Journal of Mathematical Physics
Feedback | Help | Exit
   
 
 
 
Previous Article
Orthogonal transitivity, invertibility and null geodesic separability in type D vacuum solutions of Einstein's field equations with cosmological constant
It is shown without explicit integration that all Petrov type-D vacuum solutions with cosmological constant admit at least a two-parameter, abelian, orthogonally transitive group of local isometries. ...
Next Article
Theory of nonlinear Langevin equation with quadratic noise
A generalized Langevin equation with a multiplicative linear and quadratic Ornstein–Uhlenbeck stochastic noise is investigated. An exact generalized Fokker–Planck equation for this Langevi...

NUT 4-momenta are forever

J. Math. Phys. 23, 2168 (1982); doi:10.1063/1.525274

Issue Date: November 1982

You are not logged in to this journal. Log in

Abhay Ashtekar
Physics Department, Syracuse University, Syracuse, New York 13210 and Département de Physique, Université de Clermont-Fd., 63170 Aubière, France

Amitabha Sen
Department of Physics and Astronomy, University of Maryland, College Park, Maryland 20742
The asymptotic structure of the gravitational field at null infinity is re-examined by allowing certain potentials to develop ``wire singularities'', keeping the physical fields smooth. This relaxation of the regularity conditions leads to the introduction of the Newman–Unti–Tamburino (NUT) 4-momentum which is the ``magnetic'' or the ``dual'' counterpart of the Bondi–Sachs 4-momentum. It is shown that, unlike the Bondi–Sachs 4-momentum, the NUT 4-vector is absolutely conserved even in the presence of gravitational radiation. Thus, while the gravitational field resembles the nonabelian Yang–Mills fields in its ``electric'' properties, it is analogous to the abelian Maxwell field in its ``magnetic'' properties. It is pointed out that gravitational fields with nonvanishing NUT 4-momenta may have a substantial role in quantum gravity even though they are not physically significant in classical general relativity. Journal of Mathematical Physics is copyrighted by The American Institute of Physics.
History: Received 8 April 1982; accepted 21 May 1982
Permalink: http://link.aip.org/link/?JMAPAQ/23/2168/1
BUY THIS ARTICLE   (US$28)
Download PDF (1002 kB) View Cart

KEYWORDS and PACS

Keywords
PACS
  • 04.50.+h
    Relativity and gravitation Unified field theories and other theories of gravitation
  • YEAR: 1982

PUBLICATION DATA

ISSN:
0022-2488 (print)   1089-7658 (online)
Publisher:
AIP is a member of CrossRef AIP

REFERENCES (25)
View in Separate Window/Tab
For access to fully linked references, you need to log in. For access to fully linked references, you need to Log in.
  1. E. T. Newman, L. Tamburino, and T. Unti, J. Math. Phys. 4, 915 (1963).
  2. C. W. Misner, J. Math. Phys. 4, 924 (1963).
  3. S. Ramaswamy and A. Sen, J. Math. Phys. 22, 2612 (1981).
  4. H. Bondi, M. G. J. Van der Burg, and A. W. K. Metzner, Proc. R. Soc. (London) Ser. A 269, 21 (1962);
    5. R. K. Sachs, ibid. 270, 103 (1962).
  5. Note also that our expression for the NUT 4-momentum contains terms involving Bondi news and is therefore not the “obvious” generalization of Eq. (21) of Ref. 3. Also, we shall find that, in presence of news, supermomenta cannot be introduced in a satisfactory manner.
  6. A. Ashtekar, Phys. Rev. Lett. 46, 573 (1981);
    7. in Quantum Gravity 2, edited by C. J. Isham, R. Penrose, and D. W. Sciama (Oxford U.P., Oxford, 1975).
  7. For details, see A. Ashtekar, J. Math. Phys. 22, 2885 (1981).
  8. See, e.g., A. Magnon-Ashtekar, J. Math. Phys. 22, 2012 (1981).
  9. See, e.g., S. W. Hawking, D. Page, and C. N. Pope, Nucl. Phys. B 170, 283 (1980).
  10. B. G. Schmidt, M. Walker, and P. D. Sommers, Gen. Relativ. Gravit. 6, 489 (1975).
  11. In the Newman-Penrose notation, Nabmamb = −2sigma and the five independent components of *Kab correspond to psi<sub>4</sub><sup>0</sup>, psi<sub>3</sub><sup>0</sup> and psi<sub>2</sub><sup>0</sup>. The two degrees of freedom in {D} are coded in sigma. For details, see C. N. Kozomeh and E. T. Newman, “A Note on Asymptotically Flat Spaces II” (preprint). See also Sec. III of Ref. 3 or Appendix A of Ref. 7.
  12. R. Geroch, in Asymptotic Structure of Space-time, edited by P. Esposito and L. Witten (Plenum, New York, 1977).
  13. Our conventions are the following. Curvature tensor is defined by D[aDb]kc = (1/2)[script R]<sup>d</sup><sub>abc</sub>kd·qab has signature (0++). epsilonabc is defined by epsilonabcepsilonmnrqbnqam = 2ncn[prime] and epsilonabc by epsilonabcepsilonabc = 3!
  14. Note that there is an error of a factor of 2 in the expression of [script R]<sub>abc</sub><sup>d</sup> in Ref. 7. Equation (II.3) of the present paper corrects this error.
  15. Why do we not require that f be C[infinity]? This requirement would lead us to treat {D} and {D[prime]} as being distinct if (D<sub>a</sub><sup>[prime]</sup>−Da)kb = fqabnckc and if f has wire singularities, and add spurious degrees of freedom to the radiative modes {D}. (Note, that, even if f has wire singularities, {D[prime]}[is-an-element-of][script C]-bar iff {D}[is-an-element-of][script C]-bar. Thus, if {D} is admissible, so is {D[prime]}).
  16. Note that by a gauge transformation of Eq. (II.8) with f = −u,Aa can be transformed to −4l(1−cos theta)Daphi, which is the familiar expression of the vector potential of a magnetic monopole in Maxwell's theory.
  17. Note that, since all derivative operators satisfying Eq. (II.1) have the same action on covectors satisfying kana = 0, DaDbalpha is a C[infinity] tensor field even when {D} is nutty. Hence, in particular, µ is C[infinity].
  18. R. K. Sachs, Phys. Rev. 128, 2851 (1962);
    19. J. Winicour, J. Math. Phys. 9, 861 (1968);
    A. Ashtekar and M. Streubel, Proc. R. Soc. (London) Ser. A 376, 585 (1981);
    R. Geroch and J. Winicour, J. Math. Phys. 22, 803 (1981). Note, however, that these references contain two distinct definitions of supermomentum.
  19. Possible wire singularities in f do not cause complications in the proof because we have used the regularized integral [integral]-bar.
  20. Note that omega is any C[infinity], nowhere vanishing function on [script J], satisfying [script L]nomega = 0; unlike in the argument in the beginning of this section, omega is unrelated to alpha.
  21. E. T. Newman and R. Penrose, Proc. R. Soc. (London) Ser. A 305, 175 (1968). Since the duality transformation uses the 4-dimensional alternating tensor epsilonabcd, it is somewhat cumbersome to see its effect on quantities defined intrinsically on [script J]. Since the required manipulations are well known in the Newman-Penrose framework, we have chosen this method here.
  22. J. Winicour, J. Math. Phys. 9, 861 (1968);
    23. R. Geroch and J. Winicour, ibid. 22, 803 (1981).
  23. Note that this terminology is different from that in Ref. 7, where a configuration {D} was called a classical vacuum if it has a vanishing Nab and *Kab, i.e., if its curvature is trivial. From radiative energy considerations, the present terminology appears to be more natural.
  24. Since all connections D in [script C]-bar agree on covectors Ka satisfying Kana = 0,D-hat on [script L] is a C[infinity] connection. Hence, solutions to the differential equations that follow are C[infinity] modulo gauge.
  25. Arbitrariness in the choice of K corresponds to the freedom in the choice of connections D and D[prime] in {D} and {D[prime]}. If we make a gauge transformation Dawb-->D-barawb+fqabncwc such that Aana = Aana = 0, then v-hata-->v-hata+2Daf. One can use this gauge freedom to ensure that the function alpha of Eq. (IV.12)is C[infinity].

CITING ARTICLES
For access to citing articles, you need to log in.
For access to citing articles, you need to Log in.


Copyright © American Institute of Physics