- Email: [email protected]
Quantum thermal effect of arbitrarily accelerating black hole with electric and magnetic charges
Chin. Artron. Astrophyr. (1995) 19/l, 14-26 A translation of Acta Astron. Sin. (1994) 38/3! 246-257 Copyright @ 1995 Else&r Saence Ltd Printed in Great Britain. All rights reserved 02761062/9SS24.00+.00
02751062(95)00005-4
Quantum thermal effect of arbitrarily accelerating black hole with electric and magnetic charges T ZHU Jian-yang’
of P hy sits, Furhou
‘Department
ZHAO Zheng3
ZHANG Jian-hua2 Teachers’
College,
Jiangxi
944000
Shandong
274Olli
2Department
of P hy sits, Hete Education
College,
3Department
of Physics,
University,
Beijing
Abstract The Hawking radiation with electric and magnetic charges
Normal
Beijing 100875
for an arbitrarily accelerating black hole is investigated. Expressions for the event
horizon and the Hawking thermal spectrum are given. Both the figure and the temperature depend on both the time and the angles. The expressions reduce to the well-known results for the particular cases of zero acceleration, acceleration along a straight Key words:
line, zero electric dynamic
charge or zero magnetic
black hole-local
field.
event horizon-Hawking
radiation
1. INTRODUCTION Recently WANG Yong-jiu many magnetic monopoles rameters. study
We shall take this very general
the dynamical
gate the Hawking charge
et al. ~1 studied the gravitational field of accelerating bodies and obtained a black hole solution with six time-dependent
it carries
behaviour
type of black hole as our spacetime
of the Klein-Gordon
effect of this black hole and examine and its acceleration
particle
located
therein,
the effect of the electric
with pa-
background, hence investiand magnetic
on the radiat,ion.
2. THEORETICAL
CALCULATION
2.1 The Metric Tensor of an Arbitrarily Accelerating, Charged and Magnetic Black Hole For studying the Hawking effect, we re-write the metric of an arbitrarily accelerating, charged and magnetic black hole given in Ref. [l], in terms of the advanced EddingtonFinkelstein coordinates as follows: t Supported by National Natural Science Foundation Received 1993 March 27; revised version 1994 March 1
Black Hole
Accelerating
ds’ -
gmdv’ -I- Zg,,dvdt
A- 2g,,dvdO
-I- 2g,,dvd4
15
+ g,,dP
-t g,,dCb2,
(1)
where g,-l--+---
tm
EZ -f- Q’
Y rz - (blind +
20rcos0
CCOS+
-
-
r
bEZ +
QZcose
asin8)2,2 I
-
(bcos4
(2)
go2 -
1, r’(bsin4
go3
sinBcosB(bcos4
go1 -
-
b52 g33
-
c sin4)‘r2cos28 -
-
-
+
ccos4
a&6),
-
-
csin4)r2,
-r2, -r2sin28,
the parameters m = m(u), E = E(u), Q = Q(w) being the mass of the source, and the electric and magnetic charge it carries, a = a(v),b = b(v),c = c(o), the acceleration parameters, a representing the magnitude of the acceleration and b and c, the rate of change of the direction of Bcceleration. The metric
determinant
and the contravariant
components
are easily found:
g - --r4sin2B. 8 01 x - 1,
g I2 -
-(bsin+
g13 -
-cty@(bcos4
g2t =: -
(3)
-I- ccos4 -
-
asine),
,
csin4),
(4
r r2’ 1
g J3=L-
2.2
Equation
r?sin28’
for the
Local Event
We now use the condition
.BF r3F grr --= a xp a2v
of zero curvature
surfacer21
o
(5)
to find the local event horizon
F - F(v,t,B,4)
Horizon
- 0
of the specetime
(1). In the above,
(6)
ZHU Jian-yang et al.
16
is the equation of zero curvature surface. Let its explicit form be
From (6) and (7) we have E+fE-o, r
(8)
g+gz-o, g+yC&-0.
I
f
Writing out (5) and substituting
(8), we have
This is the equation for the local event horizon; it determines the location of the horizon, rH. Obviously, rH depends not only on V, it depends also on the angles 0 and 4. The position of horizon varies with time and possesses no symmetry whatever. 2.3
Dynamical
Equation of the Klein-Gordon
Particle
In curved spacetime, the dynamical behaviour of the scalar particle (Klein-Gordon ticle) is described by the K-G equatio&l,
4-bK&--
- i&A,)d
eAI,-
eE,+
ie A,))] @ -
&p -
0,
par-
(10)
(11)
qM,,
where e, Q are the electric and magnetic charge the particle carries, E,, M,, are the electric and magnetic Cvectors generated by the electric and magnetic charges carried by the black hole and ~0 is the rest mass of the K-G particle. Using the Lorentz condition,
(12) we can transform (10) into g” -a=@ al*
+f.+f,
2 - 80 ibar
+
2gl3 f!&
r
~+fo$+f+@p-o, ar
+
2gl3 EP_
ad+
+
g2’ i??_ a02
+
g33 * w
(13)
Accelerating
Black Hole
17
where
F,
=
2i.sAl - 2 r’
F,
=
-2i.s(A1g11 + Asg12 + A3g13 - Ao) + --g
2 11
&I”
+ ar
+ ctyBg12 I
+tlg12 + 8g13
ae -zp
fe
=
fc$ = f
2.4
=
-2ie(A1g12 + A2g22) + fg” + ctyOg22, -2i&(Arg13
+
A3g33)+ Eg13,
c2gfi”A,Av + pi
Generalized
Tortoise Coordinates
The extreme complexity of equations (13) and (14) makes a rigorous solutions impossible. However, only asymptotic behaviour near the horizon are physically meaningful. For this we make a tortoise
P*
=
transformation
of the following
r+ & ln[r - W(V> 0,
form:
t
dJ)l>
v* = v--o, 6, = 8-80,
+
(15)
4, = 6-407
where rH is the event horizon of the black hole, n: is an adjustable parameter (below we shall see that n is just the temperature function representing the Hawking radiation), invariant under the tortoise transformation, and 00, Bo,& are arbitrary constants, also invariant under the transformation.
ZHU Jian-yang et al.
18
From (15) we obtain
ar
--a
1+ E
a
a
1 241
IH)’
-
E-l+
al2
(arH ) a
lH)
241.7
-- a ah
ar*’
1
ae-ae,-a ST’
1a Yz
dr,’
a+
1 2k(1-
ln)
( -&
) ar*,
2 a2
1
zr(t - rff) I ar:
[
8 1
-
24r
2dB (
a2
s=ae:-
a2
)
241-
ahae,
4
241
.S-
-
(4
lH)2
2s(r-
ld
( ) ,v
241
a+
--
c
CrerIf)+
21E(r -
,
ah
82.
area+*
a2fn
_
a
ah a+ 2 (3 +_ L 1
32
2 31, a+
’
1
alH 2
a5,
( )cl- lH)+ w ae
a2 --a+:
al,
ae 1 a2 (-_) [ 2~(1- iH) al:
+
_ yp
a2
tH)2 arH
31,
a2
a
-
iJ& --
-
1H)
at:
2
)
lH)2
a
,
al*
alH L_,+
aflat
1 [
2r(r-
x
a2 dedr-
1+
a* I afdl,
1 2~(~
-
lH)
-
2k(l
-
lJ#)
241
-
rH)2
farH au (-1 1a~+ al: a*
1
1
_a al,
’
(%)
2r(r- lH) ae*ar, iK(1 - 1Jf) f ali\
L*+
x
.-- a2 .a+ar
[
d
(4 au
1+ L
1 2~(~
-
a*
lH)
1
al:
+
\ ae 24r-
I
_a_
rH)2’ al,
’
arH
[
1-t
x
2n(r-
1
1 1+ 2k(1-
ld
a* I ahal,
1
3 lH)
_
a2 + al:
c-1 a+
2n(r
-
1H)
alH w 2r(r-
a4 rH)2
a al,
l
AcceleratingBlack Hole
19
Substituting (16) in (13) we obtain, after some reduction,
{2rk(r -
rH)[1+2&(t-
+
[2P(9
-
[gq9+g33(4$]}$$+2
+
ll+2k(f
-
g*
+
{(I
a% rI#) -@
-
-
* -
rH)[1+2c(r
-
i(2g”
-
w
2g”[ 1 + 24 r -
-
24r
l
-
TH)l
$*
fH)f_l
+ (2g33($$)
tH)y
11 + 2r(r
+ 2g33(5]
+ 245
-
+ 2c(t -
{-gY1
hJlP
24r
-
r,)]}-’
2g”Cl + 24t
r*)]
rH)g
-
tm)1
6% > lM*&,
8’cp > w&, 3
33 a% I
{g” + 2 (&,
-
2g”(%)
-2g”(~)-tg”[(~)(r--m)+(~)I] -+ g”
+f*
K
$L$
)
(r -4+($$3]
h ( a+ > -
tff)[l
+
2r(r
-
f, ~+f@}“o* *
f,fl
+(L($)+fs(?$)
+ 2n(r
+ 21E(T -
-
r,,l>(r
tH)l-’
-
1
-f@
Id} - a@ ab
2 a!_
_f
* 84,
(17)
This is the K-G equation in tortoise coordinates.
2.5
Reduction
of the K-G Equation Near the Horizon
Our study of various types of black hole I44 showed that the K-G equation in tortoise coordinates should have the form of typical wave equation near the horizon. Likewise, as -+ 40, equation (17) can be reduced to the following f ---) w(u0,B0,40),u + vo,e -+ go,4 form: A*+2
ar:
“@
avear*
+B
8@
ae,ar,
+C
6@
abaf,
+(D+j2~,)&-o
ar,-
’
(18)
2HU Jian-yang
20
et al.
where A -
lim -‘If
-
g”[l
+ 2n(r
-
rH)l* + [29”‘(%)
( >I
+ 2g13 3.L a+ (21E(t -
-
[l +24r
-
TH)l -
+
[P(%J
+ P(%Y]}/
Qf)El+24r-rH)l),
(GA&
+ .sA,$’ + EAZ&?+ e A3gi3 + e Ado
+ eA,A”) ($$
Obviously, B, C, D, wg are finite. As for A, both the numerator hence, by l’H6pital rule. we have A = 1 if c_
-(g_(~)(~)y.?g)(~) 2 [l + a!” + 2 (2) - Zg"(%) -
At this point, neighbourhood 2.6
we have reduced
the G-K
equation
zg"
and denominator
p>
to a simple
1 ,
tend to 0;
(1% ,p,*O,~O).
wave equation
for the
of the horizon.
The Hawking Spectrum of Klein-Gordon We separate
2gyk)
Particle
the variables,
(20) and substitute
in equation
$+rDEquate
(18) to obtain
Zi(co - coo)] - -
both side to a constant,
-A:
B
(21)
Accelerating Black Hole
R” + [D + R -
2i(o
-
q)lR’
-
21
(22)
0.
,~++=I_.
(23)
When dealing with radiation we need consider only the radial equation (22). Solving the latter we obtain the ingoing and outgoing radial wave functions through the horizon, QY .rc
(24)
e--rr” * *9 -
e -iw*
&Z(co-q)r*
e-(D+A)r*. (25)
We now use a method first established by Damour and Ruffini[q and later refined by Sanna@ to find the Hawking radiation spectrum of K-G particle in the vicinity of the horizon. Near the horizon, we have I* - & ln(r
-
tJ.r).
(26)
hence the outgoing wave (25) can be rewritten as @ry _
c-im~*(r
_
rH)i?(r
_
r,)‘%,
(27)
Obviously, gr:“” is non-analytic at the horizon r~, and we have to reach the interior by analytic continuation in the complex plane around the horizon, when the variable is I?-
rHI ---c Ir -
tJfJ e-‘” -
(rH -
(28)
I)e-i**
Then, the outgoing wave inside the horizon is 2s
*J:Y((~ < TH) -
e-isr*[(rH
-IT
-irov*
-
,+a--eo)r*
hence, the scattering probability
,)e-‘=]iY
e-(D+LW*
[(rH . (D+lh c ‘77-~--F-
-
r)
exp i_,:
1
wo -1
2xK,
.
(2%
of the outgoing wave at the horizon is
,
where KB is the Boltzmann constant. the radiation temperature is T--&
2~
L+wr)x
Using Sannan’s method, we can find the intensity distribution N, - -
,-i=]
of the outgoing wave to be (31)
Equation (31) is the spectrum of Hawking radiation,
(32)
22
ZHU Jian-yang
et al.
Thus, we see that K:is a function that determines the Hawking temperature, which varies not only in time, it is not even constant over the surface of simultaneity of the horizon.
3.
DISCUSSION
Equations (31) and (32) s h ow that, for a charged, magnetic, arbitrarily accelerating black hole, the Hawking spectrum near the horizon contains two parameters 6 and wo, which in turn depend on six time-dependent parameters of the black hole, m(v), E(u), Q(u), a(u), a(u), c(w), and the electric charge e and magnetic charge m carried by the K-G particle. We now discussion some particular cases. 3.1 Schwarzschild Black Hole For Q = b = c = 0, E = & = 0, m =const., and neglecting the cosmic term, the spacetime (1) reduces to the Schwarzschild case, d+(1-~)dv’-2dt.~dr-r~diY-r~&~~d~~.
Obviously, the position of the horizon TH is independent of #,+, u, and the expressions (9) and (19) for the horizon and the temperature function give rH -
Ic--
2m,
1
Ta%-
4m’
1 8&m
.
And these are the well-known expressions for the horizon, surface gravity and temperature of a static, spherically symmetric black hole. 3.2 a = b = c = 0, m, E, Q are Nonzero Constants We have
rjj=
(E2 + Q'l, &,-- (E'+ Q')
m+ d/m2 -
/it-'
rHk4mrH -
tk -
2( E2 +
Q')]'
These are the well-known results for a static, spherical, charged and magnetic black hole. 3.3 a # 0, b = c = 0, E(v), E(w), Q(u) # 0 The line element is now dr2 ip
1 _
.k! + E2T2 Q' _ r
-
Zdvdr -
2atr
sin
2ar cost9 -
8dvdB -
rzde2 -
40
E2+Q2COSe r
a2r2sh2tl
1
do2
rzsin28d+2,
and it describes the case of a charged, magnetic black hole undergoing linear acceleration. Because of the acceleration, the black hole is no longer spherically symmetric; it is now axisymmetric. The equation for the horizon in this case is
23
Accelerating Black Hole
1 -- 2~+~-2arcos6-4a~cose+2asine.(Q)+(*)2/P-2(*)=0. For E = & = 0, it reduces to 2 (2)
-
(1 -
2arHcosQ -
2)
+ 2asin8
l
($$)
-
(.$!$)l/rz~
0,
I and the temperature
function is m
in agreement with the results of Ref. [5]. 3.4 a = 0, b(v) # 0, C(V) # 0, E = & = 0, m(w) # 0 The line element is ds2 P
1 - ?! [
r
-
Zdvdr
-
(bsin(b
+ 2(bsin
+ CCOSC#I)~~~- (bcos+
-
csin4)r2dvd~
and it describes a black hole in arbitrary *
-
t
7
and the temperature
1
rotation. (2)
2(bsinb
+ ccos+)
arn 2+
-$-&$-2($9-o,
1 ( ae
+
>
t2 1 dv2 _!
$I -I- ccos&)t’dvde
+ 2sinBcos8(bcos4
1-
csin4)‘cos28*
-
-
-
rz(dt12 + sin28d+2),
The horizon equation is 2 (2)
(bcos(b
-
csinb)ctye
function is
-2+4m+2
+ 2(bsin+
+ ccos+)
‘H -
csind+tye
.
It should be pointed out that there are three points of difference between the black holes of this paper and Ref. [6]. One, the black hole discussed here is originally spherically symmetric; because of acceleration, the spherical symmetry is lost, it may even have no axial symmetry. The black hole of Ref. [6], on the other hand, is axially symmetric to start with and that symmetry is preserved when the acceleration is along the axis. Two, The acceleration considered here is arbitrary; that in Ref. [S] is along a straight line. Three, the black hole here has an electric charge and a magnetic charge; that of Ref. [6] has not. Since in general relativity, gravitation is not an external force, it is difficult to see how a black hole with no electric or magnetic charges, such as considered in Ref. [6], can deviate from the geodesic and undergo acceleration. This difficulty does not exist here, because our
24
ZHU Jian-yang et al.
black hole is assumed to have electric and magnetic charges and electromagnetic external force in general relativity.
4. OBSERVATIONALLY
RELEVANT
ASTRONOMICAL
force isan
EXAMPLES
To clarify the issues, we should make some astronomical calculations relevant to observations. It is, however, difficult to give precise numerical description of the results obtained above. For simplicity, we consider a one solar mass black hole with electric and magnetic charges, undergoing linear acceleration (a # 0, b = c = 0) under the external, electromagnetic force. It is known that for black holes on and above the order of stellar masses, the evaporation rate is extremely slow, and this means that the horizon rH not only varies little in time, it could not vary very much in space, either. Hence, in the calculation, it is reasonable to make the following approximations:
Also, since the evaporation is low, over a short period of observation in, E, & can all be regarded as constant. Then, the horizon equation and the temperature function are simplified respectively into l-
2++2+Q2-~arcos~-~~E2+Q2cosePF~
r2
r m
EL + Q' r3
-;?-
c-
E2 + Q’
--2m f
r2
4
r
-
ac0sB+20
E”+ 0'
+ 2af c0se + 4a
(34) case
r2
E2-+ Q'case’ T
The above formulae were derived using the natural units (c = h = G = 1). For numerical calculation we go to conventional cgs units. This means we make the following substitutions: G
m+-
112
()
m,
C&
a+
AG
C’
(
“‘a, >
(35)
1 la
Ec
oc > s E,
Q-(k)L/'Q, Ic--ihe results
AG LIZ .K
()c3 are
l
Accelerating Black Hole
1 -- 2Gm+~E'+Q'
c2r
2ar
-
c+
r2
7
25
case- -4GE'+Q2acOSe-o c6 r
,
(36) Gm
--c2r2
)i-
G E2 -I- Q2 c+
a T2
-
r3
2G
casei--t6
G E2 -I- Q2
2Gm --c2r
c’
a
E2 -I- Q’ r2
+
r2
r
COse
Q2acose' i
Again, since
T
0:
PK, P = fi/K~,
we have
T&K,
T=kk.
KB
2YiKB
1. a = 0, E = & = 0, we have 2Gm
T
h’
_
rH ---p
8nKBGm
.
Taking m = mg = 1.99x1O33 g, we have rH = 2.95x lo5 cm, T = 6.2x 10e8 K. 2. a = 0, E # 0, & # 0, we have rs -
-$ [GmfJ(Gm)’
-
G(E’
mrfi - f
T_-&2zKB
Q')],
+
(E2 + Q’) $W
+
Q2)j*
Usual astrophysical objects carry no electric charge to speak of (for the sun, for example, be neglected. On the other hand, if the number of magnetic monopoles the black hole has is so large that it corresponds to a magnetic charge of Q = (Gm2)‘12, then we shall have E N lOOCoulomb = 3x 10” esu), which can therefore
rH = Gm/c2,T=
OK,
showing that the maximum magnetic charge a black hole can carry is Q,, = (Gm2)‘12. In the following calculations we continue to assume E < Q. We also set Q2 = $Gm2, m = mg, Then, we have 3Gm 2 ca = 2.21 x lo5 cm, Tl = $& = 5.5x10-“K. rf;=
1
rjj = $9
T2 < 0,
= 0.74x105cm UNREASONABLE,
‘TO BE DISCARDED.
ZHU Jian-yang et al.
26
3. a#O,E,Q#O
Let m = mO, E2 + Q2 = (3/4)Gm2. Then, the horizon equation is f2 -
2.94 X
105r +
1.62 x lOlo -
0.22 x 10-f)ar3cos8
-
0.72 X 10-‘“arcos8
-
0.
This shows that, when a < 10” cm/s a, the effect of acceleration on the location of the horizon is very small. To illustrate this, let us take a = 1014cm. Then, we have ra - 2.94x 105r + 1.62x 10” - 0.22x10-6r3~~sB
- 0.72x104rcos8
= 0,
and we find for B = Q, for 8 = z/2, for 0 = ?F,
r = 2.21x105cm,
T = 4.59x10-sK T = 5.52x10w8K
r = 1.96x105cm,
T = 6.20x10-*K
P = 2.50x105cm,
5. CONCLUDING
REMARKS
In this paper we studied the Hawking effect in one type of black hole with six time-dependent parameters including mazs, electric and magnetic charges. We found the equation that determines its local event horizon and showed that thermal radiation exists in this dynamic type of black hole. It should be noted that, because of the acceleration, the black hole is no longer spherically symmetric and is not even axisymmetric, in general. Its horizon and temperature vary not only in time, but also with the angles. It should be emphasized that the case of the black hole temperature varying with the angles has not been seen reported in the literature outside China. This is because it is very difficult to investigate the Hawking effect of a dynamic, asymmetric black hole. The conclusion that for such a black hole, the temperature varies over the surface of horizon is based on our own original method. The results reported in this paper and in Refs. [5] and [S] are hopefully of some significance in astrophysics.
References Wang Yong-jiu
and Tang Zhi-ming, Scientia Siuica (Ser. A), 1986,5, 525
Liu Zheng and Liu Liao, Acts Physica Sinks, 1991,40, 1546 131
Schwiuger J., Phys. Rev., 1975, D12, 3105
141
Zhao Zheng, Dai Xianxiu, Chinese Phys. Lett., 1991,8,548
[51
Luo Zhi-qiang and Zhao Zheng, to be published in Acta Physica Sica,
161
Wu Si and Zhao Zheng, CAA, 1993,18,247 = AAnS, 1993,34,17
[71
Damour T., RufIini Fl., Phys. Rev., 1976, D14,332
181
Sanuan S., Gen. F&l. Grav., 1988,20, 239
1993,42
02751062(95)00005-4
Quantum thermal effect of arbitrarily accelerating black hole with electric and magnetic charges T ZHU Jian-yang’
of P hy sits, Furhou
‘Department
ZHAO Zheng3
ZHANG Jian-hua2 Teachers’
College,
Jiangxi
944000
Shandong
274Olli
2Department
of P hy sits, Hete Education
College,
3Department
of Physics,
University,
Beijing
Abstract The Hawking radiation with electric and magnetic charges
Normal
Beijing 100875
for an arbitrarily accelerating black hole is investigated. Expressions for the event
horizon and the Hawking thermal spectrum are given. Both the figure and the temperature depend on both the time and the angles. The expressions reduce to the well-known results for the particular cases of zero acceleration, acceleration along a straight Key words:
line, zero electric dynamic
charge or zero magnetic
black hole-local
field.
event horizon-Hawking
radiation
1. INTRODUCTION Recently WANG Yong-jiu many magnetic monopoles rameters. study
We shall take this very general
the dynamical
gate the Hawking charge
et al. ~1 studied the gravitational field of accelerating bodies and obtained a black hole solution with six time-dependent
it carries
behaviour
type of black hole as our spacetime
of the Klein-Gordon
effect of this black hole and examine and its acceleration
particle
located
therein,
the effect of the electric
with pa-
background, hence investiand magnetic
on the radiat,ion.
2. THEORETICAL
CALCULATION
2.1 The Metric Tensor of an Arbitrarily Accelerating, Charged and Magnetic Black Hole For studying the Hawking effect, we re-write the metric of an arbitrarily accelerating, charged and magnetic black hole given in Ref. [l], in terms of the advanced EddingtonFinkelstein coordinates as follows: t Supported by National Natural Science Foundation Received 1993 March 27; revised version 1994 March 1
Black Hole
Accelerating
ds’ -
gmdv’ -I- Zg,,dvdt
A- 2g,,dvdO
-I- 2g,,dvd4
15
+ g,,dP
-t g,,dCb2,
(1)
where g,-l--+---
tm
EZ -f- Q’
Y rz - (blind +
20rcos0
CCOS+
-
-
r
bEZ +
QZcose
asin8)2,2 I
-
(bcos4
(2)
go2 -
1, r’(bsin4
go3
sinBcosB(bcos4
go1 -
-
b52 g33
-
c sin4)‘r2cos28 -
-
-
+
ccos4
a&6),
-
-
csin4)r2,
-r2, -r2sin28,
the parameters m = m(u), E = E(u), Q = Q(w) being the mass of the source, and the electric and magnetic charge it carries, a = a(v),b = b(v),c = c(o), the acceleration parameters, a representing the magnitude of the acceleration and b and c, the rate of change of the direction of Bcceleration. The metric
determinant
and the contravariant
components
are easily found:
g - --r4sin2B. 8 01 x - 1,
g I2 -
-(bsin+
g13 -
-cty@(bcos4
g2t =: -
(3)
-I- ccos4 -
-
asine),
,
csin4),
(4
r r2’ 1
g J3=L-
2.2
Equation
r?sin28’
for the
Local Event
We now use the condition
.BF r3F grr --= a xp a2v
of zero curvature
surfacer21
o
(5)
to find the local event horizon
F - F(v,t,B,4)
Horizon
- 0
of the specetime
(1). In the above,
(6)
ZHU Jian-yang et al.
16
is the equation of zero curvature surface. Let its explicit form be
From (6) and (7) we have E+fE-o, r
(8)
g+gz-o, g+yC&-0.
I
f
Writing out (5) and substituting
(8), we have
This is the equation for the local event horizon; it determines the location of the horizon, rH. Obviously, rH depends not only on V, it depends also on the angles 0 and 4. The position of horizon varies with time and possesses no symmetry whatever. 2.3
Dynamical
Equation of the Klein-Gordon
Particle
In curved spacetime, the dynamical behaviour of the scalar particle (Klein-Gordon ticle) is described by the K-G equatio&l,
4-bK&--
- i&A,)d
eAI,-
eE,+
ie A,))] @ -
&p -
0,
par-
(10)
(11)
qM,,
where e, Q are the electric and magnetic charge the particle carries, E,, M,, are the electric and magnetic Cvectors generated by the electric and magnetic charges carried by the black hole and ~0 is the rest mass of the K-G particle. Using the Lorentz condition,
(12) we can transform (10) into g” -a=@ al*
+f.+f,
2 - 80 ibar
+
2gl3 f!&
r
~+fo$+f+@p-o, ar
+
2gl3 EP_
ad+
+
g2’ i??_ a02
+
g33 * w
(13)
Accelerating
Black Hole
17
where
F,
=
2i.sAl - 2 r’
F,
=
-2i.s(A1g11 + Asg12 + A3g13 - Ao) + --g
2 11
&I”
+ ar
+ ctyBg12 I
+tlg12 + 8g13
ae -zp
fe
=
fc$ = f
2.4
=
-2ie(A1g12 + A2g22) + fg” + ctyOg22, -2i&(Arg13
+
A3g33)+ Eg13,
c2gfi”A,Av + pi
Generalized
Tortoise Coordinates
The extreme complexity of equations (13) and (14) makes a rigorous solutions impossible. However, only asymptotic behaviour near the horizon are physically meaningful. For this we make a tortoise
P*
=
transformation
of the following
r+ & ln[r - W(V> 0,
form:
t
dJ)l>
v* = v--o, 6, = 8-80,
+
(15)
4, = 6-407
where rH is the event horizon of the black hole, n: is an adjustable parameter (below we shall see that n is just the temperature function representing the Hawking radiation), invariant under the tortoise transformation, and 00, Bo,& are arbitrary constants, also invariant under the transformation.
ZHU Jian-yang et al.
18
From (15) we obtain
ar
--a
1+ E
a
a
1 241
IH)’
-
E-l+
al2
(arH ) a
lH)
241.7
-- a ah
ar*’
1
ae-ae,-a ST’
1a Yz
dr,’
a+
1 2k(1-
ln)
( -&
) ar*,
2 a2
1
zr(t - rff) I ar:
[
8 1
-
24r
2dB (
a2
s=ae:-
a2
)
241-
ahae,
4
241
.S-
-
(4
lH)2
2s(r-
ld
( ) ,v
241
a+
--
c
CrerIf)+
21E(r -
,
ah
82.
area+*
a2fn
_
a
ah a+ 2 (3 +_ L 1
32
2 31, a+
’
1
alH 2
a5,
( )cl- lH)+ w ae
a2 --a+:
al,
ae 1 a2 (-_) [ 2~(1- iH) al:
+
_ yp
a2
tH)2 arH
31,
a2
a
-
iJ& --
-
1H)
at:
2
)
lH)2
a
,
al*
alH L_,+
aflat
1 [
2r(r-
x
a2 dedr-
1+
a* I afdl,
1 2~(~
-
lH)
-
2k(l
-
lJ#)
241
-
rH)2
farH au (-1 1a~+ al: a*
1
1
_a al,
’
(%)
2r(r- lH) ae*ar, iK(1 - 1Jf) f ali\
L*+
x
.-- a2 .a+ar
[
d
(4 au
1+ L
1 2~(~
-
a*
lH)
1
al:
+
\ ae 24r-
I
_a_
rH)2’ al,
’
arH
[
1-t
x
2n(r-
1
1 1+ 2k(1-
ld
a* I ahal,
1
3 lH)
_
a2 + al:
c-1 a+
2n(r
-
1H)
alH w 2r(r-
a4 rH)2
a al,
l
AcceleratingBlack Hole
19
Substituting (16) in (13) we obtain, after some reduction,
{2rk(r -
rH)[1+2&(t-
+
[2P(9
-
[gq9+g33(4$]}$$+2
+
ll+2k(f
-
g*
+
{(I
a% rI#) -@
-
-
* -
rH)[1+2c(r
-
i(2g”
-
w
2g”[ 1 + 24 r -
-
24r
l
-
TH)l
$*
fH)f_l
+ (2g33($$)
tH)y
11 + 2r(r
+ 2g33(5]
+ 245
-
+ 2c(t -
{-gY1
hJlP
24r
-
r,)]}-’
2g”Cl + 24t
r*)]
rH)g
-
tm)1
6% > lM*&,
8’cp > w&, 3
33 a% I
{g” + 2 (&,
-
2g”(%)
-2g”(~)-tg”[(~)(r--m)+(~)I] -+ g”
+f*
K
$L$
)
(r -4+($$3]
h ( a+ > -
tff)[l
+
2r(r
-
f, ~+f@}“o* *
f,fl
+(L($)+fs(?$)
+ 2n(r
+ 21E(T -
-
r,,l>(r
tH)l-’
-
1
-f@
Id} - a@ ab
2 a!_
_f
* 84,
(17)
This is the K-G equation in tortoise coordinates.
2.5
Reduction
of the K-G Equation Near the Horizon
Our study of various types of black hole I44 showed that the K-G equation in tortoise coordinates should have the form of typical wave equation near the horizon. Likewise, as -+ 40, equation (17) can be reduced to the following f ---) w(u0,B0,40),u + vo,e -+ go,4 form: A*+2
ar:
“@
avear*
+B
8@
ae,ar,
+C
6@
abaf,
+(D+j2~,)&-o
ar,-
’
(18)
2HU Jian-yang
20
et al.
where A -
lim -‘If
-
g”[l
+ 2n(r
-
rH)l* + [29”‘(%)
( >I
+ 2g13 3.L a+ (21E(t -
-
[l +24r
-
TH)l -
+
[P(%J
+ P(%Y]}/
Qf)El+24r-rH)l),
(GA&
+ .sA,$’ + EAZ&?+ e A3gi3 + e Ado
+ eA,A”) ($$
Obviously, B, C, D, wg are finite. As for A, both the numerator hence, by l’H6pital rule. we have A = 1 if c_
-(g_(~)(~)y.?g)(~) 2 [l + a!” + 2 (2) - Zg"(%) -
At this point, neighbourhood 2.6
we have reduced
the G-K
equation
zg"
and denominator
p>
to a simple
1 ,
tend to 0;
(1% ,p,*O,~O).
wave equation
for the
of the horizon.
The Hawking Spectrum of Klein-Gordon We separate
2gyk)
Particle
the variables,
(20) and substitute
in equation
$+rDEquate
(18) to obtain
Zi(co - coo)] - -
both side to a constant,
-A:
B
(21)
Accelerating Black Hole
R” + [D + R -
2i(o
-
q)lR’
-
21
(22)
0.
,~++=I_.
(23)
When dealing with radiation we need consider only the radial equation (22). Solving the latter we obtain the ingoing and outgoing radial wave functions through the horizon, QY .rc
(24)
e--rr” * *9 -
e -iw*
&Z(co-q)r*
e-(D+A)r*. (25)
We now use a method first established by Damour and Ruffini[q and later refined by Sanna@ to find the Hawking radiation spectrum of K-G particle in the vicinity of the horizon. Near the horizon, we have I* - & ln(r
-
tJ.r).
(26)
hence the outgoing wave (25) can be rewritten as @ry _
c-im~*(r
_
rH)i?(r
_
r,)‘%,
(27)
Obviously, gr:“” is non-analytic at the horizon r~, and we have to reach the interior by analytic continuation in the complex plane around the horizon, when the variable is I?-
rHI ---c Ir -
tJfJ e-‘” -
(rH -
(28)
I)e-i**
Then, the outgoing wave inside the horizon is 2s
*J:Y((~ < TH) -
e-isr*[(rH
-IT
-irov*
-
,+a--eo)r*
hence, the scattering probability
,)e-‘=]iY
e-(D+LW*
[(rH . (D+lh c ‘77-~--F-
-
r)
exp i_,:
1
wo -1
2xK,
.
(2%
of the outgoing wave at the horizon is
,
where KB is the Boltzmann constant. the radiation temperature is T--&
2~
L+wr)x
Using Sannan’s method, we can find the intensity distribution N, - -
,-i=]
of the outgoing wave to be (31)
Equation (31) is the spectrum of Hawking radiation,
(32)
22
ZHU Jian-yang
et al.
Thus, we see that K:is a function that determines the Hawking temperature, which varies not only in time, it is not even constant over the surface of simultaneity of the horizon.
3.
DISCUSSION
Equations (31) and (32) s h ow that, for a charged, magnetic, arbitrarily accelerating black hole, the Hawking spectrum near the horizon contains two parameters 6 and wo, which in turn depend on six time-dependent parameters of the black hole, m(v), E(u), Q(u), a(u), a(u), c(w), and the electric charge e and magnetic charge m carried by the K-G particle. We now discussion some particular cases. 3.1 Schwarzschild Black Hole For Q = b = c = 0, E = & = 0, m =const., and neglecting the cosmic term, the spacetime (1) reduces to the Schwarzschild case, d+(1-~)dv’-2dt.~dr-r~diY-r~&~~d~~.
Obviously, the position of the horizon TH is independent of #,+, u, and the expressions (9) and (19) for the horizon and the temperature function give rH -
Ic--
2m,
1
Ta%-
4m’
1 8&m
.
And these are the well-known expressions for the horizon, surface gravity and temperature of a static, spherically symmetric black hole. 3.2 a = b = c = 0, m, E, Q are Nonzero Constants We have
rjj=
(E2 + Q'l, &,-- (E'+ Q')
m+ d/m2 -
/it-'
rHk4mrH -
tk -
2( E2 +
Q')]'
These are the well-known results for a static, spherical, charged and magnetic black hole. 3.3 a # 0, b = c = 0, E(v), E(w), Q(u) # 0 The line element is now dr2 ip
1 _
.k! + E2T2 Q' _ r
-
Zdvdr -
2atr
sin
2ar cost9 -
8dvdB -
rzde2 -
40
E2+Q2COSe r
a2r2sh2tl
1
do2
rzsin28d+2,
and it describes the case of a charged, magnetic black hole undergoing linear acceleration. Because of the acceleration, the black hole is no longer spherically symmetric; it is now axisymmetric. The equation for the horizon in this case is
23
Accelerating Black Hole
1 -- 2~+~-2arcos6-4a~cose+2asine.(Q)+(*)2/P-2(*)=0. For E = & = 0, it reduces to 2 (2)
-
(1 -
2arHcosQ -
2)
+ 2asin8
l
($$)
-
(.$!$)l/rz~
0,
I and the temperature
function is m
in agreement with the results of Ref. [5]. 3.4 a = 0, b(v) # 0, C(V) # 0, E = & = 0, m(w) # 0 The line element is ds2 P
1 - ?! [
r
-
Zdvdr
-
(bsin(b
+ 2(bsin
+ CCOSC#I)~~~- (bcos+
-
csin4)r2dvd~
and it describes a black hole in arbitrary *
-
t
7
and the temperature
1
rotation. (2)
2(bsinb
+ ccos+)
arn 2+
-$-&$-2($9-o,
1 ( ae
+
>
t2 1 dv2 _!
$I -I- ccos&)t’dvde
+ 2sinBcos8(bcos4
1-
csin4)‘cos28*
-
-
-
rz(dt12 + sin28d+2),
The horizon equation is 2 (2)
(bcos(b
-
csinb)ctye
function is
-2+4m+2
+ 2(bsin+
+ ccos+)
‘H -
csind+tye
.
It should be pointed out that there are three points of difference between the black holes of this paper and Ref. [6]. One, the black hole discussed here is originally spherically symmetric; because of acceleration, the spherical symmetry is lost, it may even have no axial symmetry. The black hole of Ref. [6], on the other hand, is axially symmetric to start with and that symmetry is preserved when the acceleration is along the axis. Two, The acceleration considered here is arbitrary; that in Ref. [S] is along a straight line. Three, the black hole here has an electric charge and a magnetic charge; that of Ref. [6] has not. Since in general relativity, gravitation is not an external force, it is difficult to see how a black hole with no electric or magnetic charges, such as considered in Ref. [6], can deviate from the geodesic and undergo acceleration. This difficulty does not exist here, because our
24
ZHU Jian-yang et al.
black hole is assumed to have electric and magnetic charges and electromagnetic external force in general relativity.
4. OBSERVATIONALLY
RELEVANT
ASTRONOMICAL
force isan
EXAMPLES
To clarify the issues, we should make some astronomical calculations relevant to observations. It is, however, difficult to give precise numerical description of the results obtained above. For simplicity, we consider a one solar mass black hole with electric and magnetic charges, undergoing linear acceleration (a # 0, b = c = 0) under the external, electromagnetic force. It is known that for black holes on and above the order of stellar masses, the evaporation rate is extremely slow, and this means that the horizon rH not only varies little in time, it could not vary very much in space, either. Hence, in the calculation, it is reasonable to make the following approximations:
Also, since the evaporation is low, over a short period of observation in, E, & can all be regarded as constant. Then, the horizon equation and the temperature function are simplified respectively into l-
2++2+Q2-~arcos~-~~E2+Q2cosePF~
r2
r m
EL + Q' r3
-;?-
c-
E2 + Q’
--2m f
r2
4
r
-
ac0sB+20
E”+ 0'
+ 2af c0se + 4a
(34) case
r2
E2-+ Q'case’ T
The above formulae were derived using the natural units (c = h = G = 1). For numerical calculation we go to conventional cgs units. This means we make the following substitutions: G
m+-
112
()
m,
C&
a+
AG
C’
(
“‘a, >
(35)
1 la
Ec
oc > s E,
Q-(k)L/'Q, Ic--ihe results
AG LIZ .K
()c3 are
l
Accelerating Black Hole
1 -- 2Gm+~E'+Q'
c2r
2ar
-
c+
r2
7
25
case- -4GE'+Q2acOSe-o c6 r
,
(36) Gm
--c2r2
)i-
G E2 -I- Q2 c+
a T2
-
r3
2G
casei--t6
G E2 -I- Q2
2Gm --c2r
c’
a
E2 -I- Q’ r2
+
r2
r
COse
Q2acose' i
Again, since
T
0:
PK, P = fi/K~,
we have
T&K,
T=kk.
KB
2YiKB
1. a = 0, E = & = 0, we have 2Gm
T
h’
_
rH ---p
8nKBGm
.
Taking m = mg = 1.99x1O33 g, we have rH = 2.95x lo5 cm, T = 6.2x 10e8 K. 2. a = 0, E # 0, & # 0, we have rs -
-$ [GmfJ(Gm)’
-
G(E’
mrfi - f
T_-&2zKB
Q')],
+
(E2 + Q’) $W
+
Q2)j*
Usual astrophysical objects carry no electric charge to speak of (for the sun, for example, be neglected. On the other hand, if the number of magnetic monopoles the black hole has is so large that it corresponds to a magnetic charge of Q = (Gm2)‘12, then we shall have E N lOOCoulomb = 3x 10” esu), which can therefore
rH = Gm/c2,T=
OK,
showing that the maximum magnetic charge a black hole can carry is Q,, = (Gm2)‘12. In the following calculations we continue to assume E < Q. We also set Q2 = $Gm2, m = mg, Then, we have 3Gm 2 ca = 2.21 x lo5 cm, Tl = $& = 5.5x10-“K. rf;=
1
rjj = $9
T2 < 0,
= 0.74x105cm UNREASONABLE,
‘TO BE DISCARDED.
ZHU Jian-yang et al.
26
3. a#O,E,Q#O
Let m = mO, E2 + Q2 = (3/4)Gm2. Then, the horizon equation is f2 -
2.94 X
105r +
1.62 x lOlo -
0.22 x 10-f)ar3cos8
-
0.72 X 10-‘“arcos8
-
0.
This shows that, when a < 10” cm/s a, the effect of acceleration on the location of the horizon is very small. To illustrate this, let us take a = 1014cm. Then, we have ra - 2.94x 105r + 1.62x 10” - 0.22x10-6r3~~sB
- 0.72x104rcos8
= 0,
and we find for B = Q, for 8 = z/2, for 0 = ?F,
r = 2.21x105cm,
T = 4.59x10-sK T = 5.52x10w8K
r = 1.96x105cm,
T = 6.20x10-*K
P = 2.50x105cm,
5. CONCLUDING
REMARKS
In this paper we studied the Hawking effect in one type of black hole with six time-dependent parameters including mazs, electric and magnetic charges. We found the equation that determines its local event horizon and showed that thermal radiation exists in this dynamic type of black hole. It should be noted that, because of the acceleration, the black hole is no longer spherically symmetric and is not even axisymmetric, in general. Its horizon and temperature vary not only in time, but also with the angles. It should be emphasized that the case of the black hole temperature varying with the angles has not been seen reported in the literature outside China. This is because it is very difficult to investigate the Hawking effect of a dynamic, asymmetric black hole. The conclusion that for such a black hole, the temperature varies over the surface of horizon is based on our own original method. The results reported in this paper and in Refs. [5] and [S] are hopefully of some significance in astrophysics.
References Wang Yong-jiu
and Tang Zhi-ming, Scientia Siuica (Ser. A), 1986,5, 525
Liu Zheng and Liu Liao, Acts Physica Sinks, 1991,40, 1546 131
Schwiuger J., Phys. Rev., 1975, D12, 3105
141
Zhao Zheng, Dai Xianxiu, Chinese Phys. Lett., 1991,8,548
[51
Luo Zhi-qiang and Zhao Zheng, to be published in Acta Physica Sica,
161
Wu Si and Zhao Zheng, CAA, 1993,18,247 = AAnS, 1993,34,17
[71
Damour T., RufIini Fl., Phys. Rev., 1976, D14,332
181
Sanuan S., Gen. F&l. Grav., 1988,20, 239
1993,42