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The Flatter Regions of Newman, Unti, and Tamburino's Generalized Schwarzschild Space

J. Math. Phys. 4, 924 (1963); doi:10.1063/1.1704019

Issue Date: July 1963

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Charles W. Misner
Palmer Physical Laboratory, Princeton University, Princeton, New Jersey
The ``generalized Schwarzschild'' metric discovered by Newman, Unti, and Tamburino, which is stationary and spherically symmetric, is investigated. We find that the orbit of a point under the group of time translations is a circle, rather than a line as in the Schwarzschild case. The time-like hypersurfaces r = const which are left invariant by the group of motions are topologically three-spheres S3, in contrast to the topology S2 × R (or S2 × S1) for the r = const surfaces in the Schwarzschild case. In the Schwarzschild case, the intersection of a spacelike surface t = const and an r = const surface is a sphere S2. If sigma is any spacelike hypersurface in the generalized metric, then its (two-dimensional) intersection with an r = const surface is not any closed two-dimensional manifold, that is, the generalized metric admits no reasonable spacelike surfaces. Thus, even though all curvature invariants vanish as r --> [infinity], in fact Rµnualphabeta = O(1/r3) as in the Schwarzschild case, this metric is not asymptotically flat in the sense that coordinates can be introduced for which gµnu − etaµnu = O(1/r). An apparent singularity in the metric at small values of r, which appears to be similar to the spurious Schwarzschild singularity at r = 2m, has not been studied. If this singularity should again be spurious, then the ``generalized Schwarzschild'' space would represent a terminal phase in the evolution of an entirely nonsingular cosmological model which, in other phases, contains closed spacelike hypersurfaces but no matter. ©1963 The American Institute of Physics
History: Received 17 December 1962
Permalink: http://link.aip.org/link/?JMAPAQ/4/924/1
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0022-2488 (print)   1089-7658 (online)
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REFERENCES (20)
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  1. E. Newman, L. Tamburino, and T. Unti, J. Math. Phys. 4, 915 (1963). I wish to thank these authors for sending me a preprint of their paper.
  2. M. D. Kruskal, Phys. Rev. 119, 1743 (1960).
  3. C. Fronsdal, Phys. Rev. 116, 778 (1959).
  4. I wish to thank Mr. L. Shepley for preparing this review and for correcting numerous errors in an earlier draft. We have borrowed heavily from L. Marcus' lectures on this topic at the American Mathematical Society's 1962 Summer Institute at the University of California at Santa Barbara.
  5. H. Hopf and W. Rinow, Comment. Math. Helv. 3, 209 (1931) .
    6. See also T. J. Willmore, An Introduction to Differential Geometry (Oxford University Press, England, 1959), p. 133 (although the theorem is here stated for a 2-surface, the proof given does not use this assumption);
    S. S. Chern, Differentiable Manifolds (Department of Mathematics, University of Chicago, 1959), p. 121;
    S. Sternberg, Lectures on Differential Geometry (Department of Mathematics, Harvard University; Prentice-Hall, Inc., 1961), Sec. 4–7;
    S. Helgason, Differential Geometry and Symmetric Spaces (Academic Press, Inc., New York, 1962), p. 55.
  6. See, for example, S. S. Chern (reference 5), p. 122.
  7. Avez has considered some ways of introducing a completeness idea analogous to criterion (1). As yet unpublished, they supercede his proposal in Compt. Rend. 240, 485 (1955).
  8. E. Calabi and L. Marcus, Ann. Math. 75, 63 (1962).
  9. See, for an exposition and further references, F. A. E. Pirani's article in Gravitation: an introduction to current research, edited by L. Witten (John Wiley & Sons, Inc., New York, 1962), Sec. 6-5.
  10. The notation and computational methods being used here are due to E. Cartan; see his text, Leçons sur la Geometrie des Espaces de Riemann, (Gauthier-Villars, Paris, 1951), 2nd Ed.
    11. The conceptual basis of this approach has been much clarified by C. Chevalley in his Theory of Lie Groups (Princeton University Press, Princeton, New Jersey, 1946), Chap. III.
    For an exposition, see T. J. Willmore [cf. reference 5, Chaps. V and VI].
  11. For a clear geometrical picture of the collapsing star metric of J. R. Oppenheimer and H. Snyder [Phys. Rev. 56, 455 (1939)],
    12. see D. L. Beckedorff (senior thesis, Mathematics Department, Princeton University, Princeton, New Jersey, 1962),
    and D. L. Beckedorff and C. W. Misner (in preparation).
  12. Some expository illustrations of the idea of a contra-variant vector as a differential operator, and of the Lie bracket, can be found in F. K. Manasse and C. W. Misner, J. Math. Phys. 4, 751 (1963).
  13. N. Steenrod, The Topology of Fibre Bundles (Princeton University Press, Princeton, New Jersey, 1951), Theorem 7.3.
    14. Duality of a basis omegaµ for covariant vectors and a basis eµ for contravariant vectors means that they satisfy the “inner product” relations <omegaµ,enu> = deltanuµ. These relations are verified between Eqs. (21) and (32) by evaluating them in terms of the relations <dxµ,[partial-derivative]/[partial-derivative]chinu> = deltanuµ which follow from the definition <df,v> = [partial-derivative]f/[partial-derivative]v of the gradient operator d.
  14. This is known as Frobenius' Theorem. In Willmore [cf. reference 5, Eqs. (VII-14.2) and (VII-14.8)], it is reduced to its perhaps more familiar statement about completely integrable Pfaffian systems. A complete proof can be found, for instance, in Sternberg [cf. reference 5, Sec. III-4].
  15. The axioms for defining the idea of a differentiable manifold can be found in Willmore [reference 5, Sec. VI-1].
    16. Several elementary examples of the application of these axioms can be found in C. W. Misner and J. A. Wheeler, Ann. Phys. (N.Y.) 2, 553 (1957).
  16. R. H. Bing, Ann. Math. 68, 17 (1958), Theorem 3.
    17. A theorem similar to that proven in this section has been given by A. Avez, Compt. Rend. 254, 3984 (1962).
  17. See, for example, S. Lefshetz, Topology (Chelsea Publishing Company, New York, 1956), 2nd Ed., p. 144. I wish to thank D. Lowdenslager for directing me to this proof.
  18. This metric has been given, in another coordinate system covering only the region limited by Eq. (55), by A. Taub, Ann. Math. 53, 472 (1951).
    19. This can be recognized by the similarity of Eq. (47) here and the form of Taub's metric given by O. Heckmann and E. Schüking in Eqs. (11-1.21) of their article in Gravitation: an introduction to current research, edited by L. Witten (John Wiley and Sons, Inc., New York, 1962).
    Taub (private communication) has found that the transformation theta = x1, phi = −x2, and t+4lphi = 2lx3 relates the coordinates used in Eq. (2) and those used in Eq. (7.3) of Taub's article. The time coordinates, r here, and t in Taub's article, are related in an obvious way by equating proper time along curves of fixed spatial coordinates.
  19. It has been previously noted that the curvature remains finite at the boundaries [Eq. (55)] of the cosmological region by V. Joseph, Proc. Cambridge Phil. Soc. 53, 836 (1957).
    20. I am also informed that this metric, at least in the form of Eq. (2), has been studied by J. Ehlers in his unpublished thesis “Konstructionen und Charakterisierungen der Einsteinschen Gravitationsfeldgleichungen” (Hamburg, 1957).
    Note added in proof. The spurious nature of the singularities f2 = 0 has been demonstrated by Taub and the author who note that the transformation

    [dformula t [equivalent] t[sub N] = - 2l(phi + psi) + [integral](1/f[sup 2])dr]

    leaves the metric in the form

    [dformula ds[sup 2] = - f[sup 2](2l)[sup 2](d psi + cos theta d phi)[sup 2] + 2dr(2l)d psi + cos theta d phi) + (r[sup 2] + l[sup 2])(d theta[sup 2] + sin[sup 2] theta d phi[sup 2])]

    which is analytic for −[infinity]<r<+[infinity], i.e. is everywhere analytic on the manifold R×S3 provided psithetaphi are interpreted as the Euler angle coordinates on S3, [see Corben and Stehle, Classical Mechanics, (John Wiley & Sons, Inc., New York, 1960) 2nd ed., Eqs. IV-34, for the relation of psithetaphi to quaternion coordinates wxyz]. They furthermore show that this metric is nonsingular in a newly defined sense, namely that every open geodesic arc which approaches the boundaries of the manifold (here r = ±[infinity]) is infinitely long as measured by any affine path parameter. This definition seems mathematically satisfactory since it includes both affinely complete and closed (compact) manifolds, and it implies that the manifold is inextensible. The physically singular aspects of this solution as well as the above results and various families of geodesies in this space are more fully described in a paper by Misner and Taub which will be submitted to JETP.
    Another result from the paper mentioned above is an example of an analytic solution of the Einstein equations in a neighborhood of a closed spacelike hypersurf ace which can be extended in two inequivalent ways as an analytic Riemannian manifold satisfying Rµnu = 0. Consider the metric

    [dformula ds[sup 2] = - f[sup 2](2l)[sup 2](d Psi + cos theta d phi)[sup 2] - 2dr(2l)(d Psi + cos theta d phi) + (r[sup 2] + l[sup 2])(d theta[sup 2] + sin[sup 2] theta d phi[sup 2])]

    which is everywhere analytic (−[infinity]<r<+[infinity]) and satisfies Rµnu = 0 on the manifold R×S3 provided Psithetaphi are interpreted as the Euler angle coordinates on S3. In the region f2<0, which is a neighborhood of the closed spacelike hypersurface r = 0 where f2 = −1, the two metrics above are analytically isometric under the isometry which identifies the coordinates rthetaphi for the two metrics but sets

    [dformula psi = Psi + [integral][sub 0][sup r](lf[sup 2])[sup -1]dr]

    . This isometry cannot be extended analytically to a global isometry of the two manifolds, since some of the geodesic arcs which are identified under this isometry can be extended to infinite length in one manifold but not in the other.

  20. E. Cartan [cf. reference 10, Chap. VII];
    21. T. J. Willmore [cf. reference 5, Secs. VII-16 and VII-19];
    H. Flanders, Trans. Am. Math. Soc. 75, 311 (1953).

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