General radially moving references frames in the black hole background A. V. Toporensky

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General radially moving references frames in the black hole

background

A. V. Toporensky

Sternberg Astronomical Institute, Lomonosov Moscow State University and

Kazan Federal University, Kremlevskaya 18, Kazan 420008, Russia∗

O. B. Zaslavskii

Department of Physics and Technology,

Kharkov V.N. Karazin National University,

4 Svoboda Square, Kharkov 61022, Ukraine†

We consider general radially moving frames realized in the background of nonex-

tremal black holes having causal structure similar to that of the Schwarzschild metric.

In doing so, we generalize the Lemaˆıtre approach, constructing free-falling frames

which are built from the reference particles with an arbitrary speciﬁc energy e0in-

cluding e0<0 and a special case e0= 0. The general formula of 3-velocity of a

freely falling particle with the speciﬁc energy ewith respect to a frame with e0is

presented. We point out the relation between the properties of considered frames

near a horizon and the Banados-Silk-West eﬀect of an indeﬁnite growth of energy

of particle collisions. Using our radially moving frames, we consider also nonradial

motion of test particles including the regions near the horizon and singularity. We

also point out the properties of the Lemaˆıtre time at horizons depending on the

frame and sign of particle energy.

PACS numbers: 04.20.-q; 04.20.Cv; 04.70.Bw

∗Electronic address: atopor@rambler.ru

†Electronic address: zaslav@ukr.net

arXiv:2210.03670v1 [gr-qc] 7 Oct 2022

I. INTRODUCTION

The Schwarzschild metric [1] is a part of primer of black hole physics and enters all text-

books on gravitation. In spite of this, it still remains a testing area of diﬀerent approaches,

including classes of coordinate transformations. As is known, this metric is singular in the

original (so-called curvature, or Schwazschild) coordinates. There exists completely diﬀerent

methods to remove such a seeming singularity. These approaches can be united in a one pic-

ture [2]. What is especially interesting is that if the speciﬁc energy e0of reference particles

(i.e. particle realizing a frame) is included explicitly in the coordinate transformation, the

diﬀerent standard forms can be obtained as diﬀerent limiting transitions. In this approach,

one can recover some well-known metric like the Eddington-Finkelstein ones [3], [4]. The

Lemaˆıtre metric can be also included in this scheme [5], [6]. (Alternatively, one can use a

velocity of the local Lorentz transformation instead of e0[7].) Meanwhile, there exists one

more aspect connected not only with the frames themselves but with particle dynamics in

corresponding background. One may ask, how particle motion looks like depending on e0

and particle speciﬁc energy eand relation between them.

In the previous paper [8] we considered such frames that all reference particles have

e0= 1 or e0= 0. In the present work, we make the next step and consider a more

general situation when e0is arbitrary. In particular, this includes the case of e0<0.

Motivation for such generalization is at least three-fold. (i) In the aforementioned papers,

the introduction of e0was made for frames, now we consider particle dynamics. (ii) If we

make e0a free parameter, we can trace the relation between reference particles and any other

test particles thus establishing connection between the choice of a frame and properties of

particle collisions. This is especially actual in the context of high energy particle collisions

[9]. (iii) We hope that a general approach developed in our work will be useful tool for

description of particle motion under the horizon, some concrete examples of which in the

Schwarzschild background were discussed in [10] (see also references therein).

In this work, we develop general formalism. In doing so, we suggest simple classiﬁcation

of frames based on the their character (contracting or expanding) and the sign of e0. The

applications of our formalism will be considered in a next paper. Also, we restrict ourselves

by static black holes and postpone the generalization to rotating black holes to future works.

In what follows we deal with the spherically symmetric metrics of the form

ds2=−fdt2+dr2

f+r2(dθ2+dφ2sin2θ). (1)

For the Schwarzschild metric, f= 1 −r+

rwhere r+is the radius of the event horizon.

II. REFERENCE PARTICLES AND FRAMES

One of known frames that removes the coordinate singularity on the horizon is the

Gullstrand-Painlev´e (GP) one [11], [12]. In recent years, this frame again attracts attention

in diﬀerent contexts (see [6] and references therein). Quite recently, some modiﬁcation of

the GP frame was suggested in [13]. The generalization of the original GP frame can be

obtained after introducing a new time variable via

d˜

t=e0dt +dr

fP0,(2)

where by deﬁnition

P0≡qe2

0−f. (3)

The static time tis expressed now as

dt =1

e0d˜

t−dr

fP0(4)

that gives us the metric in the form

ds2=−f

d˜

t2+2d˜

tdr

P0+dr2

+r2dω2.(5)

After introducing a new spatial variable ρvia

dρ =dr

+d˜

t(6)

the metric can be set to a synchronous form

ds2=−d˜

t2+P2

dρ2+r2dω2.(7)

In this form it is evident that the coordinate system is formed by particles, free falling

with the speciﬁc energy e0. Indeed, for a radial fall with the the speciﬁc energy ewe have

the following equations of motion:

dτ =e

f,(8)

dτ =−pe2−f≡ −P, (9)

where τis the proper time.

Now we can consider the derivative

d˜

dτ =e0

dτ +dr

dτ

f=e0e−P0P

f,(10)

where ˜

tis the e0-synchronous time, and τis the proper time of a particle with the energy e.

This entails

d˜

t=−P f

e0e−P0P(11)

and

dρ

dτ =e0(eP0−e0P)

f(12)

that gives us the rate with which the synchronous spatial coordinate ρchanges for a free

falling particle with the energy e. If e=e0, it follows that P=P0and

dρ

dτ = 0,(13)

as is should be. We see that indeed the particle with the energy e0has synchronous spatial

coordinate ρconstant during a free fall.

We can note also that for e0= 1 the coordinate ﬂow velocity dr/d˜

t=−√1−fco-

incides with the velocity of a particle with respect to a stationary frame. For a general

case the velocity of the particle having ewith respect to a stationary frame is equal to

vst =p(e2−f)/e2(see eqs.4.2, 4.4 and 6.5 in [14]), so that vst =P/e = (dr/d˜

t)/e.

Let e0→0. We can easily see that there is no smooth limit for the generalized GP

metric (5). However, as it has been pointed out in [6], we can write a regular form of the

synchronous metric if we make a redeﬁnition ρ=e0˜ρ. As now

e0d˜ρ=dr

+d˜

t, (14)

we see than in this limit ris the function of ˜

tonly and we get from (7) a homogeneous metric

under the horizon where we set f≡ −g:

ds2=−d˜

t2+g(r(˜

t))d˜ρ2+r2(˜

t)dω2. (15)

So far, we have considered only positive energy ingoing particles and frames. To include

negative eand outgoing motion, we generalize the transformation to a synchronous frame.

One can write

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GeneralradiallymovingreferencesframesintheblackholebackgroundA.V.ToporenskySternbergAstronomicalInstitute,LomonosovMoscowStateUniversityandKazanFederalUniversity,Kremlevskaya18,Kazan420008,RussiaO.B.ZaslavskiiDepartmentofPhysicsandTechnology,KharkovV.N.KarazinNationalUniversity,4SvobodaSquare,Khark...

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General radially moving references frames in the black hole background A. V. Toporensky

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