Commun. math. Phys. 8, 245--260 (1968)

Event Horizons in Static Electrovac Space-Times

Wr~.~-m~ I s t ~ L * Dublin Institute for Advanced Studies, Dublin

l~eceived November 20, 1967

Abstract. The following theorem is established. Among all static, asymptotically flat eleetrovae fields with closed, simply-connected equipotential surfaces goo = const., the only ones which have regular event horizons g00 = 0 are the t~eiss- ner-Nordstr6m family of spherisymmetric solutions with m > Gll~le]/c. In the special case where the gravitational coupling of the electromagnetic energy density is neglected (G = 0) all solutions are computed explicitly, thus extending an earlier result of G~zBv~G for a magnetic dipole in SC]~WAX~ZSCItXLD'S space-time. Possible implications for gravitational collapse are briefly discussed.

1. Introduction

Of centrM importance to the theory of gravi ta t ional collapse is the question whether event horizons are a fairly normal characteristic of very intense gravi tat ional fields, or whether t hey are merely quirks of the special highly symmetr ic solutions which have so far been studied.

I f we restrict ourselves to the class of asymptot ical ly flat, static vacuum fields, it is already known [1] t ha t the only regular event ho- rizons are spherical. More precisely: among all fields in this class with closed, s imply-connected equipotentiM surfaces g00 = c o n s t . , Schwarz- schild's solution is the only one with a regular event horizon go0 = 0. This means t h a t no static asymmetr ic per turbat ion of the Schwarzsehild field which originates f rom sources within the critical surface goo = 0 (r = 2m) can preserve a regular event horizon. (On the other hand, per- turbat ions due to exterior sources, such as dis tant masses, leave the quali tat ive character of the event horizon unaffected [2].)

Quite generally, in the case of an a rb i t ra ry asymptot ica l ly flat field, it therefore seems natura l to ask whether the regular i ty of an event horizon is destroyed by any asymmetr ic per turbat ion due to interior sources (e.g. mass quadrupole [3], magnet ic dipole field [4], outgoing gravi tat ional waves ; an exception has to be made here for ro ta t ion - - the Kerr manifold has a regular event horizon [5]). 1 I f this were true, i t would force a drastic reappraisal of our current ideas on the nature of gravi ta t ional collapse [6].

* On leave of absence from the Mathematics Department, University oi Alberta, Edmonton, Canada.

1 For instance, it might be conjectured that every vacuum field which has a regular event horizon and which is asymptoticMly flat (with an outgoing radiation condition) is algebraically special.

246 W. Is~E~:

The present paper represents a small step towards a definitive answer to this question. We shall study the class of eleetrovae spaces - - i.e. regions which are the seat of electromagnetic fields but free of charge and mass (all sources are assumed to be immured within the surface goo= 0). The main object of the paper is to prove the following result: of all static, asymptotically fiat electrovac fields with closed, simply- connected equipotential surfaces goo = eonst., the only ones which pos- sess regular event horizons go0 = 0 are the spheri-symmetric l~eissner- ~qordstrSm solutions for a charged particle. (A precise formulation is given in Scc. 4.)

In the special case where one neglects the gravitationM effect of the electromagnetic energy density [4], i t is a straightfol~ard matter to compute the solutions explicitly (See. 7).

In the general case, we proceed by reformulating the given conditions in terms of the geometry of the surfaces g0o = const. (See. 2--41), showing that the equipotential sili'faces of the electric (or magnetic) field neces- sarily coincide with these surfaces (See. 5), and finally proving that they are spheres (Sec. 6). In the interests of mathematical simplicity we shall confine ourselves to the situation where the field is purely electric or purely magnetic. However, there should be no essential difficulty in extending the proof given here to the slightly more general situation of crossed electrostatic and inagnetostatic fields.

2. Static Fields

In this section we shall deal with a general static field. Our aim is to reformulate Einstein's static field equations as conditions on the geo- metry of the equipotentiM surfaces.

The notation follows reference 1. (Signature of metric - -+÷-F . Cap- italized Latin indices (range 0--3) refer to space-time tensors. Three- dimensional and two-dimensionM subtensors are distinguished by Greek indices (range 1--3) and by lower-case Latin indices (values 2, 3). Covariant differentiation with respect to the 4-dimensionM, 3-dimen- sional, and 2-dimensional metrics is denoted by F, a stroke and a semi- colon respectively.)

A space-time is called "stat ic" if it admits a regular vector field which satisfies Kflling's equations:

0 = VA~B+ V~A=sCOcgA:~+gAca.B~C+gc.BaA~ c , (1)

is hypersurface-orthogonal: ~ VG~B~ = 0 , (2)

and time-like over some domain. With V defined by

V = ( - ~ ~A)~/~, (3) the identity a[~ (V -~ ~] ) = 0 (4)

Event Horizons 247

follows f rom (1) and (2), and shows t h a t a s imply-connected domain which has ~A ~A < 0 th roughou t admi t s a scalar field t (x A) such t h a t

V - ~ ~ t = - - aa t . (5)

I n this domain, we can therefore in t roduce "s ta t ic co-ordinates". Le t the 4-scalars x ~ be a n y three independent solutions of

~ a ~ x ~ = 0 . (6)

I n the co-ordina~e sys tem x ° ~ t, x" we derive ~ " = 0 f rom (6), t hen 2 ° = 1, 8~ = g~o = 0 f rom (3) and (5), finally agA~/~t = 0 f rom (]). Thus the metr ic is reducible (in the domain where ~A ~A < 0) to the s t andard form

ds~ = g ~ (xl' x~' xS) d x ~ d x ~ - V~dt2 '1

V = V(x 1,x 2,x 3 ) > 0 . ~ (7)

The form (7) can be decomposed fur ther . We suppose t h a t

~-~ - (VI~ VI~)~/~ (S)

vanishes nowhere in the domain of interest (cf. end of See. 4). As in- trinsic co-ordinates for the eqnipotent ia l 2-spaces V = const., t = const. in t roduce funct ions 0 ~, 02 which arc cons tan t along the or thogonal t ra- jectories: g ~ (0~0 ~) ( ~ V) = 0. Then the spat ial metr ic reduces to

g~pdx~dx ~ = g~b(V, O) dOadO b + [~(V, 0)] 2 d V 2 . (9)

Le t n be the uni t spat ia l vector normal to the eqnipotent ia l suHaces,

n~ = @a~ V = @-'lOx~(V, O)/O V , (1O)

and e(~) the tangent ia l base-vectors associated with 0%

e(a) ~ = a x e ( V , O)/O0 a , e(a) ~- gaOe(b ) . (11)

The t r iad {e(a), n} spans the 3-space a t each point , and the following decomposi t ions are der ivable f rom (9) (by making the special choice of co-ordinates x 1 = V, x ~ = 0% or otherwise [1]):

g~[J = ga%(a)~e(J + n~n ~ , (12)

V [ ~ = @-lKab e~(~) eb(~) - - @-2 (ae@) (e(e) n~ + e(c)~n~) (13)

- - ~ - - 3 ( ~ I ~ V ) % n ~ . t Iere ,

1 K ~ == -2- @-I~ga~/a V (14)

is the extrinsic curva ture of the 2-space V = const. , considered as imbedded in the 3-space t = eons~. Note the re la ted formulas

~ g ~ / a V = - - 2@K ~ , ~g~t~/~ V = g ~ t ~ K ~

a(gi/~R)/~ v = - 2gl/~ [@ ( K a~' - - Kgab)]; a~ J (15)

248 W. IsRAEL:

which can be deduced from (14); g is the 2 × 2 determinant of g~b, R ~ gabRab is the two-dimensional curvature invariant (the Gaussian curvature i s - ½R), and K-~ ga~K~ is (twice) the mean curvature. From (13) and (12), we have

Vl~l~ = @-IK-- @-~a@/a v . (16)

The imbedding relations for the three-dimensional l~icci and Einstein tensors R ~ , G~ are [1]

1 G~n~n ~ = y ( K a b K ab - - K ~ - R ) , (173)

R~e(a) ~n ~ = a a K - - Kb;b , (17b)

1 R~ze(~)~e(b)~ =.~_Rg~b + ~-1@;~ ÷ KKa~ + @-Ig~OK~/aV. (18)

Let X, Ya and Zab denote the right-hand sides of (173), (17b) and (18) respectively. Then the identities

g-~(a/~V) (gX) = @Z~b(Ka~--Kg~b)--@-~(@~Y'9,~, (19a)

g-1/2(~/OV) (gl/2y~) = [@(SaZ Cb ¢ __zb)];b + @-x(@2X).~, (19b)

are consequences of (14). The left-hand sides of (17) and (18) of course satisfy corresponding identities, whose content is merely that of the contracted Bianchi identities G~ z = 0. Thus, if (14) and (18) are regarded (in a given 3-space and for given @) as a system o~ first-order equations determining the evolution of gao and K ~ as functions of V, then (173, b) are "involutive constraints": if they are satisfied on one surface V = const. then, by virtue of (19), they are satisfied identically.

We are now ready to decompose the Einstein field equations

a ~ = - - 8 ~ 7 T ~ (20)

(7 = 7.3 × 10 -29 cm/gm is hTewton's constant of gravitation divided by c2). Under a change of spatial co-ordinates x ~' = x~'(x), the 3 × 3 sub- matrix T ~ of the energy tensor T~B transforms as a 3-tensor, and To o is invariant. The 3 + 1 split of (20) yields [1]

1 y g ~R,~ = 8JrTT ° , (213)

0 = 8~?T~0 , (21b)

G~ = - 8 : r y T ~ - - V-I(VI,, ~ - vl~l~g~). (21c)

We can immediately deduce a relativistic analogue of Poisson's equation:

V -~ VI~ I~ -- 4~r7 ( T ~ - T°) . (22)

From (16) and (22),

@-~@/0 V = K - - 4zr~ V@(T~-- T °) . (23)

Event Horizons 249

From (18), (21) and (13),

V-lg-:[/~.a(gl/2VK~)/OV "b 1 ~R(5 b

(24) 1

- -8x~,@ (T~e(a) e(b)~--~- TAgab) .

From (17), (21) and (13),

1 -~ (KabK ab - - K 2 - - R) = - - 87~y T ~ n ~ n ~ + K](~V) , (25)

~aK - - K~;b = - - 8 ~ y T~e(a)~n ~ + (aa~)/(~ 2 V) . (26)

Eqs. (14), (23) and (24) form a complete system for determining the evolution of gab, ~, Kba as functions of V. The tangential stresses Ta~e(a)~e~b) ~ can be assigned arbitrarily over the 3-space ; the conservation identities VB T Ag = 0 are satisfied automatically if the normal stresses T ~ n ~ are determined by (25) and (26).

Finally, we record the expression [1]

"-41 RA BC~oRA ~C9 = G ~ G t~ + V-~ VIt~ V[~ (27)

for the square of the four-dimensional Rieman tensor. Evaluating the second term with the aid of (13), we find

1 -~ RA Bc DR A Bc D = G~,G~'~ + ~-~ V-2 Kab Kab

(2s) + 2o~ -~ V - ~ , ~ ; ~ + ~-~ V - ~ ( ~ / O V ) ~ .

3. Static Electrovac Fields

An electromagnetic field

F.~ c = O~A c - - OcAB (29)

in a static space-time is itself static if the Lie-derivative of the 4-poten- tial A vanishes:

~c V e A B _ A c Vc~A = 0 . (30)

The field is purely electric (or, by an obvious re-interpretation, purely magnetic) i f / ~ a is a simple bivector of the form

FBC = 2 V - I ~[BEc] . (31)

The "electric vector" E~, defined by (31), may be taken without loss of generality (as long as ~B~ B # 0) to be purely spatial (~AEA = 0), and is then given explicitly by

E ~ = V - 1 ~ ~ , (32) i . e .

EA = - - V - l a A ~ , c f ~ - - A t ~ B - ' - - A o . (33) :[7 Commun. ma~h. Phys., Vol. 8

250 W. I s R ~ :

I t is readily verified t h a t

VI~FAB== ~A V B ( V - 1 E B ) = ~A V-1E~I~ (34)

so tha t , in the absence of charge and current , the Maxwell equat ions VBF A B = 0 can be wr i t ten

E : I : = ( V - l g = ~ ) l ~ = O. (35)

B y vir tue of (9), this can also be wri t ten as

Vg-1/2 (a/a v ) (gl/2 v-1 ~o) = - - (~ ~o; a), a , (36)

wi th ~p defined b y ~0/~ V = e 9 . (37)

For an electrovae field (electromagnetic field wi thout ma t t e r ) the energy tensor is g iven b y

4 ~ T ~ ~ = F ~ cFB c - - ~ ga BF~ c F - c 4 I (a8)

-~gABE~ - - E A E B + E ~ V - -~A ~B .

More explicitly, we have

4 ~ T o o = - - ~ - E ,

4z rT °= = 0 , ~ (39)

J 4:~ T ~ = + g ~ E 2 - - E ~ E ~,

with VE~ = ~ ~ n ~ - - e ( ~ ) ~ , (40)

E ~ ~ E ~ E ~ = V - 2 ( ~ ~- + ~ ; a ~ ;a ) . (41)

Hence (22) reduces to Vt~'It , = ?, V E 2 . (42)

We subst i tu te (39) into the basic Eqs. (23)--(26) of the previous section. This leads to the following complete first-order sys t em/or deter- mining the march o] the variables gab, q~, Y~, ~, Kba as ]unctions o~ V:

Geometrical equation: ag~/O V = 2 ~ K ~ . (14)

Electrostatic equations: ~/~ V : e ~o, (37)

Vg-~/2a (gl/~ V-1 ~p)/O V = - - (~ q0; a); a" (36)

Gravitational equations:

~-~0e/~ V = K - - ~ V-~(~o ~ + ~ ; a ~ ;~) , (43)

1 V-:g-x/~o (g~/~ VKba)/~ V = - - e; a ;b __ -~ ~ R 5ha (44)

+ y V - ~ q [ 2 ~ o ; ~ ; ° - ~ ( ~ + ~ ; ~ ; ~ ) ] •

Even~ Horizons 251

V = 1 - - (m/r) ÷ ~ ,

= O(r -~ ) , O ~ = O(r - 8 ) ,

where r ~- (8~x~xP)X/~.

Involutive constraints:

1 - ~ ( K a ~ K a b - - K 2 - - R ) = y V-2(~v ~ - ~0;a~ ;a) ÷ ~ - ~ V - 1 K , (45)

OAK-- Kb;b = 2~, V - ~ p qJ ; a + ~-~ V-~ ~; a . (46)

That the constraints (45), (46) are respected by the equations of evolu- tion can be verified explicitly with the aid of the identities (19).

The following result, which will be needed later, is obtained by contracting (44) and eliminating R by means of (45):

V a ( V - I K ) / a V = - - Q ; a ; a - - I ~ K 2 - - Q A a ~ A ~ b - - 2 y ~ V - 2 ~ ; ~ v ;a (47) 2

]~[ere, 1

Aab ~ Kab - - ~ Kgab (48)

iS a measure of the deviation from spherical symmetry. Combining (28) and (43), we have finally

1 -~ RABCD RABCD = G ~ G ~ ÷ ~-2 V-2KabKab - ~ 2~-d V-2~;a~;a (49)

+ e -~ V - 2 [~ V- le ( ~ + ~ ; a ~ ;a) - - K ] ~ .

A rather complicated explicit expression for the term G ~ G F'~ in terms of the field variables can be obtained from (21 e), (39) and (13). For our purposes it will be stffficient merely to note that this term is obviously non-negative.

4. Statement of Theorem

In a static space-time, let Z be any spatial hypersurface t = const., maximally extended consistent with ~A ~A < 0. We consider the class of static fields such that the following conditions are satisfied on 27:

(i) X is an electrovac space (i.e. free of charge and matter). (ii) X is regular, non-compact and "asymptotically Euclidean". The

]ast statement means that there exist co-ordinates x ~ in terms of which the metric (7) has the asymptotic form

g ~ = ~ + 0(r-~), ~ g ~ = 0(r-~) ,] m = const., I (r -+ c~) (50)

a ~ 0 ~ = 0 (r -4)

(iii) The elcctrovac field is purely electrostatic (or purely magneto- static). The asymptotic form of the electrostatic (or magnetostatie) scalar potential is

cf = (e/r) + ~, e = const., (r-~ ~) (51)

= 0(r -~) , a ~ = 0(r -a) . 17"

252 W. IsRAEL:

(iv) The equipotential surfaces V = con_st. > 0, t = const, are a regu- lar family of simply-connected closed 2-spaces.

(v) If the greatest lower bound of V on X is zero, then the geometry of the equipotential surfaces V = s approaches a limit as s -+ 0 + , corre- sponding to a closed regular 2-space of finite area.

(vi) The invariant R A ~ O D R ABOD is bounded on X.

Theorem. T h e on l y static space- t ime u, hic]~o satisf ies condi t ions (i) - (vi)

is the spher ica l ly s y m m e t r i c Re i s sner -hrords t r6m so lu t ion

d s ~. = V - 2 d r 2 + r~(d02 + sin~0 dq52) - - Vg'dt 2]

V ~ = 1 - - 2 m / r + ye~/r 2 , qD = e / r , ~ (52)

w i t h m >= yl/~ tel. (We here assume y > 0. In the exceptional case y = 0, discussed in

Sec. 7, the Theorem does not hold in quite this form.) That the Reissner-NordstrSm manifold with m >= y~/~ ]el actually does

satisfy (i)--(vi) is well-known, and can be easily verified from the explicit formulas

ga~dOadO b = r~.(dO~ + sin20 d # 2) ,

= V - l d r / d V = r~/ (mr - y e a) , ~fl = ~ e V / r ~ , (53)

Kab = V g ~ b / r ,

obtained by applying the definitions (8), (37) and (14) to the metric (52).

There are two major steps in the proof of the theorem. In Sec. 5 we show, mainly with the aid of the electrostatic Eqs. (36) and (37), that (i)--(vi) can only be satisfied if ~ is constant on the equipotential surfaces V = const.: ~ - - ~(V). This simplifies the gravitational Eqs. (43) and (44), from which it can then be deduced (Sec. 6) that the equipotcntial surfaces are spherical.

There is one trivial case for which the proof can be quickly disposed of here. Suppose tha t V has a positive lower bound. Then the (maximally extended) 3-space X is complete. From (35) and the boundary condition (51) we deduce ~ ~ 0 with the aid of Green's theorem. Eq. (42) now reduces to Laplace's equation VI,[~ = 0; together with the boundary conditions (50) this yields V ~ 1, showing tha t space-time is flat and establishing the theorem.

We may assume henceforth tha t V comes arbitrarily close to zero on X. The 2-space V = 0 ÷ then forms an inner boundary of 2: and encloses an internal "hole". By (iv), every equipotential surface is homo- topic to V = 0 ÷ . I t follows that the gradient of V cannot vanish at any interior point P of X. I f it did, then by (42) V would have a minimum at P (unless E vanishes identically) and the equipotential surfaces near P could be shrunk to a point. [If E ~- 0, P would be a point of bifurcation

Event Horizons 253

of the equipotential surfaces [1], wtfieh also contradicts (iv).] Hence remains finite, and the metric ]orm (8) regular, at all interior points o/Z. The possibi~ty that ~ -+ ~ as V-> 0 ÷ is, however, not excluded.

We conclude this section by recording the exterior and interior boundary conditions in a form convenient for later application. For the asymptotic forms (50) and (51) we find from (8), (15) and (37)

r ~ o ~ , e/r ~'-+m -1, rg-+2:~ (54)

rq~-+e, r2~o-~--e as V-->I

According to (vi) and (49), the regularity of the manifold at the inner boundary V = O + requires tha t

K ~ = O ( ~ V ) , e;~=O(e2V), ~ (55)

~ o = 0 ( V ) , ~ ; a = 0 ( V ) as V - + 0 + J Thus ~0 and ~-1 are constant on the event horizon:

(0, 0 I, 02) = ~o = const., ] ~-1 (0, 01, 03) 1/~ 0 = eonst. (possibly zero) .~ (56)

5. The Electrostatic Field

In this section, our interest wi~ center on the electrostatic Eqs. (36) and (37). A number of integral relations will be derived which enable us to show that the electrostatic and gravitational fields must be chained together by the condition ~; a = 0 if (i)--(vi) are to be satisfied, and we shall determine the function ~ = ~(V) explicitly.

Let I~'(V, q~), G(V, q)) be (for the moment, arbitrary) d~erentiable functions. From (36), (37), (43) and (15) we readily obtain the identity

g-1/~-~gl/~[V-1F(V, qg) ~ ÷ ~-IG(V, ~)]

=A(V,q~)e(~2÷q;;~9~;~)+ B(V, qJ)yJ+e-~OG/aV (57a)

where A V ~ ~G ÷ OF/O~, B ~ V-IaF/aV ÷ aG/acf. (57b)

To obtain integral conservation laws from (57), let us require that

A = B = OG/aV = 0 . (58)

The general solution of this (over-determined) linear system of differen- tial equations for ~, G is a linear combination of the three particular solutions

1 7 = 1 , G = 0 , (59a)

F = ~ 0 , G = - - 1, (59b)

E = ~,~2-t- V ~, G = - - 2 ~ . (59c)

254 W. Isl~t~L:

Taking each of these values for F, G in turn, we integrate (57~) over X - - i.e. we form f f f [(57a)] g~/2dOldO 2 d V - - noting tha t the integral of

the last term (2-divergence) vanishes when taken over any closed 2-space V = const. The results express the equality of the surface integrals of the expression in square brackets over the two boundary surfaces V = 1 and V = 0 + (both have to be understood as limits):

f (~/V) v=x 0 (60a) d S l v = 0 + = ,

f (Z/e) dSJvV=0~+ = 0 , (60b)

f [ ( V + V~2/V) 7~--2q~/e] d S V=Xv=o+ = O. (60c)

We have defined the element of area d S = gl/2 dOadO 2 and

1 V_10(V2__ y ~ 2 ) / O V = 1 - - ) ,OyqJIV. (61)

[Eqs. (60a, b) could also have been inferred somewhat more directly from (35) and (42).] Evaluating the surface integrals for V = 1 by means of (54), and taking (56) into account, we find

f (~PiV) d S = - - 4 x e e , (61~) V=O+

- - 7 % f ( y , / V ) d S + S o / ~ o = 4 : r m , (61b) V=O+

ycf~) f (~lV) d S - - 2 9 o S o l Q o = - - 4 z r e , (61c) V=O+

where S O is the area of the 2-space V = 0 + [finite and non-vanishing by (v)]. Solving (61), we find

y e ~00 = m - - ( m S - - 7 e~) a /2 , ( 6 2 )

So/~o =- 4 7~ (m 2 - - 7 e2) 11~ " (63)

(A second solution for ~o involving the opposite sign for the square root, is unacceptable because it makes So/~o negative.) For the existence of a regular event horizon it is thus necessary tha t m S > y C.

To motivate the next step, we begin by observing from (53) tha t the manifestly nonnegative expression

(q V) -1 [e~(1 - - ~ecflm ) + e Vim] 2 (64)

vanishes for spherical symmetry. Now, the expression (64) resembles the right-hand side of (57 a) in form. Accordingly, let us require

A -- V-~(1 - - yecf /m) ~ , B = 2(elm) (1 - - yeq) /m) . (65)

0 G/O V = e 2 Vim 2

Evcn~ Horizons 255

in (57a, b). The resulting linear differential equations for F, G have the particular solution

1 2 V ~ ~- 1 ~,2e~,~3_ 2~ ,em,~ "]l m 2 F = - - ~ - y e ~ - - 2 ~ w ~ Y (66)

I 2 -s 1 ?-ira ~ m2G = ~ e V - - - ~ ye2~ 2 + 2emc f +

With F, G given by (66), we have thus the identity g-~/2(~/O V) g l / S ( V - l . F v -[- @-IG) = (@ V) -1 [~ ~/)(1 - - ~e ~/Tf/~) ~- eV/m] 2

+ ~ V - l ( 1 - - ? e ~ v / m ) 2 F ; a ~ ; a - V - 1 ( F @ F : a ) ; a . (67)

Integrating over X, we deduce the inequality

f ( V - 1 F V + e - I G ) d S > f(V-1FV+ e-lG) dS, (68) 81 So

where S 0, S~ are the inner and outer boundary surfaces, V = 0 + and V = 1 (both understood as limits). Equality in (68) holds if and only if

~ ; a = 0 , i.e. ~ = ~ ( V ) , ~ (69)

e~v(1 - - 7ecf/m) + e Vim 0 J everywhere on Z. Now, the surface integrals in (68) can actually by evaluated with the aid of (54), (61a), (62) and (63). A straightforward calculation yields the value

4Jrm (9,-1 + 2 e2/m2)

for both sides of (68). Vv% conclude that (69) must be true. (This argument clearly breaks down for the special case of zero coupling: ? = 0. This case is dealt with separatc]y in Sec. 7.)

From (69) and (37),

~ ~p = d q~/d V = - - e V/ (m - - y e cf ) . (70)

Solving the differential Eq. (70) subject to ~v = (v 0 [see (62)] when V = 0, we find

7e~(V) = m - - ~¢(V) , (71) where

~(V) ~- [m s - y e ~ ( 1 - V2)]1/~" . (72)

(The other boundary condition, ~v -+ 0 when V -* 1, now shows that the parameter m introduced in (50) cannot be negative.) From (61), (69) and (71) we have the explicit formulae

• ~ (V) = m/o: ( V ) , (73)

v 2 = - - e V/~ (V). (74)

6. T h e G r a v i t a t i o n a l F i e ld

The explicit formulae just obtained for the electrostatic field, when substituted into the gravitational Eqs. (43) and (44), result in a great simplification. We are now in a position to derive two integral relations

256 W. I s ~ L :

from (43) and (44) which ~dll enable us to infer tha t only spherical surfaces V = const, are compatible with conditions (i)--(vi) of Sec. 4. This will complete the proof of our theorem.

Remembering tha t ~0 is a function of V only, we have from (47), (43), (36) and (37)

(Ola V) [(g ~le) 1/2 V-1K] = - - [g~(V)] 1t2 V -1 [2 (el/2) ;a; a (75)

1 -3/~ o;a +-ff ~ ~ ;~ + ~I/2A~A ~b]

The integration f f f (75) dOldO~d V yields Z

f (2/q)I/~(K/V) dS <= f (,~/q)I/~(K/V) tiN, (76) 8i So

with equality if and only if

Aab = 0 , Q,~ = 0 (77)

everywhere on ~. According to (54), the left-hand side of (76) has the value 8~ml/2. For the right-hand side we have from (55), (45) and (74)

1 lira (K/V) = - - y e o R ( O , 01, 0 ~) - - Yeo v~m+ (*P/V) ~ V--->O +

1 (78) - - 2 ~°R(O' 01' 02 ) - -Y# / (O°~2° ) '

where ~o ~ ~(0) = (m s - ~e2) 1/~ . (79)

Noting tha t f R(e, 0 I, 0 z) d S = - - 8~ (80)

for any closed, simply-connected 2-space V = c (Gauss-Bonnet theorem), we obtain the value

4 ~ (ml~o)l/~ Do1/~ - - ~, e~l(~o ~01/2) ] (81)

for the right-hand side of (76). We thus arrive at the inequality

~oqo ~ (m + ~0) 2 . (82)

(If Oo 1 = O, (82) is properly interpreted as an inequality for lim (~ a).) V-->0 +

We shall next derive a second inequality giving an upper bound for %~o. I f / ( V ) , h(V) are arbi trary differentiable functions, we have

(OIO V) ( /K + e-~h) = (1' + V - 1 / - - h ) K - - ~ IOK 2 + yhq V-~o 2 (83)

+ 0 - 1 h ' ~ l e ; ~ ; ~ - - / q A ~ b A '~ •

In the term involving K 2 on the right-hand side, we substitute the value

1 K ~ = A~,bA ~ b - - - (q V ) - I K - - 2 7 ~ (84) B 2 V-~

Event Horizons 257

obtained from (45), and we substitute for ~ from (74). We also note the result (easily derived from the formulae of See. 3)

0 ( g l / ~ - l ~ ) / a v = 0 . (85)

We thus obtain the identi ty

(3f8 V) [gl/20-l,~(/K ÷ e-lh)] - - ~ )~(V) / (V) gl/2 ]~

q- g'12e-l). [(1' + 3 V - ' l - - h) K ÷ e-~{h ' ÷ yVe2o:-~(h ÷ 2V-11)}] (86)

- - )~(V) ](V) gl/2 [(ln~);a a ÷ e-:q:oe:: + AobAob],

valid for arbi t rary/ (V) , h(V). Let us now choose ], h so as to make the second line of (86) vanish, i.0.

/ ' + 3 V - ~ / - - h = 0 , (87)

h' ÷ 7 Ve2~z-~'( h q- 2 V-l~) = 0 .

A particular solution is

/ ( V ) = V[~o + ~ ( V ) ] -2 , (88)

h(V) = 2[~0 + ~ ( V ) ] ~ I [~(V)~ -1 •

~Vith these values for [, h (which, it should be noted, are positive on Z) we form f f f (86) dO~dO2d V. We find, recalling (80),

X

<= f O-U~(/K + e-~h) dS- -8z f /(V);L(V)dV, (89) So 0

with equality if and only if (77) holds everywhere on Z. Evaluating the surface integrals with the aid of the boundary conditions (54), (55) and (63) yields

f . . . . 0 , f . . . . 2(0) So(qoa0) -2 = 4 z m ~ o l g o 2 . (90) $~ 8o

Using the expression (73) for ~(V), we obtain further 1

ye 2 2:¢o mq- ~0 " 0

The inequality (89) can now be reduced to

~oqo G (m + ~o) 2 . (92)

Comparing the inequalities (82) and (92), we irder tha t (77) holds everywhere on X, i.e.

e ~ ~ (V) , K ~ , ~ - 2 f f a b K ( V ) . (93)

258 W. ISR~L:

This implies t ha t the equipotential surfaces are spherical. Indeed, if we introduce a funct ion r (V) defined by ~ = e/r, we can readily deduce the formulae (53) which characterize the Reissner-NordstrSm field. We recall [remarks following (63) and (72)] t ha t the parameters had to be restr icted by m > yi/2 [e[. Our proof is thus complete.

7. Zero Coupling

We now take up the exceptional case of zero coupling between the gravi tat ional and electromagnetic fields [y = 0 in the Einstein field Eq. (20)]. This had to be excluded f rom the previous considerations [see the remarks after Eq. {69)]. I n this case, the simplest procedure is to exhibit the explicit solutions, which are in any case of interest in their own right.

Our problem is to obtain solutions of the vacuum equations G~ B = 0 and the electrostatic Eq. (35) which satisfy conditions (i)--(vi) of Sec. 4. I t is already known [1] t ha t the only vacuum space-times compatible with (i)--(vi) are the Schwarzsehild solutions with m > 0. The problem thus reduces to finding well-behaved electrostatic fields defined on the Sehwarzschild background :

~ d x ~ d x ~ = (1 - - 2re~r) - I dr ~ ÷ r~(dO 2 + sin~O dqb 2) ,

V = (1 - - 2m/r)i/~ .

The electrostatic Eq. (35), which is linear in ~, reduces to

(94)

1 a ( 0q~) 1 a2q _ 0 . sin 0 a 0 sin 0 - ~ - + sin 20 ~ q)2 (95)

Separable solutions which are regular on the axis have the form

= R(r) pM(cos0) eiMe, where R satisfies

(96)

x(1 + x) d2R/dx 2 + 2 x d R / d x - - n ( n + 1) R = 0 , (97)

x ~ ( r / 2 m ) - 1 . (98)

For n = 0, we have the spherically symmetr ic solution

~v + const. = e/r (99)

which is regular for 2m < r < co and satisfies (i)--(vi).

Event Horizons 259

For n = 1, R = x is an obvious solution of (97), and the second solu- t ion can be found by variat ion of parameters. We thus obtain the axially symmetr ic solutions (M = 0)

q~(r, O) = e l ( r - - 2m) cos0 , (100a)

~(r , O) = c 2 [-- 1 ÷ re~r-- (r /2m - - 1) ln(1 - - 2m/r)] cos0 . (100b)

The first represents a uniform electrostatic or magnetosta t ic field. The second is the static field of an electric or magnetic dipole; its asymptot ic form is

q~ ~ c 2 cosO/r 2 (r---> oo) . (101)

Results equivalent to (100 b) have been given previously by GINZBUgG [4]. For general n, (97) is reducible to Legendre 's equation, and has the

linearly independent solutions

R = % [x/(1 + x)]~/~ ~ ( 1 + 2x), R = c 2 [x/(1 + x)]~/~ E~(1 + 2x) (102)

where q3n I (z), ~ l ( z ) are the associated Legendre functions, normalized to be real for real z > 1. For the electrostatic field we thus have the two families of solutions

c 2 (1 2m \I]2

Since

~t~(z) ~ z ~, ~ln(Z ) ~ z-(~+1) (z-+ co , n >_-- 1) , (104)

only the second solution has the correct behaviour at infinity. To examine the behaviour of (103b) near r ~= 2m, we observe f rom (41) t ha t the field s t rength E is given by

E 2 = ( 1 - - ~ - ) - l [ V ~ + - ~ \ O 0 ] ~ \ - ~ - / j . (105)

Now, C~(1 + 2 x ) ~ x-~/2 as x -+ 0, so t h a t the radial factor (102) ap- proaches a constant non-zero limit. Hence, as r - > 2 m , 0g /00 and (for M 4= 0) O~0/a~b remain of order un i ty for n --> 1, and E is of order (1 - - 2 re~r)-1~2. We conclude t h a t the only electrostatic field on a Sehwarz- sehfld background which is well-behaved for 2m _-< r < oo is spherically symmetric. Nevertheless, all the solutions (103b) are compatible with (i)--(vi), since the geometry is regular at r = 2 m despite the singulari ty of the energy- density, on account of the zero coupling.

Acknowte~ements. I t is a pleasure to thank Professor J. L. S~GE for the hospitality of the Dublin Institute for Advanced Studies. I am much indebted to Mr. V. DE LA C~uz for several useful discussions and for checking some of the formulae. Part of this work was carried out during tenure of a Senior Research Eellowship from the National Research Council of Canada.

260 W. ISR~.EL: Event Horizons

References

1. ISRAEL, W. : Phys. Rev. 164, 1776 (1967). 2. ~¢IYSAK, L. A., and G. SZE~ES: Can. J. Phys. 44, 617 (1966).

ISRAEL, W., and K. A. Ktt~:~: Nuovo Cimento 33, 331 (1964). 3. DO~OSR~E~C~, A. G., ~ . B. ZEL'DOVlC~, and I. D. NOVlKOV: JETP 49, 170

(1965) [transl. as Solder Phys. JETP 22, 122 (1966)]. 4. Gn~zBc~G, V. L.: Dokl. Akad. Nauk SSSR 156, 43 (1964) [transl. as Soviet

Phys. Dotdady 9, 329 (1964)]. - - , and L. M. OZERNOI: JETP 47, 1030 (1964) [transl. as Soviet Phys. JETP

20, 689 (1965)]. 5. BoYE~, R. H., and R. W. LINDQUIST: J. ]~/[a~h. Phys. 8, 265 (1967). 6. ISRAEL, W. : Nature 216, 148 and 312 (1967).

Dr. W. ISRAEL Dublin Institute for Advanced Studies School of Theoretical Physics 64--65, Merrion Square Dublin (Ireland)