ISSN 0202-2893, Gravitation and Cosmology, 2012, Vol. 18, No. 3, pp. 168–174. c  Pleiades Publishing, Ltd., 2012. Can One See the Infinite Future of the Universe when Falling to Kerr and Reissner-Nordstr ¨ om Black Holes? A. A. Grib 1,2* and A. M. Rasulova 1,2** 1 Friedmann Lab. for Theoretical Physics, Griboedov can. 30/32, St. Peterburg 191023, Russia 2 Herzen Russian State Pedagogical University, Moika 48, St. Petersburg 191186, Russia Received July 21, 2011; in final form, January 12, 2012 Abstract—We analyze the falling time of massless and massive particles to a black hole, using timelike and null geodesics for spaces described by the Kerr and Reissner-Nordstr ¨ om metrics. It is shown that an observer falling to a black hole horizon will see only a finite future of the external universe. The formula obtained can be used in assessing the difference in arrival time of photons and massive particles from Kerr and Reissner-Nordstr ¨ om black holes to an observer on Earth. DOI: 10.1134/S020228931203005X 1. INTRODUCTION In 1916 Schwarzschild obtained the spherically symmetrical solution of Einstein’s equations for a single isolated mass. In a year, Reissner and Nord- str ¨ om found the solution for a spherically symmetric charged mass. However, only in 1963 Kerr found his axially symmetric solution for a stationary moving black hole characterized by its mass and angular momentum. Today’s observations show the existence of a large number of black holes described by Schwarzschild and Kerr metrics. The Reissner-Nordstr ¨ om solution is generally not considered in astrophysics because the charge of a black hole can be neglected [1]. How- ever, a study of the Reissner-Nordstr ¨ om solution is useful in order to better understand the properties of space-times and passing over from Schwarzschild’s solution to Kerr’s. Sometimes in popular literature (see [2, 3] and (in Russian) a comment to [4]) one can find the claim that a traveller falling to a black hole and crossing its horizon can in a short time see the infinite future of the Universe due to receiving light from distant objects in it. In [5] it was shown for the Schwarzschild case that this claim is incorrect. A traveller falling to a non- rotating electrically neutral black hole can only see a finite future of the external Universe. In this paper the same result will be obtained for the more complex case of Kerr and Reissner-Nordstr ¨ om black holes. * E-mail: andrei_grib@mail.ru ** E-mail: ARasulova@gmail.com Of course, the mere finiteness of time is evident from consideration of the Penrose-Carter diagrams. However, the formulas obtained in this paper give a numerical evaluation of this time and are significant for astrophysics. With these formulas one can answer the question of how much earlier from the source to an observer on Earth come photons than massive particles (neutrinos if their mass is not zero, particles in cosmic rays, etc.). Furthermore, studies carried out in [6, 7] have shown that close to the horizon of a rotating black hole, due to scattering of particles, one gets very high energies of the order of Grand Unification and even Planckian scale. As a result of inelastic collisions in this area, an observer can see both massive and massless particles. Note that, in the supernova SN1987A observa- tions [8], neutrinos were first detected and then pho- tons. In this case it was not necessary to discuss the region near the horizon. But in the case of radiation from active galactic nuclei, to observe particles from the area of superhigh energies of collisions, one must have formulas for the particle motion time to the Earth from the near-horizon region. Formally, these formulas are obtained from those for the falling time to the horizon by changing the sign to the opposite. This means that the results of this paper are not only methodological. We here obtain relations for the special case of equatorial geodesics, allowing for simple analytical expression. In the future we expect to explore the more complex general case. The system of units G = c =1 will be used. 168