Physics Letters B 298 (1993) 299-304 North-Holland P H Y S I C S I_ E T T E R $ B

Dilaton-axion hair for slowly rotating Kerr black holes

S. Mignemi and N.R. Stewart Laboratoire de Physique Th~orique, CNRS/URA 769, Institut Henri PoincarO, Universit~ Pierre et Marie Curie, 11, rue Pierre et Marie Curie, F- 75231 Paris Cedex 05, France

Received 10 June 1992

Campbell et al. demonstrated the existence of axion "hair" for Kerr black holes due to the non-trivial Lorentz Chern-Simons term and calculated it explicitly for the case of slow rotation. Here we consider the dilaton coupling to the axion field strength, consistent with low energy string theory and calculate the dilaton "hair" arising from this specific axion source.

In low energy string theory, the classical gravi ta t ional sector is modif ied by the inclusion of di laton and axion couplings. There has been much recent interest [ 1-3 ] in the new structures which are found to emerge in black hole solut ions as a direct result of the non-min imal coupling of fields to Einstein gravity within string gravity.

The sparseness o f black hole solut ions is a t t r ibuted to the "no hai r" conjecture which l imits the exterior field solut ions to those required by a local gauge invariance. Hence the known static Schwarzschild holes character- ized by the mass M and K e r r - N e w m a n and Re i s sne r -Nords t rom holes, character ized addi t ional ly by angular m o m e n t u m A a n d / o r charged gauge fields, Q. At tempts to enlarge the space of known solutions by explicit dependence on hair, lead to non-tr ivial solut ions which are, however, unstable against radial per turbat ions [ 4].

For scalar fields in part icular , it has been shown by many authors [5 ] that no solutions are available for min imal ly coupled fields even in the presence of non-tr ivial potentials [ 6 ]. A slightly different result is obta ined if non-min imal couplings are considered. This p roblem has been studied in the context of supergravity and K a l uza - K le in theories [7] and it has been shown that even when non-tr ivial scalar "ha i r " is present, the scalar charge is not an independen t pa ramete r but is a function of M, A and Q. The holes can therefore still be classified in terms of these three parameters and so the scalar "ha i r " does not violate the "no hair" conjecture. Similar remarks apply to string theories.

I f the mass pa ramete r of a black hole is large enough compared to the Planck mass, then the higher order curvature invar iants predic ted in the string effective action, may be neglected outside the event horizon. In part icular , for small curvature, the non-min imal coupling of the di laton to the Gauss -Bonne t invar iant of order t~' in the string tension can be neglected; moreover , the di laton gauge field strength coupling may be retained despi te the fact that it is also O ( a ' ) . This is the case discussed by Garf inkle et al. [ 1 ] and Shapere et al. [ 2 ], where magnet ic and dyonic black hole solut ions are respectively obtained.

At the lowest order in the string tension, (c~')o, the four-dimensional theory consists of Einstein gravity, coupled to a free di laton field and a d i l a ton -ax ion term:

S = ~ d4x x / - g [R--2(V~9) 2 - ½ e 4~ t12+ ...] , (1)

where we assume natural units, h = c = G = 1. It is known [8 ] that for the minimal ly coupled case, axionic black holes cor respond to static, spherically symmetr ic solutions of the Schwarzschild type with mass M and a purely topological axion charge q,

Elsevier Science Publishers B.V. 299

Volume 298, number 3,4 PHYSICS LETTERS B 14 January 1993

- |

ds2= - 1 - d t2+ 1 - 2 - - d r 2 + r 2 d ~ , B l , . = q ~ r 2 , H ~ . x = 0 , (2)

where dg2 is the line element on the surface of a two-sphere. For a pure Ka lb -Ramond field ( H = d B ) , its equa- tion of motion, d ' H = 0 implies that locally, the dual of H is given by *H= da, where a is a free massless scalar. For any Einstein-scalar field system, the only (static a n d / o r stationary) black hole solutions which exist are for constant a and other scalars. The non-minimal dilaton coupling of the string-induced action ( 1 ) does not alter this uniqueness theorem as long as the Ka lb -Ramond field is itself minimally coupled, as the exterior derivative of a two-form field. However, one of the novel features o f string theory is the fact that the three-form field Hu.x is not minimally coupled: gauge and gravitational anomalies [9] are removed from the theory through the introduction o f Lorentz and Yang-Mills Chern-Simons forms, thus

H = d B + o:( OgL -o-N) , (3)

where a is proportional to the inverse string tension a ' (in the following we put a = 1 ). In particular, the Lo- rentz Chern-Simons term, o9,, may be written in terms of the spin connection on the manifold,

O.) L =Tr(o9 A dog+ 2~o/~ o9/, o9) , (4)

with a similar structure for ogv in terms of the gauge fieldA u. In this paper we shall focus on the significant effect of COL in producing black hole "hair"; thus we shall ignore the gauge field sector in the following. It has been shown [10] that for all four-dimensional spacetimes conformal to a spacetime with a maximally symmetric two-dimensional subspace, the Lorentz Chern-Simons form is exact (i.e., OgL = dfl) so that it does not contribute to the equations o f motion. Thus for a Schwarzschild background, solution (2) holds with constant dilaton field. However, for (stationary) rotating black holes described by the Kerr metric, the Lorentz Chern-Simons term is non-trivial and acts as a source for axion hair by means of the Bianchi identity,

d H = T r R A R ¢ O . (5)

This mechanism was used by Campbell, Duncan, Kaloper and Olive [ 11 ] to generate axion field strength "hair" in the specific case of a slowly rotating black hole. In the limit of small rotation where the angular momentum A << M, the Kerr metric becomes to O (A):

( ds2= - 1 - dr2+ 1 - 2 - - drZ-r2d .Q - d tdO . (6) r

The dual of the Hirzebruch signature density, *Tr(R A R), then gives an O (A) term which acts as a source for the pseudoscalar field *H= da, the equation of motion for a being

1 fzuupa C a = - --4! ~ R~flu"RP~fl~ " ( 7 )

Campbell et al. [ 11 ] solve eq. (7) by using Green function techniques: for a solution to O(A) it is sufficient to use the Green function for the Schwarzschild metric. Moreover, the back reaction of the axion on the metric (through the energy-momentum tensor of a) can be ignored in this approximation. The result [ 11 ] is regular and finite ( r> 2M):

5A ( 4 M 2 8~'/3 7 2~//4 ~ (8) a(r, O ) - f ( r ) cos 0, f ( r ) = 4~-M5 \ ~ 7 - + ~ - + 5 ~ - ] "

In this calculation the dilaton field was neglected and effectively set to a constant. As a result of the di laton- axion in ( 1 ), the axion hair (8) will in turn act as a source for dilaton hair. Here we shall explicitly derive this

effect.

300

Volume 298, number 3,4 PHYSICS LETTERS B 14 January 1993

From the action ( 1 ) the equations of mot ion are

Ru, = 2 Vu@ V , 0 + e -40 Hu~.pH ~ -- ]g~, e -40 H e - ]gu, V~( e-40 H~a"R°a,~) + aw t,,-4o rr~ ~oo 3 - ~ 1, u ~, , ~ j , (9)

ga(e -40 H u"x) = 0 , (10)

D 0 + -~ e-4OH2 = 0 • ( 1 1 )

If we define

1 ~_g~U"z°Yp=e-4° HU'~ , (12)

then eq. (10) is satisfied ( d Y = 0 ) at least locally, for Y=db. The Bianchi identity * ( d H ) = * T r ( R A R) now becomes equivalent to eq. (7) after the rescaling

1 Oub=--~ e-4O 01,a . (13)

In terms of the field a(r, 0), the dilaton equation of motion can be rewritten

[~0"}- N e-a° g u" Oua O,a=O , (14)

where the normalization N - 3 × 6. From eqs. (8) and (14), we see that the dilaton hair is an O (A 2) effect. Consider the perturbative expansion

~=~o +A2~o(x) + .... (15)

where Do-cons tan t ( = 0) is the solution up to O (A l). The scalar field ~o (x) represents the non-trivial dilaton hair to O(A2); inserting this expansion into eq. (14) and expanding the exponential gives

1 [ZoJ_ NAZOuaOUa. (16)

We may now use the same Green function technique to solve for ~o(x) in eq. (16). From the result ofeq. (8), the source term appearing on the right hand side of eq. (16) is

1 [ r - 2 M f ( ~ ) 2 ] ,¢(r, O) = NA z ' ( r ) z cos20+ sin20 . (17)

From the static Green function,

d 3 ( x - - y ) []]G(x,y)= ~-_g , (18)

we have

09= f d3y ~ G(x, y ) j r ( y ) . (19)

Eq. (18) is solved to (A °) in a Schwarzschild background for a point source at ro=b, 0o=0=0o :

,o[( l r 2 Or r2 1 -- - - ~-r._] r 2 s i n 0 s i n 0 0-0 = r 2 ( 2 0 )

The Green function G(r, O) can be expressed in terms of the Legendre functions Pz and Qz; the details of its derivation are given in ref. [ 3 ]. We then have

301

Volume 298, number 3,4 PHYSICS LETTERS B 14 January 1993

i i 2f co= - db d0o d0o b 2 sin 0o G(r, O, O, b, 0o, Oo),~¢(b, 0o, 0o) 2 M 0 0

(21)

=-2JridbidOob2sinOoG(r,O,b, Oo).J(b, Oo), 2M 0

(22)

where in eq. (22), we have made use of the addition theorem of spherical harmonics; the ~bo integration may be performed immediately since the source .J (?., 0) does not depend on the variable ~. Substituting for G(r, O, b, 0o) [31 we obtain

o a = _ k (21-+-13i i z=o k 2M J db d00 PI(b/M- 1 )Q¢(r/M- 1 )Pl(cos 0)Pt(cos 0o)d (b, 0o)b 2 sin 00 2M 0

__ ~ (2 /+ 1"] f db i dO°Q'(b/M-1)P'(r/M-1)Pl(c°sO)Pl(c°sO°)f(b'O°)b2sinO° t=o\ 2M J J

r 0 (23)

If we consider the angular integrals first, we observe that the only non-zero contributions in G(r, O) come from the l= 0 and l= 2 terms of the Legendre series, corresponding to pointlike and quadrupole sources respectively. Thus we require the following functions:

l{3b 2 6b ) Po(COSO)=l, P2(b/M-1)=SkM2 M + 2 ,

3 1 ( 3 2 ( b - ~ / , 2 - 1) l n ( b ~ 2 ~ ) - ~ ( b - 1) . (24)

Having substituted for these we are left with eight radial integrals to evaluate over the domain, re [ 2M, ~ ) . The integrations were carried out using a MATHEMATICA program. After further algebraic manipulation, we find that the solution to eq. (16) is

1 ~o(r, 0 )= -

169344 NM 5

×~14889 (! 214)+ 9926M 2 k 2 \ r + 72 r 3

9989M 3 21 112M 4 15 176~41 s 29376M 6 31 752M 7 + - - + ?.4 5r 5 ?.6 ?.7 ?.8

+2P2(cos 0) / ~471M2 \ r 3 +

2513M 3 2656M 4 19 580~15 31 320346 31 752M7~ 1.4 ?.5 ?.6 ?.7 ~ j j ( r>2M) ,

(25)

where P2 (cos 0) --- ½ (3 cos20 - 1 ). Thus oo(r, O) is the O(A 2) dilaton "hair" around a slowly rotating Kerr black hole. It is important to stress

that the axion and dilaton "hair" arise without the presence of a net axion/dilaton source; the rotation of the hole itself is the source through the Lorentz Chern-Simons coupling. The shape of the dilaton field ~o is shown in fig. ( 1 ): o)(r, O) is a monotonic decreasing function of ?. and is finite at the horizon. The quadrupole term gives rise to a small angular dependence which is more appreciable in the vicinity of the horizon.

Unlike the axion, the dilaton "hair" is associated with a charge D where

302

Volume 298, number 3,4 PHYSICS LETTERS B 14 January 1993

0.015 1

w N

0.01

0.00

-10 10

Fig. 1. Dilaton hair, o~N (in polar coordinates) exterior to horizon; M= 1.

1 } d2XU VI~ (26) D = 4 ~

and the integral is taken over a two-sphere at spatial infinity. Evaluating D using co in eq. (25) gives

1 709 A2 D = - N 8 0 6 4 2M 5 " (27)

The negative sign implies that the di la ton co corresponds to a long range at tract ive force between weakly rotat ing black holes. This charge is not, however, a new free parameter since it is de termined by the mass M and angular m o m e n t u m A of the black hole. The integral of the source depends on A 2 and so this charge vanishes as A ~ 0 in the Schwarzschild limit.

We have shown that the effective string theory can give rise to non-tr ivial di laton hair through the coupling due to the Lorentz Che rn -S imons term. As in the other cases of non-min imal coupling of a scalar field with gravitat ion, the di laton charge is not an independent parameter and so the "no-ha i r" conjecture still holds.

We stress that, in view of the Che rn -S imons coupling in eq. (3) , this effect is of order c~' 2 and therefore is the leading correct ion for the di laton in the effective string theory only if the Gauss -Bonne t terms, which arise in the effective act ion at order c~', can be neglected [ 12 ].

In this paper, we have not considered the back reaction on the gravi tat ional field due to the mat ter fields. This would be impor tan t in order to see how the causal structure o f space t ime is modif ied near the horizon. A rigor- ous discussion of this topic, however, would require an exact solution of the field equations, which seems at the moment a highly non-tr ivial task.

303

Volume 298, number 3,4 PHYSICS LETTERS B 14 January 1993

We not ice , however , that s o m e exact so lu t ions have been found for a ro ta t ing d i la ton black hole in special

cases o f the ef fec t ive str ing theory [ 13 ], but where the Loren tz C h e r n - S i m o n s t e rm is neglected.

We wou ld like to thank B. Line t for helpful discussions. The work o f N.R.S. was suppor ted by a Royal Socie ty

Fel lowship . The work o f S.M. was suppor t ed by a fe l lowship f rom the Minis t6re de la Reche rche et Technologie .

References

[ 1 ] D. Garfinkle, G. Horowitz and A. Strominger, Phys. Rev. D 43 ( 1991 ) 3140; G. Gibbons and K. Maeda, Nucl. Phys. B 298 (1988) 741; T. Koikawa and M. Yosbimura, Phys. Lett. B 189 (1987) 29.

[ 2 ] A. Shapere, S. Trivedi and F. Wilczek, Mod. Phys. Lett. A 6 ( 1991 ) 2677. [ 3 ] B. Campbell, N. Kaloper and K.A. Olive, Phys. Lett. B 263 ( 1991 ) 364. [4] P. Bizon, O.T. Popp, Class. Quantum Gray. 9 (1992) 193, and references therein. [5] J.D. Bekenstein, Phys. Rev. D 5 (1972) 1239;

J.E. Chase, Commun. Math. Phys. 19 (1970) 276. [6] S. Mignemi and D.L. Wiltshire, Phys. Rev. D 46 (1992) 1475. [7] G.W. Gibbons, Nucl. Phys. B 207 (1982) 337;

G.W. Gibbons and D.L. Wiltshire, Ann. Phys. 167 ( 1986 ) 201. [8] M.J. Bowick, S.B. Giddings, J.A. Harvey, G.T. Horowitz and A. Strominger, Phys. Rev. Lett. 61 (1988) 2823. [9 ] See e.g.M. Green, J. Schwarz and E. Witten, Superstring theory (Cambridge U.P., Cambridge, 1987).

[ 10] B.A. Campbell, M.J. Duncan, N. Kaloper and K.A. Olive, Nucl. Phys. B 351 ( 1991 ) 778. [ 11 ] B.A. Campbell, M.J. Duncan, N. Kaloper and K.A. Olive, Phys. Lett. B 251 (1990) 34. [ 12 ] B.A. Campbell, N. Kaloper and K.A. Olive, Phys. Lett. B 285 ( 1992 ) 199. [ 13 ] J.H. Horne and G.T. Horowitz, preprint UCSBTH-92- l 1 ;

A. Sen, preprint TIFR/TH/92-20.

304