On the possibility of observation of the future for movement in the field of black holes of different types

0.13 MB·English

by  Yu. V. Pavlov

On the possibility of observation of the future for
movement in the field of black holes of different types
Yuri V. Pavlov
3
Abstract It is shown for a spherically symmetric black hole of general type
1
that it is impossible to observe the infinite future of the Universe external
0
2 to the hole during the finite proper time interval of the free fall. Quantita-
tive evaluations of the effect of time dilatation for circular orbits around the
n
a Kerr black hole are obtained and it is shown that the effect is essential for
J ultrarelativistic energies of the rotating particle.

7
Keywords Black holes Kerr metric Circular orbits

· ·
]
c PACS 04.70.-s 04.70.Bw 97.60.Lf
· ·
q
-
r
g 1 Introduction
[

2 It is well known an observer falling radially into a black hole will reach the
v horizoninfinite propertime,butthecoordinatetime inthe Schwarzschildco-
0
ordinate system is infinite [1,2]. This leads to an illusion of the possibility for
0
a falling in the black hole by a cosmonaut so as to observe the infinite future
0
4 oftheUniverseexternaltotheblackhole(see,forexample,[3,4]).Theimpos-
. sibility ofsuchan observationis shownin [5,6]. Note that the impossibility of
3
theobservationoftheinfinite futureforthe radialfallingtotheSchwarzschild
0
2 blackholeinthefour-dimensionalspace-timeis evidentfromthe propertiesof
1 the Kruskal–Szekerescoordinate system [7,8].
v: The possibility of observing the infinite future when falling on the black
i hole is analyzed in Sect.2 for spherically symmetrical black holes of general
X
type: black holes with electrical charge, with nonzero cosmological constant,
r
dirty blackholes (those with nonzerostressenergyoutside ofstatic horizons),
a
and multidimensional black holes. In the general case, there are no explicit
Yu.V.Pavlov
InstituteofProblemsinMechanical Engineering,RussianAcademyofSciences,
Bol’shoypr.61,V.O.,SaintPetersburg199178, Russia
E-mail:[email protected]
2 YuriV.Pavlov
analytical expressions for Kruskal–Szekeres coordinates and the analysis of
space-time properties in such cases is difficult even for the radial falling on a
blackhole.Inthispaperitisshownfromthe analysisofthe nullandtime-like
geodesicsfor sphericallysymmetric black holes ofthe generaltype that, if the
proper time of the fall is finite, then the time intervalof observationof events
in the point of the beginning of the fall is also finite. Quantitative evaluations
are given for the Schwarzschild black holes.
Another the possibility to observe the far future of the external Universe
is also discussed in the literature, namely due to time dilatation near a black
hole. So in [9], p. 92, it is stated “However, a more prudent astronaut who
managed to get into the closest possible orbit around a rapidly spinning hole
withoutfallingintoitwouldalsohaveinterestingexperiences:space-timeisso
distortedtherethathisclockwouldrunarbitraryslowandhecould,therefore,
in subjectively short period, view an immensely long future timespan in the
externaluniverse”.InSect.3ofthepaperquantitativeevaluationsforthetime
dilatation on the circular orbits around the rotating black hole are obtained
and it is shown that the effect becomes essential for ultrarelativistic energies
of the rotating object.
2 Observation of the future when falling on the spherically
symmetric black holes
Consider a spherically symmetric black hole with the metric
dr2
ds2 =A(r)c2dt2 r2dΩ2 , (1)
− A(r) − N−2
where c is the light velocity, dΩ — the angle element in space-time of
N−2
the dimension N 4, A(r) — a certain function of the radial coordinate r
≥
which is zero on the event horizon r of the black hole: A(r ) = 0. For the
H H
Schwarzschildblack hole it holds [10]
r 2GM
g
A(r)=1 , r = , (2)
− r g c2
where G is the gravitational constant, M — the mass of the black hole. For
an electrically charged nonrotating black hole in vacuum we have [11,12]
r q2
g
A(r)=1 + , (3)
− r r2
where q is the charge of the black hole. For nonrotating black holes with
nonzero cosmologicalconstant Λ, one has Kottler [13] solution
r Λr2
g
A(r)=1 . (4)
− r − 3
Onthepossibilityofobservation 3
Formultidimensionalnonrotatingchargedblackholes[14]withacosmological
constant, one has
r N−3 2Λr2 q2
g
A(r)=1 + , (5)
− r − (N 1)(N 2) r2(N−3)
(cid:16) (cid:17) − −
where
1/(N−3)
def 2GNM
r = , (6)
g (N 3)c2
(cid:20) − (cid:21)
providedG —theN-dimensionalgravitationalconstantisnormalizedsothat
N
theN-dimensionalNewtonlawinnon-relativisticapproximationpossessesthe
following form
mM
F =G . (7)
NrN−2
Equations for geodesics in metric (1) can be written as
dt dϕ L
A(r) =ε, = , (8)
dτ cdτ r2
dr 2 L2
=ε2 A(r) κ+ , (9)
cdτ − r2
(cid:18) (cid:19) (cid:18) (cid:19)
where κ = 1 is for timelike geodesics and κ = 0 is for the null geodesics.
For a particle with the rest mass m, the parameter τ is the proper time,
εmc2 = const is its energy in the gravitational field (1); and Lmc = const is
the projection of the angular momentum on the axis orthogonal to the plane
of movement in the four-dimensional case.
From Eqs. (8), (9) for the intervals of the coordinate time t t and
f 0
−
proper ∆τ time of movement of the particle from the point with the radial
coordinate r to the point with coordinate r <r , one has
0 f 0
r0
1 dr
t t = , (10)
f 0
− c
Z A(r) L2
rf A(r) 1 κ+
s − ε2 r2
(cid:18) (cid:19)
r0
1 dr
∆τ = . (11)
c
Z L2
rf ε2 A(r) 1+
s − r2
(cid:18) (cid:19)
Asonecanseefrom(10),thesmallestcoordinatetimeofmovementisrealized
for photons with zero angular momentum. It is equal to
r0
1 dr
t t = , (12)
f s
− c A(r)
Z
rf
4 YuriV.Pavlov
where t is the starting time for radial movement of the photon from the
s
point r .
0
Subtracting (12) from (10) for κ=1, one finds an answer to the question:
how much later are the events in points with the same value of the radial
coordinateasinthebeginningofthefallwhichcanbeobservedbytheobserver
falling up to the point r ?
f
r0 1 1+ L2 dr
1 ε2 r2
t t = . (13)
s− 0 c (cid:16) (cid:17)
rZf 1−Aε(2r) 1+Lr22 1+ 1−Aε(2r) 1+Lr22
r (cid:16) (cid:17)(cid:20) r (cid:16) (cid:17)(cid:21)
From(11)and(13), onearrivesatthe followingconclusion:ifthe propertime
interval is finite, then the time interval of observation of the future events at
the point of the beginning of the fall in the process of falling is also finite.
This conclusion is generalized for the case of movement of the charged
particle. For the metric (5) this can be obtained by the transformation ε
→
ε (qQ/rN−3),whereQisthespecificchargeofthemovingparticleinEqs.(8)–
−
(11),(13).Iftheenergy,angularmomentumandthe chargeoftheparticleare
such that the proper time of the fall on the black hole is finite, then the
observation of the infinite future of the external Universe is impossible.
In Fig. 1 the results of calculations of the ratio of the possible time of
observationofthefutureatthepointofthefalltothepropertime intervalfor
the observer radially falling in the Schwarzschild black hole are given. As we
Ht -t L(cid:144)DΤ
s 0
1.3
1.2
1.1
r
€€€0€€€€
r
50 100 150 g
0.9
Fig.1 Theratio(ts−t0)/∆τ fortheobserverfallingfromrestatthepointr0tothehorizon
oftheSchwarzschildblackhole.
can see from (11), (13) for the Schwarzschildblack hole (see explicit formulas
(6), (9) from [5]) the following asymptotic behaviour holds (t t )/∆τ 1
s 0
− →
forr .Ifthefallbeginsfromrestatthepointclosetotheeventhorizon,
0
→∞
then
t t r
s 0 g
− log2 , r r . (14)
0 g
∆τ ∼ r r →∞ →
0 g
r −
Onthepossibilityofobservation 5
But in this case t t (r /c)2log2, i.e., the possible time interval of the
s 0 g
− ≈
observation of the future is small.
For any nonradial fall of the nonrelativistic particle on the Schwarzschild
blackholeitisalsoimpossibletohavealargeintervalofthefuturetimewhich
can be seen from Fig. 2:
Ht -t L(cid:144)DΤ
s 0
2
1.8
1.6
1.4
1.2
L
€€€€€€€€€€€
r c
0.5 1 1.5 2 g
0.8
0.6
Fig. 2 Thedependence(ts−t0)/∆τ ontheangularmomentumoftheparticlewithε=1,
r0=3rg fallingtothehorizonoftheSchwarzschildblackhole.
3 Time dilatation on circular orbits of the Kerr black hole
Kerr’smetric[15]oftherotatingblackholeinBoyer-Lindquist[16]coordinates
has the form
2Mr(dt asin2θdϕ)2
ds2 =dt2 − (a2cos2θ
− r2+a2cos2θ −
dr2
+r2) +dθ2 (r2+a2)sin2θdϕ2, (15)
∆ −
(cid:16) (cid:17)
where
∆=r2 2Mr+a2, (16)
−
M is the mass of the black hole, aM — its angular momentum. Here we use
the units: c=G=1.For a=0, the metric (15) describes a nonrotatingblack
hole in Schwarzschild coordinates. The event horizon of the Kerr’s black hole
corresponds to the radial coordinate
r =r M + M2 a2. (17)
H
≡ −
Equatorial (θ = π/2) geodesics in Kperr’s metric (15) are defined by the
equations (see [17], Sect.61):
dt 1 2Ma2 2Ma
= r2+a2+ ε L , (18)
dτ ∆ r − r
(cid:20)(cid:18) (cid:19) (cid:21)
6 YuriV.Pavlov
dϕ 1 2Ma 2M
= ε+ 1 L , (19)
dτ ∆ r − r
(cid:20) (cid:18) (cid:19) (cid:21)
dr 2 2M a2ε2 L2 ∆
=ε2+ (aε L)2+ − κ, (20)
dτ r3 − r2 − r2
(cid:18) (cid:19)
where εm = const is the energy of the particle with the rest mass m in the
gravitationalfield(15);Lm=constistheprojectionoftheangularmomentum
of the particle on the rotation axis of the black hole.
Let us define the effective potential of the particle in the field of the black
hole by
2
1 dr
V = . (21)
eff
−2 dτ
(cid:18) (cid:19)
Then d2r/dτ2 = dV /dr and the necessary conditions for the existence of
eff
−
circular orbits in equatorial plane are
dV
eff
V =0, =0. (22)
eff
dr
It is sufficient for the existence of stable circular orbits that
dV d2V
eff eff
V =0, =0, >0. (23)
eff dr dr2
The solutions of Eqs. (22) can be written in the form [18]
x3/2 2√x A
ε= − ± , (24)
x(x2 3x 2A√x)
− ±
p
x2 2A√x+A2
l = ∓ , (25)
± x(x2 3x 2A√x)
− ±
where the upper signcorrespopndsto the directorbits(i.e., the orbitalangular
momentum of a particle is parallel to the angular momentum of the black
hole), the lower sign corresponds to retrograde orbits,
r a L
x= , A= , l= . (26)
M M M
The circular orbits exist from r = up to minimal value corresponding to
∞
the photon circular orbit r defined by the roots of the denominator (24),
ph
(25) equal to [18]
2
r± =2M 1+cos arccos( A) . (27)
ph 3 ∓
(cid:20) (cid:18) (cid:19)(cid:21)
The minimal radius of the stable circular orbit is equal to [18]
x± =3+Z (3 Z )(3+Z +2Z ), (28)
ms 2∓ − 1 1 2
p
Onthepossibilityofobservation 7
where
Z =1+ 1 A2 1/3 (1+A)1/3+(1 A)1/3 , Z = 3A2+Z2. (29)
1 − − 2 1
(cid:0) (cid:1) h i q
The specific energy of the particle on such a limiting stable orbit is
ε= 1 (2/3x ).
ms
−
Theminimalradiusoftheboundedorbit(i.e.,orbitwithε<1)isobtained
p
for ε=1 (the particle is nonrelativistic at infinity) and is equal to [18]:
x± =2 1+√1 A A. (30)
mb ∓ ∓
(cid:16) (cid:17)
In this case, we have
l± = 2 1+√1 A . (31)
mb ± ∓
(cid:16) (cid:17)
This orbit is nonstable.
From (18), (24), (25) one obtains
dt x3/2 A
= ± . (32)
dτ x3/2(x3/2 3√x 2A)
− ±
p
Due to the fact that t is the time of the observer resting at infinity from the
blackhole,τ isthepropertimeoftheobservermovingalongthegeodesics,one
can consider this value to be “time dilatation” for the corresponding circular
orbit.
For the Schwarzschild black hole (A = 0), the time dilatation on the lim-
iting bounded circular orbit (x = 4) is equal to 2 (the limiting horizontal
mb
dashed-line on Fig. 2 correspondsto this value). Onthe minimal stable circu-
larorbit(x =6),thetimedilatationisonly√2.However,forcircularorbits
ms
close to photon orbit r = 3M, from (24), (32) for A = 0, using the series
ph
expansion in ε−1 one gets
dt 1 1
=3ε+ +O , ε , (33)
dτ 6ε ε3 →∞
(cid:18) (cid:19)
i.e., the time dilatation can be as large as possible.
Note that in Minkowski space-time one gets from the special relativity
dt/dτ = ε. So in the case of movement around the Schwarzschild black hole,
the time dilatation for relativistic energies of the objects can be enlarged at
most by a factor of three.
For circular orbits with ε around the rotating black holes formu-
→ ∞
las (24), (27), (32) give
dt 2A
3 ε. (34)
dτ ∼ ± 2cos arccos(∓A) A
3 ∓
 (cid:16) (cid:17) 
8 YuriV.Pavlov
For direct orbits close to the rapidly rotating black holes with A 1, r+
→ ph →
M, it holds
dt 6 1
+ +O(√1 A) ε, (35)
dτ ∼ r1−A 3 − !
for retrograde orbits r− 4M,
ph →
dt 7
+O(1 A) ε. (36)
dτ ∼ 3 −
(cid:18) (cid:19)
The time dilatation on the minimal stable circular orbit of the rapidly
rotatingblackhole(A 1)canbeobtainedfrom(24),(28)–(32)andisequal,
→
for direct orbit x+ 1,
ms →
dt 24/3
(x+ )= 1+O(√31 A) , (37)
dτ ms √3(1 A)1/3 −
− h i
for the retrograde orbit x− 9, dt/dτ 13/(6√3). For minimally bound
direct circular orbit, x+ ms1,≈we have ≈
mb →
dt 2
(x+ )= 1+O(√1 A) . (38)
dτ mb √1 A −
− h i
For the retrograde minimally bound orbit: x− 3+2√2, dt/dτ 3 √2.
mb ≈ ≈ −
Let us give the values of time dilatation on circular orbits for the black
holewiththe Thorne’slimitforastrophysicalblackholesA=0.998(see[19]):
dt dt
(x+ )=43.8, (x+ )=10.8. (39)
dτ mb dτ ms
For particles with large specific energy rotating in the direction of rotation of
the black hole for orbits close to circular photon orbit one has
dt
A=0.998 55.12ε, ε . (40)
⇒ dτ ∼ →∞
So the relativistic effect of the time dilatation close to a rotating black hole
may be 55 times larger!
Acknowledgements The author is indebted to Prof. A.A. Grib for the interest in the
paperandusefuldiscussions.TheresearchisdoneincollaborationwithCopernicusCenter
for Interdisciplinary Studies, Krak´ow, Poland and supported by the grant from The John
TempletonFoundation.
Onthepossibilityofobservation 9
References
1. LandauL.D.andLifshitzE.M.:TheClassicalTheoryofFields.PergamonPress,Oxford
(1975)
2. Misner C.W., Thorne K.S. and Wheeler J.A.: Gravitation. Freeman, San Francisco
(1973)
3. ReggeT.:CronacheDell’Universo.Boringhieri,Torino(1981)[TranslatedintoRussian.
Mir,Moscow,1985]
4. CherepashchukA.M.:BlackHolesintheUniverse(inRussian).Vek2,Fryazino(2005)
5. Grib A.A. and Pavlov Yu.V.: Is it possible to see the infinite future of the Universe
when falling into a black hole? UspekhiFiz.Nauk179,279(2009) [English transl.:
Physics–Uspekhi52,257(2009)]
6. Krasnikov S.: Falling into the Schwarzschild black hole. Important details.
Grav.Cosmol.14,362(2008)
7. KruskalM.D.:MaximalextensionofSchwarzschildmetric.Phys.Rev.119,1743(1960)
8. Szekeres G.: On the singularities of a Riemannian manifold. Publ. Mat. Debrecen 7,
285(1960)
9. ReesM.:OurCosmicHabitat.PrincetonUniversityPress,Princeton(2001)
10. SchwarzschildK.:U¨berdasgravitationsfeldeinesmassenpunktesnachderEinsteinschen
theorie.Sitz.Preuss.Akad.Wiss.Berlin,189(1916)
11. Reissner H.: U¨ber die eigengravitation des elektrischen feldes nach der Einsteinschen
theorie.Ann.Physik355,106(1916)
12. Nordstr¨om G.: On the energy of the gravitation field in Einstein’s theory.
Kon.Nederland.Akad.Wet.Proc.20,1238(1918)
13. Kottler F.: U¨ber die physikalischen grundlagen der Einsteinschen gravitationstheorie.
Ann.Phys.361,401(1918)
14. Tangherlini F.R.: Schwarzschild field in n dimensions and the dimensionality of space
problem.NuovoCimento27,636(1963)
15. KerrR.P.:Gravitationalfieldofaspinningmassasanexampleofalgebraicallyspecial
metrics.Phys.Rev.Lett.11,237(1963)
16. Boyer R.H. and Lindquist R.W.: Maximal analytic extension of the Kerr metric.
J.Math.Phys.8,265(1967)
17. ChandrasekharS.:TheMathematical TheoryofBlackHoles.OxfordUniversityPress,
Oxford(1983)
18. Bardeen J.M., Press W.H. and Teukolsky S.A.: Rotating black holes: lo-
cally nonrotating frames, energy extraction, and scalar synchrotron radiation.
Astrophys.J.178,347(1972)
19. Thorne K.S.: Disk-accretion onto a black hole. II. Evolution of the hole.
Astrophys.J.191,507(1974)