PHYSICAL REVIEW D VOLUME 7 , NUMBER 4 1 5 FEBRUARY 1 9 7 3

G y r o m a g n e t i c R a t i o of a Massive Body:':

Jeffrey M. Coheni Physics Department, University of Pennsylvania, Philadelphia, Pennsylvania 19104

and Goddard Space Flight Center, Greenbelt, Maryland 20771

and

Jayme Tiomno3 The Institute for Advanced Study, Princeton, New Jersey 08540

and

Robert M. Wald Joseph Henry Laboratories, Princeton University, Princeton, New Jersey 08540

(Received 10 July 1972)

It i s well hown that the gyromagnetic ratio ( g factor) of a classical, slowly rotating body whose charge density is proportional to its mass density must be equal to unity. However, if the body is very massive the spacetime curvature effects of general relativity become im- portant and the result g = l is no longer valid. We calculate here the gyromagnetic ratio of a slowly rotating, massive shell with uniform charge density. When the shell is large com- pared with the Schwarzschild radius we have g = l , but as the shell becomes more massive the g factor increases. In the limit as the shell approaches its Schwarzschild radius we ob- tain g - 2 (the same value as for an electron).

I. INTRODUCTION

The gyromagnetic rat io of a body is defined a s the rat io of i t s magnetic dipole moment to i t s an- gu la r momentum. In flat space-t ime the magnetic moment of a body i n stationary motion is given by the formula1 (in units G = c = 1)

i + where e , , r i , and ti a r e , respectively, the charge, position, and velocity of the i th part ic le in the body. If I<< 1, the angular momentum 5 of the body is2

+

J =zil/liFi x t i , (2) I

where M i is the m a s s of the ith particle. Thus, if the par t i c les in the body al l have the s a m e charge- to -mass rat io (i.e., if the charge and m a s s den- s i t i es a r e proportional) one obtains

where q is the total charge and nz is the total m a s s of the body. If we define the g fac tor (also re fe r red t o as the gyromagnetic rat io) of a body by the formula

we obtain the following well-known result: In f la t space-t ime, a c lass ica l body in slow, s t a -

tionary motion whose charge density is propor- tional to i t s m a s s density h a s a gyromagnetic rzt io (g factor) equal to unity.

However, in curved space-t ime (i.e., if strong gravitational fields a r e present) Eqs. (1) and (2) a r e no longer valid; thus the resul t g = 1 does not follow. In th i s paper we calculate the gyromagnet- i c rat io of a slowly rotating spherical shell having a uniform charge and m a s s distribution (with smal l charge-to-mass ratio). When the shell is l a r g e compared with its Schwarzschild radius, we obtain g= 1, a s expected. However, a s the shell becomes m o r e massive, g inc reases , and in the l imit a s the shel l approaches its own Schwarzschild radius, we obtain g - 2, the s a m e g value a s f o r a Di rac particle.

A Kerr-Newman (charged, rotating) black hole a l so has3 a g factor of 2 . Thus, a s the shell approaches i t s Schwarzschild radius, to lowest o rder in charge and angular momentum the elec- t romagnet ic field of the shell approaches the Ker r - Newman electromagnetic field. This must occur if the collapse of the shel l is to resul t in a black hole r a t h e r than a naked singularity.* We show that th i s approach to the Kerr-Newman electromagnetic field is not special to the assumed uniform charge distribution of the shel l but a l so holds f o r an un- charged rotating shel l with a point charge and mag- netic dipole placed at i t s center.

In Sec. II we employ previous resu l t s of Cohen5 to derive the expression f o r the gyromagnetic rat io

7 - G Y R O M A G N E T I C R A T I O O F A M A S S I V E B O D Y 999

of a massive, charged shell. The approach t o the F = e l w O ~ w1 +h1u2r\ w 3 + h z w 3 ~ w l , ( 1 5 ) Kerr-Newman electromagnetic field is discussed in Sec. 111.

11. GYROMAGNETIC RATIO OF MASSIVE, SLOWLY ROTATING, CHARGED SHELL

The electromagnetic field of a slowly rotating, charged shell placed inside a slowly rotating m a s - sive shel l h a s been analyzed by Cohena5 The pr i - m a r y concern in that reference was with Machian effects occurr ing inside the mass ive shell. H e r e we use Cohen's analysis, in the l imit where the charged and mass ive shel ls coincide, to calculate the electromagnetic field at l a r g e dis tances f r o m the shel l . This is done in o rder to determine i t s gyromagnetic rat io .

T o f i r s t o r d e r in angular momentum, the space- t i m e m e t r i c of a rotating, mass ive shell is given by6

where

The m a s s m and angular momentum J of the shell a r e given by7

A convenient s e t of orthonormal f r a m e s f o r the m e t r i c (5) is

w0 = Vdt , ( 1 1 )

If the rotating shel l is given a small , uniform charge density uF(r - 7,) , then to lowest o r d e r in charge and rotation the electromagnet ic field t ensor takes the f o r m

where

e l = e l @ ) , ( 1 6 )

h l = n ( r ) c o s e , (17 )

12, = P ( Y ) s i n e . ( 1 8 )

H e r e A denotes ex te r io r product. Maxwell's equa- tions imply that e l , n , and p satisfy5

Cohen5 h a s shown that fo r r < r, the genera l solu- tion of axw well's equations (19 ) - (21 ) fo r n is

and that f o r r > r, the general solution f o r n is

where

and

where q is the total charge,

Regularity of n at r = O implies B =0, and the requirement that n - 0 a s r - implies D = O . The constants A and C may then be evaluated f r o m the jump conditions f o r the electromagnetic field, which a r e obtained by integration of Maxwell's equations a c r o s s the shell. One finds that n is continuous a c r o s s the shell,

but the r derivative of n undergoes a jump given by

Substitution f r o m Eq. ( 2 3 ) yields the following equation, which determines the constant C:

1000 C O H E N , T I O M N O , A N D W A L D

where f' denotes the derivative o f f . Solving f o r C and substituting f o r a, w , and no in t e r m s of m and J , we obtain

Since f o r l a r g e r we have

[where f is defined by Eq. (24)], we s e e f rom Eq. (23) that a t l a r g e dis tances f rom the body the radial component of the magnetic field is

Thus, the magnetic dipole moment of the body is

Hence, f rom Eqs. (25) and (30), we find that the g factor of the shell [defined by Eq. (4)] is given

by

F o r yo >> m , since f (yo) = l/rO3 We obtain

a s the arguments of Sec. I showed must be the case. However, a s the shel l approaches its own Schwarz- schild radius, yo - f m , we have f(r,) - and Eq. (34) yields

A plot of g v s ro is given in Fig. 1. Thus, a s the shell approaches i t s Schwarzschild

radius, i t s magnetic moment is related to its mass , angular momentum, and charge by the formula

This is precisely the s a m e formula a s applies to the magnetic dipole moment of an electron and of a Kerr-Newman black hole.3 In th i s o r d e r of ap- proximation, the electromagnetic field (as the shell tends t o i t s Schwarzschild radius) becomes

IOm 5 m

'0

FIG. 1. Gyromagnetic g factor versus Schwarzschild isotropic coordinate radius ro of a massive, slowly rotating, charged spherical shell. For ro >> m , g ap- proaches the classical value unity, but a s the shell ap- proaches i ts Schwarzschild radius, ro- i m , one obtains g - 2 .

identical to the Kerr-Newman electromagnetic field. In the next section we show that this ap- proach t o the Kerr-Newman field is not special t o the case of a uniformly charged shell t reated above, but a l so occurs when a point charge and magnetic dipole a r e placed at the center of an uncharged rotating shell.

111. APPROACH TO THE KERR -NEWMAN ELECTROMAGNETIC FIELD

We now compute the magnetic dipole moment of the following configurations: A point charge q and a magnetic dipole of strength po a r e placed at the center of a slowly rotating, massive, uncharged shell.

According t o Eq. (22), inside the mass ive shell, n is given by

while outside the shel l we have

where A and C a r e constants. The jump condi- tion a c r o s s the shell, Eq. (28), yields f o r the case of an uncharged shell

Thus,

7 - G Y R O M A G N E T I C R A T I O O F A M A S S I V E BODY

Solving for C and using Eq. (33), we find that the magnetic dipole moment i s given by

3 = [ 2my;+$f(y0) - y$+:f'(yo) I

6 + 2(a/r0) + L 2 - 2 m r o 2 ~ ~ ( r o ) - ~ ~ ~ ~ ~ f ~ ( r ~ ) I qJ . (42)

Thus, the magnetic dipole moment consists of two te rms: The f i r s t t e rm ar i ses from the in- t r insic magnetic dipole moment, and, inthis order of approximation, is the same a s the dipole mo- ment one would obtain if the shell were nonrotat- ing.g The second t e rm gives the dipole moment induced by the rotation of the massive shell around the charge. (The same sor t of splitting of the magnetic dipole field can also be given for the field of the rotating, massive, charged shell treated in Sec. 11.')

For yo >> m , the intrinsic dipole moment t e rm i s simply

P intrinsic = PO (YO

The induced dipole moment i s given by

in agreement with the results of Ehlers and Rind- ler." However, a s the shell approaches i t s Schwarzschild radius, yo- $m, the intrinsic di- pole t e rm goes to zero,g

Pint r ins ic -0 ( yo - im) , (45)

and the induced dipole moment approaches the Kerr-Newman value,

qJ ( ~ ~ - $ m ) . P induced - Hence, a s the shell approaches i t s Schwarzschild radius, the electromagnetic field outside the shell approaches the Kerr-Newman electromagnetic field. Thus, we conclude that in the strong-field limit, g - 2 independent of the source configura- tion.

*Worl< supported in part under the National Science Foundation Grant No. GP 30799X and U. S. Atomic Ener- gy Commission Grant No. 2171T, and by the National Aeronautics and Space Administration.

TNational Academy of Science - NRC Senior Research Associate.

$Present Address: Av. Alexandre Fe r r e i r a 291, Apt. 102, Jardim Botanico, Rio de Janeiro, G. B. , Brazil.

'see, e.g. , L. Landau and E. Lifshitz, The Classical Theory of Fields (Addison-Wesley, Reading, Mass. , 1962) ,+p. 119.

'1f / v i / i s not assumed to be small we have j =zi yim, Fi x t i , where y , = (1 - u i ')-'/' and mi i s the proper (rest-frame) mass.

3~ Carter , Phys. Rev. fi, 1559 (1968). 4 ~ e e R. M. Wald, J . Math. Phys. 13, 490 (1972) and the

references cited therein. 5 ~ . M. Cohen, Phys. Rev. 148, 1264 (1966). 6 ~ . Brill and J. M. Cohen, Phys. Rev. 143, 1011 (1966). 'For the definition of mass and angular momentum, see ,

e.g., J. M. Cohen and R. M. Wald, J. Math. Phys. 2, 523 (1972) and the references cited therein.

' ~ o t e that the problem treated in Sec. I1 can be viewed as a special limiting case of the more general problem treated in Sec. 111. Namely, the electromagnetic field

outside a slowly rotating charged shell placed in the (flat-space) interior of a slowly rotating massive shell (and concentric with it) i s the same as that of a charge q and magnetic dipole

placed at the center of the shell, where w, is the angular velocity of the charged shell a s seen from infinity [and thus (a, - no)/Vo i s the proper angular velocity with re- spect to the inertial frame], and r, i s the radial coor- dinate of the charged shell (and thus r,$Jo2 i s i t s proper radius). But the problem treated in Sec. II i s precisely the problem in the l imit where the charged and massive shells coincide. Thus, we may obtain the solution for the magnetic dipole field of a massive, rotating charged shell [see Eq. (34)l by setting

in Eq. (42). 'R. M. Wald, Phys. Rev. D 2 , 1476 (1972).

'OJ. Ehlers and W. Rindler, Phys. Rev. D 5, 3543 (1971).