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Fig. 3.23 The geodesics of Poincaré metric in the upper plane compared to the geodesics of the Euclidian metric, namely the straight lines
With this result Lobachevskij geometry is easily visualized. Examples of planar figures with sides that are arcs of geodesics are presented in Fig. 3.23.
References
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2.Riemann, G.F.B.: Ueber die Hypothesen, welche der Geometrie zu Grunde liegen. In: Gesammelte Mathematische Werke (1866)
3.Christoffel, E.B.: Über die Transformation der homogenen Differentialausdrücke zweiten Grades. J. Reine Angew. Math. 70, 46–70 (1869)
4.Levi Civita, T., Ricci, G.: Méthodes de calcul differential absolu et leurs applications. Math. Ann. B 54, 125–201 (1900)
5.Klein, F.: Vergleichende Betrachtungen ber neuere geometrische Forschungen. Math. Ann. 43, 63–100 (1893). Also: Gesammelte Abh. Vol. 1, pp. 460–497. Springer (1921)
6.Ricci, G., Atti R. Inst. Venelo 53(2), 1233–1239 (1903–1904)
7.Ricci, G.: Résumé de quelques travaux sur le systémes variable de fonctions associées a une forme diffé rentielle quadratique. Bull. Sci. Math. (1892)
8.Bianchi, L.: Sugli spazii a tre dimensioni che ammettono un gruppo continuo di movimenti. (On the spaces of three dimensions that admit a continuous group of movements. Soc. Ital. Sci. Mem. di Mat. 11, 267 (1898)
9.Bianchi Rend, L.: Accad. Naz. Lincei 11, 3 (1902)
10.Cartan, E.: Sur l’integration des systémes d’équations aux differentielles totales. Ann. Éc. Norm. Instit. 18, 241–311 (1901)
11.Frenet, J.F.: Sur quelques proprétés des courbes à double courbure. J. Math. Pures Appl. (1852)
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12.Serret, J.A.: Sur quelques formules relatives à la théorie des courbes à double courbure. J. De Math. 16 (1851)
13.Killing, W.K.J.: Die Zusammensetzung der stetigen/endlichen Transformationsgruppen. Math. Ann. 31(2), 252–290 (1888)
14.Killing, W.K.J.: Die Zusammensetzung der stetigen/endlichen Transformationsgruppen. Math. Ann. 33(1), 1–48 (1888)
15.Killing, W.K.J.: Die Zusammensetzung der stetigen/endlichen Transformationsgruppen. Math. Ann. 34(1), 57–122 (1889)
16.Killing, W.K.J.: Die Zusammensetzung der stetigen/endlichen Transformationsgruppen. Math. Ann. 36(2), 161–189 (1890)
17.Cartan, E.: Über die einfachen Transformationgruppen, pp. 395–420. Leipz. Ber. (1893)
18.Cartan, E.: Sur la structure des groupes de transformations finis et continus. Thése, Paris, Nony (1894)
19.Cartan, E.: Sur la structure des groupes infinis de transformations. Ann. Sci. de l’ENS 21, 153–206 (1904)
20.Maurer, L.W.: Ueber continuirliche Transformationsgruppen. Math. Ann. 39, 409–440 (1891)
21.Cartan, E.: Sur les equations de la gravitation d’Einstein. J. Math. Pures Appl. 9(1), 93–161 (1922)
22.Dirac, P.: Quantised singularities in the electromagnetic field. Proc. R. Soc. Lond. Ser. A 133, 60 (1931)
23.Sylvester, J.J.: A demonstration of the theorem that every homogeneous quadratic poly-
nomial is reducible by real orthogonal substitutions to the form of a sum of positive and negative squares. Philos. Mag. IV, 138–142 (1852). http://www.maths.ed.ac.uk/~aar/ sylv/inertia.pdf
Chapter 4
Motion of a Test Particle in the Schwarzschild
Metric
The most incomprehensible thing about the Universe is that it is comprehensible. . .
Albert Einstein
4.1 Introduction
It is fair to say that modern physics started with the Copernican Revolution and with the remarkable conceptual synthesis of Johannes Kepler (his portrait is given in Fig. 4.1) who summarized two millennia of astronomical observations of the solar system into three simple laws that describe the orbits of planets and their periods of revolution.
Indeed Newton’s theory of gravitational interactions and Newtonian mechanics were just invented to explain Kepler’s laws within a unified theory of all possible motions.
It is quite obvious that General Relativity which aims at replacing Newton’s with a more profound and consistent theory of gravity should reproduce Kepler’s laws, at least in first approximation. Clearly we expect some modifications and some new effects but, in order to make the new theory successful, they have to be extremely tiny in physical systems of the size of the solar system. On the contrary the same effects are allowed to become very large and even dominant in extremely narrow astrophysical systems like those provided by binaries of compact stars that are presently accessible to the astronomical observation and could not be even suspected at the time of Kepler or Newton.
In this chapter we show that these mostly desirable features characterize the timelike and space-like geodesics of a particular one-parameter space-time metric with Minkowskian signature that is named the Schwarzschild metric after his discoverer.
The significance of this result is appreciated through the following reasoning. Once we accept the geometrical model of space-time as the pair (M , g) where M is a differentiable manifold and g is a pseudo-Riemannian metric, the fundamental problem of mechanics, namely the determination of physical trajectories of pointparticles given the forces that act on them, is replaced by the geometrical problem of calculating the geodesics for the metric g. There are three kinds of these latter:
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Fig. 4.1 Tycho Brahe (1546–1601) on the left and Johannes Kepler (1571–1630) on the right. Tycho Brahe, born Tyge Ottesen Brahe was a Danish nobleman who received the support of the King of Denmark to pursue his systematic naked-eye astronomical observations by means of various instruments and state built installations on the island of Hven. He studied astronomy at the University of Copenhagen and when he began his measurements of planetary parallax he achieved unprecedented precisions, accurate to the arcminute. He was the first to reveal a new star in the sky. On 11 November 1572 Tycho observed a very bright star, now named SN 1572, which unexpectedly appeared in the constellation of Cassiopea. The title of his publication De stella nova is responsible for the introduction of the term nova in astronomy. As we know today, SN 1572, was actually a supernova of type Ia, whose remnant is still observable. Because of a disagreement with the new King of Denmark, in 1597 Tycho Brahe left his country accepting the invitation to Prague of the King of Bohemia Rudolph II who became Emperor of the Holy Roman Empire. In Prague, Brahe had as student and scientific heir, Johannes Kepler. Born in Weil der Stadt, near Stuttgart, Kepler had noble ancestors but the wealth of his family had declined by the time of his coming to this world and his mother was the daughter of a simple inn-keeper. In later years she was accused of witchcraft and escaped burning at the stake just for her courage to deny all charges under torture. Interest in astronomy was raised in Kepler precisely by such a mother who showed him the 1577 comet. Johannes university education was in Tubingen where he came in touch with Copernican theories and elaborated his personal persuasion of their correctness. His first publication dates back to 1597. In the Mysterium Cosmographicum he attempted a first systematic description of the order of the Universe. In 1599 he become assistant of Tycho Brahe in Prague and when the latter died two years later he inherited all of his precious observational data that were the basis for the formulation of his famous three laws. The first two appeared in 1609 in the Astronomia Nova, while he discovered the third in 1618 and published it the next here in Harmonice Mundi. Just as his master Brahe, also Kepler had the venture of observing a supernova in 1604. Also SN 1604 was of type Ia and it has been the last so far observed galactic supernova to the present time (see Fig. 4.2). Imperial Astronomer, notwithstanding his crucial discoveries that eventually led to Newton’s theory of gravitation mixed science, theology and metaphysics in his work trying to find a divine order in the laws of motion of celestial bodies. He died in 1630 in Regensburg
1.the time-like geodesics are the possible world-lines followed by massive particles,
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Fig. 4.2 The remnant of the supernova SN 1572 (on the left) and of the supernova SN 1604 (on the right) respectively observed by Tycho Brahe and Johannes Kepler. They were both of type Ia, namely they were caused by the explosion of a white dwarf that reached the critical Chandrasekhar mass limit (see later chapters) by swallowing material from the companion normal star in a binary system. SN 1572 is at a distance of 7500 light years from the Earth in the Cassiopea Constellation. Kepler’s star SN 1604 is instead at a distance of 20000 light-years in the constellation Ophiuchus
2.the light-like geodesics are the possible world-lines followed by massless particles such as the photons,
3.the space-like geodesics cannot be world-lines for any physical particle since you can travel along them only at a speed larger than the speed of light.
Hence the metric g is a substitute for the concept of force field and the calculation of time-like geodesics is a substitute for the solution of the fundamental problem of mechanics in this force field. Retrieving almost Keplerian orbits from the time-like geodesics of the Schwarzschild metric shows that this latter is a correct replacement for Newton’s law of gravitation. Indeed the one parameter occurring in the Schwarzschild metric can be identified with the mass M of the Newtonian source.
General Relativity is a field theory for the space-time metric gμν (x) and it will be a correct theory of gravity if the Schwarzschild metric is a solution of its field equations, actually the unique solution with the symmetry corresponding to Kepler’s problem, namely spherical symmetry. This is what we show in later chapters once Einstein’s field equations have been introduced.
In the present chapter we simply assume the Schwarzschild metric and we work out all of its consequences. After a review (see Sect. 4.2) of Kepler’s problem in the context of Newtonian mechanics, in Sect. 4.3 we study the equations for time-like geodesics in Schwarzschild geometry and we show how such a geometrical problem admits a one-to-one map into the previous one. In particular the first integrals of the Newtonian motion, the energy E and the angular momentum are mapped into their relativistic analogues E and L which are constant along the geodesics and are associated with the same symmetries in both cases: time-independence and spherical symmetry of the gravitational field, respectively. Writing the differential equation of the orbit we find that it is formally identical to that of Newton’s theory but with a modified central potential Veff (r). In addition to the attractive term −1/r and to the centrifugal barrier 1/r2 there is a third attractive term −1/r3 that is responsible for all the deviations from purely Keplerian motion. At large distances this new term is completely negligible and this explains why Newton’s theory works so well,