Regular frames and particle's rotation near a black hole
aa r X i v : . [ g r- q c ] M a y General Relativity and Gravitation (2019) 51: 60 https://doi.org/10.1007/s10714-019-2544-z
Regular frames and particle’s rotation near a black hole
Yuri V. Pavlov , · Oleg B. Zaslavskii , Received: 28 September 2018 / Accepted: 29 April 2019
Abstract
We consider a particle moving towards a rotating black hole. Weare interested in the number of its revolution n around a black hole. Inour previous work (Pavlov and Zaslavskii in Gen Relativ Gravit 50: 14, 2018.arXiv:1707.02860) we considered this issue in the Boyer-Lindquist type of co-ordinates with a subsequent procedure of subtraction. Now, we reconsider thisissue using from the very beginning the frames regular on the horizon. For anonextremal black hole, regularity of a coordinate frame leads to the finitenessof a number of revolutions around a black hole without a subtraction proce-dure. Meanwhile, for extremal black holes comparison of n calculated in theregular frame with some subtraction procedures used by us earlier shows thatthe results can be different. Keywords
Black holes · Rotating Frames · Nonextremal and extremalhorizons · Critical particles · Rotational analogue of Eddington—Finkelsteinframes
Yu. V. PavlovE-mail: [email protected]. B. ZaslavskiiE-mail: [email protected] Institute of Problems in Mechanical Engineering, Russian Academy of Sciences,61 Bol’shoy pr., St. Petersburg 199178, Russia N.I. Lobachevsky Institute of Mathematics and Mechanics, Kazan Federal Univer-sity, 18 Kremlyovskaya St., Kazan 420008, Russia; Department of Physics and Technology, Kharkov V.N. Karazin National University,4 Svoboda Square, Kharkov 61022, Ukraine Yu.V. Pavlov, O.B. Zaslavskii
It is well known that if a particle falls towards a black hole, there is a sharpcontrast between the time intervals measured by an observer at infinity ( t )and an observer comoving with the particle (the proper time τ ). To reach theevent horizon, a particle needs an infinite t . Meanwhile, τ (except from somespecial situations) is finite. The similar relation exists between the numberof revolutions n around a rotating black hole [1]. For a remote observer, it isalways infinite. For a comoving observer it can be finite or infinite.In the previous work [1] we showed that the relationship between these twocharacteristics can be quite nontrivial. In particular, it can depend not only onthe type of black hole and a trajectory but also on a way the measurement areperformed. We considered (i) the measurement for a free-falling particle withrespect to another free-falling one, (ii) measurements with respect to a particlethat is not in free fall, (iii) with respect to a black hole itself. In all three casesit was impossible to view the effect using one fixed coordinate frame.The divergences of n are connected with the fact that the original coordi-nates (like the Boyer-Lindquist ones for the Kerr metric) fail near the horizon,so both t and the polar angle variable φ diverge. Meanwhile, in recent years,coordinate frames regular near the rotating black holes were constructed [2]–[5] that enabled one to trace rather subtle details of particle behavior near thehorizon [6], [7] or give physical interpretation of metrics in terms of rotatingfluid [8]. The aim of the present work is to consider the properties of revolutionof a particle around a black hole using this type of angle variable. Then, thereis no need to develop special schemes for subtraction of the contribution to n from the reference frame since this subtraction is already implicitly containedin the definition of a “good” angle variable. We will see that this simple anddirect procedure agrees with the aforementioned method (i) that, however, canbe different from (ii) and (iii).It turned out that overlap between different but related issues (descriptionof geometry near the horizon and description of particle revolution around it)lead us to construction of the rotational analogue of the contracting/expandingEddington-Finkelstein (or Lemaˆıtre) frames for black/white holes. This gener-alizes corresponding construction, known for the Kerr or Kerr-Newman metric. Let us consider the metric ds = − N dt + g φ ( dφ − ωdt ) + dr A + g θ dθ . (1)Here, the event horizon is described by N = 0. We assume that the metriccoefficients do not depend on t and φ , so that the energy E and angularmomentum L of a particle are conserved. We also assume N = α∆, A = ∆ρ , (2) egular frames and particle’s rotation near a black hole 3 where ∆ = ∆ ( r ), α > ρ > r and θ . The maximumzero r = r + of ∆ corresponds to the event horizon, α and ρ are supposed toremain finite and nonzero at r = r + .In what follows, we use notation X = E − ωL. (3)It can be written in a more general form as X = − mu µ ( κ µ + ωζ µ ), where κ µ isthe Killing vector responsible for time translations and ζ µ is that responsiblefor rotations. We restrict ourselves by motion in the equatorial plane θ = π/ m dtdτ = XN , (4) m dφdτ = Lg φ + ωXN , (5) mρ √ α drdτ = − Z, Z = s X − N (cid:18) L g φ + m (cid:19) , (6)where we assumed that a particle moves towards a black hole, so dr/dτ < m is a particle’s mass, τ being a proper time. It follows from (5), (6)that dφdr = − ρ √ αZ (cid:18) Lg φ + ωXN (cid:19) , (7) t = − Z r Xρ dr ′ √ α∆Z . (8)We imply that the so-called forward-in-time condition dt/dτ > X ≥ In [5], a general approach was suggested that enables one to build framesregular near the horizon. This includes previous known coordinate frames,in particular, Painlev´e-Gullstrand ones for the Kerr and Kerr-Newman met-rics [2]–[4]. To make presentation self-contained, we repeat below some generalformulas from [5], where a reader can find further details. For the metric (1),transformations within the equatorial plane have the form dt = d ¯ t + z ( r ) dr∆ , (9) dφ = d ¯ φ + ξ ( r, θ ) dr∆ + δ ( r, θ ) dθ. (10) Yu.V. Pavlov, O.B. Zaslavskii
The key property of barred coordinates consist in that (apart from some ex-ceptional cases) the trajectories of particles falling on a black hole reach thehorizon for a finite time ¯ t and perform a finite number of revolutions. This isdiscussed below in Sec. 7.We restrict ourselves by motion of particles in the equatorial plane θ =const = π/
2, so the last term in (10) is irrelevant and put δ = 0. It followsfrom (9), (10) that dφ − ωdt = d ¯ φ − ωd ¯ t + hdr, (11) g rr = µ + g φ h , (12)where we introduced new functions h and µ according to h∆ = ξ − ωz, (13) µ∆ = ρ − z α. (14)In new variables, the metric has the form ds = − N d ¯ t − zαdrd ¯ t + g φ ( d ¯ φ − ωd ¯ t + hdr + dθδ ) + µdr . (15)We want to kill the divergences in the metric coefficient g rr . To this end,we choose h and µ finite on the horizon. If we specify some functions z ( r ) and h ( r, θ ) where z ( r ) is also finite on the horizon, it follows from (13), (14) andthe fact that ∆ = 0 on the horizon, that( ξ − ωz ) r = r + = 0 , (16) (cid:0) z α − ρ (cid:1) r = r + = 0 . (17)Then, d ¯ φdr = − ξ∆ − ρ √ αZ (cid:18) Lg φ + ωXN (cid:19) = − ω∆ (cid:18) z + XρZ √ α (cid:19) − h − ρ √ αZ Lg φ . (18)The number of revolutions that particle experiences during travel betweenpoints 1 and 2 is equal to n = ∆ ¯ φ π , ∆ ¯ φ = ¯ φ − ¯ φ . (19)It is worth paying attention to the choice z = − ρ √ − α∆α , (20) h = ωρ z , µ = αρ , (21) ξ = ωρ zα . (22)Then, for a particle moving with L = 0, E = m we see the angle ¯ φ = const.This is the generalization of the corresponding property [6] inherent to the Kerrmetric in coordinates of Ref. [2]. One reservation is in order. The aforemen-tioned choice implies that the combination (20) does not contain the angle θ .This is valid for the Kerr metric but not necessarily in a general case. However,for motion in the plane θ = π/ egular frames and particle’s rotation near a black hole 5 We want to elucidate, whether the change of the angle variable ∆ ¯ φ during aparticle fall into a black hole is infinite or finite. To this end, we examine thebehavior of the time variable ¯ t and show that in cases under consideration itremains finite. Then, we calculate the horizon limit of the quantity d ¯ φd ¯ t . As ¯ t changes in finite limits, a finite (cid:16) d ¯ φd ¯ t (cid:17) H is quite sufficient to conclude that ∆ ¯ φ is finite as well (hereafter, subscript “ H ” means that a corresponding quantityis calculated on the horizon). Alternatively, one may evaluate the quantity (cid:16) d ¯ φdr (cid:17) H since r is supposed to change within a finite interval between someinitial value r and the horizon radius r + .Let us consider now different types of black holes and of particles separately. Now, near the horizon the expansion of ω takes the form [9] ω = ω H − B N + O ( N ) . (23)For the Kerr-Newman metric, B > X = X H + B N L + O ( N ) , (24) Z ≈ X − N X H (cid:18) L g H + m (cid:19) . (25)We have from (17) that near the horizon either z + ρ/ √ α ≈ z − ρ/ √ α ≈ z + ρ √ α ≈ − b∆, (26)where g H = g φ ( N = 0) and b is the model-dependent coefficient. Expan-sion (26) agrees with (17). Then, (cid:18) d ¯ φdr (cid:19) H = ω H b − h H − ρ H √ α H X H Lg H − ω H ρ H √ α H X H (cid:18) L g H + m (cid:19) . (27)It is finite. It follows from the above equations that the quantity ¯ φ obtainedby the integration of the right hand side of (18) is also finite. Thus the anglechanges at a finite value during infall of a particle to a black hole. Yu.V. Pavlov, O.B. Zaslavskii
Now N = D ( r − r + ) + O (( r − r + ) ) , (28)where D > ω takes the form [9] ω = ω H − B N + O ( N ) . (29)Below, we use classification of particles according to which a particle with X H > X H = 0 is called critical. Here, X isused according to definition (3).6.1 Usual particlesThen, one can see that the expression for (27) on the horizon retains its validity,so ¯ φ is finite.6.2 Critical particlesUsing (29), one obtains X = B LN + O ( N ) , (30) Z = Z N + O ( N ) , Z = s E ω H (cid:18) B − g H (cid:19) − m . (31)As a result, n ≈ ρ H √ α H ω H (cid:16) B Eω H − Z (cid:17) πZ D ( r − r + ) , (32)so (32) diverges. Equation (32) corresponds to Eq. (48) of [1]. We saw that, after introducing a regular frame, the number of revolutionsbecomes a finite, except a rather special case of the critical particle movingaround the extremal horizon. Meanwhile, one can notice here a rather in-teresting peculiarity. Near the horizon, Eq. (17) admits two branches for z .Regularization of n leads to a finite result for a quite definite choice of a signof z near the horizon. However, if instead of (26) we take z ≈ + ρ/ √ α , thequantity n remains infinite, although the metric itself looks regular since (17)is satisfied. How can it happen? egular frames and particle’s rotation near a black hole 7 To elucidate this issue, let us consider Eq. (9). It follows from it that¯ t = t − Z rr z ( r ′ ) dr ′ ∆ ( r ′ ) , (33)where r is some constant. Let a free particle fall towards a black hole. Usingequation of motion (8), we have, choosing the integration constant properly,¯ t = − Z rr dr ′ ∆ (cid:18) z + XρZ √ α (cid:19) , (34)where ¯ t ( r ) = 0. A particle moves with decreasing r , so further r < r , t > X/Z ≈ t . In other words, a particle movingfrom the outside towards a horizon, reaches it for a finite interval of ¯ t , makinga finite number of revolutions.In a similar way, we can consider a particle that moves outward. Then, dr/dt > t = − Z rr dr ′ ∆ (cid:18) z − XρZ √ α (cid:19) . (35)With the same choice (26) we obtain divergent ¯ t . Also, the angle variablediverges if r → r + .This has a clear analogy with the contracting and expanding Eddington-Finkelstein or Lemaˆıtre frames (see, e.g. Sec. 33 in [10], Sec. 2.4 and 2.5 of [12]).The contracting frame describes properly a history of particles falling underthe horizon (black hole) but is unable to describe the history of particle ap-pearing from the inner region (white hole). For the expanding system, thesituation is opposite. The aforementioned frames are suited for the descrip-tion of radial motion. Meanwhile, now we dealt with a similar problem for arotational motion.The finiteness of ¯ t gives one more simple explanation of finiteness of thenumber of revolutions. It follows from Eq. (27) that d ¯ φdr is finite. As drd ¯ t is finiteas well, d ¯ φd ¯ t is finite too. Then, the barred angle changes at a finite value duringparticle travel to the horizon. We want to stress the following interesting property of a particle that is scat-tered by a black hole. Let such a particle move from large r to some minimumdistance r f and, afterwards, escapes with the same parameters m, E, L to in-finity again. Then, the change of the angle from r to r f is equal to ∆ φ = + Z r r f η ( r ′ ) dr ′ , (36) Yu.V. Pavlov, O.B. Zaslavskii where η = ρ √ αZ (cid:16) Lg φ + ωXN (cid:17) according to (7). From r f to r a particle moveswith dr/dt >
0, so that the sign in Eq. (7) changes. Therefore, if a particlereturns to the same r , it receives an additional change expressed by the sameformula (36). As a result, the full change ∆φ = 2 ∆ φ . If r f → r + , ∆φ → ∞ due to the factor N in the denominator of the second term in η .If, instead of φ , one uses ¯ φ , one can easily see from(10) that the change ofthe angle ¯ φ for moving from r to r f and return is ∆ ¯ φ = Z r r f (cid:18) ξ∆ + η ( r ′ ) (cid:19) dr ′ + Z r f r (cid:18) ξ∆ − η ( r ′ ) (cid:19) dr ′ . (37)In (37) two integrals containing ξ mutually cancel, and again we obtain ∆ ¯ φ =2 ∆ φ , where ∆ φ is given by Eq. (36). In this sense, the considered coordinatetransformation does not change this result. What is especially interesting isthat the dragging effect persists and is described by the same formula ∆ ¯ φ =2 ∆ φ even if, by choosing appropriate functions z ( r ), h ( r ) , we can achieve¯ φ = const for the falling in equatorial plane. (But, in the chosen coordinates,the angle variable will not remain constant during motion in the outwarddirection.) Let us consider the Kerr metric as an explicit example. Then, in the Boyer-Lindquist coordinates ∆ = r − M r + a , α = 1 Σ , g φ = Σ, g θ = ρ , (38) ρ = r + a cos θ, Σ = r + a + 2 M ra ρ sin θ, (39) ω = 2 M arρ Σ , ρ Σ = ( r + a ) − ∆a sin θ, (40)where M is a black hole mass, aM being its angular momentum. The eventhorizon lies at r = r + = M + √ M − a . The black hole angular velocity ω H = ω | ∆ =0 = ar + a . (41)The choice z = − p M r ( r + a ) , (42) h = ωρ z = − a √ M rΣ √ r + a , (43) ξ = − √ M ra √ r + a (44)in which we took into account (13) corresponds to Ref. [2]. egular frames and particle’s rotation near a black hole 9 Fig. 1
The angles of rotation for particles falling onto the black hole with a = 0 . M fromthe point r = 9 M with E = m , L = 2 mM (red lines), L = 0 (green lines), and L = − mM (blue lines) for the coordinates of Nat´ario (the top lines in groups), Doran (the middle linesin groups), and Kerr (the lower lines in groups). The choice h = 0 , z = − p M r ( r + a ) ,ξ = − M ar p M r ( r + a ) ρ Σ , µ = ρ Σ (45)corresponds to the Natario’s frame [3]. In the plane θ = π/ ξ = − M ar p M r ( r + a ) r + a + Ma r . (46)The choice z = − ( r + a ) , ξ = − a (47)corresponds to so-called Kerr coordinates — see Sec. 33.2 of [10] or pp. 163,164 of [11].It is also instructive to trace how changes the angle ¯ φ during motion of aparticle when it approaches the black hole horizon. See Fig. 1.
10 Discussion and conclusion
It is instructive to compare the above results with those from [1]. For nonex-tremal black holes, the situation unambiguously agrees with Sec. 5 of [1] andthe corresponding line in Table 1 there. For extremal black holes, the situationis more subtle. If a particle is usual, it is shown in [1] that the result dependson a way the angle is measured (see line 3 in Table 1 there). Now, we sawthat n is finite. For critical particles, n was found in [1] to be infinite in allcases. However, the asymptotic behavior is different depending on the proce-dure of measurement of the angle (either as ( r − r + ) − or ln( r − r + )). Now, itbehaves like ( r − r + ) − that agrees with Eq. (48) of [1]. Thus we see that the behavior of n (19) in terms of ¯ φ coincides exactlywith that obtained in [1] for measurement of relative angles φ of two particles(this is column n − n in Table 1 there). Now ¯ φ is a well-behaved singlecoordinate, so one does not need to use the subtraction procedure. Meanwhile,now disagreement is possible between (19) and other methods of computing n , as is seen from the same Table 1 in Ref. [1]. In this sense, the procedure ofsubtraction used in the relative measurement in [1] is the most natural and isconfirmed now by direct calculations of the coordinate behavior.Also, even without referring to previous results [1], we can formulate animportant observation. There is an ultimate connection between two proper-ties: (i) the regularity of a coordinate system that can penetrate inside acrossthe horizon, (ii) a number of revolutions performed by a free falling parti-cle around the horizon. For a nonextremal black hole, a frame can be chosenin such a way that n is always finite. It would be of interest to extend theapproach and results of the present work to the case of nonequatorial orbits.We want to stress that our main result concerns not the properties ofparticle motion as such but has rather conceptual nature. We generalized so-called Kerr coordinates in that our approach is applicable to generic stationaryaxially symmetric black/white holes. There are two different entities: (i) theregularity of the metric near the horizon and (ii) the finiteness of a numberof revolution during particle motion. We showed that both properties becomeregular (finite) simultaneously, if a coordinate transformation is chosen prop-erly.In general, the barred angles discussed in our paper involve the functionof r such as z, ξ , h . In this sense, the quantity ¯ φ is not gauge-invariant as wellas n . However, what is indeed gauge-invariant is the mutual cancellation of alldivergences for generic nonextremal black holes.For spherically symmetric black holes such as the Schwarzschild one, thereexist coordinates in which a metric is regular near the horizon, so that the cor-responding frame is able to describe trajectories of falling particles (say, thecontracting Eddington-Finkelstein or Painlev´e-Gullstrand coordinates). Mean-while, for the description of particles appearing from a white hole region aexpanding version of such coordinates is needed. In doing so, the new coor-dinates represent some combinations of previous time and radial coordinates.In the present paper, we used the angle counterpart of such constructions anddemonstrated that the number of revolutions around of a black hole is, as arule, finite (some exceptional cases are also described). We hope that, apartfrom the behavior of angular variables, the corresponding frames will be use-ful for description of more subtle characteristics in the spirit of the ”river ofspace” [13]. For example, this will help us to build description of kinematics ofparticle collision in terms of peculiar velocities that was done for radial motionin [14]. This is supposed to be considered elsewhere. Acknowledgements
This work was supported by the Russian Government Program ofCompetitive Growth of Kazan Federal University. The work of Yu. P. was supported alsoegular frames and particle’s rotation near a black hole 11by the Russian Foundation for Basic Research, grant No. 18-02-00461-a. The work of O. Z.was also supported by SFFR, Ukraine, Project No. 32367.
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