Integrating the geodesic equations in the Schwarzschild and Kerr space-times using Beltrami’s “geometrical” method. (English) Zbl 1093.83008
Summary: We revisit a little known theorem due to Beltrami, through which the integration of the geodesic equations of a curved manifold is accomplished by a method which, even if inspired by the Hamilton-Jacobi method, is purely geometric. The application of this theorem to the Schwarzschild and Kerr metrics leads straightforwardly to the general solution of their geodesic equations. This way of dealing with the problem is, in our opinion, very much in keeping with the geometric spirit of general relativity. In fact, thanks to this theorem we can integrate the geodesic equations by a geometrical method and then verify that the classical conservation laws follow from these equations.
References:
| [1] | Wintner, A.: The Analytical Foundations of Celestial Mechanics. Princeton University Press, Princeton (1941) · Zbl 0026.02302 |
| [2] | The paper Sulla Teorica Generale dei Parametri Differenziali by Eugenio Beltrami was published in Memorie dell’Accademia delle Scienze dell’Istituto di Bologna, serie II, tomo VIII, pp. 551–590 (1868) and reprinted in Opere Matematiche, vol. II, pp. 74–118. Hoepli, Milano (1904). For the theorem considered here, which generalizes to n-dimensions previous results relative to ordinary surfaces, see pp. 366–373 op. cit. vol. I, and also A.R. Forsyth, Lectures on the Differential Geometry of Curves and Surfaces, pp. 163–165, (Cambridge University Press, 1912). We summarize the Beltrami contribution: in the first paper Beltrami follows Jacobi up to the remark that the results obtained can be expressed using a partial differential equation via the differential parameter ({\(\delta\)}1 U = 1) that he had introduced in previous work studying surfaces. This correspondence and a theorem of Gauss allows one to obtain the geodesics by differentiation alone (as we have summarized in Sect. 2). In the second article Beltrami extends the introduction of the differential parameters to n-dimensional Riemann spaces. After this he tries and succeeds to extend the theorem for the integration of geodesics to an n-dimensional space. This method is included in a long article as an application of the differential parameter. This may be the reason it is not as well known as it deserves to be, like other contributions of Beltrami. |
| [3] | Bianchi Luigi, Lezioni di Geometria Differenziale, 2nd edition, vol. I, II, (Spoerri, Pisa, 1902); German translation by M.Lukat (B.G. Teubner, Leipzig 1910). The theorem is discussed in pp. 336–338 of vol. I. In the 3rd edition of Bianchi’s treatise (Zanichelli, Bologna, 1924) the theorem is given for two dimensions in vol. I, p. 299, and for n dimensions in pp. 423–426 of vol. II, part 2. For the theorem in two dimensions Bianchi credits Jacobi. As far as n dimensions are concerned he credits Beltrami for the introduction of the differential parameter and the theorem is considered as an important consequence of this introduction, but without reference to any author for this theorem. |
| [4] | Eisenhart, L.P.: Riemannian Geometry (Princeton University Press, Princeton, 1964). The theorem is quoted in pp. 59–60. In his exposition Eisenhart follows Bianchi, with the extension to non-definite metrics. Forsyth (see [2] above) correctly quotes the 1868 Beltrami paper and credits Beltrami with having introduced the definition of the differential parameter. |
| [5] | Misner, C., Thorne, K.S., Wheeler, J.W.: Gravitation. W.H. Freeman and Company, San Francisco (1973) |
| [6] | Chandrasekhar, S.: The Mathematical Theory of Black Holes (Oxford University Press, 1983); see also: Kerr, R.P.: Phys. Rev. Letters 11, 237–238 (1963); Carter, B.: Phys. Rev. 174, 1559–1571 (1968) |
| [7] | do Carmo, M.P.: Differential Geometry of Curves and Surfaces Prentice Hall, New York (1976); O’Neill, B.: Semi-Riemannian Geometry Academic Press, New York (1983) |
| [8] | As has been recently pointed out, the metric ([34]) universally known as the Schwarzschild metric was really found by G. Droste who published his paper some months after the publication of Schwarzschild’s paper. The reader can find an exhaustive study of the question in: Antoci, S., Liebscher, D.E.: ”On the gravitational field of a mass point according to Einstein’s theory by K. Schwarzschild” Gen. Rel. Grav., 35 p. 945, 2003). It turns out that it was D. Hilbert who created and passed on the phrase Schwarzschild metric. The English translation of the original Schwarzschild paper is published in Gen. Rel. Grav., 35, p. 951 (2003) and the reprint of Droste’s paper in Gen. Rel. Grav. 34, p. 1545 (2002). |
| [9] | Landau, L., Lifshitz, E.: The Classical Theory of Fields, 4th English Edition Pergamon Press (1975) · Zbl 0178.28704 |
| [10] | Boccaletti, D., Pucacco, G.: Theory of Orbits, 3rd Corrected Printing Springer-Verlag, Berlin (2004), vol. I · Zbl 0927.70002 |
| [11] | Bergmann, P. G.: Introduction to the Theory of Relativity (Dover, 1976), Chap. XIV |
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