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. This can be remedied In a similar way, we have for a photon moving in the with the use of standard KS coordinates or its generalization. outward direction Let us introduce coordinates U and V , where κU Q − ω kU = (Q − ω ) = 0 , (26) f 0 FκV U =−exp(−κu), V = exp(κv), (12) κV ω + Q = − ∗,v= + ∗; kV = (ω + Q) = 0 . (27) u t r t r (13) 0 κ r f FU ∗ dr r = (14) λ = f Let be the affine parameter. On the future horizon U 0, U = dU = = k dλ 0 kV , which agrees with (26) if we put V is the tortoise coordinate. It is seen from (12) that ω0 = 0 = f = Q in the right hand side. Thus only k ∗ remains nonzero. It follows from the geodesic equations that UV =−exp(2κr ). (15) on the horizon Then the metric reads dkV 2 =− V kV . λ VV (28) ds2 = FdUdV + r 2dω2. (16) d V ∼ ∂ F ∼ ∂ F ∂r ∼ = Here the Christoffel symbol VV ∂V ∂r ∂V f 0 Here, on the horizon. Therefore, kV = const along the horizon du dv generator. We have from (8), (24) on the horizon F =−f . (17) dU dV 1 ω =− FkV uU , (29) For the transformation (12), 2 f and we have F = , (18) κ2 E UV ω = 2 kV . (30) Vmκ F = 0 on the horizon. It is clear from (1), (15) that F = F(r). = For example, for the Schwarzschild metric, Let observer 1 cross the horizon at some V V1 and the same for observer 2 but later, at V = V2. It follows from (12), 3 4r+ − r (13) that V > V . Assuming that observers are identical in F =− e r+ , (19) 2 1 r that they have the same values of E and m, we obtain provided the constant of integration is chosen in (14) in such ω2 V1 ∗ r−r+ = < 1. (31) = + + ω a way that r r r ln r+ . 1 V2 We consider the vicinity of the future horizon, on which This agrees with Eq. (A16) of [3] obtained for the = U 0. Along this horizon, V takes finite values. Schwarzschild metric by another method. Near the horizon It is also instructive to check that indeed ω0 = 0. By ∗ 1 r − r+ definition, ω is a constant Killing frequency, r ≈ ln + const, (20) 0 2κ r+ μ ω =−kμξ , (32) ( − ) 0 ≈ r r+ , UV C0 (21) ξ μ r+ where is the Killing vector. In the original coordinates (1), where C0 is a constant. In the Schwarzschild case, C0 = e.It μ is instructive to rewrite the equations of motion for massive ξ = (1, 0, 0, 0), ξμ = (− f, 0, 0, 0). (33) particles outside the event horizon (3), (4) in terms of the Passing to KS coordinates, one obtains KS coordinates and take the horizon limit afterwards. For an observer moving inward they read ξU =−κU,ξV = κV. (34) 123 179 Page 4 of 8 Eur. Phys. J. C (2017) 77:179
Then we see from (32) that l = r2 − r1. (39) V U U V V U ω0 =−F(k ξ + k ξ ) = Fκ(k U − k V ). (35) It is also instructive to calculate the velocity. Let, say, point 1 be fixed and let us focus on the velocity of free fall v = dl On the future horizon, kU = 0 and U = 0, so we see that dτ of a particle with E = m, where r ≡ r changes depending indeed ω = 0. 2 0 on time. Then it is easy to find from (4), (39) that Also, it is easy to check that for a photon propagating along the horizon l = 0. Indeed, if we write down the condition v =− − . μ 1 f (40) kμk = 0 on the future horizon, we obtain kφ = 0. This v agrees with previous observations concerning the properties Taking the derivative once more, we obtain d = √1 d f . dr 2 1− f dr of trajectories on the horizon [10,11]. On the horizon, this gives us
dv = κ, (41) 4 Generalized Lemaitre frame dr H where we took into account that for our metric the surface 4.1 Form of metric gravity κ = 1 d f . The subscript “H” means that the 2 dr H It is instructive to reformulate the redshift value in the corresponding quantity is calculated on the horizon. Equation Lemaitre-like coordinates ρ,τ. In contrast to the Kruskal (41) will be used below. It is worth noting that for the extremal κ = dv = ones, this frame is based on free falling particles. The horizon ( 0) we have also dr H 0. Lemaitre frame is well known for the Schwarzschild met- ric. Now, we suggest its generalization valid for the metric 4.2 Redshift: from Kruskal coordinates to Lemaitre ones (1). The general theory of transformations that make the metric The above frame is especially useful for the presentation of of a spherically symmetric black hole regular, was developed the redshift (31). On the horizon, f = 0. Then, in its vicinity, in [12]. For our goals, it is sufficient to find a particular class we obtain from (13), (36), (37) that on the horizon of transformations that (i) makes the metric regular on the v = τ + = ρ + , horizon, (ii) generalize the Lemaitre metric (in particular, the C1 C2 (42) metric should have gττ =−1). We make the transformation where C1,2 are constants. As a result, we obtain from (31) dr ∗ ρ = t + √ , (36) ω − 2 = (κ(τ − τ )) = (κ(ρ − ρ )). 1 f exp 1 2 exp 1 2 (43) ω1 ∗ τ = t + dr 1 − f , (37) Thus the Lemaitre frame allows us to present the resulting redshift along the horizon in a simple and intuitively clear where r ∗ isgivenby(14). Equations (36), (37) are direct picture—the redshift grows (and, consequently, the emitter generalizations of Eqs. 102.1 of [8]. Then it is easy to check looks dimmer) exponentially with respect to Lemaitre time that that passes from emitting to observation. ds2 =−dτ 2 + (1 − f )dρ2 + r 2(ρ, τ)(dθ 2 + sin2 θdφ2). In the last paragraph of Sect. 2, we listed the general con- (38) dition for the geometrical optic to be valid. Now, we can express it in another way. Since a physical wave packet has On the horizon, f = 0, the metric coefficient is regular, a finite length, parts of it will move away from the black gρρ = 1. In the particular case of the Schwarzschild metric, hole horizon even if its center is located exactly on the hori- = − r+ f 1 r and we return to the standard formula for the zon. Since the equations of light geodesics√ in the generalized Lemaitre metric, when r is expressed in terms of ρ and τ. Lemaitre frame reads dr/dτ = 1 − 1 − f for outward The coordinates (36), (37) are suitable for the description propagation, the Lemaitre time needed to leave the vicinity of a black hole including both the outer R region and the of the horizon r = r+ diverges as |ln(r/r+ − 1)|. Suppose, contracting T one [13]. In a similar way, one can use the the emitter radiates light with the wavelength λ. Since in expanding version that would result in a changing sign at τ. any case the wave packets cannot be smaller than λ, we can Now, we want to pay attention to some nice properties roughly estimate initial scale as r − r+ ∼ λ. Then we find τ/ ∼ /λ ∼ ω of the metric (38). The proper distance between √ points 1 that after the Lemaitre time r+ ln r+ ln 0r+ the and 2 calculated for a given τ is equal to l = dρ 1 − f . wave packet will reach the scale of black hole horizon, the Requiring dτ = 0in(37) and substituting dt into (36), we geometric optic approximation fails and, in particular, Eq. obtain from (14), (36) (43) evidently breaks down. 123 Eur. Phys. J. C (2017) 77:179 Page 5 of 8 179
∂v > 5 Photon emitted at the inner horizon seen from (13), (14) that, for a fixed u, ∂r 0. However, it is seen from (44) that event 2, which takes place after 1, Let us consider a situation similar to that considered has r2 < r1. As a result, v2
U = exp(−κ−u), (48) In the previous subsection we have shown that the blueshift at the inner horizon (and, consequently, the energy absorbed V = exp(κ−v), (49) by the observer) grows exponentially with the Lemaitre time where r ∗ is given by the formula between the moments of emission and observation. Here we compare this interesting effect with others known in the lit- ∗ dr r = . (50) erature. g If two particles collide, their energy Ec.m. in the center of The metric takes the form (16) with mass frame can be defined on the point of collision according g to F =− , (51) κ2 UV − 2 =− μ, Ec.m. Pμ P (54) where κ− is the surface gravity associated with the inner μ μ Pμ = p + p being the total momentum of two particles. horizon and F = 0 is finite there. 1 2 μ If particle 1 is massive and particle 2 is massless, p = muμ Repeating the calculations step by step, we arrive at the μ 1 and p = kμ, where we put the Planck constant to unity. As same formula (31), 2 a result, ω2 V1 = . (52) 2 2 E . . = m + 2mω. (55) ω1 V2 c m
Now, we would like to pay attention to the fact that, according In the example under discussion, if V1 = O(1) and V2 → to (12), V is a monotonically increasing function of v.Itis 0, the frequency ω2 →∞according to (52). Then Ec.m. → 123 179 Page 6 of 8 Eur. Phys. J. C (2017) 77:179
∞ as well and we encounter a counterpart of the BSW effect For the Reissner–Nordström–de Sitter metric, such a prob- near the inner horizon. But V = 0 on the future horizon lem was considered in Sect. IV b of [1]. It follows from the U = 0 is nothing else than the bifurcation point [5] (see corresponding results that different cases are possible here: also below for more details). Thus the present results for the ω2 <ω1, ω2 = ω1, ω2 >ω1. On the first glance, this dis- blueshift agree with the previous ones in the limit when the agrees with our results described above since we obtained bifurcation point is reached. either a redshift (for the event horizon of black hole or inner There is also another issue to which we can compare horizon of a white hole) or a blueshift (for the inner one in a the present consideration. As is well known, near the inner black hole or event horizon of a white hole). Fortunately, this (Cauchy) horizon an instability develops inside black holes. contradiction is illusory. Now, we will explain how one can This happens when a decaying flux of radiation coming from obtain the results for the bifurcation point from ours. To this infinity crosses the event horizon and concentrates near the end, we compare (i) the generic situation and (ii) that with inner one—see, e.g. Chapter 14.3.1 in [6]. (For a modern crossing the bifurcation point and trace how (ii) arises from review of the subject see [7].) However, now we consider (i) within the limiting transition. radiation which is not coming from infinity but is emitted For our purposes, it is sufficient to discuss the simplest by an observer who crosses the inner horizon. The resulting metric that possesses the bifurcation point, so we can imply energy flux from an emitter at the inner horizon appears to it to be, say, the Schwarzschild one. We assume that emitter 1 be finite, though it is not restricted from above if V → 0. moves from the inner expanding T + region (i.e. white hole) Thus as far as the radiation near the inner horizon is con- [13], crosses the past horizon and enters the R region. After- cerned, we have three situations: (i) the analog of the BSW wards, it crosses the event horizon falling into a black hole. effect (relevant near the bifurcation point), (ii) blueshift of a Let, as before, the emitter and receiver have equal masses photon in the situation under discussion (relevant near any m1 = m2 = m. However, now we cannot put E1 = E2. This point of the inner horizon, the blueshift is in general finite), is because a particle with E = m would escape to infinity (iii) the instability of the inner horizon (infinite blueshift due instead of falling into a black hole. We remind the reader to concentration of radiation along the horizon). Cases (i) and that, up to now, in all our considerations an emitter and an (ii) are closely related in the sense that in the limit when the observer are set to be at rest in infinity. However, the most point where a photon is absorbed approaches the bifurcation general case can easily be obtained by adding correspond- point, one obtains (i) from (ii). Meanwhile, in case (iii) the ing Lorentz boosts. In the present subsection we meet the effect is unbounded and this points to a potential pathology situation where this procedure is needed. connected with the nature of the inner horizon. Therefore, we must use a more general formula based on (30),
ω2 E2 V1 6 Special case: emission at the bifurcation point = . (56) ω1 E1 V2 In Sect. 5, we discussed briefly spacetimes that contain T + The first factor can be interpreted as a Lorentz boost respon- = regions (white holes). Then the intersection between the sible for the Doppler effect. For E1 E2 we return to the future and past horizons forms the so-called bifurcation point case considered by us above but now the first factor is not (sphere, if the angle variables are taken into account), where equal to 1 and plays now a crucial role. it is possible to pass from the white hole region to the black If, by assumption, particle 1 falls into a black hole, this one. White holes and bifurcation points do not arise in the means that it must bounce from the potential barrier in the = situation when a black hole is formed due to gravitational turning point r r0. According to the equation of motion (4), this means that collapse and in this sense they are not feasible astrophysi- cally. However, they are inevitably present in the full picture E = m f (r0). (57) of an eternal black–white hole. Therefore, we consider such → ( ) → → objects for theoretical reasons and for completeness. In par- If r0 r+, f r0 0, so E 0 as well. More pre- cisely, it is seen from (2), (21) that ticular, in Sect. 5, we saw that accounting for the bifurcation point arises naturally in the connection between our problem E ∼ r0 − r+ ∼ |U| V . (58) and the BSW effect. In doing so, it is a receiver that passes As a result, near the bifurcation point. In the present section, we consider another case, when it ω2 αE2 |V | is an emitter that passes through this point at the moment ∼ ,α≡ 1 . (59) ω V U of radiation. Consideration of the frequency shift when a 1 2 1 photon emitted from the bifurcation point is a separate case In the limit when the trajectory of particle 1 passes closer that does not follow directly from the previous formulas. and closer to the bifurcation point U = 0 = V , α remains 123 Eur. Phys. J. C (2017) 77:179 Page 7 of 8 179
finite. Using the equations of motion in the T region (see the In a sense, this is quite natural. Indeed, the extremal hori- previous section), it is easy to show that in the limit V → 0, zon is the double one. The inner and outer horizons merge. U → 0, the component of the velocity uU contains just this But for an inner horizon we had a blueshift, for the outer factor α. one we had a redshift. Together, they mutually cancel and Thus depending on the relation between α and V2 one can produce no effect. obtain any result for ω2 (redshift, blueshift, the absence of The absence of the redshift or blueshift formally agrees the frequency shift). In this sense, the general formula (56) with (43) if one puts κ = 0 there. However, for (ultra)extremal reproduces both the “standard” fall of the emitter in a black black holes the Kruskal-like transformation looks very dif- hole and the behavior of the emitter that passes through the ferent, so we could not use Eq. (43) directly. Therefore, it bifurcation point. was not obvious in advance whether or not the redshift for the extremal horizon can be obtained as the extremal limit of a nonextremal one. Now, we see that this is the case. 7 Extremal horizon
Let an observer cross the (ultra) extremal horizon r+.By 8 Summary definition, this means that near it the metric function is
n Thus we showed that for emission along the outer horizon f ∼ (r − r+) (60) redshift occurs and we derived a simple formula that gen- where n = 2 in the extremal case and n = 3, 4 ... in the eralized the one previously found in the literature. We also ultraextremal one. The difference from the nonextremal case showed that along the inner horizon blueshift occurs and consists in a different nature of the transformation making found its relation with the BSW effect. We also showed how the metric regular. Let the two-dimensional part of the metric the previously known results for the emission at the bifurca- have the same form as in (1). The subsequent procedure is tion point are reproduced from a general formula and lead well known—see, e.g., [14,15] (Sect. 3.5.1). We use the same to a diversity of situations (redshift, blueshift or the absence coordinates u, v and want to find appropriate coordinates U, of frequency shift). For (ultra)extremal horizons the effect is V , absent. These observations have a quite general character in agree- V = V (v), U = U(u). (61) ment with the universality of black hole physics. We also Now, we are interested in the situation with emission of a generalized the Lemaitre frame and in this frame derived a photon exactly along the horizon. Near the horizon it follows simple and instructive formula for a redshift along the hori- for the tortoise coordinate (14) that zon in terms of the Lemaitre time and the surface gravity.
1 ∗ − Acknowledgements The work was supported by the Russian Govern- r − r+ ∼ r 1 n , (62) ment Program of Competitive Growth of Kazan Federal University. ∗ n f ∼ r 1−n . (63) Open Access This article is distributed under the terms of the Creative We consider the metric near the future horizon where v is Commons Attribution 4.0 International License (http://creativecomm ∗ →−∞ = v − ∗ →+∞ ons.org/licenses/by/4.0/), which permits unrestricted use, distribution, finite, r , u 2r .Wehave and reproduction in any medium, provided you give appropriate credit n to the original author(s) and the source, provide a link to the Creative ∼ 1−n . f u (64) Commons license, and indicate if changes were made. Funded by SCOAP3. We try a transformation that behaves like
− 1 U ∼ u n−1 , (65) References so that U → 0. Then it is easy to check that the metric has the form (16) where F = 0 is finite on the horizon. To find the 1. K. Lake, Reissner-NorBstrom-de Sitter metric, the third law, and frequency, we must use the expression for uU (22) in which cosmic censorship. Phys. Rev. D 19, 421 (1979) n 2. A.T. Augousti, M. Gawełczyk, A. Siwek, A. Radosz, Touching now (65) is valid, so dU ∼ u 1−n . It is seen from (64) that du ghosts: observing free fall from an infalling frame of reference U → u const on the horizon and it does not contain V . Taking into a Schwarzschild black hole. Eur. J. Phys. 33, 1–11 (2012) into account that kV is a constant along the horizon generator 3. K. Kassner, Why ghosts don’t touch: a tale of two adventurers as before, we come to the conclusion that V drops out and falling one after another into a black hole. Eur. J. Phys. 38, 015605 ω2 (2017). arXiv:1608.07511 ω = const. We see that in the horizon limit the quantity 1 ω 4. M. Bañados, J. Silk, S.M. West, Kerr Black holes as particle accel- V does not enter the frequency. In this sense, 2 does not ω1 erators to arbitrarily high energy. Phys. Rev. Lett. 103, 111102 change along the horizon, so redshift or blueshift is absent. (2009). arXiv:0909.0169 123 179 Page 8 of 8 Eur. Phys. J. C (2017) 77:179
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