PHYSICAL REVIEW D VOLUME 42, NUMBER 2 15 JULY 1990

Gyromagnetic ratio of a black hole

David Garfinkle Department of Physics, University of California, Santa Barbara, California 93106

Jennie Traschen Department of Physics and Astronomy, University of Massachusetts, Amherst, Massachusetts 01003

(Received 23 February 1990)

We examine the properties of a rotating loop of charged matter in the presence of a static charged black hole. The behavior of the gyromagnetic ratio is examined in the limit as the radius of the loop becomes very large and in the limit as the loop radius approaches the radius of the black hole's event horizon. The implications of these results for the gyromagnetic ratio of a black hole are dis- cussed. For large radii of the loop the gyromagnetic ratio reduces to the value computed for a loop in classical electromagnetism, which has g = 1. As the radius of the loop approaches the horizon, the ratio approaches that for a black hole which has g =2, like an electron. The latter result is also true for a neutral loop; hence, magnetic-monopole black holes that accrete matter with angular momentum have gravitationally induced electric fields.

I. INTRODUCTION

In classical electrodynamics a steadily rotating charge distribution creates a time-independent magnetic field. Far from the source the leading behavior of the magnetic field is determined by a single quantity: the magnetic mo- ment p . The gauge potential is given by A , = ( p / r)sin26v,4. Let q, m , and J denote, respectively, the to- tal charge, total mass, and total angular momentum of the distribution. Define the gyromagnetic ratio y of the system as

Now suppose that the charge distribution is made of matter with a constant charge-to-mass ratio. Then y satisfies the relation

The magnetic moment of the system can also be characterized in terms of the energy change undergone by the system when it is placed in a constant magnetic field. A magnetic moment, and thus a gyromagnetic ra- tio, can then be assigned to the electron from its behavior in a magnetic field. From the nonrelativistic limit of the Dirac equation it is seen that the gyromagnetic ratio of the electron satisfies the relation

Thus the gyromagnetic ratio for an electron is twice as large as it is for classical charged matter; that is, the elec- tron has a "g factor" equal to two.

A charged, rotating black hole can be assigned a mag- netic moment p from the asymptotic (i.e., far from the hole) behavior of its magnetic field. Similarly the hole

can be assigned a charge Q from the asymptotic electric field and a mass M and angular momentum J from the asymptotic gravitational field. The state of the black hole is completely determined by the quantities M, J, and Q. Thus there is a three-parameter family of black holes. For all the black holes in this family, the gyromagnetic ratio is found to satisfy the relation

The black-hole gyromagnetic ratio is thus twice as large as that of classical matter and the same as that of an elec- tron.

If a distribution of classical charged matter collapsed to form a black hole, the system would initially have a gyromagnetic ratio y = q / ( 2 m , 1, and evolve to a state with y = q / m f . How can this happen? A priori one could imagine that the magnetic moment could change in the process of collapse and/or the system could radiate away energy and angular momentum. However, one could prevent angular momentum from being radiated by making the initial matter distribution axisymmetric. One could also, with an appropriate initial configuration, en- sure that very little energy was radiated in the collapse. In the limit where the change in mass is negligible, the magnetic moment of the system changes by a factor of 2 in the process of collapse to a black hole. It is curious that y must contrive to change by precisely a factor of 2. It would be interesting to examine the details of such a collapse and see how the change in magnetic moment arises. Unfortunately the gravitational collapse of a dis- tribution of charged matter is a very complicated process and difficult to treat. Instead we will examine a much simpler system: a black hole with a loop of rotating charged matter around it. Instead of treating gravita- tional collapse, we will examine the behavior of the sys- tem as the radius of the loop is changed from large values

419 @ 1990 The American Physical Society

420 DAVID GARFINKLE AND JENNIE TRASCHEN 42

to the size of the horizon. We consider the case when the black hole carries charge and mass, but no angular momentum, so all the angular momentum of the system is carried by the current loop and the electromagnetic field. When the radius of the loop is large enough that gravity is unimportant, the gyromagnetic ratio of the sys- tem should reduce to that for a charged rotating loop in classical electromagnetism. As the radius approaches the horizon, the far field must be indistinguishable from that of a charged, rotating black hole, and hence should have y for a black hole. Indeed we shall see explicitly that y interpolates between the two appropriate values, and dis- cuss the physical mechanism by which this happens.

Previous authors have treated the behavior of a mass- less rotating charged loop in the presence of a charged nonrotating black hole' and in the presence of an un- charged rotating black hole.2 In addition the behavior of a point charge in the presence of a rotating uncharged black hole3 and the behavior of a slowly rotating charged shell4 have been studied. In this paper we will treat a massive rotating charged loop in the presence of a charged nonrotating black hole. The main difference be- tween the previous work and our approach is that we consider the magnetic effects of moving masses in an elec- tric field. In Sec. I1 we set up the equations governing the behavior of the electromagnetic and gravitational fields around the black hole. These equations are then solved for the fields of the loop and the magnetic moment and angular momentum of the system are found. The behav- ior of the quantities p, J, and y is then found in the limit of large loop radius and in the limit as the loop radius ap- proaches the horizon size. The implications of these re- sults are discussed in Sec. 111. We use the metric signa- ture ( -, +, +, +) and units where G =c= l .

11. EINSTEIN-MAXWELL EQUATIONS

The metric of a static charged black hole is the Reissner-Nordstrom metric5

&,:;I= - f d t 2 + f ' d r 2 + r 2 ( d 6 2 + ~ i n 2 6 d ~ 2 ) , (2.1)

where

Here M and Q are, respectively, the black hole's charge and mass. The hole's electromagnetic gauge potential is

We will find the electromagnetic and gravitational field of a loop of charged matter that is rotating with angular velocity R around the black hole. Let the loop be located at r =b, 6=n-/2 and carry total charge q so that its current vector is

Since the system is stationary and axisymmetric, the spacetime still has two Killing vector fields: P = ( a / a t ) ' and = ( a /a4 1".

We assume that the matter is charged dust, with a charge-to-mass ratio of q /m. We must place the dust in- side a hollow static tube, otherwise it would not stay in circular orbits at r = b (except for special values of R when the orbits are those of free charged particles). This can be seen from the fact that T & + Tit,,, the sum of the electromagnetic and dust stress energies is not, in general, conserved. The stress energy of the loop is most easily expressed in terms of the four-velocity of the orbiting charges:

with ~ ( r ) = ; [ f (r)-C12r2]1'2. Then

The exact form of the stress energy of the tube is unim- portant for our purposes, since the static tube does not contribute to either the magnetic moment or the angular momentum of the system. The tensor Tf,b,, must, howev- er, be chosen in such a way that the total stress-energy tensor of the system is conserved. We choose Tf&, to have the form

(2.7) where a, and a , are constants. By an appropriate choice of a, and a , it is always possible to impose the conditions that V, r U h = 0 and that Tf,b,, satisfy the weak energy condition.

The electromagnetic field stress energy is given by

where FQb =Va Ab - Vb Aa. Hence we wish to solve the Einstein-Maxwell equations

Gab = ST( ToEbM + Tdust ab ) 9 (2.9)

vaFUb= -4n-jb . (2.10)

Actually (fortunately) we do not need to know all the components of the metric and gauge fields to find the gyromagnetic ratio. Let

U = A, and X=g,* . (2.11)

The magnetic moment p is defined by the far field behav- ior of the magnetic field:

42 GYROMAGNETIC RATIO OF A BLACK HOLE 42 1 -

Further, the total angular momentum is given by6 (2.16) and (2.17) amazingly simplify.' Let u = U l ( r ) and x = X , ( r ) . Then the equations for n = 1 can be written as

Here S is the two-sphere of constant r and t in the limit as r - a. From this it is straightforward to show that

where c is an arbitrary constant, and Therefore to find the gyromagnetic ratio for a current loop at any radius b,

what we need is the leading behavior of X and U as r-+ o3. Note that both these quantities vanish in the background spacetime. This facilitates a perturbative solution in q / Q and m /M. To first order one finds that the "t-#J component" of the Einstein equation and the I$

component of the Maxwell equation decouple from the rest of the system for the unknown functions U ( r , 8 ) and X ( r , 8 ) . R,, is easily found from the relation VaVa$f = - R C , v for v a Killing vector. From Eq. (2.8) one finds (to first order)

We can actually solve this system exactly. For arbitrary c, the solution to Eq. (2.18) is u = f c Q 2 / ~ r +uh where uh is a solution of Eq. (2.18) with c=O. For a choice of u, Eq. (2.19) is now a first-order equation for X . The whole system is equivalent to a first-order system for four un- known functions, so there are four linearly independent solutions for the pair ( u , ~ ) We choose the solutions so that two of them are well behaved on the horizon (inner solutions) and two (outer solutions) are well behaved as r + w . The inner solutions are E M - Q a u Q 2 8rT,, - 2 T f + 7 X

r a r r

Then, to first order,

The outer solutions are

u 4 = 3 i i ( r ) J m f ( r ' ) i i 2 ( r ' ) dr ' '

Note that ~ : i ~ ~ = O so the stress energy of the tube has no effect on the solutions of these equations. U and X can be expanded, U = z U , ( r ) Z , ( 8 ) and X = z X n ( r ) Z n ( 8 ) , where

where u ( r ) = r 2 - 3 Q 2 + ( 2 Q 4 / ~ r ) . The limiting behav- ior of u4 and x4 is given by

s i n d 8 I ] - n i . s i n 8 d 8 + l )Zn . 1 3 M -+-7, r - a , r 2 r

This is solved by

where P, is a Legendre polynomial. In particular, 1 , (8)=sin28.

As we are studying an isolated system, we want U and X to go to zero as r goes to infinity. For each n there is a solution that grows and one that decays at large r; we choose the decaying solution, which has the behavior Un - r -" as r -+ a . Hence to compute y we need only the n = 1 terms. Since the sources are 6 functions, the system (2.16) and (2.17) is equivalent to a homogeneous system with jump conditions at r = b . For n = 1 Eqs.

Here rH = M + d k f 2 - Q 2 is the value of r on the hor- izon, and ~ , = 3 r f , [ 2 ( r , - M ) Q ( ~ , ) I - ' is constant.

Actually, u3 and x3 are also well behaved on the hor- izon. However, the condition that all the angular momentum comes from the loop and not from the hole it- self' excludes this solution from the region r < b.

The solution we seek is a linear combination of

422 DAVID GARFINKLE AND JENNIE TRASCHEN 42 -

(2.22a)-(2.22d) that is continuous at r = b and satisfies the jump conditions (2.20) and (2.2 1 ). Hence,

and similarly for u ( r ) . The matching conditions consti- tute four equations for the four unknown c,.

From the large-r behavior of the solutions we see that

sin26 U + ( c 3 +c,)---

r

and

as r - co. Hence from our previous discussion, p = c 3 +c4, J =Mc3/Q, and

The angular momentum of the dust is given by5

where C is a t =const surface with normal n,. This im plies

which allows us to express in terms of L. Doing the matching yields c 3 and c,, or equivalently p and J.

Now solving Eqs. (2.20) and (2.21) for c3 and c4 and us- ing Eq. (2.28) some straightforward but tedious algebra leads to the following expressions for J and p:

The gyromagnetic ratio of the system for a loop for any radius b > rH is given by the ratio p / J . We now examine the behavior of the gyromagnetic ratio for large and small values of the loop radius b. As b approaches the horizon radius r,, the quantity u 4 ( b ) diverges as In( r - rH ) and f ( b ) approaches zero as r - r,. Therefore

lim f ' / ' ( b )u , (b )=0 . b - r ~

It then follows from Eqs. (2.29) and (2.30) that limb-,H y = Q /M. Keeping the next-leading terms, one

finds

where

Thus as the radius of the loop approaches the horizon ra- dius, the gyromagnetic ratio of the system approaches the gyromagnetic ratio of a rotating charged black hole.

As b approaches infinity, the quantities f ( b ) and u , (b) behave as follows: limb+, f (b)=limb,, b u 4 ( b ) = 1. It

then follows from Eqs. (2.29) and (2.30) that limb-, y =q/2m. Thus as the radius of the loop gets large, the gyromagnetic ratio of the system approaches the gyromagnetic ratio that the loop would have in the absence of the black hole. Keeping terms to order l /b we find that for large b the gyromagnetic ratio is well ap- proximated by

Equations (2.31) and (2.33) are the main result of this pa- per.

For a black hole formed by gravitational collapse the charge is typically much less than the mass. If we ap- proximate our expressions for J and ,u by keeping terms only to order Q / M in Eqs. (2.29) and (2.30), then we find that J -- L and that

Within this approximation, the term proportional to q/2m vanishes as b approaches the horizon and 2Q /b z Q /M.

42 GYROMAGNETIC RATIO OF A BLACK HOLE 423

111. CONCLUSIONS

In classical electromagnetism, a moving charge pro- duces a magnetic field. In general relativity the equations for the electromagnetic and gravitational fields are cou- pled. Therefore in the presence of a charged black hole a moving mass, even if it is electrically neutral, produces not only a gravitational field but also an electromagnetic field. 8 , 7 , 9 In this calculation the moving masses induce a g,* component of the metric, which then requires a mag- netic field. Thus there are two effects that contribute to the magnetic moment of our charged massive loop: (1 ) magnetic fields produced by moving charges and (2) mag- netic fields produced by moving masses. As the size of the loop gets very large the equations for electromagne- tism and gravity nearly decouple in the region near the loop. Then the first effect dominates: The moving masses make only a negligible contribution to the mag- netic moment.

As the radius of the loop approaches the horizon ra- dius the second effect dominates. This is because, for fixed angular momentum, as the loop approaches the hor- izon the angular velocity i2 approaches zero. In the limit as the loop approaches the horizon the contribution of the mass to the magnetic moment is proportional to the angular momentum while the contribution of the charge is proportional to the angular velocity. Thus the charge makes a negligible contribution to the magnetic moment.

So we see that there are weak- and strong-gravity re- gimes for the behavior of y . The present calculation shows how the system interpolates between the two values as it is changed quasistatically. This was a pertur- bative calculation in q / Q and m / M ; but as the solution in two limits is known exactly, and this calculation con- nects them, we feel confident that it correctly indicates what is happening.

Finally, consider an uncharged massive loop rotating in the presence of a charged static black hole. With only moving masses and no moving charges there is still a magnetic field. The angular momentum and magnetic moment of the system are given by Eqs. (2.29) and (2.30),

respectively, with q=0. An analogous effect occurs in the case of a black hole with magnetic charge. Given a solution (gab,Fab ) of the Einstein-Maxwell equations with ja=O, there is another solution given by (gab, + E ~ ~ ~ ~ F ~ ~ 1, in other words the same metric and the dual of the Maxwell tensor. The second solution is said to be related to the first by a duality transformation. Applying a duali- ty transformation to an electrically charged nonrotating black hole, we obtain a black hole with magnetic charge, i.e., a nonrotating black hole with a radial magnetic field. One could produce such an object by dropping a magnet- ic monopole into an uncharged nonrotating black hole. Now consider the fields produced by a rotating loop of uncharged matter in the presence of a black hole with magnetic charge Q,. These fields are related by a duality transformation to the fields of the same loop of un- charged matter in the presence of an electrically charged black hole, which we have just calculated. Thus the loop of matter in the presence of a magnetically charged black hole will produce an electric field with Emag=-Belec. Hence to first order in Q , / M the electric dipole moment p~ is given by

Astrophysically this means that if magnetic monopoles exist and have been trapped in compact objects with strong gravitational fields, an electric dipole field will be produced as matter accretes onto the monopole star.

ACKNOWLEDGMENTS

We would like to thank Curt Cutler, Steve Detweiler, John Donaghue, Lee Lindblom, and Beth Norton for helpful discussions. We would also like to thank The As- pen Center for Physics for hospitality. This work was supported by NSF grants to The University of Mas- sachusetts, The University of Florida (Grant No. PHY- 850498), and the University of California (Grant No. PHY-85-06686).

'J. Bicak and L. Dvorak, Phys. Rev. D 22,2933 (1980). *D. M. Chitre and C. V. Vishveshwara, Phys. Rev. D 12, 1538

(1975). 3 ~ . M. Cohen, L. Kegeles, C. V. Vishveshwara, and R. M. Wald,

Ann. Phys. (N.Y.) 82, 597 (1974). 4 ~ . M. Cohen, J. Tiomno, and R. M. Wald, Phys. Rev. D 7, 998

(1973). 5 ~ . Kramer, H. Stephani, E. Herlt, and M. MacCallum, Exact

Solutions of Einstein S Field Equations (Cambridge University Press, Cambridge, England, 1980).

6R. M. Wald, General Relativity (University of Chicago Press, Chicago, 1984).

'v. Moncrief, Phys. Rev. D 9, 2707 (1974). 8 ~ . Zerilli, Phys. Rev. D 9, 860 (1974). 9D. W. Olson and W. G. Unruh, Phys. Rev. Lett. 33, 1116

(1974).