Limiting masses of solutions of the Einstein-Yang/Mills equations

ELqEVIER Physica D 93 (1996) 123-136

PHYSICA

Limiting masses of solutions of the Einstein-Yang/Mills equations J,A. Smoller*" 1,2, A.G. Wasserman l

Mathematics Department, University of Michigan, Ann Arbor, M148109-1003, USA

Received 7 March 1995; revised 18 September 1995; accepted 13 November 1995 Communicated by C.K.R.T. Jones

Abstract

The first main result of this paper is that the masses of any sequence of distinct, globally defined, spherically symmetric solutions of the SU(2) Einstein-Yang/Mills equations (i.e., particle-like solutions) converge to 2. In addition, if one considers black-hole solutions of these equations, defined for r > p, where p is the radius of the black-hole (the event horizon), then it is shown that the masses of any sequence of distinct black-hole solutions converge to p + l /p if p > 1, and to 2, if p < 1.

O. Introduction

The equations of general relativity are based on

Einstein's hypotheses:

(E]) The gravitational field is a metric gij in four-

dimensional space-t ime.

(E2) At each point in space-t ime, one can diagonalize

gij as d i a g ( - I , 1, 1, 1).

(E3) The equations which determine the gravitational

field should be "covariant"; i.e., independent of

the coordinate system.

The hypothesis (El) is Einstein's brilliant insight,

whereby he "geometrizes" the gravitational field. (E2)

means that special relativity is included in general

relativity, and (E3) implies that the gravitational field

equations must be tensor equations.

* Corresponding author. E-mail: [email protected]. edu.

I Research supported in part by the N.S.E, Contract No. G-DMS-9501128.

2 Research supported in part by the ONR, Contract No. N00014-94-1-0691.

The equations of general relativity for the metric

are extremely complicated nonlinear equations, and it

is very difficult to find solutions. The first solution

found was the Schwarzschild solution [1], which is a

static (time-independent), spherically symmetric solu-

tion, where the metric is

ds 2 1

= - T - 2 ( r ) dt 2 + - - dr 2 A(r)

+ r2(d02 + sin20 dq02), (o.1)

and A(r) = (1 - 2 m / r ) = T-2(r ) . Note that when

A = 0, the metric becomes singular. This occurs at

r = 2m, and is called the "event horizon".

To incorporate an electromagnetic field into the eq-

uations, one couples Einstein's equations to Maxwel l ' s

equations, and to make the theory coordinate invari-

ant, the electromagnetic field is constructed from a

U(1) connection on a vector bundle. The resulting

Einste in/Maxwell equations admit a static spherically

symmetric solution called the Reissner-Nords t r rm

(RN) solution. Here the metric is the same form as

(0.1) above, with A (r) = (1 - ( p + , o - l ) / r + l / r 2) =

0167-2789/96/$15.00 © 1996 Elsevier Science B.V. All rights reserved SSDI 0 1 67-2789(95)0029 1 -X

124 J.A. Smoller. A.G. Wasserman/Physica D 93 (1996) 123-136

lim r(1 - A(r)). r---+ o o

The Schwarzschild solution to the Einstein equations

is also a solution to the EYM equations (when w(r ) --

1), and the mass of the Schwarzschild solution is 2m.

Similarly, the RN solution to the Einstein/Maxwell equations is also a solution to the EYM equations (when w(r ) ~ 0), and the mass is given by p + p - l .

In [7] it was proved that the masses of all particle-

like solutions are uniformly bounded. One can mod- ify that argument to show that for any fixed event

horizon p > 0, the masses of the black-hole solutions are also uniformly bounded. Numerical values for the first few masses were obtained in [4,7]. In this

paper we approach the problem of determining the

masses indirectly, and we obtain more precise state-

ments. We show that any sequence of distinct solu-

tions with fixed event horizon p > 0 converges to a

Reissner-Nordstrrm solution.

If (A(r) , T(r) , w(r ) ) is a solution to the EYM equa-

tions, and A(p) = 0 = At(p) for some p, then the so-

lution is called an "extreme" black-hole. (Physically,

this corresponds to zero surface temperature of the

black-hole; see [15].) The RN solution with p = 1 is

an extreme black-hole. In fact it was proved in [ l l ]

that the only extreme SU(2) static, spherically sym-

metric black-hole solution of the EYM equations is

the extreme RN solution.

The main results of this paper are (see Section 2):

(a) For fixed event horizon p, 0 < p _< 1, any se-

quence of distinct solutions of the EYM equations

converges to the extreme Reissner-Nordstr6m

(ERN) solution, uniformly on compact intervals

of the form [a, b], where a > 1. (p = 0cor re -

sponds to particle-like solutions; see [3].)

(b) For any fixed event horizon p > 1, any sequence

of distinct black-hole solutions converges to the

RN solution with event horizon p, uniformly on

compact intervals of the form [a, b], where a > p.

(c) For any sequence of distinct solutions (wn(r) ,

An (r), Tn (r)), of the EYM equations having event

horizon p >_ 0, the mass functions, p~n(r) -=

r(1 - An(r)) converge, as n ---> c~, to the mass

function of the associated RN solution. In addi-

tion the limit of the invariant quantities tZn(C~)

(the (ADM) mass) is equal to the mass of the as-

sociated RN solution.

The proofs of these results rely on an important

technical fact; namely, for any solution of the EYM

equations defined on an interval (a, b], the rotation

of the projection of the orbit in the (w-w' ) -p lane

is bounded. This result is false for intervals of the

form [a, b); see [5,9]. We pose the question: Is there a

physical interpretation of the nodal class of an EYM solution?

It is interesting to note that for essentially all solutions, the (ADM) mass exterior to the black-hole is less than 1/p, if p > 1, and less than ( 2 - p), if p < 1 ; see Corollary 2.5.

T -2 ( r ) , p > 1/p (see [1]). We note that A(p) --- O,

and thus p is the event horizon for the Reissner-

Nordstrrm solution. To incorporate nuclear forces, one couples the

Einstein equations to a Yang-Mills field, and to make

the theory coordinate invariant, the Yang-Mills field

is again constructed from a connection on a vector

bundle. In this paper, we consider SU(2) bundles. The

equations for static, spherically symmetric solutions

of the Einstein/Yang-Mills (EYM) equations were

first derived by Bartnik and McKinnon [3] and are

a set of three coupled ordinary differential equations

for the metric coefficients, A, T, and a connection

coefficient w (see Eqs. (1.3)-(1.5)). They discov-

ered global non-singular solutions - these are called

"particle-like" solutions. There are in fact, an infinite

family of such solutions [8].

Black-hole solutions of the EYM equations, analo-

gous to the Schwarzschild solution, were discovered

by Bizon [4], Kunzle and Masood-ul-Alam [7], and

others. In fact, for any event horizon, there are an in-

finite number of such solutions [12]; see also [14].

Associated to every particle-like, or black-hole so-

lution of the EYM equations is an invariant quantity

called the (ADM) mass; see [2]. In other words, num-

bers representing masses are a direct consequence of

the existence of such solutions, and are not prescribed

in an ad hoc manner. In the static, spherically sym-

metric case, the metric is of the form (0.1), and the

mass is (cf. [10])

J.A. Smoller, A.G. Wasserman/Physica D 93 (1996) 123-136 125

The results of this paper were announced in gr-

qc/9405003. Theorem 2.1 of this paper corrects Theo-

rem 4.1 of [9]. In [6], Breitenlohner and Maison have

shown (using different methods) that solutions con-

verge to the ERN solution in the case where p < 1.

1 . P r e l i m i n a r i e s

The EYM equations with gauge group SU(2) can

be written in the form [2]

Gij = ~ Tij, d* Fij = O, i, j = 0 . . . . . 3.

Here T/j denotes the stress--energy tensor associated

to the su(2)-valued Fang /Mi l l s curvature two-form

Fij, Gij = Rij - l Rgij is the Einstein tensor com-

puted with respect to the sought-for metric g i j , * de-

notes the Hodge star operator, and o- is a universal

constant. If we consider static, symmetric solutions,

i.e., solutions depending only on r, then we may write

the metric [3,13] as

ds 2 = - T ( r ) -2 dt 2 + A(r ) -1 dr 2

+ rZ(d02 + sin 2 0 d~b2), (1.1)

and the Yang/Mil ls curvature two-form as

F -- w ' r l d r / x dO + w'r2 d r / x (s in0 dcp)

- (1 - w2)r3 d0 /x (s in0 &p). (1.2)

Here (T, A) and w = w(r) denote the unknown metric

and connection coefficients, respectively, and r l , r2, r3

form a (suitably normalized) basis for su(2). As dis-

cussed in [3,13], the EYM equations in this framework

take the following form as a system of three ordinary

differential equations:

rAt + (1 + 2 w ' 2 ) A = 1 - ( 1 - -w2)2 / r 2, (1.3)

r2Aw '' + [r(1 - A) - (1 - w2)2/r]w '

+ w(1 - w e ) = 0, (1.4)

2 r A ( T ' / T ) = (1 -- W2)2/r 2 + (1 -- 2w'Z)A - 1.

(1.5)

T; thus we shall mainly concentrate on Eqs. (1.3) and

(1.4).

If the mass function tz(r) is defined by 3

# ( r ) = r ( l - A(r)) , (1.6)

then we require of our solutions that they have finite

(ADM) mass (cf. [2]), namely that

lira /z(r) < ~ . (1.7) r - - - + OG

In [8,12], we proved that for every p > 0,

there exists a sequence of solutions An(r) = (Wn (r), wtn (r), An (r), r) which satisfy the following:

lira A n ( r ) = 1, (1.8) F - ~ " OG

lira /~,,(r) < ~ , (1.9) J l ----~ O 0

lim (w2(r), w'( r ) ) = (1,0) . (1.10) r 1 ~ o G

Furthermore, if p > 0 , and we define the functions

4~ and u by

• (A, w, r) = r ( l - A(r)) - u2(r)/r ,

u(r) = I - w2(r),

then if A(p) = O,

~(0 , w(p), p )w ' (p) + (uw)(p) = 0: ( l . l 1)

that is, the "initial" values (w(p) , w'(p)) = (?,, ~)

must lie on the curve C(p) defined by

C(p) = {(w, w') ~ ~2: ~(0 , w, p )u / + uw = O,

w > 0, w' < 0}. (1.12)

In [12, Proposition 2.2] we proved the following local

existence result, for black hole solutions.

Proposition 1.1. For each ?, satisfying ?,2 < I, and

p q_ ?,2 5& 1 (i.e., 4~(p) 5~ 0), there exists a unique

solution (the ?'-orbit)

A(r, ?,) -~ (A(r, ?,), w(r, ?,), w'(r, ?,), r) (1.13)

of (1.3), (1.4) satisfying the initial conditions

A(p , ?,) = O, w(p, ?,) = ?,, (1.14) w'(p, •) = ~ = ~(p, ?,)

Since (1.3) and (1.4) do not involve T, we can use

these to solve for A and w, and then solve (1.5) for 3 Our mass functions are half those usually encountered in the

literature.

126 J.A. Smoller, A.G. Wasserman/Physica D 93 (1996) 123-136

defined and analytic on some interval 0 < (r - p) <

s(y). The solutions form a continuous one parameter

family (see [12, Definition 2.1]).

Next, for p > 0, we define the region F C ~4 by

f ' = {(A, w, w ~, r): W 2 < 1, A > 0,

r > p, (w, w') • (0,0)},

and we let re(y) be the smallest r > p for which the

y-orbit exits F ; re(y) = oo if the P-orbit stays in

F for all r > p. If the y-orbit exits F via A = 0,

we say that the y-orbit crashes. If O(r, y) is defined by O(r, y) = t an- l (w ' ( r , y) /w(r , y)), and - l y r <

O(p, y) < O, we set

,.Q (y ) = --(1/zr )IO (re(y ), y) -- O(p, y)];

~ ( y ) is called the rotation number (or nodal class

if (1.8) and (1.10) hold) of the y-orbit. The major result

in [12] is the following global existence theorem.

Theorem 1.2. Given any p > 0, there exists a se-

quence (Yn, fin, P) satisfying p + y~ ¢ 1 and (1.11) such that the corresponding solution A(r, Yn) of (1.3), (1.4) with these data lies in F for all r > p,

and I-2(yn) = n.

That is, for every value of the event horizon p >

0, there exists a sequence of initial values (Yn, fin) lying on the curve C (p) which yield regular black-hole

solutions having rotation number n. These solutions

have uniformly bounded masses, outside of the event

horizon [7].

Similarly, if p = 0, we proved in [8] the following

theorem.

Theorem 1.3. There exists a sequence )~n, 0 < ~-n < 2

such that the solution

A(r, ),n) = (A(r, Xn), w(r, )~n), w' (r, ~.n), rn)

of (1.3), (1.4) satisfying the initial conditions

a ( 0 , ~-n) = 1, w(0 , Xn) = 1,

w'(0, xn) = 0, w"(0, zn) = - z , , ,

£20~n) = -(1/zr)O(reO~n), kn),

where reO~n) is the smallest r > 0 for which the ~-n-

orbit exits/- ' , and

O(r, ~-n) = tan-I (w'(r,)~n)/W(r, ~n)),

with 0(0, ~-n) = 0.)

That is, there exists a sequence of"particle-like" so-

lutions having rotation number n, n --- 1,2 . . . . These

solutions have uniformly bounded masses [7].

2. L imi t ing masse s

The main results of this paper are the following:

A. Let An(r) = (wn(r), wrn(r), An(r), r) be a sequence of distinct solutions of the EYM equa-

tions (1.3), (1.4) defined for all r > 0. Then

(i)A limn~oo An(r) : AERN(r), for each r > 1, where AERN(r) is the extreme Reissner-

Nordstr6m (ERN) solution,

AERN(r) ---- (0, 0, ((r -- 1) /r) 2, r).

(ii)A The convergence is uniform on all inter-

vals [a, b], a > 1.

(iii)A The (ADM) masses, / zn (~) , converge to 2,

the (ADM) mass of the ERN solution, as n --+

oo. Here IZn(r) = r ( l - An(r)). (iv)A The metric coefficients (An(r), Tn-Z(r)) con-

verge uniformly to the extreme Reissner-

Nordstr6m metric coefficients, (((r - 1) /r) 2,

((r - 1)/r)2), on all intervals of the form [a, oo),a > 1.

B. Let 0 < O < 1, and let An(r) = (wn(r), w'n(r), An (r), r) be a sequence of distinct black-hole solu-

tions of the EYM equations (1.3), (1.4) with fixed

event horizon p. Then p

(i)B limn-~oo An(r) = AERN(r), for each r > p, where A p is the extreme Reissner- ERN

Nordstr6m solution,

A~RN(r) = (0, 0, ((r -- 1) /r) 2, r).

lies in F for all r > 0, and satisfies £2 (;~n) = n. (Here ~(~-n) is defined by

(ii)B The convergence is uniform on all intervals [a, b], a > p.

J.A. Smoller, A.G. Wasserman/Physica D 93 (1996) 123-136

(iii)B The (ADM) masses,/.£ n (OO), converge to p +

1/p, the (ADM) mass of the ERN solution P

AERN, as n --~ oo.

(iv)B The metric coefficients (An (r), Tn-2(r)) con-

verge uniformly to the extreme Reissner-

NordstrOm metric coefficients (((r - 1) /r) 2,

((r - 1) / r ) 2) on all intervals of the form

[a, oo),a > p. C. Let p > I, and let An (r) = (wn (r), w~n (r), An (r),

r) be a sequence of distinct black-hole solutions

of the EYM equations (1.3), (1.4) with fixed event

horizon p. Then

(i)c l i m n ~ An(r) = A~N(r), for each r > p. (ii)c The convergence is uniform on all inter-

vals [a, b], a > p.

(iii)c The (ADM) masses,/Zn ( ~ ) , converge to p +

1/p, the (ADM) mass of the RN solution A~N,

as n -+ ~ .

(iv)¢ The metric coefficients (An (r), Tn-2(r)) con-

verge uniformly to the Reissner-NordstrOm metric coefficients ( l - ( p + p - l ) / r + l / r 2, 1 - (p + p - l ) / r + l / r2) , on all intervals of the

form [a, c~), a > p.

The proofs of these main results are given in The-

orems 2.1, 2.3, and 2.6.

Theorem 2.1. Let p > 0, and suppose that An(r) = (wn(r), W'n(r), An(r), r) is a sequence of solutions (1.3), (1.4), defined for all r > p, where

pA(p) = 0. Assume moreover that the rotation num-

bers S2(An) ---> ~ as n ~ o0. Then

lim A n ( r ) = { AERN(r) f o r a l l r > l i f p < l , n~o~ ARN(r) for all r > p if p > 1,

(2.1)

127

I 2 - 1/r (extreme), /ZRN(r) I p + 1 / p - 1/r (non-extreme, p > l).

(2.2)

We thus have the following corollary.

Corollary 2.2. Under the hypotheses of Theorem 2. l,

the (ADM) mass functions /Zn(r) = r ( l - An(r)) satisfy

lim #n(r ) n - - ~ O 0

2 - 1/r = p + p - 1 _ l / r

for all r > 1, if p _< 1,

for all r > p, if p > 1.

(2.3)

The convergence is uniform on compact r-intervals.

Note that from (2.1),

lim IZn(r) < /ZRN(OO) ----- / 2 i f p ~ 1, n~o~ -- | p + p- t i f p > 1,

so that the mass functions satisfy limn~oo IZn(C~) _<

/ZRN (OO). In fact, these quantities are equal, as the next

result shows.

Theorem 2.3. For any p > 0, given any sequence

An(r) = (Wn(r), w~n(r), An(r), r), of black-hole so-

lutions, (particle-like if p = 0), of (1.3), (1.4), hav-

ing unbounded rotation numbers, then the limit of the

(ADM) masses of the solutions is equal to the (ADM)

mass of the associated Reissner-NordstrOm solution;

that is

lim #n(OO) = I 2 'if p < 1, n--,oo [ p + l / p i f p > 1.

where AERN(r) is the extreme Reissner-NordstrOm

solution, AERN(r) = [0, 0, ((r -- 1) /r) 2, r], and

ARN (r) is the Reissner-NordstrOm solution, ARN (r) = [0, O, 1 -- (p + p - l ) / r + l / r 2, r]. The convergence

is uniform on compact r-intervals.

Note that the Reissner-NordstrOm mass functions 4 for black-holes are

4 See footnote 3.

(Note that this together with (2.1) implies

lim lim /z,(r)= iim lim IZn(r).) /'---~" O0 ~---~ O0 ~---Y 0~) F--')" O0

We also have the following corollary.

Corollary 2.4. Fix p > 0, and let {An(r)} be a sequence of distinct solutions of (1.3), (1.4) defined for all r > p, where pA(p) = 0. Then (2.1) holds. In

particular, the mass functions satisfy (2.3).


The proof of Corol lary 2.4 follows from a result

in [5], where it was shown that there are at most a finite

number of solutions of (1.3), (1.4) in each nodal class.

This implies that the rotation numbers ~ ( A n ) ~ cx~

as n ~ c~, so that the hypotheses of Theorem 2.1 are

satisfied.

Corollary 2.5. Given any e > 0 and any sequence

of black-hole solutions of (1.3), (1.4), having event

hor izon p, there are at most a finite n u m b e r of solut ions

having (ADM) mass exterior to the black-hole greater

than to- l + e, if /9 > 1, or greater than (2 - to + e), if

p < l .

Proof. If p > 1, we have from Theorem 2.3,

l im /zn(c~) =/ZRN(~x~) = p + p - l ; n- - -~ O ~

hence

1 l im (#n ( ~ ) - Un (P)) = l im (/Zn (c~) - p) = - .

If p _< 1, we have again from Theorem 2.3,

l im /Zn (c~) =/Z~RN (~X~) = 2, ?/----~ (3~

so that

l im (/Zn(C~) - tZn(p)) = l im (#n(C~) - to) n - ~ - O ~ n----~ O ~

= 2 - - p []

Before giving the proofs, we need a pre l iminary

result, which is important in its own right, and also

provides a crucial step in proving Theorem 2.1.

Theorem 2.6. If A ( r ) = (w(r ) , w1(r), A( r ) , r ) , a <

r < b, is an orbit segment in F , then l imr'~a(O(b ) -

O(r)) < cx~.

and crashes at r = a (i.e., l imr~ a A(r ) = 0), then

limr"~ a (A w I2) (r) = 0.

Remarks . It was proved in [13, Proposi t ion 5.1] that

A w I2 is bounded on all orbit segments of (1.3), (1.4)

satisfying either black-hole or particle-like initial con-

ditions. The fact that A w 12 was bounded enabled us to

show that A w ' --+ 0 at crash. This was an important

ingredient in the papers [12,13]. Lem m a 2.7 shows

that l imr ,~a(Aw12)(r) = 0 at crash if an orbit has in-

finite rotation. This s tatement is true even without the

hypothesis of infinite rotation if a > 1, and either r s a

or r"~a, but we only prove it here in the case r"~a > 1,

and the orbit has infinite rotation.

Proo f o f Lemma 2.7. Set f = Aw'2; then f satisfies

the equat ion

r Z f ÷ ( 2 r f + ~ ) W '2 + 2W(1 -- W2)W 1 = 0. (2.4)

Since the orbit has infinite rotation, it follows that

0 = l im in f r~ a f ( r ) . Now suppose lira supr '~a f ( r ) =

~r > 0; we shall show that this leads to a contradict ion

(which will prove the lemma). Choose e, with cr >

e > 0; then there exists a sequence rn~a such that

f ( r n ) = ~r - e, f l ( r n ) > O, and Iw'( r , ) [ ~ cx~ (since

[w1(rn)[ = ~/ f ( r n ) / a ( r n ) = ~/ (a - e ) / a ( r n ) , and

A(rn) ~ 0). Now for r > a we have # ( r ) = r ( l -

A ( r ) ) , so /z(a) = a, and since #1 > 0 (see [3]), it

follows that for r > a , / z ( r ) > /z(a) = a > 1. Thus if

r > a ,

(1 - w2(r)) 2 1 1 • ( r ) = I z ( r ) > a - - > a - - > 0 ;

r r a

i.e.,

q~(r) > 0 i f r > a. (2.5)

We remark that this result is false in the "forward"

direction; i.e., there are solutions of (1.3), (1.4) for

which l imr lb (O(b ) -- O(r)) = --cx~; see [5,9].

The proof of Theorem 2.6 is broken up into two

cases: a > 1 and a < 1. The case a > 1 requires a

pre l iminary lemma; namely

L e m m a 2.7. If an orbit segment A (r ) = ( w ( r ) , w ' (r ) ,

A ( r ) , r ) , 1 < a < r < b, has infinite rotation,

Thus 2 r n f ( r n ) + ~(rn) > 2(or - e). Then from (2.4)

we get the contradict ion

0 = r 2 f ' ( r n ) + ( 2 r n f ( r n ) q- clg(rn))W'(rn) 2

q- 2 ( w w ' ) ( r n ) ( 1 - w2(rn))

> r 2 f ' ( r n ) + 2 (a - ~)w'(rn) 2

+ 2 ( w w ' ) ( r n ) ( l -- wZ(rn)) > 0

for large n, since Iwt(rn)l ~ ~ . []


W------C

, 4

W e

/ / / /

,4 '

W = l g

Fig. 1.

Proof o f Theorem 2.6. First, i f limr-~ a A(r) > 0, then

the solution extends to a - e, for some E > 0 and

hence O(r) is a cont inuous funct ion on a < r < b, so

it is bounded. Thus we may assume l imr~ a A(r) = O.

The proof is d ivided into two cases: a > 1 and a < 1.

We first assume a > 1.

We define the funct ion h(r) by

h(r) = (Aw'2)(r) + w2(2 - w2) /2r 2, (2.6)

where A, w, and w ' refer to the A-orb i t segment . Note

that h(r) > 0 for a < r < b, because A(r ) C F, and

hence A(r) > 0 and (w(r) , w ' (r ) ) :~ (0, 0). S ince

h'(r) - - 2 A w ' 4 dp w2(2 - w 2) - w '2 ( 2 . 7 )

r r r 3 '

(2.5) shows that

h'(r) < 0 i f r > a , (2.8)

and

h(b) > 0. (2.9)

We now show that the assumption that A(r ) has

infinite rotation leads to a contradict ion. Thus, un-

der this assumption, l imin f r ' ,aW2(r ) = 0, and

f rom L e m m a 2.4, l imr~a(Aw'Z)(r) = 0. Therefore

l im inf,..,., h(r) = 0, and thus for some rL, a < rl <

b, h(r j ) < h(b), and this contradicts (2.8). This

proves Theo rem 2.6 in the case a >_ 1.

We turn now to the case a < 1. This is d iv ided into

two subcases: a > 0 and a = 0. We begin with the

case a > O.

) W

Case 1: a > 0. In this case we first c la im that

l im SUpr~aClg(r ) > O. (2.10)

Thus, i f l im SUPr',a~(r) = - c 2 < O, then f rom (1.3),

which we write in the form

rA ' + 2w12A = ~ / r , (2.11)

we see that A ' = cI9/r 2 - 2 w ' 2 / 2 < ~ / r 2, so

l imsuPr \aA ' ( r ) < - c 2. Thus for r near a , r > a,

a(b) - A(r) = a ' (~) (b - r)

< lim sup A'(r) (b - a) < 0;

here ~ is in termediate to b and r. It fo l lows that

for these r , A ( r ) > A(b) > 0. On the other hand

l imr~ a A(r) = 0 < A(b), and this is a contradict ion.

Thus (2.10) holds.

N o w since a < l , a - 1/a = -2~r < 0. If we

cons ider dp(r, A, w) =-- r - rA - (1 - w2) / r , as an

abstract cont inuous funct ion o f three-variables, then

~ ( r , A, w) < r - (1 - w2)2/r , and as @(a, A,O) <

- 2 a , there exists an e > 0 such that

@ < - ~ r i f l w ] < s a n d l r - a ] < s . (2.12)

N o w cons ider the region a~ U M' where (cf. Fig. 1 )

a / = {(w, w ' ) : - s _< w _< 0, w ' > 0},

a / ' = {(w, w ' ) : 0 _< w < s, w ' < 0}.

We now cla im that

w~(r) is bounded in a~ tO ,~'. (*)


To see this note that (2.12) implies that in the region

s~ U M ~, 4~ < -o r if Ir - al < e. The claim ( . ) now

follows from [13, Lemma 5.13].

We next show that

The orbit A ( r ) cannot stay in the region Iwl < e

for all r satisfying 0 < r - a < e. (**)

To see this, suppose that Iw(r)[ < s for all r , 0 <

r - a < s. Now for small 6 > 0, we can write

a ( a + ½e) - a ( a + 3) = a ' ( ~ ) ( ½ e - 3),

for some intermediate point ~, ~ < ~ - a < ½e. Letting

~ 0 shows A'(~) >_ 0 because limr,~ a A ( r ) = O.

Then from (2.11) we see that ~2A'(~) < q~(~). If

Iw(¢)l < e, then 4~(~) < - a so 0 < ~2A'(~) <

4,(~) < 0. This contradiction establishes (**).

Now as w' is bounded in the region ,~ U ~ ' , say

Iw'l _< M, and the orbit has infinite rotation, every

time the orbit goes through ~ /U ~l', the "time", Ar ,

it stays in ~t tO ~/' satisfies A r > e / M . Thus, as the

orbit enters ~/U ~ ' infinitely many times, it spends an

infinite t ime in ~/U ~ ' , thereby violating b - a < oo.

This contradiction completes the proof of Theorem 2.6

in the case where a > 0.

l i m s u p w ( r ) = 1 and l i m i n f w ( r ) > - 1 , (2.15) r"~0 r"~0

l i m s u p w ( r ) < 1 and l i m i n f w ( r ) = - 1 , (2.16) r "~ 0 r"~0

l i m s u p w ( r ) = 1 and l i m i n f w ( r ) = - 1 . (2.17) r ' -0 r~'0

Since (2.15) and (2.16) have similar proofs, we

shall only consider cases (2.15) and (2.17). We be-

with (2.15). Let e < 1 be chosen so small that gin

w ( r ) > - 1 + e for r near 0. Since A ( r ) >_ 0, ~ ( r ) <

r - u Z ( r ) / r , u = (1 - w2), so in the region Se =

{ w : - I + e _< w _< 1 - e}, we can find a constant

c < 0 such that

q)(r) < - c . (2.18)

Now claim that in the region Se,

( A w ' 2 ) ( r ) < 2 u 2 ( r ) / r 2.

To see this, let f = A w ~2, and set

g( r ) = r2 f (r) - 2u2(r) .

(2.19)

Since (2.13) holds, there exist null sequences rn < Sn,

such that w'(rn) = 0 = w~(Sn), and w ' ( r ) > 0 i f rn <

r < Sn. Thus we have

Case 2: a = 0. Thus suppose in this case that

lira (O(b) - O(r)) = e~; (2.13) r----~0

we will show that this leads to a contradiction.

Now if both of the following hold:

l i m s u p w ( r ) < 1 and l i m i n f w ( r ) > - 1 , (2.14) r"~0 r~'0

then there is an r / > 0 such that

lim inf(1 - w2(r)) > r/. r"~0

Then from (1.3), we obtain

r A ' ( r ) < 1 - (1 - w 2 ) 2 / r 2 < 1 - rl2/r 2,

and this implies that limr,~ 0 A ( r ) > 1. Such behavior

violates the assumption limr-~ 0 A ( r ) = 0. Thus (2.14)

cannot hold, so we are left with the following three

cases:

g(rn) < O. (2.20)

We shall show that for rn < r < Sn, and n large

g( r ) < 0 if w ( r ) ~ Se. (2.21)

Thus fix n, and let F be the first zero of g such that

Sn > F > rn, and w(F) is in the region S~. Then from

(2.4), we find,

gt(F) = (u /F) (6 -- 3w I2) -- r w ~2 + 6 u w w t. (2.22)

Now in Se, we have a bound on u from below, say

u > r / > 0. Thus for F near zero (i.e., n large), uZ/F >

r//F > 2, so f ( F ) = (Aw'Z)(F) = 2u2/F > 4. It

follows that wr2(F) > 4. Then from (2.22), we have

at F,

g'(F) < (u2/?)(6 - 3w '2) + 6w'

< ( u 2 / F ) ( - 3 t o '2) + 6w '

< - 6 w I2 + 6w' = 6 w ' ( l -- w') < O.

J.A. Smol ler , A . G . W a s s e r m a n / P h y s i c a D 93 ( 1 9 9 6 ) 1 2 3 - 1 3 6 131

Thus we have shown that g ' (7) < 0 if g(7) = 0, and

w(7) ~ Se (provided that n is large). This implies

(2.21), and hence in Se, (2.19) holds so

A w t2 < 2u2 / r 2. (2.23)

Next, if w(r ) > l / c and w ( r ) belongs to Se, then

(1.4) gives

el ) to t - - UtO C W t - - UlI) CtO j - - 1 - _> > - - > 0 ,

wt~ ( r ) r2 A r2 A r2 A

where we have used (2.18). Now on the interval

0 < w < ¼, w ~ cannot be bounded; otherwise A r =

A w / w ' ( ~ ) > constant, and (2.13) would give the

contradict ion b = e~. Thus w' > 1/c somewhere

in this interval, say at @. Then at if), w ~' > 0 and

w' cont inues to exceed 1/c, so w 't > 0. Thus in

the region S~, for w > tb, w ~ is increasing, so w t is

increasing from w - ¼ to w = 1 - e. Furthermore,

w' cannot be bounded in any region on which the

orbit goes through infinitely often; otherwise we can

again bound A r from below by a positive constant. In

part icular w'(r l /4) ~ oc as the orbit rotates (where

w(r l /4) = ¼) and so w' must tend to infinity in the

region ¼ < w < 1 - e. Thus we may assume that r

is so close to zero that in the region } < w < 1 - e,

- u w > ~ w ' . (2.24)

Now from (1.4), if ¼ < w _< 1 - e, (2.22) and (2.24)

give

But as w~(rt/4) ~ ~ , we see that we have a contra-

diction. This contradict ion shows that (2.15) cannot

occur. (As we have remarked above, a similar proof

shows that (2.16) cannot occur.)

Now suppose that (2.17) holds. We begin by proving

that, i f w ' > 0 , a n d - I < w _ < 0 ,

(Aw'2) ( r ) < u 2 / r 2 + 2; (2.25)

(cf. (2.19)). To see this, again write f = A w '2, and

let us change the g from the proof of (2.21) to

~(r) = r 2 f - u 2 - 2r 2.

Now when w' = 0 , ~ < 0. As in the last case, we

show that ~(F) = 0 implies ~ '(~) < 0, so ~ is always

negative. We have, in the region w' > 0, - 1 < w < 0,

~ ' (7) = 2 F f - ( 2 F f + ~ ) t o t 2 - 2 u w w '

+ 4 u w w t - 47

= 2 7 f ( 1 - w ' 2 ) - 7 w '2 + r A w '2

-t- ( u 2 / r ) w '2 q - 2uww' - 4 7

= T w ' 2 ( 2 A - 2 A w '2 - I + A + u2 /F 2)

+ 2 u w w ~ - 47

= 7wtZ(3A - 1 - 2u2 /~ 2 - 4 + ?i2/72)

+ 2 u w w ' - 47

= T w ' Z ( 3 A - 5 - - ~t-2/72) + 2UWW' -- 47 < 0,

• w' - u w - 0 . 9 q ~ w t - 0 . 9 ~ w t2 It 1/)1 1/) - - ~> - -

r2 A - r2 A r2 A w t2

- 0 . 9 q ~ w I2 > W t" - 2u 2

But in this region ¢' < - c (by (2.18)), and u 2 < 1.

Thus we obtain

w t' > k w t3 if 1 < w < 1 - e.

Then integrat ing the inequal i ty w " / w t2 > k w ' from

rl/4 to r l -E ( w ( r i - E ) = 1 - e), we get

1 1 1

wt(r l /4) -- wt(r l /4) w t ( r l - e )

> k ( l - e - 1 ) _ ~ > 0 .

and this proves (2.25).

Now i f 0 < E < ¼, let T~ = { ( w , w ' ) : - l + c <

w < 0, w ' > 0}. In Te, we again have a bound on 4~

of the form (2.18). Moreover, as above, w' cannot be

bounded on - 1 + e _< w < - 3, and if w' > 1/c, w" >

0. We can thus assume that wt(r_3/4) ~ cx3 as the 3 orbit rotates through this region; (w(r -3 /4 ) ---- - ~ ) .

Furthermore, if r is sufficiently small, a bound of the 3 form (2.24) holds in Te. Then from (1.4), on - ~ _<

w < O . w ' > O ,

- -0 .9@w' --0.94~w '3 - 0 . 9 4 ~ w ~3 l/3/f > - - >

r2A r 2 A w r2 u 2 + r 2

- 0 . 9 4 ~ w ~ > >_ k w p3, - - 7 2


where k > 0 is a constant. Thus tOrt/tO t2 > kw' , and

integrating this from r_3/4 to ro (here w(ro) = 0), we

get

1 , 1 > - - + > k ,

wt(r-3/4) -- wt(ro) w'(r-3/4) --

and as w~(r_3/4) ~ cx~, we obtain the desired con-

tradiction. This proves that (2.17) cannot occur, and

completes the proof of Theorem 2.6. []

Proof o f Theorem 2.1. Fix p >_ 0, and R > max(l , p).

Now from [13, Lemma 4.2], there exists an r / = r/(R)

such that for all n,

1 > An(R) > rl.

Now consider the sequence of points

!

A n ( R ) = (wn(R) ,Wn(R) ,An(R) ,R) , n = 1 , 2 ; . . .

From [12, Proposition 2.5], there exists a r > 0, such

that IW'n(R)[ < r, for all n. Since w2(R) < 1, the

sequence { An (R) } contains a convergent subsequence

An~(R) ~ P =-- ( ~ , ~ , A, R),

where ~ 2 < 1, [~t[ < r, and A > r/.

We now show that

e = (0, 0, A, R). (2.26)

To see this, we first show that P ¢ F . Suppose that

P E F , and let A(r ) = (w(r), wl(r), A(r), r) be the

unique orbit through P. It then follows as in the proof

of Theorem 4.1 in [12], that the orbit A(r ) can be

continued back for r > p and has unbounded rotation;

this contradicts Theorem 2.6. Thus P ¢ F , i.e., either P = (4-1,0, A, R) or P = (0, 0, A, R).

Now in order to prove (2.26), we shall show that

numbers m, 3, and w0, 0 < w0 < 1 (all independent of n) such that if w 2 < w(r) 2 < 1, then

a( r ) >_ ~ and Iw'(r)[ _< m. (2.29)

Thus, since the An-orbits are connecting orbits, we

may assume, without loss of generality, that for Ank -

orbit, we have

?

lira (Wnk(r), Wnk(r), Ank(r)) = (1, 0, 1). r ---~ o o

Let rk denote the largest r for which Wnk (rk) = wO.

Now consider the points

Qk = (wo, Wlnk (rl~), Ank (rk), rk).

Since rk < R for each k, it follows from (2.29) that

the sequence Qk has a convergent subsequence which

converges to a point

Q = (w0, w6, Ao, r0),

where [w6l _< m, and A0 _> 3; thus Q 6 F . Then by

considering the backwards orbit through Q (defined

for r < r0), we obtain the same contradiction to The-

orem 2.6 as before; i.e., as above, the backwards orbit

through Q, defined for r < ro, would have infinite

rotation, and crashes at r = p. It follows that (2.26)

holds; thus

Ank (R) ~ (0, O, A, R).

We next calculate A. To do this, we note that as

A > 0, solutions through (0, 0, A, R) are unique. But

the Reissner-Nordstr6m solution

w(r) -= 0, A(r) = 1 - c / r -4- 1/r 2 (2.30)

satisfies (1.3) and (1.4). Hence it is the unique solu-

tion through (0, 0, A, R). Here c is determined by the equation 1 - c / R + 1 /R 2 = -A.

We shall now verify some claims about this solution.

D

P ~ ( - t - I ,O ,A ,R ) . (2.27) Claim1. c 2 ~ 4.

Indeed, suppose that

e = (1,0, A, R) (2.28)

(the proof in the symmetric case is similar). We recall from [8] that if w 2 is near 1 then w' is bounded, and

A is bounded away from zero. That is, there exist

Proof If C 2 < 4, then A(r) > 0 for all r, and we

can find an rl, 0 < rl < R, such that A(rl) > 1; cf. Fig. 2. Thus since lim Ank (R) = A(R) , it follows

by "continuous dependence on initial conditions", that

Ank(rl) --+ A(rl) . Since Ank(rl) ~ A(rl) > 1, we get a contradiction. []


1 .....................

{

r I R

A(r')

Proof If A(r) has a single root, then ¢2 = 4 so by

Claim 2, c = 2 and thus A(r) = ( ( r - l ) / r )2 ; cf.

Fig. 3. In this case, Ank(R) --+ A ( R ) so Ank(P)

A(p) . This implies Ank(P) --~ A(p) , which is impos-

sible since Ank (P) = 0 but A(p) > O. []

) r Thus A(r) has two roots, 0 and 0 -~, where 0 >

l > 0-1', cf. Fig. 4.

Fig. 2. c 2 < 4. Claim 4. 0 = p.

1 .....................

l p R

) r"

Fig. 3. c 2 = 4 .

Remark. To invoke "continuous dependence on initial

conditions", we must have A(r) > 0 on rl < r < R.

Claim 2. c > O.

Proof Otherwise A(r) > 0 if r > 0 and we get a

similar contradiction as in Claim 1. []

Proof I f p > 0 , t h e n A ( r ) > 0 o n p < r < R. Then

by "continuous dependence", lim Ank (/9) = A(p), and

this gives the same contradiction as in Claim 3. Thus

p < 0. Suppose p < 0; we will show that this is im-

possible. Thus, if p < 0, then for each r 6 [0, R],

there is an r/(r) > 0 such that An~(r) > r/(r); this fol-

lows from [9, Lemma 4.2], where, it is easy to check

that o(r) is a continuous function of r. Thus, by com-

pactness, we can find an ~ > 0, independent of k,

such that Ank(r) > -~ if 0 < r < R. Since A(O) =

O, A(O + e) < ~ for small e > 0. By "continuous de-

pendence" on the interval 0 ÷ e < r < R, we obtain

the contradiction

< limAnk(O + e ) = A(O + e ) < ~;

thus p = 0. []

We now consider separately the cases p > 1 and p < l .

Case 1. p > 1.

Claim 3. A(r) = 0 has two roots.

It follows from Claim 4, that if p > 1, c = p + 1//9,

and so c does not depend on the subsequence chosen.

This implies that the entire sequence {An(R)} con-

verges to (0, 0, A, R), where A is uniquely determined

by the equation A = 1 - (p ÷ p I ) / R ÷ R -2. Thus

A(r) = l - ( p + 1 / p ) / r + l / r 2 s o i f r > p, An(r) --+

1 ..............

1 r R 1

A(r)

) 1"

Fig. 4.


A(r) , or 1 - IZn(r)/r -+ 1 - (p + 1 / p ) / r + 1/r 2, so

lZn(r) --+ (p + I / p ) - 1 /r . This proves (4.1) in the

case p > 1. We now consider

Case 2. p < 1.

Claim 5. A(r ) = 0 has one root; i.e., c = 2.

Proof. Suppose A(r) had two roots 0 - l , 0 . 0 > 1 >

0 -1 ; cf. Fig. 4. Then R > 0 is impossible because as

above, we can choose small E > 0, and using 0 + E

we obtain contradiction as in Claim 4; thus R < 0.

Now as p < 1, and we are considering only those r

for which r > p, we have A ( R ) < 0, but Ank(R) >

0 > 0; this contradiction implies that R < 0 is also

impossible. []

Thus if p _< 1, A(r) = ((r - 1) / r )2; as in Case 1,

this representation shows that the entire sequence

{An(R)} converges to (0, 0, A, R). Now if r > 1,

An(r) -+ A(r ) so 1 - I-tn(r)/r ---> 1 - 2 / r + 1/r 2, or

tXn(r) ---> 2 - 1 /r . This proves (2.1) in the case p _%< 1.

To complete the proof of Theorem 2.1, we must

show that the convergence in (2.1) is uniform

on compact r-intervals. Thus let 1 = [ a ,b ] C

(max(p, 1), c¢). Then using the fact that A(r) > 0 on

I , if the An-orbits converge to the A-orbi t at a point

in I , it follows again by "continuous dependence

on initial conditions", that. the An-orbits converge

uniformly to the A-orbi t on the entire interval. In

particular An(r) ---> A(r ) uniformly for r c I . Thus

#n ( r ) ~ # ( r ) uniformly on I . This completes the

proof of Theorem 2.1. []

We can now turn to the proof of Theorem 2.3. We

begin with the following lemma.

Lemma 2.8. There exist positive numbers M and R,

both independent of n, such that

[ r W ' n ( r ) l < M i f r > R . (2.31)

Proof From (1.4), we obtain

( ~ ) w( l - w 2 ) (rw' ) ' = w' 1 - - ~ r a '

so at a critical point for rw' ,

w(1 - W 2) rw ' -- (2.32)

A(1 - ~ / r A )

Now from a result in [7] (done for particle-like so-

lutions, but easily extended to black-hole solutions),

there exists a number M1 > 0 (independent of n, but

depending on p), such that for every r > 0, #n (r) _<

M1; then ~n(r ) < M1, for every r > 0. Thus we can

find a 3 > 0 such that

An(r) = 1 - t xn(r ) / r >_ 1 - M 1 / r > 6

if r is sufficiently large, and thus for such r ,

qbn/rAn < M l / r 6 ,

s o

1 -- ~ n / r A n > 1 - M1/r8 > 1,

i f r is sufficiently large, say r > R > max( l , p). Thus

at a critical point for rw' , (2.31) implies

, wn( l - w n) 2 rWn(r) < < - i f r > R. (2.33)

I An(r )

Now from [12, Lemma 4.1], r w ' ( r ) ~ 0 as r

o0. Moreover, since (wn(R) , w'n(R)) --+ (0, 0) (proof

of Theorem 2.1) we see that we can find a constant

M2 > 0 such that I R w ' ( R ) I < M2 for all n. This

together with (2.33) shows that (2.31) holds with M =

max(M2, 2/3). []

We can now complete the proof of Theorem 2.3.

For some constant c > 0, and R > R we have

O0 - / # n ( O 0 ) -- l z n ( R ) = # t n ( r ) d r

-fi

(1 - - 1/)2) 2 = ( 2 A n w ~ + rT ) d r

o o

dr c <_c

Also ~RN(R) -- ~,RN(e~) = - -1 /R . Now let e > 0

be given. Choose R > R such that (1 + c ) / R < e/2 .


For this R, choose N so large that n > N implies that

I#n(R) - #RN(R)[ < e/2; this can be done because

of Theorem 2.1. Then n > N implies

I/zn (ec) - / Z R N ( O O ) < [/Zn ((3~) - - ~ n ( ' e ) [

+ I#n(e) -/ZRN(R)I

+ I#RN(R) -- #RN(C~)I C e 1

< = + + : < e . R 2 R

This completes the proof of Theorem 2.3. []

lZn(r) >/Zn(rl) > /ZRN(rl) -- e /2

> #RN(OO) -- ~" > /ZRN(r) -- 6.

Thus, if r > rl and n > max(Ni, N2),

[/Zn(r) --/ZRN(r)l < ~,

so the functions {#n(r)} converge uniformly

[rl, oe). Then if r >_ rl and n >_ max(Ni, N2),

[An(r) - ARN(r)I _< (l/r)llzn(r) -- #RN(r)[

<_ (1/rl)[#~(r) --/ZRN(r)l,

o n

To complete the proofs of the main results A, B, and

C, above, we must show that the metric coefficients

An and Tn converge to AERN if 0 < p < 1, or to

A~N if p > 1, uniformly on an interval [a, ~ ) , where

a > max(p, 1).

We begin with the Ans. To prove the uniform con-

vergence of the Ans, we shall first show that the mass

functions #n (r) = r (I - An (r)) converge uniformly.

Let e > 0 be given, and let /ZRN(OO) denote either

the mass of the ERN solution or the RN solution with

event horizon p.

From (iii)A or (iii)B or (iii)c, we can find N1 such

that for n > NI

]An(OO ) - - /ZRN(OO ) < e/2.

Since/z ' > 0~/zn (r) < / zn (~x~), so

and this implies that the Ans converge uniformly on

[rl, ~x~). Now in [12] we have shown that for each n,

Tn-2(r) : An (r) exp(2 Qn (r)), (2.36)

where Q~ = 2w'n2/r. Thus, in order to show that Tn -2

coverges to ARN uniformly on an interval [a, cx~), a >

max(p, 1), it suffices to show that {Qn (r)} converges

uniformly to 0 on such an interval.

Now we normalize T,, so that T,, (c~) = 1, for each

n, and as An(cX~) : !, (2.36) requires Qn(oo) = 0.

Then

Qn(r) = - ~l 2w'2(s)- ds. J S F

But using (2.31), we see that i f r > max(rl , R) = r2,

we have

#n(r) < #RN(OC) + e/2.

We can find rl > max(p, 1), such that

/ZRN(OO) --/ZRN(rl) < e/2, (2.34)

because ~RN is continuous. Then if r > rl ,

/ZRN(O0) -- /ZRN(r) < /ZRN(OO) -- /ZRN(rl) < e/2,

SO if r _> r I and n > Ni,

oo

I Qn(r)l _< j s3 ds = = r---T-,

F

SO Qn(r) ~ O, uniformly on an interval of the form

[a, c~). This completes the proof.

Finally, we remark that there is no uniform conver-

gence of the connection coefficients wn on any inter-

val [a, c~), since for each n, w2(r) --+ 1 as r ~ cx~,

while WRN(r) = 0 for every r > 0.

IZn(r) < /ZRN((X0 %- e /2 < /ZRN(r) -I- E. (2.35)

On the other hand, because of Theorem 2.1, we can

find N2 such that n > N2 implies

/Zn(rl) > /ZRN(rl) -- ~/2.

Thus, if n > N2 and r > r l , using (2.34), we get

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Limiting masses of solutions of the Einstein-Yang/Mills equations