Il Nuovo Cimento (1955-1965)

, Volume 25, Issue 5, pp 1081–1105

# Postulational approach to schwarzschild’s exterior solution with application to a class of interior solutions

• F. R. Tangherlini
Article

## Summary

In order to clarify the physical ideas underlying Schwarzschild’s exterior solution, a postulational derivation is given that does not make use of the field equations. Basically this amounts to replacing two field equations by two postulates, one of which is a strong version of the principle of equivalence and the other, Newton’s inverse square law. These postulates are more general than the approach would indicate and there is actually a class of solutions for static systems with spherical symmetry which satisfy them. The energy-stress tensors producing these solutions have the important property that energy density equals radial stress. Two examples of interior solutions that fulfill these postulates are given: a solid sphere and a hollow shell. The basic properties of these solutions are described and compared with those of Schwarzschild’s interior solution. The solid sphere solution is used to complete a previous discussion of the clock paradox. In an Appendix, the field equations for static systems with spherical symmetry are written in a form that indicates the limitations of the postulates.

## Riassunto

Si deriva il campo di Schwarzschild per mezzo di postulati diversi con lo scopo di mettere in chiaro alcuni aspetti fisici di questa soluzione. I postulati particolari sono: una versione forte del principio di equivalenza e la legge « quadrato inverso » di Newton. Si danno due soluzioni interne che soddisfano questi postulati: una sfera solida e un guscio cavo. Si discute la proprietà fondamentale di queste soluzioni e si fa una comparazione con la soluzione interna di Schwarzschild. Si dà un’applicazione della soluzione per la sfera al paradosso dell’orologio. Nell’Appendice si scrive l’equazione del campo di gravitazione nella forma che indica le limitazioni dei postulati.

## Preview

Unable to display preview. Download preview PDF.

### References

1. (1).
Phys. Rev. Lett.,6, 147 (1961). This paper, hereafter referred to as (I), deals with a point mass source for the Schwarzschild field.Suppl. Nuovo Cimento,20, 1 (1961), hereafter referred to as (II). A third paper (III) (submitted for publication) will deal with the Poincaré compensating stresses for the classical point electron. A brief description is given inBull. Am. Phys. Soc.,7, 31 (1962).Google Scholar
2. (2).
See the discussion in Einstein’s essay:Notes on the Origin of the General Theory of Relativity inEssays in Science (New York, 1953), p. 78, § 7.Google Scholar
3. (3).
A. Einstein:Ann. Phys.,49, 769 (1916).
4. (4).
See the arguments byW. Lenz, as described inA. Sommerfeld:Lectures on Theoretical Physics, vol.3 (New York, 1952).Google Scholar
5. (13).
However a discontinuity ing 11,r will produce a discontinuity in the radial acceleration as described by (2.9). Note that some authors impose as one of the requirements for « admissible co-ordinates » thatg μν be of classC 1; seee.g.,J. L. Synge:Relativity, The General Theory (Amsterdam, 1960), pp. 1, 268, 273;A. Lichnerowicz:Théories Relativistes de la Gravitation et de l’Electromagnetisme (Paris, 1955), p. 5;W. Israel:Proc. Roy. Soc., A248, 404 (1958).Google Scholar
6. (14).
This is the kind of symmetry between clocks whichH. Dingle and others have emphasized should exist in a complete relativity theory, but one must go outside the domain of special relativity to obtain it.Google Scholar
7. (16).
Since there is no gravitational pseudo-stress to contribute to the force. CompareA. Einstein:Phys. Zeit.,19, 115 (1918).
8. (18).
Seee.g.,A. S. Eddington:Mathematical Theory of Relativity (Cambridge 1924), 2nd ed., p. 94. Also,L. Landau andE. Lifschitz:The Classical Theory of Fields (Cambridge, Mass., 1951), p. 311.Google Scholar
9. (19).
Equation (A.8) can alternatively be derived from a more general expression due toT. Levi-Civita:The Absolute Differential Calculus (Glasgow, 1926), p. 382. See also the generalization of Gauss’s theorem byE. T. Whittaker:Proc. Roy. Soc., A149, 384 (1935). Note that some of the examples given byWhittaker violate the contracted Bianchi identities.Google Scholar

© Società Italiana di Fisica 1962

## Authors and Affiliations

• F. R. Tangherlini
• 1
1. 1.Duke UniversityDurham