killing symmetry with respect to that coordinate.
9. where gives
10.C. W. Misner, K. S. Thorne, and J. A. Wheeler, Gravitation (Freeman, New York, 1973), p. 841.
11.Suggested to me by Sebastiano Sonego.
12.W. Rindler, Relativity: Special, General and Cosmological (Oxford U.P., Oxford, 2001), pp. 267–272.
13.W. Rindler, Essential Relativity: Special, General and Cosmological (Springer-Verlag, New York, 1977), pp. 204–207.
14.This may well violate the energy conditions.
15.D. Marolf, “Spacetime Embedding Diagrams for Black Holes,” Gen. Relativ. Gravit. 31, 919–944 (1999).
16.L. C. Epstein, Relativity Visualized (Insight, San Francisco, 1994), Chaps. 10–12.
17.R. Jonsson, “Embedding Spacetime via a Geodesically Equivalent Metric of Euclidean Signature,” Gen. Relativ. Gravit. 33, 1207–1235 (2000).
18.S. W. Hawking and G. F. R. Ellis, The Large Scale Structure of Space-time (Cambridge U.P., Cambridge, 1973), p. 39.
19.T. Dray, “The twin paradox revisited,” Am. J. Phys. 58, 822–825 (1989).
20.The Frobenius condition in explicit form reads:
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