Transcription

1 Max-Planck-Institut fu Mathematik in den Natuwissenschaften Leizig Absence of stationay, sheically symmetic black hole solutions fo Einstein-Diac-Yang/Mills equations with angula momentum by Felix Finste, Joel Smolle, and Shing-Tung Yau Peint no.: 36 2

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3 Absence of Stationay, Sheically Symmetic Black Hole Solutions fo Einstein-Diac-Yang/Mills Equations with Angula Momentum Felix Finste, Joel Smolle,andShing-Tung Yau y May 2 Abstact We study a stationay, sheically symmetic system of (2j + ) massive Diac aticles, each having angula momentum j, j = 2 :::,ina classical gavitational and SU(2) Yang-Mills eld. We show that fo any black hole solution of the associated Einstein-Diac-Yang/Mills equations, the sinos must vanish identically outside of the event hoizon. Intoduction Recently the Einstein-Diac-Yang/Mills (EDYM) equations wee studied fo a static, sheically symmetic system of a Diac aticle inteacting with both a gavitational eld and an SU(2) Yang-Mills eld [, 2]. In these aes, the Diac aticle had no angula momentum, and we could make a consistent ansatz fo the Diac wave function involving two eal sino functions. In the esent ae, we allow the Diac aticles to have non-zeo angula momentum j, j = 2 :::. Simila to [3], we can build u a sheically symmetic system out of (2j + ) such Diac aticles. In this case howeve, a eduction to eal 2-sinos is no longe ossible, but we can obtain a consistent ansatz involving fou eal sino functions. We show that the only black hole solutions of ou 4-sino EDYM equations ae those fo which the sinos vanish identically outside the black hole thus these EDYM equations admit only the Batnik-McKinnon (BM) black hole solutions of the SU(2) Einstein- Yang/Mills equations [4, 5]. This esult extends ou wok in [2] to the case with angula momentum it again means hysically that the Diac aticles must eithe ente the black hole o escae to innity. This genealization comes as a suise because if one thinks of the classical limit, then classical oint aticles with angula momentum can \otate aound" the black holeonastableobit. Ou esult thus shows that the non-existence of black hole solutions is actually a quantum mechanical eect. In Section 2, we deive the stationay, sheically symmetic SU(2) EDYM equations with non-zeo angula momentum. By assuming the BM ansatz fo the YM otential (the vanishing of the electic comonent), the esulting system consists of 4 st-ode equations fo the sinos, two st-ode Einstein equations, and a second-ode equation Reseach suoted in at by the NSF, Gant No. DMS-G y Reseach suoted in at by the NSF, Gant No

4 fo the YM otential. This EDYM system is much moe comlicated than the system consideed in [2], and in ode to make ossible a igoous mathematical analysis of the equations, we often assume (as in [3]) a owe ansatz fo the metic functions and the YM otential. Ou analysis combines both geometical and analytic techniques. 2 Deivation of the EDYM Equations We begin with the seaation of vaiables fo the Diac equation in a static, sheically symmetic EYM backgound. As in [], we choose the line element and the YM otential A in the fom ds 2 = T () 2 dt2 ; A() d2 ; 2 d# 2 ; 2 sin 2 #d' 2 (2.) A = w() d# + (cos # 3 + w() sin # 2 ) d' (2.2) with two metic functions A, T, and the YM otential w. The Diac oeato was comuted in [, Section 2] to be G = it t + i A@ + i ( A ; ) ; i T A + i # + i ' 2 T + 2i (w ; ) (~~ ; ) : (2.3) This Diac oeato acts on 8-comonent wave functions, which as in [] we denote by ( ua ) u a= 2, whee ae the two sin oientations, u coesonds to the ue and lowe comonents of the Diac sino (usually called the \lage" and \small" comonents, esectively), and a is the YM index. As exlained in [], the Diac oeato (2.3) commutes with the \total angula momentum oeatos" ~J = ~ L + ~ S + ~ (2.4) whee L ~ is angula momentum, S ~ the sin oeato, and ~ the standad basis of su(2) YM. Thus the Diac oeato is invaiant on the eigensaces of total angula momentum, and we can seaate out the angula deendence by esticting the Diac oeato to suitable eigensaces of the oeatos J. ~ Since (2.4) can be egaded as the addition of angula momentum and two sins 2, the eigenvalues of J ~ ae integes. In [], the Diac equation was consideed on the kenel of the oeato J 2 this leads to the two-comonent Diac equation [, (2.23),(2.24)]. Hee we want to study the eect of angula momentum and shall thus concentate on the eigensaces of J 2 with eigenvalues j(j +), j = 2 :::. Since the eigenvalues of J z meely descibe the oientation of the wave function in sace, it is futhemoe sucient to estict attention to the eigensace of J z coesonding to the highest ossible eigenvalue. Thus we shall conside the Diac equation on the wave functions with J 2 = j(j +) and J z = j (j = 2 :::): (2.5) Since (2.5) involves only angula oeatos, it is convenient to analyze these equations on sinos a (# ') on S 2. Let us st detemine the dimension of the sace sanned by thevectos satisfying (2.5). Using the well-known decomosition of two sins 2 into a singlet and a tilet, we choose a sino basis st with s = and;s t s satisfying ( ~ S + ~) 2 st = s(s +) st (S z + z ) st = t st : 2

5 The sheical hamonics (Y lk ) l ;lkl, on the othe hand, ae a basis of L 2 (S 2 ). Using the ules fo the addition of angula momentum[6], the wave functions satisfying (2.5) must be linea combinations of the following vectos, Y jj (2.6) Y j; j; (2.7) Y jj; Y jj (2.8) Y j+ j; Y j+ j Y j+ j+ ; : (2.9) These vectos all satisfy the second equation in (2.5), but they ae not necessaily eigenfunctions of J 2. We now use the fact that a vecto 6= satisfying the equation J z =j is an eigenstate of J 2 with eigenvalue j(j +) if and only if it is in the kenel of the oeato J + = J x + ij y. Thus the dimension of the eigensace (2.5) coincides with the dimension of the kenel of J +, esticted to the sace sanned by the vectos (2.6)- (2.9). A simle calculation shows that this dimension is fou (fo examle, we have J + (Y jj; ) = Y jj = J + (Y jj ), and thus J + alied to the vectos (2.8) has a one-dimensional kenel). We next constuct a convenient basis fo the angula functions satisfying (2.5). We denote the vecto (2.6) by. It is uniquely chaacteized by the conditions L 2 = j(j +) L z = j ( ~ S + ~) = k k S 2 = : We fom the emaining thee basis vectos by multilying with sheically symmetic combinations of the sin and angula momentum oeatos, namely = 2S = ;2 2 = 2 c (~ S ~ L) = ; 2 c (~ ~ L) 3 = 4 c S ( ~ S ~ L) = ; 4 c (~ ~ L) whee c = q j(j +) 6= : Since the oeatos S,, and ( ~ S ~ L)commute with ~ J, it is clea that the vectos ::: 3 satisfy (2.5). Futhemoe, using the standad commutation elations between the oeatos ~ L, ~x, and ~ S [6], we obtain the elations ( ~ S ~ L) 2 = 4 L2 + i 2 jkl S l L j L k = 4 L2 ; 4 jkl S l jkm L m = 4 L2 ; 2 ~ S ~ L ( ~ S~) 2 = 4 2 ; 2 ~ S~ = 3 6 ; 2 ~ S~ 2S = ;2 = 2S = ;2 = 2S 2 = 2 2 = 3 2S 3 = 2 3 = 2 (~S~) = ;S 2 = ; 3 4 3

6 ( ~ S~) = 2S k k S = ;2S k (~x ~ S) S k = 2S 2 S ; S k x k = 2 S = 4 ( ~ S~) 2 = 2 c (~ S~)( ~ S ~ L) = c (~ L~) ; 2 c (~ S ~ L)( ~ S~) = ; c (~ S ~ L) + 3 2c (~ S ~ L) = 4 2 ( ~ S~) 3 = 4 c (~ S~) S ( ~ S~) = 2 c ( ~ S ~ L) ; 4 c S ( ~ S~) 2 = 2 3 ; 2 S 2 = 4 3 ( S ~ L) ~ = c 2 2 ( S ~ L) ~ = 2( S ~ L) ~ S = 2 S j [L j S ] + 2 fs j S g L j ; 2 S ( S ~ L) ~ = ;2iS j jkl x k S l ; c 2 3 = ; jkl x j klm S m ; c 2 3 = ; ; c 2 3 ( S ~ L) ~ 2 = 2 c (~ S L) ~ 2 = 2 c 4 L2 ; S 2 ~ L ~ = c 2 ; 2 2 ( ~ S ~ L) 3 = 4 c (~ S ~ L) S ( ~ S ~ L) = ; 2 c S ( ~ S ~ L) ; cs and thus = ; 2 3 ; c 2 2( ~ S~ ; S ) = ; 2 2( ~ S~ ; S ) = 2 2( ~ S~ ; S ) 2 = 2( ~ S~ ; S ) 3 = : Using these elations, it is easy to veify that the vectos ::: 3 ae othonomal on L 2 (S 2 ). We takefothewave function the ansatz T () ua (t # ') = e ;i!t (() a (# ') u + () a 2 (# ') u + i() a (# ') u 2 + i() a 3 (# ') u 2 ) (2.) with eal functions,,, and, whee!> is the enegy of the Diac aticle. This ansatz gives a consistent set of ODEs, and the Diac equation educes to the following system of ODEs fo the fou-comonent wave function := ( ), w c ;!T ; m A!T ; m ; w ; c = c : (2.) B ;!T ; ; c C A!T ; m 4

7 Hee m is the est mass of the Diac aticle, which we assume to be ositive (m > ). Substituting the ansatz (2.), (2.2), and (2.) into the Einstein and YM equations [], we get the following system of ODEs, A = ; A ; e 2 ( ; w 2 ) 2 2 2A T T = A ; + ( ; w 2 ) 2 e 2 2 +T ; 2!T 2 ( ) ; 2Aw2 e 2 (2.2) ; 2!T 2 ( ) ; 2Aw2 e 2 2m ( 2 ; ; 2 )+ 4c ( + ) + 4w 2 Aw = ;w( ; w 2 ) + e 2 T ; 2 2 A w + 2 AT w T (2.3) : (2.4) Hee (2.2) and (2.3) ae the Einstein equations, and (2.4) is the YM equation. Notice that the YM equation does not deend on and moeove the lowe two ows in the Diac equation (2.) ae indeendent of w. This means that the Diac aticles coule to the YM eld only via the sino functions and. Indeed, a main diculty hee as comaed to the two-sino oblem [2] will be to contol the behavio of and. Fo late use, we also give the equations fo the following comosite functions, 2 (Aw ) = ;w( ; w 2 ) + e 2 T + (AT 2 ) 2 2 T 2 w (2.5) (AT 2 ) = ; 4! T 4 ( ) ; 4AT2 w 2 e 2 +T 3 2m ( 2 ; ; 2 )+ 4c ( + ) + 4w : (2.6) Also, it is quite emakable and will be useful late that fo! =, the squaed Diac equation slits into seaate equations fo ( ) and ( ) namely fom (2.), A@ (! A@ ) = m 2 + c A B ; w 2 ; c 2 ; ; w c 2 2 c 2 ; c 3 Non-Existence Results C A + w 2 wc 2 2 w 2 wc 2 2 wc 2 wc 2 3 C7 A5 : (2.7) As in [2], we conside the situation whee = > is the event hoizon of a black hole, i.e. A() =, and A() > if>. We again make (cf. []) suitable assumtions on the egulaity of the event hoizon: q (A I ) The volume element j det g ij j = j sin #j 2 A ; T ;2 is smooth and non-zeo on the hoizon i.e. T ;2 A ; T 2 A 2 C ([ )) : (A II ) The stength of the Yang-Mills eld F ij is given by T(F ij F ij ) = 2Aw2 2 + ( ; w2 ) 2 4 : 5

8 We assume that it is bounded nea the hoizon i.e. w and Aw 2 ae bounded fo <<+ ". (3.) Futhemoe, the sinos should be nomalizable outside and away fom the event hoizon, i.e. Z jj 2 T A < fo evey ">: (3.2) +" Finally, we assume that the metic functions and the YM otential satisfy a owe ansatz nea the event hoizon. Moe ecisely, setting u ; we assume the ansatz A() = A u s + o(u s ) (3.3) w ; w = w u + o(u ) (3.4) with eal coecients A 6=andw,owes s > and w = lim & w(). Hee and in what follows, f(u) = o(u ) means that 9 > with lim su ju ;; f(u)j < : & Also, we shall always assume that the deivatives of a function in o(u )have the natual decay oeties moe ecisely, f(u) = o(u ) imlies that f (n) (u) = o(u ;n ) : Accoding to A I, (3.3) yields that T also satises a owe law, moe ecisely Ou main esult is the following. T () u ; s 2 + o(u ; s 2 ) : (3.5) Theoem 3. Unde the above assumtions, the only black hole solutions of the EDYM equations (2.){(2.4) ae eithe the Batnik-McKinnon black hole solutions of the EYM equations, o s = 4 and = 3 3 : (3.6) In (3.6), the so-called excetional case, the sinos behave nea the hoizon like ()() u 3 + o(u 3 ) < ()() < : (3.7) Ou method fo the oof of this theoem is to assume a black hole solution with 6, and to show that this imlies (3.6) and (3.7). The oof, which is slit u into seveal ats, is given in Sections 4{7. In Section 8, we will analyze the excetional case. It is shown numeically that the ansatz (3.6),(3.7) does not yield global solutions of the EDYM equations. Fom this we conclude that fo all black hole solutions of ou EDYM system, the Diac sinos must vanish identically outside of the event hoizon. 6

9 4 Poof that! = Let us assume that thee is a solution of the EDYM equations whee the sinos ae not identically zeo, 6. In this section we will show that then! must be zeo. Fist we shall ove that the nom of the sinos jj is bounded fom above and below nea the event hoizon. We distinguish between the two cases whee A ; 2 is o is not integable nea the event hoizon. Lemma 4. If A ; 2 is integable nea the event hoizon =, then thee ae ositive constant c and " such that c j()j2 c if <<+ ". (4.) Poof: Witing (2.) as A = M, we have d A 2 d jj2 = 2 t (M + M ) = w (2 + 2 ) ; 2m ( + ) + 2c ( ; ) w (2 ; 2 ) + m ( ) + 2c ( ; ) c jj 2 : (4.2) Hee the constant c is indeendent of 2 ( + ], since w is bounded nea the hoizon accoding to assumtion A II. Since we ae assuming that 6 in>, the uniqueness theoem fo solutions of ODEs yields that jj 2 > on ( +]. Then dividing (4.2) by 2 Ajj 2 and integating fom to 2, < < 2,we get log j( 2 )j 2 ; log j( )j 2 2c Z 2 A ; 2 () d : Taking the limit & in this last inequality gives the desied esult. Lemma 4.2 If A ; 2 is not integable nea the event hoizon = and! 6=, then thee ae ositive constants c and " such that Poof: Dene the matix J by c j()j2 c if <<+ ". (4.3) ; m!t ; w J =!T ; c!t ; w!t + m!t ; c!t ; c!t ; c!t ; m!t + m!t C A 7

10 and notice that, since T ()! as!, J is close to the identity matix fo nea. If we let F () = <() J()()> then a staightfowad calculation yields that F = <() J ()()> : In a manne simila to that in [2], we can ove that jj j is integable nea =, and as in [2], it follows that (4.3) holds. Lemma 4.3 If 6 fo >, then! =. Poof: Assume that! 6=. We wite the (AT 2 ) equation (2.6) as (AT 2 ) = ;4! T 4 jj 2 + 2m ( 2 ; ; 2 )+ 4 w 4c + ( + ) T 3 ; 4Aw2 e 2 T 2 : (4.4) Accoding to hyothesis A II, the left side of this equation is bounded nea the event hoizon. The Lemmas 4. and 4.2 togethe with A II imly that the coecients of T 4, T 3, and T 2 in this equation ae all all bounded, and that the coecient of T 4 is bounded away fom zeo nea =. Assumtion A II imlies that T ()!as &. Hence the ight side of (4.4) diveges as &. This is a contadiction. 5 Reduction to the Case () =, () 6= Since! =, the Diac equation (2.) educes to A = w= ;m c= ;m ;w= ;c= c= ;m ;c= ;m C M : (5.) A The following Lemma gives some global infomation on the behavio of the solutions to (5.). Lemma 5. The function ( + ) is stictly ositive, deceasing, and tends to zeo as!. Poof: A staightfowad calculation gives A ( + ) = ;m jj 2 so that ( + )() is a stictly deceasing function, and thus has a (ossibly innite) limit as!. Since jj 2 2 j + j, we see that the nomalization condition (3.2) holds only if this limit is zeo. It follows that ( + ) is stictly ositive. 8

11 Next we want to show that the sinos have a (ossibly innite) limit as &. When A ; 2 is integable nea the event hoizon, it is an immediate consequence of Lemma 4. that this limit exists and is even nite. Coollay 5.2 If A ; 2 is integable nea the hoizon, then has a nite limit fo &. Poof: We can integate (5.) fom to 2, < < 2, ( 2 ) ; ( ) = Z 2 A ; 2 () M()() d : Lemma 4. yields that the ight sideconveges as &, and hence has a nite limit. In the case when A ; 2 is not integable nea the hoizon, we ague as follows. Accoding to the owe ansatz (3.4), the matix in (5.) has a nite limit on the hoizon. Exactly as shown in [7, Section 5] using the stable manifold theoem, thee ae fundamental solutions of the Diac equation which behave nea the event hoizon exonentially like R ex( j A ; 2 ), whee j 2 IR ae the eigenvalues fo & of the matix in (5.) (notice that the j ae eal since they ae the eigenvalues of a symmetic matix). Thus fo any linea combination of these fundamental solutions, the sino functions ae monotone in a neighbohood of the event hoizon, and hence as &, hasalimit in IR [fg. We set () = lim () ()() = lim()() : & & Poosition 5.3 ()() =. Poof: We conside the (Aw ) equation (2.5) with! =, 2 (Aw ) = ;w( ; w 2 ) + e 2 ( AT ) A + 2 (AT 2 ) 2AT 2 : (5.2) Suose that ()() > : (5.3) Fom hyotheses A I and and A II, we se that the coecient of A ; 2 is ositive nea =, as ae the othe tems on the ight side of (5.2). Thus we may wite (2.5) in the fom (Aw ) (() = () + (5.4) A() whee is bounded and > nea. Thus we can nd constants, satisfying (Aw ) > + A() > (5.5) fo nea. Then exactly as in [2, Section 3], it follows that the sinos must vanish in >. If on the othe hand ()() < : (5.6) 9

12 then (5.4) holds with () < nea. Thus Setting ~w = ;w, (5.7) becomes ;(Aw ) = ;() ; (() A() : (5.7) (A ~w ) = ;() ; (() A() whee ; () > fo nea. Thus we see that (5.4) holds fo w elaced by ~w. This again leads to a contadiction. The next oosition ules out the case that both and vanish on the event hoizon. Poosition 5.4 Eithe () =, () 6= o () 6=, () =. Poof: Suose that () = = () : (5.8) Using (5.), we have fo nea, A = c + o() (5.9) A = ; c + o() : (5.) If A ; 2 is not integable nea the event hoizon, these equations show that () and() ae nite (othewise multilying (5.9) and (5.) by A ; 2 and integating would contadict (5.8)) if A ; 2 is integable nea, Coollay 5.2 shows that () and() ae again nite. Fom (2.6) with! =wehave (AT 2 ) = 2m ( 2 ; ; 2 )+ 4w + 4c ( + ) T 3 ; 4 e 2 (Aw2 ) T 2 : (5.) Since the coecients of T 3 and T 2 ae bounded, as is the left-hand side, we conclude that, since T ()!as &, the coecient oft 3 must vanish on the hoizon, 2m ( 2 ; ; 2 )+ 4w 4c + = ( + ) = : (5.2) As a consequence, () 2 = () 2, and Lemma 5. yields that Futhemoe fom (5.9) and (5.), fo nea, () = () 6= : (5.3) sgn () = sgn () and sgn () = ;sgn () : (5.4) Fom (5.3) and (5.4), we see that fo nea, the sinos must lie in the shaded aeas in one of the two conguations (I) o (II) in Figue. Now we claim that in eithe conguation (I) o (II), the shaded egions ae invaiant. Fo the oof, we conside the Diac equation (5.). One easily checks that the shaded egions in the =-lots ae

13 (I) (II) Figue : Invaiant Regions fo the Sinos invaiant, ovided that and ae as deicted in thei shaded egions. Similaly, one veies that the shaded egions in the =-lots ae invaiant, ovided that and lie in the shaded egions. Moeove, Lemma 5. shows that the sinos cannot leave thei egions simultaneously (i.e. fo the same ). This oves the claim. Next we conside the situation fo lage. In the limit!, the matix M in (5.) goes ove to the matix S given by S = ;m ;m ;m ; ;m In S, the non-zeo 2 2 ue and lowe tiangula blocks,! ;m ;m C A : have eigenvectos ( ) t and ( ;) t with coesonding eigenvalues ;m and m, esectively. Since the system of ODEs A = S slits into seaate equations fo ( ) and ( ), we see that (() ()) must be a linea combination of e ;c() ( ) t and e d() ( ;) t, whee the functions c and d ae close to m. Since the sinos ae assumed to be nomalizable (i.e. (3.2) holds), and ae non-zeo fo >, it follows that fo lage, the sinos ae close to a constant multile of e ;c() ( ) t, and thus fo lage, sgn () = sgn (). Similaly, fo lage, sgn () = sgn (). This is a contadiction to the shaded invaiant egions of Figue. The two cases in Poosition 5.4 can be teated vey similaly. Theefoe we shall in what follows estict attention to the st case. Futhemoe, we know fom Lemma 5.

14 and Poosition 5.3 that ()() >. Using lineaity of the Diac equation, we can assume that both () and () ae ositive. Hence the emaining oblem is to conside the case whee () = () 6= () () > : (5.5) 6 Poof that A ; 2 is Integable Nea the Event Hoizon In this section we shall assume that A ; 2 is not integable nea the event hoizon and deduce a contadiction. We wok with the owe ansatz (3.3),(3.4) and thus assume that s 2. We st conside the case w 6=. The st comonent of the squaed Diac equation (2.7) is A@ ( " A@ ) = m 2 + c2 + w # w A + c (w ; A) 2 : (6.) The squae backet is bounded accoding to A II. Since () = and () >, ou assumtion w 6= imlies that the ight side of (6.) is bounded away fom zeo nea the event hoizon, i.e. thee ae constants " with A@ ( A@ ) fo <<+ " whee \" coesonds to the two cases w > and w <, esectively. We multily this inequality by A ; 2 and integate fom to 2, < < 2, Z A@ 2 2 A ; 2 : The ight side diveges as &, and thus lim & A@ =. Hence nea the event A ; 2, and integating once again yields that lim & =, in contadiction to () =. Suose now thatw =. We st conside the A-equation (2.2), which since! = becomes A = ; A ; ( ; w 2 ) 2 e 2 2 ; 2 e 2 Aw2 : (6.2) Emloying the owe ansatz (3.3),(3.4) gives Hee and it what follows, O(u s; ) = + O(u s ) ; e O(u2 ) + O(u s+2;2 ) : (6.3) f(u) = O(u ) means that lim & u ; f(u) is nite and non-zeo also we omit the exessions \o(u )." The constant tem in (6.3) must vanish, and thus e 2 2 =. Using also that O(u s ) is of highe ode, (6.3) educes to O(u s; ) = O(u) + O(u 2 ) + O(u s+2;2 ) : (6.4) Suose st that s > 2. Then (6.4) yields that = 2. Substituting ou owe ansatz into the Aw -equation (2.5) gives O(u s; 3 2 ) = O(u 2 ) + e 2 T 2

15 and thus = O(u +s 2 ). Since () 6=,we conclude that thee ae constants c > with jj c u +s 2 fo <<+ : (6.5) Fom this one sees that the st summand on the ight side of (6.) is of highe ode moe ecisely, A@ ( A@ ) = O(u 2 ) : Multilying by A ; 2 and integating twice, we conclude that = O(u 5 2 ;s ) and this contadicts (6.5). The nal case to conside is w =ands =2. Now theaw -equation (2.5) gives O(u ) = O(u ) + e 2 T and thus = o(u). This gives a contadiction in (6.) unless w ; A = o(u), and we conclude that =. Now conside the Diac equation (5.). Since w() =, the eigenvalues of the matix in (5.) on the hoizon ae = m 2 + c 2 = 2. As a consequence, the fundamental solutions behave nea the hoizon u m 2 +c 2 = 2. The bounday conditions (5.5) imly that u + m 2 +c 2 = 2, wheeas u ; m 2 +c 2 = 2,andwe conclude that ( + )() > : (6.6) Next we conside the AT 2 -equation (2.6), which fo! = takes the fom (5.). It is convenient tointoduce fo the squae backet the shot notation [] = 2m ( 2 ; ; 2 )+ 4w 4c + ( + ) : (6.7) We dene the matix B by B = A shot calculation shows that m w= c= w= ;m c= c= m c= ;m [] = 2 < B > C A : and futhemoe, using the Diac equation (5.), [] = 2 < B w > = 4 ; 4c ( + ) 2 = 4w 4w 4c ; 2 ; ( + ) 2 : (6.8) Since ()() = and ( + )() > accoding to (6.6), ;[] c 2 fo <<+ andaconstant c 2 >.Integating on both sides shows that j[]j c 3 u fo <<+ with c 3 >. As a consequence, the st summand in (5.) diveges fo &, wheeas the left side and the second summand on the ight ae bounded in this limit. This is a contadiction. We conclude that A ; 2 mustbeintegable nea the event hoizon, and so s<2. 3

16 7 Poof of the Main Theoem In this section we shall analyze the EDYM equations with the owe ansatz (3.3),(3.4) nea the event hoizon. We will deive estictions fo the owes s and k until only the excetional case (3.6) of Theoem 3. emains. So fa, we know fom Section 6 that s<2. A simle lowe bound follows fom the A-equation (2.2) which fo! = simlies to (6.2). Namely in view of hyothesis A II, the ight-hand side of (6.2) is bounded, and thus s. The case s = is excluded just as in [2] by matching the sinos acoss the hoizon and alying a adial ux agument. Thus it emains only to conside s in the ange < s < 2 : (7.) We begin by deiving a owe exansion fo nea the event hoizon. Lemma 7. Suose that w 6= o 6= s=2. Then the function behaves nea the hoizon as = u + o(u ) 6= (7.2) whee the owe is eithe o Poof: We set = = su ( ( = ; s 2 2 ; s if w 6= 2 ; s +min( s=2) if w =. : lim su ju ; ()j < & ) (7.3) (7.4) : (7.5) Suose st that <. Then fo evey <thee ae constants c> and "> with j()j < cu fo <<+ ": (7.6) We conside the st comonent of the squaed Diac equation (6.) and wite it in the fom A@ ( A@ ) = f+ g (7.7) whee f stands fo the squae backet and g fo the last summand in (6.), esectively. Multilying by A ; 2 and integating gives A@ () = Z A ; 2 (f+ g) + C with an integation constant C. We again multily by A ; 2 and integate. Since () =, we obtain Z Z s Z () = A ; 2 (s) ds A ; 2 (f + g) + C A ; 2 : (7.8) Note that the function f, intoduced as an abbeviation fo the squae backet in (6.), is bounded nea the hoizon. Hence (7.6) yields a olynomial bound fo jfj. Each multilication with A ; 2 and integation inceases the owe by ; s 2,andthustheeisa constant c with Z Z s A ; 2 (s) ds A ; 2 jf j c u 2;s+ fo <<+ ": (7.9) 4

17 Since 2 ; s>, (7.9) is of the ode o(u ; s 2 + ), and thus (7.8) can be witten as () = Z Z s Z A ; 2 (s) ds A ; 2 g + C A ; 2 + o(u ; s 2 + ) : (7.) Conside the behavio of the st two summands in (7.). The function g stands fo the last summand in (6.). If w 6=,it has a non-zeo limit on the hoizon. If on the othe hand w =, then g u. Substituting into (7.) and integating, one sees that the st summand in (7.) is u with given by (7.4). The second summand in (7.) vanishes if C =,andis u with as in (7.3). Accoding to (7.), ; s 2 < 2;s <2;s+min( s=2). Thus the values of in (7.3) and (7.4) ae dieent, and so the st two summands in (7.) cannot cancel each othe. If we choose so lage that ; s 2 +, (7.) yields the Lemma. Suose now that given by (7.5) is innite. Then choosing = max ; s 2 2 ; s + min( s=2) we see that the st two summands in (7.) ae of the ode O(u s ) with s accoding to (7.3) and (7.4), esectively, and the last summand is of highe ode. Thus (7.) imlies that as dened by (7.5) is nite (namely, equal to the minimum of (7.3) and (7.4)), giving a contadiction. Thus is indeed nite. In the oof of Poosition 5.4, we aleady obseved that the squae backet in the AT 2 -equation (2.6) vanishes on the hoizon (5.2). Let us now analyze this squae backet in moe detail, whee we use again the notation (6.7). Poosition 7.2 < and with as in Lemma 7.. [] = O(u + ) + O(u) (7.) Poof: The deivative of the squae backet is again given by (6.8). Now =, 6= and fom Lemma 5., + 6= thus using (3.3), (3.4), and (7.2), we get,fo nea, [] = O(u ;+ ) + u +() + O() (7.2) whee we again omitted the exessions \o(u : )" and we use the notation () = ( if w = if w 6= : Integating (7.2) and using that [ ] = = accoding to (5.2), we obtain that [] = O(u + ) + u +()+ + O(u) : (7.3) Suose. Then + >and +()+>, and (7.3) becomes [] = O(u) : 5

18 We wite the AT 2 -equation (5.) as (AT 2 ) = T 3 [] ; 4AT2 w 2 e 2 : (7.4) Since (AT 2 ) is bounded and T 3 = O(u ; 3s 2 ) (by vitue of hyothesis A I ), (7.4) behaves nea the event hoizon like u = O(u ; 3s 2 ) + O(u 2;2 ) : (7.5) Since 2 ; 2 and ; 3s 2 <, the ight side of (7.5) is unbounded as &, givinga contadiction. We conclude that <. Fo <, the second summand in (7.2) is of highe ode, and we get (7.). In the emainde of this section, we shall substitute the owe exansions (3.3){(3.5) and (7.2) into the EDYM equations and evaluate the leading tems (i.e. the lowest owes in u). This will amount to a athe lengthy consideation of seveal cases, each of which has seveal subcases. We begin with the case w 6=. The A-equation (2.2) simlies to (6.2). The AT 2 -equation (2.6) fo! = takes the fom (5.), and we can fo the squae backet use the exansion of Poosition 7.2. Finally, we also conside the Aw -equation (2.5). Using the egulaity assumtion A I,we obtain A-eqn: O(u s; ) = ; ( ; w2 )2 e O(u ) + O(u s+2;2 ) (7.6) AT 2 -eqn: u 3s +; = O(u 2 ) + O(u ; 3s 2 ) + O(u 2;2 ) (7.7) Aw -eqn: O(u s+;2 ) = w ( ; w 2 ) + O(u ) + O(u ; s 2 ) : (7.8) Fist conside (7.6). Accoding to A II, s +2 ; 2, and so all owes in (7.6) ae ositive. We distinguish between the cases whee the owe s +2 ; 2 is lage, smalle, o equal to the othe owes on the ight of (7.6). Making sue in each case that the tems of leading owes may cancel each othe, we obtain the cases and conditions (a) <s+2 ; 2 =) w 2 = e = s ; s 3 2 (b) = s +2 ; 2 =) w 2 = e =2; s s 3 2 (c) >s+2 ; 2 > =) w 2 = e = 2 s<3 2 (7.9) (7.2) (7.2) (d) s +2 ; 2= =) =; s 2 : (7.22) In Case (a), the elations in (7.9) imly that ; 3s 2 Hence (7.7) yields ; 3s=2 = + ; 3s=2, so < 2s ; 4 = 2 ; 2 : = ; = 2 ; s : (7.23) 6

19 This is consistent with Lemma 7.. But we get a contadiction in (7.8) as follows. Since = s ;, we have s + ; 2=2s ; 3 > on the othe hand, ; s 2 = 2 ; s ; s 2 = 2 ; 3s 2 < : Thus the left-hand side of (7.8) is bounded, but the ight-hand side is unbounded as &. This comletes the oof in Case (a). In Case (b), (7.8)yields that s 2 : (7.24) We conside the two cases (7.3) and (7.4) in Lemma 7.. In the st case, (7.24) yields that s, contadicting (7.). In the second case, (7.24) imlies that s 4 3. This contadicts the inequality in(7.2),and thus comletes the oof in Case (b). In Case (c), the elations in (7.2) give s + ; 2=s ; 3 2 <, and thus (7.8) imlies that s + ; 2= ; s 2,so = 3 2 (s ; ). Accoding to Lemma 7., =2; s o =; s 2. In the st of these cases, we conclude that s = 7 5 and = 3 5. Substituting these owes into (7.7), we get u = O(u ; ) + O(u ; ) + O(u ; ) which clealy yields a contadiction. Thus =; s 2, giving s = 5 4 = 2 = 3 8 : (7.25) This case is uled out in Lemma 7.3 below. In Case (d), we conside (7.8). Since s + ; 2=; <, we obtain that s + ; 2= ; 2 and thus = s ;. Lemma 7. yields the two cases s = 4 3 = 3 = 3 and (7.26) s = 3 2 = 4 = 2 : (7.27) The st of these cases is the excetional case of Theoem 3., and the second case is excluded in Lemma 7.4 below. This concludes the oof of Theoem 3. in the case w 6=. We next conside the case w =. Then the exansions (7.6){(7.8) must be modied to A-eqn: O(u s; ) = + O(u 2 ) + O(u s+2;2 ) (7.28) AT 2 -eqn: u 3s +; = O(u 2 ) + O(u ; 3s 2 ) + O(u 2;2 ) (7.29) Aw -eqn: O(u s+;2 ) = O(u ) + O(u ; s 2 ) : (7.3) One sees immediately that, in ode to comensate the constant tem in (7.28), s +2 ; 2 must be zeo. Hence s + ; 2=; <, and (7.3) yields that s + ; 2= ; s 2 and thus = s ;. Now conside Lemma 7.. In case (7.3), we get the excetional case of Theoem 3., wheeas case (7.4) yields that s = 3 2 = 4 = 2 : 7

20 This case is uled out in Lemma 7.4 below, concluding the oof of Theoem 3. in the case w =. The nal case to conside is w =. In this case, the exansions coesonding to (7.6){(7.8) ae A-eqn: O(u s; ) = ; e O(u2 ) + O(u s+2;2 ) (7.3) AT 2 -eqn: u 3s +; = O(u 2 ) + O(u ; 3s 2 ) + O(u 2;2 ) (7.32) Aw -eqn: O(u s+;2 ) = O(u ) + O(u ; s 2 ) : (7.33) If s +2 ; 2=,we obtain exactly as in the case w 6= above that = s ;. It follows that > s 2, and Lemma 7. yields eithe the excetional case of Theoem 3., o s =2,contadicting (7.). If on the othe hand s +2 ; 2 >, we can in (7.3) use the inequality s +2 ; 2 < 2 to conclude that s ; =s +2 ; 2andthus = 2. Now < s 2, and Lemma 7. togethe with (7.33) yields the two cases s = 5 4 = 2 = 3 8 and s = 8 5 = 2 = 9 : The st case is uled out in Lemma 7.3 below, wheeas the second case leads to a contadiction in (7.32). This concludes the oof of Theoem 3., excet fo the secial cases teated in the following two lemmas. Lemma 7.3 Thee is no solution of the EDYM equations satisfying the owe ansatz (3.3), (3.4), and (7.2) with s = 5 4 = 2 = 3 8 : Poof: Suose that thee is a solution of the EDYM equations with A() = A u o(u 5 4 ) w() = w u 2 + o(u 2 ) with aametes A w 6=. Conside the A-equation (6.2). Theleftsideisofthe ode ( ; ) 4. Thus the constant tems on the ight side must cancel each othe. Then the ight side is also of the ode u 4. Comaing the coecients gives 5 4 A = ; 2e 2 A w 2 : This equation yields a contadiction because both sides have oosite sign. Lemma 7.4 Thee is no solution of the EDYM equations satisfying the owe ansatz (3.3), (3.4), and (7.2) with s = 3 2 = 4 = 2 : (7.34) 8

21 Poof: Accoding to (3.3), we can wite the function A as A = u f(u) with f = o(u 3 4 ): (7.35) Emloying the ansatz (3.3),(3.4) into the A-equation (6.2), one sees that We solvefo(w ) ; and substitute (7.35) to obtain Aw 2 = u + u 4 + o(u 4 ) : (7.36) w = u u + c f + o(u) (7.37) with a eal constant c. Now conside the AT 2 -equation (5.), which we wite again in the fom (7.4) and multily by A, A(AT 2 ) = (AT 2 ) 3 2 A ; 2 [] ; 4 e 2 (AT 2 )(Aw 2 ) : (7.38) As in the oof of Poosition 7.2, a good exansion fo the squae backet is obtained by integating its deivative. Namely, accoding to (6.8), and thus A ; 2 ()[] = 4 A; 2 [] = 4 w + u + o(u ) Z (w )(s) ds + u 4 + o(u 4 ) : Substituting into (7.38) and using A I and (7.36), we obtain A ; 2 () Z w = u + u 4 + o(u 4 ) : We multily by A, substitute (7.35) and dieentiate, w = u ; 4 + u + c 2 f + o(u ) : Multilying by =w and using (7.37) gives the following exansion fo, = u 2 + u c 3 u 3 4 f + c 4 u ; 4 f + o(u 3 4 ) : (7.39) Next we multily the Aw -equation (2.5) by A and wite it as 2 A ( A ( Aw )) = e 2 ( AT )() + u o(u 3 4 ) : We alya I and substitute (7.35), (7.36), and (7.39). This gives an equation of the fom (modulo highe ode tems), u 2 + u u 3 4 f + uf + u ; 4 f + u = u 2 + u u 3 4 f + u ; 4 f : The constant tem u must vanish since all the othe tems tend to zeo as u!. Futhemoe, the u 2 tems must cancel because all the othe tems ae o(u 2 ). We thus obtain u 3 4 f + u ; 3 4 f = u 4 + uf + f 9

22 so that uf + f = u + u 5 4 f + u 4 f = u + o(u) and we nd that f satises an equation of the fom d uf + d 2 f = d 3 u + o(u) : Astaightfowad but tedious calculation yields that the coecients d and d 2 both vanish, and that d 3 is non-zeo. This is a contadiction. 8 The Excetional Case In this section, we conside the excetional case s = 4 3 = 3 = 3 : By emloying the owe ansatz (3.3), (3.4), and (7.2) into the EDYM equations and comaing coecients (using Mathematica), we nd that the solution nea the event hoizon is detemined by the ve fee aametes ( m c ). The emaining aametes ae given by = w = ; c A = 9 2 ; 2 + 2c m = ;3 m ; c A w = 9 2 A s m ; 2cm + c ; ( ; w 2 )2 Exanding to highe ode, we obtain afte an aduous calculation two futhe constaints on the fee aametes, thus educing the oblem to one involving only thee aametes. We investigated this thee-aamete sace numeically and found stong evidence that no global black hole solutions exist. Indeed, eithe the owe ansatz was inconsistent nea the event hoizon, o else the solution could not be extended to all values of >. Acknowledgments: JS would like to thank the Max Planck Institute fo Mathematics in the Sciences, Leizig, and in aticula Pofesso E. Zeidle, fo thei hositality and geneous suot. FF and JS ae gateful to the Havad Univesity Mathematics Deatment fo suot. Refeences [] F. Finste, J. Smolle, and S.-T. Yau, \The inteaction of Diac aticles with nonabelian gauge elds and gavity { bound states," g-qc/67 2

23 [2] F. Finste, J. Smolle, and S.-T. Yau, \The inteaction of Diac aticles with nonabelian gauge elds and gavity { black holes," g-qc/9947, to aea in Mich. Math. J. (2) [3] Finste, F., Smolle, J., and Yau, S.-T., \Non-existence of black hole solutions fo a sheically symmetic, static Einstein-Diac-Maxwell system," g-qc/9848, Commun. Math. Phys. 25 (999) [4] Kunzle, H.P. and Masood-ul-Alam, A.K.M., \Sheically symmetic static SU(2) Einstein-Yang-Mills elds," J. Math. Phys. 3 (99) [5] Smolle, J., Wasseman, A. and Yau, S.-T., \Existence of black hole solutions fo the Einstein-Yang/Mills equations," Commun. Math. Phys. 54 (993) [6] Landau, L.D., Lifshitz, E.M., \Quantum Mechanics," Pegamon Pess (977) [7] Finste, F., Smolle, J., and Yau, S.-T., \Non-existence of time-eiodic solutions of the Diac equation in a Reissne-Nodstom black hole backgound," g-qc/9855, J. Math. Phys. 4 (2) Felix Finste Joel Smolle Shing-Tung Yau Max Planck Institute MIS Mathematics Deatment Mathematics Deatment Inselst The Univesity ofmichigan Havad Univesity 43 Leizig, Gemany Ann Abo, MI 489, USA Cambidge, MA 238, USA Felix.Finste@mis.mg.de smolle@umich.edu yau@math.havad.edu 2