We provide a simple derivation of the Schwarzschild solution in general relativity based on an approach by Weyl, but generalized to include Birkhoff’s theorem. This theorem states that the Schwarzschild mass must be constant in time. Our procedure is illustrated by a parallel derivation of the Coulomb field and the constancy of the electric charge in electrodynamics. We also explain the basis of Birkhoff’s theorem and note that even the original Weyl approach can be used to illuminate the special role played by the Schwarzschild coordinates.
REFERENCES
1.
and in Space-Time-Matter, translated by H. L. Brose (Methuen, London, 1922);
See also
J.
Droste
, “The field of a single centre in Einstein’s theory of gravitation
,” Proc. R. Acad. Sci. Amsterdam
XIX
, 197
–215
(1917
);D.
Hilbert
, “Die Grundlagen der Physik. (Zweitte Mitteilung)
,” Nachr. Ges. Wiss. Goettingen, Math.-Phys. Kl.
53
–76
(1917
).2.
R. S.
Palais
, “Principle of symmetric criticality
,” Commun. Math. Phys.
69
, 19
–30
(1979
).3.
M. E.
Fels
and C. G.
Torre
, “The principle of symmetric criticality in general relativity
,” Class. Quantum Grav.
19
, 641
–676
(2002
), gr-qc/0108033.4.
S.
Deser
and B.
Tekin
, “Shortcuts to high symmetry solutions in gravitational theories
,” Class. Quantum Grav.
20
, 4877
–4884
(2003
), gr-qc/0306114.5.
The theorem was obtained independently by J. T. Jebsen, “’Úber die Allgemeinen Kugelsymmetrischen Lösungen der Einsteinschen Gravitationsgleichungen im Vakuum,” Ark. f Mat., Astron. och Fys. 15 (18), 9 pages (1921), J. Eisland, “The group of motions of an Einstein space,” Trans. A.M.S. 27, 213–245 (1925), and W. Alexandrow, “’Úber den Kugelsymmetrischen Vakuumvorgang in der Einstenschen Gravitationstheorie,” Ann. Phys. (Leipzig) 72, 141–152 (1923). Birkhoff’s version is in Relativity and Modern Physics (Harvard University Press, Cambridge, 1923). A translation (with comments) of the Jebsen paper will appear in JGRG, Nov. 2004.
6.
The advantage of this method speaks for itself. For Maxwell four vector potentials, plus their 16 gradients and 40 second derivatives are reduced to just one component and its radial derivative. For gravity, the savings are even more dramatic: 10 metric components, their 40 first and 100 second derivatives are reduced to just two functions and their radial derivatives.
7.
S.
Deser
, J.
Franklin
, and B.
Tekin
, “Shortcuts to spherically symmetric solutions: A cautionary note
,” Class. Quantum Grav.
21
, 5295
(2004
).8.
R. Arnowitt, S. Deser, and C. W. Misner, “The dynamics of general relativity,” in Gravitation: An Introduction to Current Research, edited by L. Witten (Wiley, New York, 1962). Reprinted as gr-qc/0405109.
This content is only available via PDF.
© 2005 American Association of Physics Teachers.
2005
American Association of Physics Teachers
AAPT members receive access to the American Journal of Physics and The Physics Teacher as a member benefit. To learn more about this member benefit and becoming an AAPT member, visit the Joining AAPT page.
