Eur. Phys. J. C (2017) 77:179 DOI 10.1140/epjc/s10052-017-4747-3 Regular Article - Theoretical Physics Redshift of a photon emitted along the black hole horizon A. V. Toporensky1,2,a, O. B. Zaslavskii2,3,b 1 Sternberg Astronomical Institute, Lomonosov Moscow State University, Moscow, Russia 2 Kazan Federal University, Kremlevskaya 18, Kazan 420008, Russia 3 Department of Physics and Technology, Kharkov V.N. Karazin National University, 4 Svoboda Square, Kharkov 61022, Ukraine Received: 30 December 2016 / Accepted: 8 March 2017 © The Author(s) 2017. This article is an open access publication Abstract In this work we derive some general features of communicating by radio signals with another falling one. the redshift measured by radially moving observers in the In doing so, some incorrect statements were made in [2] black hole background. Let observer 1 cross the black hole about “ghosts” of the first observer supposedly waiting horizon emitting a photon, while observer 2 crossing the for the second one on the horizon. As was argued in same horizon later receives it. We show that if (i) the horizon [3], there are no such ghosts at all, although the second is the outer one (event horizon) and (ii) it is nonextremal, observer does receive a signal from the first one emit- the received frequency is redshifted. This generalizes recent ted at the moment of crossing the horizon. This required results in the literature. For the inner horizon (like in the detailed calculation of the frequency shift for a photon Reissner–Nordström metric) the frequency is blueshifted. If propagating along the horizon with the result that a finite the horizon is extremal, the frequency does not change. We redshift occurs in this case. derive explicit formulas describing the frequency shift in gen- In the present work, we generalize these observations, eralized Kruskal- and Lemaitre-like coordinates. demonstrate that such a redshift is present for any horizon of a spherically symmetric nonextremal black hole and find its value. 1 Introduction (ii) If the metric contains an inner horizon (say, for the Reissner–Nordström black hole), the calculation of the Redshift is one of the well-known effects of gravity; it plays frequency shift for a photon emitted along such a horizon an essential role in relativistic astrophysics. Its description is of special interest. We demonstrate that now, instead entered many textbooks. In particles, it concerns propagation of a redshift, a blueshift occurs. In the limit when a pho- of light in the black hole background. Thus, much attention is ton is received near the bifurcation point, the blueshift focused on the properties of light outside the event horizon. becomes infinite. This establishes the connection of the Less these properties inside the horizon are discussed. Also, issue under discussion with the analog of the Bañados– strange as it may seem, the question of the redshift of a photon Silk–West (BSW) effect [4]. It consists in the infinite moving along the horizon dropped out from consideration growth of the energy of colliding particles in the center almost completely. There is general discussion of this issue of mass frame. Originally, it was found near the event in [1] for the Reissner–Nordström–de Sitter metric but the horizon but, later on, it turned out that another similar relation between redshift or blueshift and the nature of the effect is valid also near the inner horizon (see [5] and the horizon was not revealed there. references therein). On the other hand, the issue under Meanwhile, there are several points that can serve as a discussion can be considered as an effect supplemental motivation for such a consideration. to a well-known instability of the inner horizon [6,7]. (iii) In addition to the propagation along the nonextremal hori- (i) Recently, important methodic issues were discussed in zon, there is also the question of what happens in the [2,3] concerning properties of the world visible by an extremal case. We argue that, by contrast to the two pre- observer falling into the Schwarzschild black hole and vious ones, now the frequency shift is absent. (iv) In Refs. [2,3] the Kruskal–Szekerez (KS) coordinate system was used. We also exploit it. In addition, it is a e-mail: [email protected] of interest to compare the results using another pow- b e-mail: [email protected] 123 179 Page 2 of 8 Eur. Phys. J. C (2017) 77:179 erful system—the Lemaitre one. We construct such a Here, ε =+1 if both objects (the observer and the photon) system for a whole class of metrics that includes the move in the same direction and ε =−1 if they do this in Schwarzchild one as a particular case. opposite ones. If ε =−1, this corresponds to a head-on collision between a massive and a massless particles and this In the present work, we restrict ourselves to the simplest case means that either an observer receives a photon emitted from of radially moving observers. a smaller value of r or a falling observer emits a photon in the backward direction. Near the horizon, f 1. Let ε =+1 and the observer 2 Motion outside the nonextremal event horizon looks back. Near the horizon, he will see the frequency We consider the metric 1 El2 ω m 2 ω ≈ + 0 . (10) 2 2 dr 2 2 2 2 2 ds =−f dt + + r (dθ + sin θdφ ). (1) 2 mω0r+ E f Wesuppose that the metric has the event horizon atr = r+, If ε =−1, so f (r+) = 0. (For simplicity, we assumed that g00g11 =−1 but this condition can be relaxed easily.) We consider now a nonextremal black hole. Near the horizon, ω ≈ 2E . ω (11) f ≈ κ(r − r+), (2) 0 mfem ( ) where κ = f r+ is the surface gravity. 2 μ Here f is the value of the metric function in the point Let an observer have the four-velocity uμ = dx , where τ em dτ of emission. Let the observer emit in his frame a photon is the proper time. We restrict ourselves to the radial motion having the frequency ω, traveling to infinity. Then at infinity of a massive particle (we call it “observer”). Then the four- it will be received with the frequency ω having the order velocity uμ = (t˙, r˙). Here, a dot denotes the derivative with 0 f . Equation (11) agrees with the standard result for the respect to τ. em Schwarzschild metric (see Sect. XII.102 of [8], especially The geodesic equations of motion for such a particle read Eq. 102.10). E It is instructive to compare this with another situation mt˙ = , (3) f usually discussed in textbooks when the emitter is not free falling but is static. In the latter case, the frequency at infinity mr˙ =−Z, Z = E2 − fm2, (4) √ √ ω0 = ω f (see, e.g. Eq. 88.6 of [8]) has the order fem. E =−mut is the conserved energy of a particle. Obviously, the difference can be attributed to the motion of For a photon having the wave vector kμ, the equations of an emitter (the Doppler effect makes the redshift stronger). motion are Some reservations are in order. Throughout the paper, we ω assume that the geometric optics is a reasonable approxima- t = 0 , k (5) tion for propagation of light waves. As usual, this implies that f λ the wavelength λ satisfies the relations L, R, where φ l 2π k = , (6) λ is the wave length, L is a typical scale characterizing a gφ wave packet and R−2 is the typical component of the Rie- l2 mann tensor (see Eq. 22.23c in [9]). For the Schwarzschild kr =−Q, Q = ω2 − f , (7) 0 r 2 metric, this entails, in terms of the frequency, the condition ωM 1; M is the black hole mass. In the more general case ω =− = φ where 0 k0 is the conserved frequency, l k is the of, say, the Reissner–Nordström metric, this gives ωr+ 1, conserved angular momentum. where r+ is the radius of the horizon. Let a free falling observer emit or receive a photon. Its Also, we assume that backreaction of the photon on the frequency measured by this observer is equal to metric is negligible. This implies that, in any case, its energy μ ω =−kμu . (8) is much less than the ADM mass of a black hole, so in nat- ural geometric units ω M. Thus we have the double −1 Taking into account (3)–(7), we find after straightforward inequality r+ ω M. As for the Reissner–Nordstr calculation: öm metric M ≤ r+ ≤ 2M, the conditions of validity of Eω − εZQ this approximation are practically the same for the Reissner– ω = 0 . (9) mf Nordström and Schwarzschild metrics. 123 Eur. Phys. J. C (2017) 77:179 Page 3 of 8 179 E + Z dU 3 Photon emitted at the horizon muU = , (22) f du We see from (11) that, as the point of emission of a photon is E − Z dV muV = . (23) approaching the horizon, the frequency measured at infinity f dv becomes smaller and smaller. However, this formula does Taking also into account (12) and (17), we have not describe what happens if the photon is emitted exactly ω = = = (E + Z) on the horizon. Then 0 0 and l 0, fem 0 from the mU˙ =− , (24) very beginning (see also below), the photon does not reach FVκ (E − Z) infinity at all and moves along the leg of the horizon. Here, the mV˙ = . (25) original coordinates (1) are not applicable since the metric UFκ becomes degenerate on the horizon.