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The electrostatic and gravitational field of spherically symmetric objects
Volume 62A, number 2
PHYSICS LETTERS
25 July 1977
T H E E L E C T R O S T A T I C AND G R A V I T A T I O N A L F I E L D OF SPHERICALLY SYMMETRIC OBJECTS * Michael SOFFEL, Berndt MULLER and Walter GREINER lnstitut ffir Theoretische Physik, Johann Wolfgang Goethe Universitd't, Frankfurt am Main, W.-Germany Received 18 May 1977 The electrostatic and gravitational fields of an extended spherically symmetric object is presented. The limit to a point-like object is discussed for Born-Infeld type of electrodynamics and it is shown, that the extreme ReissnerNordstr4m field, where no event horizon occurs, is unphysical.
The gravitational (G) field of a point-like object is usually described by the Reissner-Nordstr~bm metric [1,21: (guy = diag(eU' - e x ' -r2, - r 2 sin20)) e u = e - h = 1 - 2GM+ 4nGQ 2 c2r c4r 2
(1)
1
-8rtG [T~ ) + T(MAX)1 c2 uv J
(2b)
{FuulX) = 0 = FuulX + Fxulv + Fuxbu
(2c)
=
where the matter tensors for the Maxwell field and the fluid matter are given by: 1 T(MAX)uv = [FuaF~ + ¼ g u v F ~ F~t3] -'~
Eq. (1) possesses the interesting feature, that for Q2/ GM2 > 1/4rr no event horizon occurs and we face a naked singularity. However, the electrostatic (ES) field entering into eq. (1) is derived from the Maxwell equations, and it is well known that classical Maxwell electrodynamics of point charges is not free of divergences and contradictory features. The physical meaning of (1) can therefore be understood only by first considering the case of an extended (spherically symmetric) charged objects and taking the limit to a point charge or by directly attaching the point charge case with a finite (Born-Infeld type) theory of electrodynamics. (Extended charged drops). As model for the extended object we take the incompressible fluid sphere [3-5] and suppose that within the drop (r < r0) a constant mass density O0 and constant charge density r/0 are independent from each other. The coupled Einstein-Maxwell equations for this system read: Guu = R~v - ~guv R =
(_g)-l/2 [(_g)l/2 Ft~V] Iv 47r/U
P(r),h '--~P(r)r2 '--~P(r)r2 sin20) T.~ ~=diag\~,0 F )I ^ ~v '--~--~
(3)
(4)
respectively. In free space, outside the matter distribution, we obtain the Reissner-Nordstrcm metric and its corresponding ES-field. Within the fluid drop we set:
] 0 = nO e-(V+x)/2, 11 =12 =/3 = O, (r (r0).
(5)
The ES-field should have only a nonvanlshing radial component. The ansatz:
Fuv = E(r)
0 0 0
(6)
0 0 0
immediately leads to: E(r) = e(v+x)/2 41rr/0
3
r,
(r
(7)
(2a) T v(MAX) - (4rn10)2 r2 diag(1,1, - 1 , -1).
u * Supported by the Bundesministedum fiir Forschung und Technologie (BMFT), and by the Gesellschaft ffir Schwerionenforschung (GSI).
(8)
1 8c 2
Requiring E(r) to be smooth at r = r 0, we obtain: _4
Q - ~ Irr/o r03.
(9) 67
Volume 62A, number 2
PHYSICS LETTERS
25 July 1977
Now the remaining independent Einstein equations can be written as follows: 8rrG, + ~r2) = e _ h / l _ ) t ' ~ c-~ tP0 \r 2 r
1 r2
]
8rrG(P-~r2)-I -e-X/1
/
06
(10a)
+~)
Q)
('qr)
liO ;
0.6
0.4 ® 02
(10b)
~ ro4.O ro • Z5 ® ~°"z z6 F
rf)t?kp c 2 !xc2
]
if r 4
2
,
)_] 2
Z
'
6
'
lO-
eb(r)
r (10c)
where ~ - (4~rr/0)2/18c 2. The first equation can be integrated right away:
++1/ A
e - ~ = 1 - r2/R 2 - r4/'R 4
2
6 +
(11)
with the abbreviations:/~2 -3c2/87rGPo ' ~'4 _= 45c4/ 4*rG(47rr/0)2. The requirement, that e - x should be smooth at r = r 0 gives us an expression f o r M ( s e e eq. (1)) as a function of P0, r/0 and r0:
M={rr Por30 + 5~2 1 (4rrrto)2r5 =MF 4 12~Q2
0
(12)
5roc2
where the second term represents the energy MES = 47r f~* T#(MAX)r2 dr of the ES-field of a uniformly charged sphere. (10a) and (10b) tell us that: 8~rG(po c2
+P) c2
-e7- x= _ _ (~" + ~)
_ 487rG D' (r) -~ 7 - ~eV/2r
(14a)
e~ (eV/2(r)) ' =--~ [rD(r) + e v/2 (e-X)']
(146)
where D(r) is defined via:
+P - D(r)c2 e-V/2 c2 81rG
(15)
(14a) and (14b) were integrated numerically from r = r 0 to r = 0. Some results are shown in fig. ( l a - c ) , 4 where we f i x e d M F =Trr POrO3 and Q=~rrrlor3o . 68
J
)'¼'6
0.15p(r)
1
O.lO
~
0o5
\
0
+
,o.Z.s
r
L
8
I0= c)
r,40
{,
ro-25
r 3
Fig. 1. The metric eV/2(r), eh(r) and pressure p(r) for an extended charged liquid drop is shown for various values of r0. (GMF/C 2 = 1, G1/2Q/c2 = 0.5).
(13)
which requires ~' + X' > 0 for a physically meaningful theory. In the incompressible fluid model we have P0 = const., so that two coupled equations remain:
P0
® r,.3.o
IL\
It is remarkable that the radius r 0 of the drop has always to be larger than a critical value, say r c, otherwise our system of coordinates would break down and an event horizon would emerge, r e is given by: re
=
GMF c2
+
((GMFI2 \\
c2 ]
,~lrtr G ,1/2 +-~-~4Q 2 ] . 5c
(16)
(Note, that we fixed M F and not M!) This results from the decomposition of M into M F + M E s , see eq. (12). We conclude that an event horizon occurs for arbitrary values of Q if r 0 <~ r c. (The situation, in which Q2 > M 2 here never occurs. Nonlinear theory of electrodynarnics). We now turn to the problem of charged point-like objects, where electrodynamics is governed by a Born-Infeld theory [ 6 - 9 ] . Instead of (2b) we now have:
Volume 62A, number 2 ( _ g ) - l / 2 [(_g)l/2
PHYSICS LETTERS
FUU(1 +~F2/E2)-l/2 ] Iv = 47r/ta (17)
where F 2 = FaoF a# and E 0 denotes the limiting field strength in the Born-Infeld theory. We wish to treat E 0 as a freely adjustable parameter. For a point charge the solution is:
E(r) = e(V + x)/2 ~22f(r)
(1 8)
25 July 1977
all values ofr, Meff(r ) is always positive i f M F > 0, and the field contains an event horizon. Thus an arbitrary amount of charge cannot prevent the occurrence of an event horizon, even in the Maxwell limit E 0 -~ oo. The misleading property of the commonly used Reissner-Nordstr4m solution comes from the decomposition: e v = 1 - 2__G_G[(MF +MEs(oO)) _(MEs(~O) _ MES(r))]
c2r
with:
f(r) = [1 + (F/r)4 ] -1/2
(19)
and ?2 = Q/Eo" f(r) effectively plays the role of a form factor. The matter tensor can then be evaluated as: T~(BI) E2 0 d i a g ( f - l(r) - 1, f - 1 (r) - 1, f(r) - 1, f(r) c2
(20)
- 1)
leading to three Einstein equations, which require that: X = -v
(21)
(24)
= 1 -- 2 G [Mre n _ ~ E S ( r ) ] .
c2r
The total asymptotic mass M F + MES(oo) is renolmalized to Mre n which in general corresponds to a negative "bare" mass M F. For fields which have developed by collapse from an astrophysical object, this procedure is not permissible. We therefore conclude that a collapsing charged star (with zero angular momentum) will - according to the classical field theory of gravitation - always hide behind an event horizon and no naked singularities will occur.
and
References
(reV)' -
8-~--4G E2r2(f-l(r ) - 1) + 1
(22)
the integration o f which yields: ev = l
2G -C2--7
E~ F+4rr 7 -
f (r f _ l ( r ' ) - l ) 0
r'2dr
'1 (23)
-- 1 - 2 G [MF +MEs(r)]"
c2r
This result can easily be understood physically, since MES(r ) is just the mass o f the electromagnetic field which is contained inside the sphere with radius r. Meff(r ) = M F + MES(r ) can be looked upon as the effective gravitating mass when the observer is at the radial coordinate r. It contains the mass of the matter distribution (or the point-like object) and that part of the mass of the ES-field, which is inside o f r . Now for
[1] A. Reissner, Ann. Physik 50 (1916) 106. [2] G. Nordstr4m, Verhandl. Koninkl. Ned. Akad. Wetenschap. Afdel. Natuurk., Amsterdam 26 (1918) 1201. [3] A. Adler, M. Bazin and M. Schiffer, Introduction to general relativity (McGraw Hill, 1965). [4] K. Schwarzschild, Sitzungsber. Preuss. Akad. Wiss. Berlin (1916) 424. [5] H. Bauer, Sitzber. Akad. Wiss. Wien (1918), Abt. IIa, 127 2141. [6] M. Born and L. Infeld, Proc. Roy. Soe. A144 (1934) 425. For a recent discussion of the viability of nonlinear electromagnetic theories see: [7] J. Rafelski, L.P. Fulcher and W. Greiner, Phys. Rev. Lett. 27 (1971) 958. [8] J. Rafelski, L.P. Fulcher and W. Greiner, Nuovo Cirn 13B (1973) 135. [9] G. Soft, J. Rafelski and W. Greiner, Phys. Rev. A7 (1973) 903.
69
PHYSICS LETTERS
25 July 1977
T H E E L E C T R O S T A T I C AND G R A V I T A T I O N A L F I E L D OF SPHERICALLY SYMMETRIC OBJECTS * Michael SOFFEL, Berndt MULLER and Walter GREINER lnstitut ffir Theoretische Physik, Johann Wolfgang Goethe Universitd't, Frankfurt am Main, W.-Germany Received 18 May 1977 The electrostatic and gravitational fields of an extended spherically symmetric object is presented. The limit to a point-like object is discussed for Born-Infeld type of electrodynamics and it is shown, that the extreme ReissnerNordstr4m field, where no event horizon occurs, is unphysical.
The gravitational (G) field of a point-like object is usually described by the Reissner-Nordstr~bm metric [1,21: (guy = diag(eU' - e x ' -r2, - r 2 sin20)) e u = e - h = 1 - 2GM+ 4nGQ 2 c2r c4r 2
(1)
1
-8rtG [T~ ) + T(MAX)1 c2 uv J
(2b)
{FuulX) = 0 = FuulX + Fxulv + Fuxbu
(2c)
=
where the matter tensors for the Maxwell field and the fluid matter are given by: 1 T(MAX)uv = [FuaF~ + ¼ g u v F ~ F~t3] -'~
Eq. (1) possesses the interesting feature, that for Q2/ GM2 > 1/4rr no event horizon occurs and we face a naked singularity. However, the electrostatic (ES) field entering into eq. (1) is derived from the Maxwell equations, and it is well known that classical Maxwell electrodynamics of point charges is not free of divergences and contradictory features. The physical meaning of (1) can therefore be understood only by first considering the case of an extended (spherically symmetric) charged objects and taking the limit to a point charge or by directly attaching the point charge case with a finite (Born-Infeld type) theory of electrodynamics. (Extended charged drops). As model for the extended object we take the incompressible fluid sphere [3-5] and suppose that within the drop (r < r0) a constant mass density O0 and constant charge density r/0 are independent from each other. The coupled Einstein-Maxwell equations for this system read: Guu = R~v - ~guv R =
(_g)-l/2 [(_g)l/2 Ft~V] Iv 47r/U
P(r),h '--~P(r)r2 '--~P(r)r2 sin20) T.~ ~=diag\~,0 F )I ^ ~v '--~--~
(3)
(4)
respectively. In free space, outside the matter distribution, we obtain the Reissner-Nordstrcm metric and its corresponding ES-field. Within the fluid drop we set:
] 0 = nO e-(V+x)/2, 11 =12 =/3 = O, (r (r0).
(5)
The ES-field should have only a nonvanlshing radial component. The ansatz:
Fuv = E(r)
0 0 0
(6)
0 0 0
immediately leads to: E(r) = e(v+x)/2 41rr/0
3
r,
(r
(7)
(2a) T v(MAX) - (4rn10)2 r2 diag(1,1, - 1 , -1).
u * Supported by the Bundesministedum fiir Forschung und Technologie (BMFT), and by the Gesellschaft ffir Schwerionenforschung (GSI).
(8)
1 8c 2
Requiring E(r) to be smooth at r = r 0, we obtain: _4
Q - ~ Irr/o r03.
(9) 67
Volume 62A, number 2
PHYSICS LETTERS
25 July 1977
Now the remaining independent Einstein equations can be written as follows: 8rrG, + ~r2) = e _ h / l _ ) t ' ~ c-~ tP0 \r 2 r
1 r2
]
8rrG(P-~r2)-I -e-X/1
/
06
(10a)
+~)
Q)
('qr)
liO ;
0.6
0.4 ® 02
(10b)
~ ro4.O ro • Z5 ® ~°"z z6 F
rf)t?kp c 2 !xc2
]
if r 4
2
,
)_] 2
Z
'
6
'
lO-
eb(r)
r (10c)
where ~ - (4~rr/0)2/18c 2. The first equation can be integrated right away:
++1/ A
e - ~ = 1 - r2/R 2 - r4/'R 4
2
6 +
(11)
with the abbreviations:/~2 -3c2/87rGPo ' ~'4 _= 45c4/ 4*rG(47rr/0)2. The requirement, that e - x should be smooth at r = r 0 gives us an expression f o r M ( s e e eq. (1)) as a function of P0, r/0 and r0:
M={rr Por30 + 5~2 1 (4rrrto)2r5 =MF 4 12~Q2
0
(12)
5roc2
where the second term represents the energy MES = 47r f~* T#(MAX)r2 dr of the ES-field of a uniformly charged sphere. (10a) and (10b) tell us that: 8~rG(po c2
+P) c2
-e7- x= _ _ (~" + ~)
_ 487rG D' (r) -~ 7 - ~eV/2r
(14a)
e~ (eV/2(r)) ' =--~ [rD(r) + e v/2 (e-X)']
(146)
where D(r) is defined via:
+P - D(r)c2 e-V/2 c2 81rG
(15)
(14a) and (14b) were integrated numerically from r = r 0 to r = 0. Some results are shown in fig. ( l a - c ) , 4 where we f i x e d M F =Trr POrO3 and Q=~rrrlor3o . 68
J
)'¼'6
0.15p(r)
1
O.lO
~
0o5
\
0
+
,o.Z.s
r
L
8
I0= c)
r,40
{,
ro-25
r 3
Fig. 1. The metric eV/2(r), eh(r) and pressure p(r) for an extended charged liquid drop is shown for various values of r0. (GMF/C 2 = 1, G1/2Q/c2 = 0.5).
(13)
which requires ~' + X' > 0 for a physically meaningful theory. In the incompressible fluid model we have P0 = const., so that two coupled equations remain:
P0
® r,.3.o
IL\
It is remarkable that the radius r 0 of the drop has always to be larger than a critical value, say r c, otherwise our system of coordinates would break down and an event horizon would emerge, r e is given by: re
=
GMF c2
+
((GMFI2 \\
c2 ]
,~lrtr G ,1/2 +-~-~4Q 2 ] . 5c
(16)
(Note, that we fixed M F and not M!) This results from the decomposition of M into M F + M E s , see eq. (12). We conclude that an event horizon occurs for arbitrary values of Q if r 0 <~ r c. (The situation, in which Q2 > M 2 here never occurs. Nonlinear theory of electrodynarnics). We now turn to the problem of charged point-like objects, where electrodynamics is governed by a Born-Infeld theory [ 6 - 9 ] . Instead of (2b) we now have:
Volume 62A, number 2 ( _ g ) - l / 2 [(_g)l/2
PHYSICS LETTERS
FUU(1 +~F2/E2)-l/2 ] Iv = 47r/ta (17)
where F 2 = FaoF a# and E 0 denotes the limiting field strength in the Born-Infeld theory. We wish to treat E 0 as a freely adjustable parameter. For a point charge the solution is:
E(r) = e(V + x)/2 ~22f(r)
(1 8)
25 July 1977
all values ofr, Meff(r ) is always positive i f M F > 0, and the field contains an event horizon. Thus an arbitrary amount of charge cannot prevent the occurrence of an event horizon, even in the Maxwell limit E 0 -~ oo. The misleading property of the commonly used Reissner-Nordstr4m solution comes from the decomposition: e v = 1 - 2__G_G[(MF +MEs(oO)) _(MEs(~O) _ MES(r))]
c2r
with:
f(r) = [1 + (F/r)4 ] -1/2
(19)
and ?2 = Q/Eo" f(r) effectively plays the role of a form factor. The matter tensor can then be evaluated as: T~(BI) E2 0 d i a g ( f - l(r) - 1, f - 1 (r) - 1, f(r) - 1, f(r) c2
(20)
- 1)
leading to three Einstein equations, which require that: X = -v
(21)
(24)
= 1 -- 2 G [Mre n _ ~ E S ( r ) ] .
c2r
The total asymptotic mass M F + MES(oo) is renolmalized to Mre n which in general corresponds to a negative "bare" mass M F. For fields which have developed by collapse from an astrophysical object, this procedure is not permissible. We therefore conclude that a collapsing charged star (with zero angular momentum) will - according to the classical field theory of gravitation - always hide behind an event horizon and no naked singularities will occur.
and
References
(reV)' -
8-~--4G E2r2(f-l(r ) - 1) + 1
(22)
the integration o f which yields: ev = l
2G -C2--7
E~ F+4rr 7 -
f (r f _ l ( r ' ) - l ) 0
r'2dr
'1 (23)
-- 1 - 2 G [MF +MEs(r)]"
c2r
This result can easily be understood physically, since MES(r ) is just the mass o f the electromagnetic field which is contained inside the sphere with radius r. Meff(r ) = M F + MES(r ) can be looked upon as the effective gravitating mass when the observer is at the radial coordinate r. It contains the mass of the matter distribution (or the point-like object) and that part of the mass of the ES-field, which is inside o f r . Now for
[1] A. Reissner, Ann. Physik 50 (1916) 106. [2] G. Nordstr4m, Verhandl. Koninkl. Ned. Akad. Wetenschap. Afdel. Natuurk., Amsterdam 26 (1918) 1201. [3] A. Adler, M. Bazin and M. Schiffer, Introduction to general relativity (McGraw Hill, 1965). [4] K. Schwarzschild, Sitzungsber. Preuss. Akad. Wiss. Berlin (1916) 424. [5] H. Bauer, Sitzber. Akad. Wiss. Wien (1918), Abt. IIa, 127 2141. [6] M. Born and L. Infeld, Proc. Roy. Soe. A144 (1934) 425. For a recent discussion of the viability of nonlinear electromagnetic theories see: [7] J. Rafelski, L.P. Fulcher and W. Greiner, Phys. Rev. Lett. 27 (1971) 958. [8] J. Rafelski, L.P. Fulcher and W. Greiner, Nuovo Cirn 13B (1973) 135. [9] G. Soft, J. Rafelski and W. Greiner, Phys. Rev. A7 (1973) 903.
69