P H YS ICAL REVIEW VOLUM E 174, NUM B ER 5 25 OCTOBER 1968

Global Structure of the Kerr Family of Gravitational Fields

BRANDON CARTER*

Department of Applied Mathematics arid Theoretical Physics, Umieersity of Cambridge, Cambridge, Prtglartd

(Received 29 March 1968)

The Kerr family of solutions of the Einstein and Einstein-Maxwell equations is the most general class

of solutions known at present which could represent the field of a rotating neutral or electrically charged

body in asymptotically flat space. When the charge and specific angular momentum are small compared

with the mass, the part of the manifold which is stationary in the strict sense is incomplete at a Killing

horizon. Analytically extended manifolds are constructed in order to remove this incompleteness. Some

general methods for the analysis of causal behavior are described and applied. It is shown that in all exceptthe spherically symmetric cases there is nontrivial causality violation, i.e., there are closed timelike lines

which are not removable by taking a covering space; moreover, when the charge or angular momentum isso large that there are no Killing horizons, this causality violation is of the most flagrant possible kind in

that it is possible to connect any event to any other by a future-directed timelike line. Although the sym-

metries provide only three constants of the motion, a fourth one turns out to be obtainable from the un-

expected separability of the Hamilton-Jacobi equation, with the result that the equations, not only «geodesics but also of charged-particle orbits, can be integrated completely in terms of explicit quadratures.This makes it possible to prove that in the extended manifolds all geodesics which do not reach the central

ring singularities are complete, and also that those timelike or null geodesics which do reach the singularities

are entirely confined to the equator, with the further restriction, in the charged case, that they be null with

a certain uniquely determined direction. The physical significance of these results is briefly discussed.

INTRODUCTION' PROBABLY the most important problems in general

relativity today are those concerning the singulari-ties and other pathological features arising in gravita-tional collapse. Because of the scanty nature of theexperimental evidence in its favor, the acceptability orthe unacceptability of Einstein's theory must dependlargely on whether its theoretical predictions seemreasonable or not.

A great deal is now known about the gravitationalcollapse to a curvature singularity of a sphericallysymmetric body. However, virtually nothing is knownabout collapse in more general circumstances, whereangular momentum is present for example, except thatby the results of Penrose' and Hawking' 4 singularbehavior of some sort must be expected to remain.

For this reason it is interesting to examine the proper-ties of the Kerr family of gravitational fields from thispoint of view since these are the only solutions ofEinstein's equations known at present which couldrepresent the exterior field of a rotating body inasymptotically Oat space.

It will be shown that the ringlike curvature singulari-ties in the inner parts of the Kerr fields are comparativelyinnocuous (they are in fact invisible except in theequatorial direction) in contrast with the all-embracingcurvature singularity in the Schwarzschild solution. Onthe other hand, there is a very complicated topologicalbehavior and a complete and unavoidable breakdownof the causality principle.

*Present address: Institute of Theoretical Astronomy, Cam-bridge, England.' R. Penrose, Phys. Rev. Letters 14, 57 (1965).' S. W. Hawking, Proc. Roy. Soc. (London) A294, 511 (1966).' S. W. Hawking, Proc. Roy. Soc. (London) A295, 490 (1966).' S. W. Hawking, Proc. Roy. Soc. (London) A300, 187 (1967).

174

The significance of these results depends on the as

yet unanswered question whether exterior fields of theKerr type could or would result as the final state in adynamical treatment of the collapse of a rotating body.A hint that this question may have a positive answer

comes from the recent demonstration by Israel' thatthe Schwarzschild solution is unique among asymp-totically-Rat static-vacuum solutions in being bounded

by a simple nonsingular Killing horizon ("simple"

meaning that the constant-time cross sections are topo-logically spherical), which suggests that the family ofstationary axisymmetric asymptotically Rat vacuumsolutions with the same property may also be veryrestricted. It can be conjectured that the low-angular-

momentum Kerr fields may be the only examples. Ifthis is the case, or even if there are other examples

provided that these also have pathological behaviorsimilar to that of the Kerr fields, then grave doubt will

have been cast on the validity of Einstein's theory in

its present form.

1. PHYSICAL AND TOPOLOGICAL STRUCTURE

A. Metric Form

The original and, for many purposes, the most useful

form of the Kerr family of solutions of the source-freeEinstein-Maxwell equations is given in terms of co-ordinates I, r, 0, and y which can be interpreted mostsimply and naturally on a manifold formed by takingthe topological product of a 2-plane on which I and rare Cartesian coordinates running from —oa to + oa

and a 2-sphere on which 8 and p are ordinary sphericalcoordinates (y is periodic with period 2sr, and 0 runsfrom 0 to sr). The covariant form of the metric tensor is

5 W. Israel, Phys. Rev. 164, 1776 (1967).

1559

i560 B RAN DON CARTE R

expressed in terms of three parameters, m, e, and a, by

ds'= p'd0' 2a—sin'0drd q7+2drdu

+p s[(r'+a')' —Aa' sin'0$ sin'0dp'—2ap '(2mr —e') sin'0d pdu

—[1—p '(2mr —e') ]du', (1)

and the corresponding covariant form of the electro-magnetic field tensor is

F= 2ep '[(r' as coss0—)dr n du 2a—'r cos0 sin0d0 n, du—a sin'0(r' —a' cos'8)dr A dq

+2ar(r'+a') cos0 sin0d0A dy j, (2)

where the abbreviations

fore are not hypersurface-orthogonal. It was by makinguse of these structural properties of the Acyl tensor,and speci6cally looking for non-hypersurface-orthogonalsolutions, that the empty space metrics of the familywere derived by Kerr. ' Subsequently these metricswere derived by Kerr and Schild, ' from a systematicstudy of empty solutions whose metric tensor is(locally) the sum of a Rat-space metric tensor and thetensor product of a null vector with itself. The chargedsolutions are also of this form, as can be seen by makingthe coordinate transformation

x+iy= (r+sa)e'" sin0, z=r cos0, t=u r,—(6)

which gives the metric tensor as

p'= r'+ a' cos'0

6=r' 2mr+ a'+ e—'

have been used, and where the usual symbol, n, hasbeen used for the operation of taking the antisym-metrized tensor product. (When e=o the electromag-netic 6eld vanishes and the metric satis6es the vacuumEinstein equations. )

These solutions are clearly stationary and axisym-metric with Killing vectors 0/0u and 0/Bq, and it isalso apparent that they are invariant under the discretetransformation of inversion about the equatorial hyper-plane tI=-,'~. Both the Inetric and the electromagneticfield forms are analytic except on the stationary ringr=0, 0= —,'x, where p' vanishes. In fact, the curvatureitself becomes singular as p' —+ 0 except in the specialcase where e and m both vanish. In this special casethere must still be a singularity of the geometry atp'=0, although the metric is then Qat everywhere else.In all cases the metric and the electromagnetic fieldare well behaved throughout the rest of the manifold,except for the usual trivial degeneracy of sphericalcoordinates at 0=0, 0=x.

In all these spaces the Weyl tensor is of type D inthe Petrov-Pirani classification, the two double prin-cipal null vectors being given by

0/Br, —(r'+ a') 0/Bu+ a0/0 (p+60/0r

(5)

By the Kundt and Trumper generalization' of theGoldberg-Sachs theorem~ these are integrable to givetwo shear-free null geodesic congruences. The first ofthese (which is ingoing in the sense that r decreases inthe time direction determined by increasing u) consistssimply of the curves on which I, 0, and q, are all con-stant, while the outgoing congruence is less simple inthese coordinates. The principal null congruences havenonzero rotation (except when a vanishes, in whichcase the solutions are spherically symmetric) and there-

6 W. Kundt and M. Trumper, Akad. Wiss. Lit. Mainz 12 (1962).7 J. N. Goldberg and R. K. Sachs, Acta Phys. Polon. 22, Suppl.

13 (1962).

25$t' —8ds'= dx'+dy'+dz' dt'+ —r'

r4+a2z2

(r(xdx+ydy) a(xdy ydx—) sds-xj + +dt, (7)

r +as

where r is determined implicitly in terms of x, y, s, by

r4 —(x'+ y'+ s' —a') r' —a'z'= 0.

However, this Kerr-Schild form of coordinate systemis rather awkward for studying global structures, be-cause (as the price of imposing a flat-space backgroundmetric on a manifold with the topology described above)each set of values of the x, y, 2,, t coordinates corre-sponds to two different points, distinguished by the twodifferent real values of r determined by (7). Thesecoordinates have the further disadvantage that theaxis symmetry is no longer manifest, but there is thecompensating advantage that the degeneracy on theaxis itself is removed.

The generalization of the solutions to include an elec-tromagnetic field was originally achieved not by asystematic logical method but by an algebraic trickdiscovered by Newman and Janis's who succeeded inobtaining the empty-space Kerr solutions by transfor-mation from the Schwarzschild solution (to which theyreduce in the case when a vanishes). The chargedgeneralization of the empty-space Kerr metrics wasobtained by Newman, Couch, Chinnapared, Kxton,Prakash, and Torrence" who applied an analogous trans-formation to the charged spherical solution of Reissnerand Nordstrom (which is likewise the limiting case towhich the charged solutions reduce when a vanishes).

Alternative systematic derivations of these solutionsfrom diferent points of view, and with more explicit

R. P. Kerr, Phys. Rev. Letters 11, 237 (1963).11 R. P. Kerr and A. Schild, Am. Math. Soc. Symposium, New

York, 1964.' E. T. Newman and A. I. Janis, J. Math. Phys. 6, 915 (1965).'E. T. Newman, E. Couch, R. Chinnapared, A. Exton, A,

Prakash, and R. Torrence, J. Math. Phys. 6, 918 i1965).

STRUCTURE OF KERR F IELDS 156i

information about the curvature, have been given byCarter" and Ernst. "

B. Rotating Body Interpretation

Despite its many advantages the coordinate system(1) (which will subsequently be referred to as the Kerr-Newman form) has the drawback that it does not dis-

play the full symmetry of the space.Papapetrou'4has shown that any connected stationary

axisymmetric solution of Einstein s empty-space equa-tions must have an additional discrete symmetry undersimultaneous inversion of the axial and stationaryKilling vectors, while Boyer and Lindquist" have inde-pendently discovered a specific transformation whichcasts the empty-space Kerr metrics into a form whichis manifestly invariant under such an inversion. Carter"has shown that Papapetrou's result can be generalizedto include cases where the space is nonempty, providedthat the matter tensor is itself invariant under simul-taneous inversion of the time and axial angle and thatthis situation holds automatically if the only contribu-tion to the matter tensor comes from a source-freeelectromagnetic field. (It would not necessarily holdin the presence of a perfect fiuid. ) Thus Papapetrou'sresult generalizes directly to the solutions of the source-free Maxwell-Einstein equations, and hence applies tothe charged Kerr solutions.

The specific transformation needed to obtain a mani-festly invertible form is an immediate generalization ofthe one given by Boyer and Lindquist; thus introducingnew time and angle coordinates t and p dehned by

dt =du —(r'+ a') 6 'dr, —

dq"=dy —aA 'dr,

we obtain the metric tensor form as

ds'= p 6 'dr'+ p'd8'+p ' sin'8Ladt —(r'+a')d&pf'

p 'amdt asin—'8d-Pj', —(10)

where the cross terms between the ignorable coordinatesand the others have been eliminated. The electromag-netic Geld tensor now takes the form

F=2ep 4(r acos 8)dr A ddt—asin 8 dP)—4ep 4ar c—os8 sin8d8A stadt (r'+a')dP j. (—11)

In this system (which will be subsequently referred toas the Boyer-Lindquist form) it is immediately clearhow the metrics reduce to the familiar forms of theSchwarzschild and Reissner-Nordstrom solutions whenu vanishes. (That the metrics are fiat when both e andm vanish can be seen more easily from the Kerr-Schildform unless one is familiar with spheroidal coordinates. )

"B.Carter, J. Math. Phys. (to be published)."F. J. Ernst, Phys. Rev. 167, 1175 (1968); 168, 1415 (1968)."A. Papapetrou, Ann. Inst. H. Poincar6 4, 83 (1966)."R. H. Boyer and R. W. Lindquist, J. Math. Phys. 8, 265(1967)."B.Carter, Comm. Math. Phys. (to be published).

It is also clear that the spaces are asymptotically Qat, inboth the local and the global sense, in the limits of largepositive or negative values of r. The Boyer-Lindquistform is ideal for the examination of the asymptoticbehavior of the fields, on which the physical interpreta-tion of the parameters is based.

It can be easily seen from (10) and (11) by analogywith the Schwarzschild and Reissner-Nordstrom solu-tions that (in unrationalized units with Newton s con-stant G and the speed of light e both set equal to unity)m represents the mass and e the charge in the limit oflarge positive r, and that the mass and charge are,respectively, —m and —e in the limit of large negative r.There is no loss of generality in assuming, as we shall dofrom now on, that m is positive; this is simply equivalentto choosing which of the two asymptotically Qat regionswe shall label with positive values of r.

The interpretation of the parameter u requires morecare, since its effects are of asymptotically higher order.In con6rmation of a remark originally made by Kerr, 'Boyer and Price" have shown, by a careful examinationof the geodesics in the equatroial plane in the unchargedcase, that it gives rise to Coriolis-type forces which areasylnptotically identical to those which one wouldexpect from a rotating body with angular momentumma in the weak-field limit (cf. also Cohen. "). As theeGects of the charge on the metric are of asymptoticallyhigher order than those of the mass, it can be seen thatthis conclusion still stands in the charged case. Thus u iswhat Inay be called the specific angular moInentum.The metric form used here has been adjusted so that apositive value of a corresponds to a positive sense ofrotation (it turned out to be the other way round in theform used by Boyer and Price).

It is the presence of rotational effects which gives theKerr solutions their importance. This family includesall solutions yet known which could represent the ex-terior fields of rotating charged or uncharged bodies,other asymptotically Qat solutions such as those ofWeyP' or Papapetrou" being either static or massless.Whether physically natural interior material solutionsexist (e.g. , a simply rotating perfect-fluid body) ofwhich these are the exterior fields is not yet known.Boyer" has given conditions which the surface of aperfect-Quid interior would have to satisfy. Zel'dovichand Novikov" attempted to argue from the apparentabsence of simultaneous inversion symmetry (at a timewhen the Boyer-Lindquist transformation had not yetbeen published) that such a body must contain meri-dional circulation. Now that the inversion symmetry is

'7 R. H. Boyer and T. G. Price, Proc. Camb. Phil. Soc. 61, 531(1965).

J. M. Cohen, J. Math. Phys. 8, 1477 (1967).+ W. Weyl, Ann. Physik 54, 117 (1917).'0 A. Papapetrou, Ann. Physik 12, 309 (1953).2' R. H. Boyer, Proc. Camb. Phil. Soc. 61, 527 (1965).22 Ya. B. Zel'dovich and I. D. Novikov, Zh. Eksperim i Teor.

49, 170 (1965) LEnglish transl. : Soviet Phys. —JETP 22, 122(1966)3.

1562 8 RAN DON CARTE R

known, one might be tempted to argue the other wayround. However, the results of Papapetrou and Carterwhich have just been mentioned show that no suchdeductions can be made at all since the invertibilityof the exterior is inevitable in any case.

Just as the parameter a couples with the mass to givethe angular momentum, also (as can be seen from theform (11) of the field] it couples with the charge to givean asymptotic magnetic dipole moment ea. There is nofreedom of variation of the gyromagnetic ratio whichis simply e/m. It is noteworthy that this is exactly thesame as the gyromagnetic ratio predicted for a spinningparticle by the simple Dirac equation, which is obeyedto quite a high accuracy by the electron. Therefore,despite the fact that the parameters of the solutionscontain only two adjustable ratios, it is possible tochoose them in such a way that they agree with thecorresponding parameters for an electron, for which, inunits with A=1, the mass, angular momentum, andsquared charge are given by m=10 ",ma= —'„e'=1/137from which we obtain u=~m '=10", e=—,', . The valueof the length scale determined by a is therefore quitelarge, in fact, about the same as the Compton radius.On the other hand, the value of m is so small that thefield differs very little from the limiting case m=0,with e and u as the only parameters.

Despite its great elegance the Boyer-Lindquist formunfortunately fails where 0 vanishes. This will occurwhenever m is greater than the critical value

m'= a'+e' (12)

in which case 5 has a zero at each of the two values ofr (both positive) defined by

r =m&(m' a' e')—'"— (13)

and is negative in between them. In the intermediateregion the solution changes character. It can be seenclearly from the form (10) that this region cannot beregarded as stationary in the strict sense since there areno longer any timelike vectors in the planes (r= const,8= const) of the Killing vectors, but instead r has takenover the role of a tirnelike variable. As a consequenceof the simultaneous inversion symmetry, the hyper-surfaces bounding this region must be null, by a theoremgiven by Carter" and must, moreover, satisfy the strictdefinition of a Killing horizon given in that reference.In the limiting case when u and e both vanish, the innerhorizon collapses onto the central singularity and theouter horizon becomes the well-known Schwarzschildhorizon at r=2m. When the equality (12) is satisfiedthe two horizons coalesce at r=m. When a'+e')m'there are no Killing horizons and, as will subsequentlybe proved, the manifold is geodesically complete exceptfor those geodesics which reach the central singularityat p'=0. However, when a'+e'(m' although the localfailure of the metric can be cured by reverting to theKerr-Newman coordinate system, the manifold defined

above remains incomplete as r tends to r+ since thereare geodesics for which the coordinate 0 becomes un-bounded within a finite affine distance. In Sec. C, theanalytic extensions required to remedy this defect willbe discussed and, subsequently, when the geodesicequations have beep. integrated, it will be shown thatthe extended manifolds so obtained are in fact geode-sically complete, again with the exception of thegeodesics which reach the singularities at p'=0.

dt= —dw+ (r'+a')6 'dr,

d(p= d(p+aA—'dr

The resulting form for the metric is

ds'= p'd8' 2a sin—'8dr dj+2dr dw

(14)

+p '[(r'+a')' Aa' sin'8] —sin'8d8—2ap '(2mr —e') sin'8d &p dw

—L1—p '(2mr —e')]dw', (15)

which is formally identical to the original Kerr-Newmanform. The transformation between the two Kerr-Newman. forms can be given directly as

du+dw=2(r'+a')6 'dr,

d&p+dp=2ad 'dr.

(16)

(»)In the case when the zeros of & coincide, i.e., when

a'+e2=m' so that r+ r=m, we ca—n—proceed directlyto the extended manifold. The transformations (16)and (17) give rise to complete ranges of the new coordi-nates in each of the regions r)m and r(m, and theytherefore describe two distinct extensions applying toeach of these regions separately. By performing thesetwo types of extension alternately, one can build up anextended manifold consisting of an infinite sequence of(N,r,8, y) patches, labelled . , (ri, —), (v+1, —),linked transversely by a symmetrically arranged se-quence of (w, r,8, p) patches, labelled (— N)

(—,v+1), with overlaps alternately in the regionsr (m and r) m. The overlap region between (e, —) and

(—,l) will be denoted (N, l). By adjusting the relativevalues of e and l one can arrange that the Donemptyoverlaps (n, l) are such that N= l (for a region r(m) orri= l+1 (for a region r) m). It will be shown in a sub-sequent section that the manifold so obtained is in factgeodesically complete, except for those geodesics which

C. Maximal Analytic Extension

The most suitable basic unit for building up the ex-tended manifold is the original Kerr-Newman coordi-nate patch (1) which is connected to the invertibleBoyer-Lindquist form (10) by the transformation (9).The starting point for the extension is the remark thatthe invertible form can be extended in a symmetricmanner in an inverted direction in terms of new timeand angle coordinates m and g by the transformation

STRUCTURE OF KERR FIELDS 1563

reach the singularities at P2=1, i.e., it is a maximalextension.

In the general case when the zeros of & are distinct,i.e., when a'+e'(m', the extension is more difficult.One can start in the same way as in the previous case,except that (16) and (17) now give rise to three distincttransformations instead of two, corresponding to theregions I; r&r+, II: r+&r) r; and III: r &r; it istherefore natural to build up an extended manifoldagain consisting of an infinite sequence of (n,«,8, &p)

patches labelled, (n, —), (n+1, —), over-

lapping a symmetric sequence of (w, «, 8,p) patcheslabelled, (—,n), (—,n+1), is such a way thatthe overlap region (n, l) is nonempty only if n=l (inwhich case it is of the type II) or if n= 1+1 (in whichcase it is of the type I if m is odd and l even, and oftype III if it is the other way around) (cf. the illustra-tions given by Carter" in a discussion of the restrictionof this manifold to the axis of symmetry).

However, the manifold just described still has an in-

completeness associated with the Killing horizons; thereare 2-surfaces missing where u and m both tend toinfinity together. The crux of the extension program isthe construction of new coordinate patches to includethese missing 2-surfaces. In preparation for this con-struction we introduce the I, and zv coordinates simul-

taneously, and drop r as a coordinate, instead treating

where by (16) we have

F(r)=2r+K+ 'ln~r «y~+—K 'ln~« —»~

(19)

with the constants ~~ defined by

~+=5(»+'+~') '(»+ —«+). (20)

We also use the device, introduced by Boyer andLindquist" for dealing with the uncharged case, ofdefining a new angle variable, constant along thetrajectories of that particular Killing vector field whichcoincides with the null generators on the Killing horizon.From (5) we see that this Killing vector field is(«~'+a')B/Bn+aB/By in terms of the original Kerr-Newman coordinates (1), which unfortunately dependson which of the Killing horizons r =r~ is under consider-ation. Thus we shall need two alternative new anglecoordinates which we shall denote by p+ and whichcan be defined by

2d y+= dip dj a(«+'—+a')—'(du dw) . —(21)

Thus we obtain the symmetric double quasi-null-metricfoITl1:

it as a function of n and w given implicitly (once theregion I, II, or III has been specified) by

P(«) = I+w,

tp' p~' ) (»'—«+')u' sin'8 P P+

ds2= p ~g~ + I(dN +dw )+p 6 + —(dl dw)+p2d82

(«2+ g2 «2+ g2) («2+ g2) (» 2+g2) —(«'+ii')' (V+&')'-

—P 2ha sin28 a sin28dq +— P+2 r 2—r2— ~

«2

(dl —dw) dq++p 2 sin28 a i2(du —dw) —(»2+ g2)dq+«+'+a' «+'+ a' (22)

where the obvious abbreviation p~' ——«~'+a' cos'8 hasbeen introduced.

This form is in itself even more limited in range thanthe Kerr-Newman form from which we started; in. fact,it covers the same patches as the Boyer-Lindquistform, depending on which of the regions I, II, and IIIthe solution of (18) is specified to lie in. However, weare now in a position to give a direct generalization ofthe method used by Carter" for the symmetry axis.When this method was originally devised a rather com-plicated transformation was used whose purpose wasnot only to cover the missing regions separately, butalso to cover the whole manifold by a single coordinatepatch. This was worthwhile when the symmetry axisalone was under consideration; it could also be donehere (except that there would remain the trivial de-generacy at 8=0, 8= m and the curvature singularities atp'=0) but it would be a messy process because theangular coordinates p+ required at the two horizonsare different, so that the angular coordinate would

"B.Carter, Phys. Rev. 141, 1242 (1966).

have t,o be gradually changed in between, which woulddestroy the manifest axisymmetry of the manifold.Instead of doing this we shall be content with coveringthe missing pieces one at a time. With this more limitedobjective it is possible to choose a coordinate transfor-mation which is very simple indeed.

We introduce new coordinates x, y and construct apatch with coordinates (x,y, 8,y+) to cover the four(u,w, 8, q+) patches adjacent to a missing region at«=«+. LTwo of these patches will have the form(n, n+1), (n+1, n) and will cover regions of type II,and the other two will have the form (n,n), (n+ 1,n+1)and will cover regions of type I or type III according towhether «+ or» is under consideration. ] The newcoordinates are defined by the simple transformation

x= (W)e"+, y= (~)e"+ (23)where the sign (+) in the definition of x changesbetween the two (u, «,8, y) patches involved, and thesign (~) in the definition of y changes between the two(w, «, 8, &p) patches involved. We shall choose the signs sothat the product xy is positive in the two regions where

8 RAN DON CARTE R

xy= (r—r~)G~—'(r), (24)

r r+—is positive, and negative in the other two. (Thisstill leaves an arbitrary choice of sign in the definitionof x and y but because of the inversion symmetry itwill not be necessary to make it explicitly. )

By (18) and (19), r will now be determined in termsof x and yby

where G~(r) is defined by

Gy(r) = s "y"[y —y (25)

Thus r is an analytic function of x and y since in thewhole (x,y) plane r lies between y~ and & ee. Thus we

obtain the new metric form

p+' ~ (y—«+)(y+y+)a»n'~ ( P P+d~ =p +

~

«+'G+'(r)s (X'dx'+x'dy")+p 'I +r'+a' r +a') (r'+a') (r '+a ) ((«2+ a2)2 (y 2+a2)s)

p2

)& (r—r~) s+sG~(r) s (dx dy)+ p'd8' —p 'a sin'|l~ Da sinsedqr+ — (r r~)G~(—r)s~ (ydx xdy) ~d—y+

( y'+yy+p 'sin'8~ (r'+a')dq++a lrgGg(r)-', (ydx —xdy)

~. (26)

rye+a'

This metric is clearly analytic everywhere on the

(x,y, 8, &p+) patch except at the curvature singularitiesp'=0. It must also be checked that it is nondegenerateon the Killing horizons at x=0 and y= 0 since the trans-formation we have used is singular there. This is also

immediately verifiable. It follows that it is nondegen-

erate everywhere except at the curvature singularitiesand (trivially) at 8=0, 8=s.

With the additional points on the 2-surfaces @=0,y= 0 in the new coordinate patches, the extension thatwe have obtained is maximal, since it is now geodesically

complete except for geodesics which reach the ring

singularity. We shall be able to prove this in Sec. 38after the integrals of the geodesic equations have been

obtained.Although the fact that it exists is of importance, the

form (26) is too complicated to have much practicaluse. However, it does clearly show the existence of

spacelike hypersurfaces x=E'y, where E is any real

nonvanishing constant, which extend right across themanifold, one such hypersurface passing through each

point of the regions r&r+. Analogous hypersurfaces in

the regions r(r do not exist, because of the curvature

singularity at p'= 0.

2. CAUSALITY

A. Causally Well-Behaved Parts of the Kerr Solutions

In this discussion the causality principle means thecondition that there exist no closed. causal (i.e., time-

like or null) curves in the space under consideration.

This condition can be violated in two ways: We shall

refer to the violation as trivial if none of the closed

causal curves are homotopic to zero, since in this case

we may construct a covering space in which the

causality principle is satisfied, and we may use the

covering space for purposes of physical interpretation;we shall refer to causality violation where there exist

closed timelike lines homotopic to zero as unavoidablesince in this case it could only be removed by alteringthe local structure of the space, not merely its globalcon.nectivity properties. Some of the possibilities oftrivial causality violation in members of this familyhave been discussed previously. Fuller and Wheeler'4considered the possibility of causality violation re-sulting from multiconnectedness introduced when thetwo asymptotically Qat backgrounds in the analyticallyextended Schwarzschild space are identified so thatthe Kruskal throat becomes a wormhole; they showedthat, in fact, causality violation cannot arise in this way.On the other hand, the author"" pointed out thatidentifications of this kind could lead to causalityviolations in the Reissner-Nordstrom and Kerr solu-tions (the argument in the latter case depending purelyon the properties of the sytnmetry axis). However, inall these cases the causality violation being contem-plated results from unnecessary identifications whichproduce multiconnectedness. In this section, we shall beconsidering causality violation of the unavoidable kindfirst studied in Godel's universe.

We have seen in Sec. 1 that, by the mode of itsconstruction, the extended spaces consist of a com-bination of patches of type I (y)r~) or III (r(r )in which the surfaces of transitivity are everywheretimelike and of type II (r &r&r+) in which the sur-faces of transitivity are everywhere spacelike, and thatthese patches are separated by null hypersurfaces-the Killing horizons. Provided we do not unnecessarilyidentify some of these patches, but piece them togetherexactly in the manner described in Sec. 1 and illustratedin the diagrams of Ref. 23, a causal curve which leavesone of these patches can never re-enter. It follows thatinsofar as we are considering only nontrivial causality

"R.W. Fuller and J. A. Wheeler, Phys. Rev. 128, 919 (1962)."B. Carter, Phys. Letters 21, 423 (1966).

STRUCTURE OF KE RR F I EL DS iS6S

violation we can consider each of these patchesseparately.

A very useful criterion for the nonexistence of closedcausal curves in a time-oriented space is the presenceof a spacelike hypersurface which is a properly im-

mersed submanifold in the sense of Sternberg" sinceit is impossible for a closed causal curve to intersectsuch a hypersurface if the curve is homotopic to zeroand therefore impossible altogether iii a simply con-nected space. (This follows from the fact that sincethe immersion is proper, the number of times the curvecrosses the hypersurface can change only by two at atime under the homotopy, so that if the homotopystarts from zero there will at all stages be a one-onecorrespondence between the crossings in the forwardand backward time directions, whereas a causal curvecan cross only in the forward time direction. ) Hawking'and Geroch'7 have given (different) constructions bywhich a covering space can be constructed which pre-serves the topology of such a spacelike hypersurfacebut at the same time removes all closed causal curvesthrough it by unwinding them. This shows explicitlythat a space with a properly immersed spacelike hyper-surface through each point cannot have nontrivialcausality violation.

Ke can apply this criterion to the Kerr solutions.Thus we saw in Sec. 1 that when a'+e'&m' such aspacelike hypersurface exists through any point inregion I (r) r+) and it is clear from the Boyer-Lindquistform (10) that the hypersurfaces r=const through anypoint of the regions II (r (r(r~) also satisfy therequired conditions. Therefore each connected regionr&r in the manifold we have constructed can haveno nontrivial causality violation; furthermore, sinceeach such region is simply connected, the possibilityof trivial causality violation does not arise and hencethe whole of each region r&r is causally well behaved.If a'+e'=re' the surfaces t=const in the Boyer-Lind-quist form also satisfy the required condition, so thatwe can conclude in this case also that each regionr)r (=m) is causally well behaved.

On the other hand, even when a'+e'&~mm we cannotdraw such conclusions for the regions r(r, and wecannot apply this criterion anywhere when a'+ e') m2.

B. Causality Violation in the Kerr Solutions

In a separate paper" the author has derived a cri-terion for causal bad behavior which is applicable toany space with an Abelian isometry group which iseverywhere transitive over timelike surfaces. Thiscriterion states that if there does not exist a Lie algebracovector (i.e., a linear map of the Lie algebra onto thereal numbers) such that the corresponding differential

'6 S. Sternberg, Iectures on Differential Geometry (Prentice-Hall,Inc. , Englewood CliGs, ¹ J., 1964).

'7 R. Geroch, J. Math. Phys. S, 782 (1967).B. Carter, Ph.D. thesis, University of Cambridge, England,

1967 (unpublished).

form in each surface of transitivity is everywhere space-like or null, then the whole space is a single nontriviallyvicious set. In a terminology which generalizes theconcept of a closed timelike curve (considered as avicious cycle), a vicious set is defined as a set in whichany point can be connected to any other point by botha future and a past directed timelike curve, i.e., it isone in which the causality principle is violated in themost Qagrant conceivable manner; by nontriviallyvicious it is meant that the same property holds in anycovering space so that the implied causality violation isnontrivial in the sense used in the previous section.

Now the group generated by 8/gt, g/8 j in the Boyer-Lindquist form satisfies the required conditions forapplying this criterion in the regions r &r whenu'+e'~& m' and in the whole manifold when a'+e') m')i.e., in the regions where the previous criterion failed.Choosing a Lie algebra covector means in effect choosinga differential form ~=Kdt+Ld j on the surfaces oftransitivity where E and L are arbitrary constants.The criterion will be satisfied if it is not possible tochoose E, L so that co is everywhere spacelike withrespect to the induced metric in the surfaces of transi-tivity. It is easy to see that the most obvious choice, titself, is spacelike everywhere except in the subregionwhere

t'2+a2+p 2(2etr e2)a2 sin2g &—0. (27)

However, it can easily be checked that no choice of +satisfies the required conditions over the whole regionexcept in the spherically symmetric cases. Thereforein the case when a'+e'~&m' (a/0) each region r &ris a vicious set and the boundaries r=r are causalityhorizons; in the case when a'+e') m' (a&0) the wholespace is a single vicious set. The essential details of thiscausality violation may be understood as follows.

In the uncharged case, condition (27) is satisfied ina small region of negative r in the immediate neighbor-hood of the singularity p'=0, and in the charged caseit is satisfied in a larger region including positive valuesof ~, although never extending beyond a point wherer' is equal to e' on the positive r side, or where r' isequal to the greatest of u', e', or 4m' on the negative rside. In this region the vector g/gq is timelike, so thecircles i=const, r=const, 0=const, are themselvesclosed timelike lines. However, although it is necessarythat any closed timelike lines should enter the regiondefined by (27), our application of the criterion ofRef. 28 shows that they are by no means restricted toit but can extend to any part of one of the regionsr&r or over the whole space when u'+e')m'. Thiscriterion also implies that they cannot be removed bytaking a covering space.

Actually, since the manifold as a whole, as it has beendescribed so far, is simply connected, there is no propercovering space, but because the geometry is singular atp'=0, one might just as well consider the manifoldfrom which these ring singularities have been excluded,

i566 8 RAN DON CARTER

as far as the physics is concerned. Since there wouldthen be curves not homotopic to zero, looping aroundthe ring singularities, it would be possible to constructnumerous covering spaces by partially or totally un-

winding them. When a'+e') m' the universal coveringspace will consist of a simple infinite linear sequence,but when a'+e'~&m' it will have an extremely com-

plicated unendingly branching topology. However,since the closed timelike lines do not need to loop roundthe ring, their existence is not affected by this process.

A more drastic way of obtaining a covering spacewould be to cut out the symmetry axis 0=0, 8=x andupwind the remaining space by treating q as a non-

periodic coordinate. This would not be a physicallyreasonable process because it would create an artificialsingularity in the limit 0 —+ 0 or 8 —+ m. However, eventhis would not be sufficient to remove the closed time-like lines, because the impossibility of finding a suitablecombination co does not depend on the presence of theaxis. (The simple circles described above would ofcourse cease to exist. )

It is fairly easy to understand the general nature ofthe more complicated closed timelike lines. Outside theregion (27) the coordinate t must increase continuallyalong a timelike line, although r and 0 may be variedin any direction at will in the region under considera-tion (r&r or the whole space for a'+e') nz') as may Palso except in a limited region satisfying the condition

p'+ e'—2mr &0, (2g)

where cl/8& becomes spacelike. In order to make upliterally for lost time the path must enter the region

(27). Here time can be gained, but only at the expenseof clocking up a large change (negative for a) 0) in the

angle y. It can be seen that in all cases the least upperbound to the time that can be saved per unit change in

angle is ~a~. However, this does not prevent the line

from being closed even when the symmetry axis is

removed and the coordinate y made nonperiodic, be-

cause by letting the line proceed to within a suKcientlysmall but finite distance from the symmetry axis, all thelost angle can be made up at a very small cost in time.

To sum up, in the case when a'+e') m', the centralregion has the properties of a time machine. It is

possible, starting from any point in the outer regionsof the space, to travel into the interior, move back-wards in time (t) as far as desired, at a rate up to

2' a~ per revolution about the axis, and then returnto the original position. (By keeping the motion at all

stages suKciently close to the light cone, the propertime involved in the process could be kept below anygiven nonzero limit, although this would not be possibleif some sort of bound were to be placed on the allowedacceleration. )

In the case when a'+e'~&m' the outer parts of the

space on the positive r side are causally well behaved,and there is even a partial Cauchy surface. However,

the null hypersurface r=r is a causality horizon, forby (irreversibly) crossing it a timelike path can entera region where causality is violated just as in theprevious case.

3. GEODESICS AND ORBITS

L= ', g,,x'x'+-eA;x' (3o)

where the covariant vector potential 3 has been intro-duced, satisfying

F=2dA, (31)

and where a dot over a symbol denotes ordinary differ-entation with respect to an alone parameter A. In orderto obtain (29), X must be related to the proper time by

(32)

which is equivalent to imposing the normalizingcondition

g;;S'X'= —p, . (33)

By taking zero and negative values of p,' in (33) and

setting &= 0, the same Lagrangian (30) can be used togive null and spacelike geodesics. [When there is nocharge, the actual value of the mass has no significance,and so we may obtain timelike and spacelike geodesicswith X as a metric parameter by setting p'= &1 in (33).]

In order to transform to a Hamiltonian formulation,we introduce the momenta obtained from (30) as

p;=g;,&+eA,

and thus obtain the Hamiltonian

(34)

II= ,'g'&(p, eA;) (p, cA;) —. ——(35)

Since it does not depend explicitly on P, the Hamil-tonian is automatically a constant of the motion, and itis apparent that it is this constant which is determinedby the normalizing condition (33). Thus we have

(36)

We shall work with the Kerr-Newman form of themetric since it is simple and since the correspondingcoordinate patches cover the whole manifold exceptfor the 2-surfaces (x=0, y =0) in the form (26) .

The simplest vector potential giving rise to the field(2) by (») is

A = ep 'r (dn —u sin'Od y) . (37)

A. Integration of Geodesic and Orbit Equations

The equations of motion of a test particle of mass p,

and charge e are given by

D'x'/Dr'= (e/p)Fg'(Dx'/Dr), (29)

where D/Dr denotes covariant differentiation withrespect to the proper time r, and F is the electromag-netic field tensor. These equations may be derived fromthe Lagrangian

174 STRUCTURE OF KERR FIELDS i567

Thus from (1) we obtain the mornenta

p„= —$1 —p '—(2mr —e')5u —ap '(2mr —e')

&&sin'0 j+r'+ep 'r,

p„=—ap—'(2mr —e') sin'0 u

+p '$(r'+a')' —Aa sin'0] sin'0 j—a sin'oi —ep 'ar sin'0,

p„=u—a sin'0 j,Pe= p'0

The inverse of the metric (1) is

(B/Bs) =p (B/Bg)'+2p '(r'+a') (B/Br) (B/Bu)

+2p 'a(-B/Br) (B/Bq )+2p 'a(B/-Bu)(B/Bq )+p 2a sin2g(B/Bu)2 +p 2 sin20(B/B~)2

+p 'a(B/Br)'-

from which we obtain the Hamiltonian

choice of gauge (37). (The method would also work inin the Boyer-Lindquist coordinates with the analogouschoice of vector potential, but a transformation in-volving the nonignorable coordinates r, 0 would destroythe separability. )

By (35) the general form of the Hamilton-Jacobiequation is

BS/B) =2 g"$(BS/Bx') e—A;)$(BS/Bx') e—A;$, (46)

S= ,'p2) Eu—+—Cy—+Se+S„ (47)

where S& and S„are, respectively, functions of 0 and r(42) only. Inserting this in (43), we see that the equation

can in fact be separated in the form

(40)

(41) where S is the Jacobi action.If there is a separable solution, then in terms of the

already known constants of the motion it must takethe form

'p '&~p'+-2t (r'+a')P +ap «r)P-+pP+ $a sin8p„+sin 'gp~)') . (43)

From the symmetries we immediately obtain twoconstants of the motion corresponding to conservationof energy, E, and angular momentum about the sym-rnetry axis, 4; thus we have

(45)

In addition, we automatically have the constant of themotion given by (36), corresponding to conservation ofrest mass.

These three first integrals are suf6cient to determinethe motion only when some restriction is imposed whichreduces the problem effectively to three or fewer di-mensions. This situation holds for the spherical cases,for which a thorough analysis in the uncharged casehas been carried out by Darwin, ""and for which adiscussion of the charged cases has been given byGraves and Brill."It also applies to suitable subspacesin the fully general cases, namely, the symmetry axis,which has been analyzed by Carter, "and the equatorialsymmetry plane, which has been analyzed by Boyerand Price" in the asymptotically Qat limit, and byBoyer and Lindquist" in the inner regions (all theseapplying to the empty-space cases only).

In order to tackle the general case, a fourth firstintegral of the motion is needed which cannot comefrom the obvious symmetries of the metric. However,it turns out that it is possible to obtain such an integralby taking advantage of the unexpected fact that theHamilton-Jacobi equation can be solved by separationof variables in the coordinate system (1), and with the

pg'+ (aE sing —C sin '0)'+a'p, ' cos'0= X

Ap, ' 2$(r'+a')E—aC. +«r]p„—+p,'r = —X. (5p)

These together with (44) and (45) provide a completeset of first integrals of the motion. )It is easy to verifydirectly, without considering the action, that expressions(49) and (50) are indeed constant, since it is almostimmediately apparent that their Poisson brackets withthe Hamiltonian (43) vanish. ]

Equation (48) can be solved completely by quadra-tures. It splits up to give two ordinary differentialequations:

dSe/dg=+0,

dS,/dr= 6 '(P+gR),(51)

(52)

where the functions O(8), P(r), R(r) are defined by

O~ =Q—cos20ga2(p2 —E2)+CP sin ~0)

P=E(r'+a') Ca+ «r, —

R=P' A(p'r'+ X)—(53)

(54)

(55)

(dSg/dg)'+ a'p, ' cos'0

+ (aE sing —C sin '8)'= h(dS„/d—r)'+2t (r'+a')E aC+ «r]d—S„/dr p,'r'. (4g—)

Thus both sides must be equal to a new constant of themotion, which we shall denote by X. It can be seenfrom the form of the right-hand side that X must bepositive whenever p is real, i.e., for all particle orbitsand timelike or null geodesics. Using the relationsP&= BS/Bg and P„=BS/Br, it may be related directly tothe momenta in the form

(56)

and where it has been convenient to define a new con-

"S. C. D i, Pm . Ro . So . (Lo do ) A249, 180 (1958)stant Q related to the others by

S. C. Darwin, Proc. Roy. Soc. (I,ondon) A263, 39 (1961)."J.C. Graves and D. R. Brill, Phys. Rev. 120, 1507 (1960). Q =X—(C —aE)'.

B RAN DON CARTE R

Thus the final solution for the Jacobi action is

5= ——'p'A —Eu+4 p

8 r r

(ge)d8+ a iPdr+ a i(QR)dr, (57)

QQia (58)

where the signs of the two square roots are independentof each other, and where the lower limits of integrationneed not be specified, since only changes of the actionare important.

The integrated forms of the geodesic and orbit equa-tions can now be obtained automatically by using thefact that the partial derivatives of the Jacobi action withrespect to the constants of the motion are themselvesconstant.

Thus by differentiating with respect to X, p, , E, C,we obtain, respectively,

3. Geodesic Completeness

%e are now in a position to demonstrate that theanalytic extensions obtained in Sec. 1.C are indeedmaximal, in the sense that the only geodesics which areincomplete are those which reach the ring singularity,so that they cannot possibly be imbedded as subspacesof any larger manifold.

A geodesic is complete if it can be extended to un-boUnded values of the alone parameter X. It is apparentthat any geodesic can be extended indefinitely unless itreaches the singularity or unless one of the integralsin the Eqs. (58), (60), or (61) diverges. The latter canoccur only where 6 has a zero, or where O~ or R has adouble zero.

If 0 or R has a double zero, then the integrals for X

will diverge, and ) itself will be unbounded except inthe cases of geodesics which reach the singularity, forwhich the divergent integrals for X may be able tocancel each other out, This can be seen more easilyfrom the form

~ c' cos'Od8

QO(59)

de% =p&(d8/QQte )

dX =p'(dr/QR),

(66)

(67)

a(aE sin'8 —C—')d8

' —(aE—C sin '8)d8

"at P+ —

~

1— dr, (61)a(aE sin'8 —C) —r'+a'(

dl= d8+I

1— — ~«(68)QQss gR)where +O~ and gR may take either sign independently,

but where, once a choice has been made, it must beused consistently in all four equations, and where thelower limits of integration may be chosen quite inde-

pendently in each term.For many purposes this information is more con-

veniently expressed in terms of the first-order differ-

ential system:

—(aE—C sin '8) a P )d8+—1— ~dr.

These equations can be reexpressed in terms of the(m, r, 8, tp) coordinates given by (16) and (17) as

(62) +a(aE sin'8 —C) r'+a'f P ~dm = d8+ —

i 1+ — idr, (70)p28 QQst

p'r =QR, (63)

of the Eqs. (62) and (63), than from (58) and (59)directly. Thus although this coupled form is not suitablefor explicit evaluation, it shows clearly that no questionof incompleteness can arise where O~ or R has a double

)g» (gp) zero except for geodesics reaching the singularity p =0)

Therefore in considering incompleteness away fromsingularity we need only consider the cases where d hasa zero, which can occur only for strictly positive valuesof r. Possible divergences occur only in the equationsfor u and q, which may be written in diGerential form as

p2g, = —a(aE sin'8 —C)+ (r'+a')&-'$(gR) —Pj, (64) +(aE—C sin '8) a f Pd8+—

~ 1+ dr. (71)p'tp= —(aE Csin '8)+ad 'L(QR) ——Pj, (65)

which may be obtained either from the explicitlyintegrated form t'(58) to (61)) or else directly from

(44), (45), (48), and (50), and where again the signs ofQO~ and gR may be chosen independently, but oncechosen must be used consistently.

Now provided that I' is nonzero where 6=0, we obtainfrom (54) and (55) the expansion

(72)

STRUCTURE OF KERR F IELDS i569

where the sign depends on which choice of gR is underconsideration. It can be seen from this that only onepair of the expressions (68) and (69), or (70) and (71),contains a genuine divergence at 6=0; in the otherpair the divergent terms cancel out. Thus, although thegeodesic leaves one of the Kerr-Newman coordinatepatches, it can be continued on an overlapping patch.

The case in which P vanishes where 6=0 remainsto be considered. In this case it follows from (55) thatE must have at least a single zero there. If 8 has adouble zero, there is no problem, because as we haveseen the integral for ) will then diverge anyway; thismust necessarily be the case if 6 has a double zero, whichshows why an analytic extension consisting only ofKerr-Newman patches is sufhcient in this case.

Thus we can restrict our attention to the case whereP vanishes where 6=0 and where 6 and R have only asingle root there. In this case we have

P/QR= O()V~2) (73)

in the limit as the zero of 6 is approached, so the co-ordinates N and w diverge to +~ or —~ together.This simply means that the geodesic reaches one of thepoints x=0, y=0 in one of the (x, y, 8, v2+) patches(26). Since the immediate neighborhoods of these pointsare well behaved, such a geodesic can straightforwardlybe continued on the other side.

By working with the geodesic equations in the Kerr-Newman coordinate system, we have left out of accountthe possibility that there may be goedesics confined en-tirely to the 2-surfaces x= 0, y=0. In fact, it is obviousfrom the symmetry of the form (26) that such geodesicsdo exist. Nevertheless no question of incompletenessarises because the surfaces @=0, y=0 are topologically2-spheres, and, as can be seen at once from (26), theyare spacelike. It is well known that a compact spacelikemanifold cannot possibly be incomplete.

This completes our demonstration that the analyticextensions of Sec. 1C. are maximal, and that only geod-esics which strike the singularity are incomplete. As aby-product we have shown that the charged-particleorbits have the same property. It is widely conjecturedthat this ought to follow automatically from the com-pleteness of the geodesics, but no rigorous theorem aboutthis question is known to the author, so it is perhapsworth mentioning that this is not a counterexample.

C. Some Qiialitative Properties of Geodesics and Orbits

It is possible to see quite easily how the 8 coordinatevaries during the geodesic and orbital motions, due tothe remarkable simplicity of the function 0 by whichthe variation of 8 is governed. It can be seen from (53)that not only is the form of 0' quite independent of thepresence of electric charge either on the test particleor in the field, but it is even independent of the massparameter of the field. In other words, 0~ is identical

with the function obtained in the limit of field-freeQat space.

Useful information about the orbits, and restrictionson the values which can be taken by the constants ofthe motion, may be obtained by examining the extentof the allowed regions where 0~ is non-negative. Theresults may be summarized as follows:

a2(E2 ~2))@2 (74)

If this is satisfied, 0 varies over a range touching theequator on one side or other. The range extends to theaxis of symmetry if and only if 4 =0.

The only other case where there are real solutions isthat in which C = 0 and a'(E' —&2) =0, when 8 may takeany constant value whatsoever.

Case (3) Q(0In this case there are no real solutions at all unless

(74) is satisfied, and, in addition,

Q) —(La'(E' —~')5'"—I~'I )2

~

If (75) is satisfied as a strict inequality, 0 varies over arange which does not touch the equatorial plane andwhich ext'ends to the symmetry axis if, and only if,4=0. If equality holds in (75), then 0 takes a fixedvalue which lies strictly between the equatorial planeand the symxnetry axis, except when C =0 in which caseit lies on the symmetry axis.

It is not easy to give such a complete description ofthe motion of the r coordinate, because the correspond-ing governing function R(r) is a quartic in the full sense:The odd-power terms do not drop out as they do for0~(8). Nevertheless, it is possible without much troubleto reach some interesting conclusions.

The function R(r) may be expanded in the form

R= (E' p2)r'+ 2 (p22—22+ eeE)r'

+Pa2E2 —c2+ e2 (e2—p2) —a2p2 —Q5r2

+2$2N (aE 4)'+ sea(a—E C')+2NQ5r-—"(aE-C)'—(a'+e')Q (76)

Case (1), Q) 0I

In this case there are always real solutions in which 8ranges over a region straddling the equator, cos8=0.This region extends to the axis of symmetry sin8=0,if and only if C =0 and Q+ a'(E' —p2) & 0.

In addition there is a solution in which 8 is constantat the axis value, sin8=0, when C =0 and

Q+ a2(E2 &2) —0

Case (Z), Q= 0

In this case there are always real solutions in which8 is constant at the equatorial value, cos8= 0.

There are real solutions in which 8 varies if and onlyif the energy is su@ciently high, i.e.,

1570 BRAN DON CA RTE R

From the form of the quartic term it can be seen thatno orbit or geodesic can escape to the asymptoticallyOat regions of large positive or negative r if it has lessthan the escape energy, i.e., if E'(p', as one wouldexpect.

From the form of the constant term in (76) it can beseen that no geodesic or orbit can possibly cross thehypersurface r=0 which extends across the mouth ofthe singular ring if Q is positive. Nor can it do so if it isconfined to the equator since the way would be blockedby the ring singularity itself. Therefore the hypersurfacer=0 cannot be crossed unless the inequality (74) issatisfied.

Thus we reach the conclusion that no orbit or geodesiccan pass through the ring between regions of positiveand negative r unless its energy is greater than someminimum which is certainly not less than the escapeenergy.

This repulsive property of the gravitational fieldacross the mouth of the ring has already been noted byCarter" insofar as it applies to the symmetry axis, andthe exact height of the energy barrier on the symmetryaxis is calculated in this reference. In general, theminimum energy for passing through the ring will de-

pend on the angular momentum, etc. , possibly in acomplicated way. We shall not investigate the matterfurther here, but only remark that geodesics andparticles with suKciently high energy can clearly passthrough without difhculty.

D. Geodesic Structure of Ring Singularity

The results of Sec. 3C can be used to give verystong restrictions on the geodesics and orbits which mayreach the singularity p'=0. Thus we have seen that rcannot reach the value zero if Q is positive and thatcosg cannot reach the value zero if Q is negative, andhence a necessary condition for an orbit or geodesic toreach the singularity is

=0. (77)

Moreover, from the form of the constant term in (76)it is apparent that when Q is zero it will still be im-

possible for r to reach the value zero unless either theequality

(78)

is satisfied, or alternatively the charge e of the solutionvanishes. In the timelike and null cases, i.e., when

p'~) 0, (78) is incompatible with (74), and therefore ifthe motion is not to be confined to the equator (78) maynot hold, but instead the charge e must vanish. Nowunder these circumstances the remaining coefficients in

(76) are all strictly positive, and therefore the parameterr can only reach zero by approaching from and returningto asymptotically large values on the positive side, andthe integral on the right-hand side of (58) remainsfinite during this process; on the other hand, theintegral on the left-hand side of (58) diverges as cosg

approaches zero, with the implication that the geodesicor orbit only approaches the equator asymptotically asr tends to infinity.

Thus we reach the conclusion that a timelike or nullgeodesic or orbit cannot reach the singularity under anycircumstances except in the case where it is confinedto the equator, cosa=0.

The restriction can be carried even further than this.An examination of the equatorial geodesics iD the casewhere the solution is uncharged (e=O) has alreadybeen made by Boyer and Lindquist, "who have shownthat there is in general a finite range of angular momen-tum within which a geodesic of a given sufficiently highenergy from a general point on the positive-r side of theequator can reach the singularity. However, when thesolution is charged (or if one is considering approachfrom the negative-r side) the restriction. is considerablymore severe because (78) must be satisfied. In otherwords, a geodesic or charged particle orbit with agiven energy can only reach the singularity if it has auniquely determined angular momentum. Even this isnot quite sufficient as can be seen from the form towhich (76) reduces when (77) and (78) are satisfied,which is

e'& (1+a'/e') p'. (80)

If this holds as a strict inequality, it is a sufhcient condi-tion for the singularity to be attainable from suKcientlyclose points on the equator on either side, whereas ifit holds as an equality the singularity will in general beattainable from one side only (although there will beexceptional cases when the energy is such that eitherthe quartic or the cubic term vanishes). In the par-ticular case of timelike geodesics (e=O, p')0), theinequality (80) cannot be satisfied at all except in theSchwarzschild limit when e and a both vanish. In thecase of null geodesics (e=O, p=O), strict equality holdsin (80) but the singularity can be reached from eitherside because the cubic term in (79) vanishes.

Thus we conclude that when the solution is charged,no timelike geodesics can reach the singularity, whileDull geodesics reach the singularity if, and only if, theylie in the equator and have a uniquely determinedangular momentum given by (78). Even when thesolution is uncharged, the only timelike or null geodesicswhich can reach the singularity are those confined tothe equator, but as Boyer and Lindquist have shown,in this case both nuB and timelike ones reach thesingularity and their angular momentum may lie in afinite range.

R= (E' p')r4+—2 (p'm+ peE)r'

y L$2 (e2 p2) g2p2]$2 (79)

It is clear that aD additional necessary condition for thesingularity to be attainable is that the coefficient of thequadratic term be Don-negative; in other words, thecharge on the test particle must be large enough tosatisfy

STRUCTURE OF KERR FIELDS 1571

The significance of this for an observer studying thesingularity visually, i.e., by receiving photons whichhave come out from the singularity along null geodesics,is as follows: If he observes from a point on the equatorthen when the field is uncharged the singularity isvisible as a finite one-dimensional line, as one wouldexpect for a ring seen edgewise on (except that by theresults of Boyer and Lindquist" the line may sometimesconsist of two disconnected parts); however, if thefield is charged then the singularity is visible from theequator only as a point, and in either case if the observermoves off the equator the singularity will becometotally invisible to him.

4. IMPLICATIONS

The fact that there are closed timelike lines loopingthrough the interior does got affect the reasonablenessof interpreting the Kerr solutions as the exterior fieldsof rotating bodies, since a source body might be ex-pected to block off these regions in any case. However,it does hint (although it certainly does not prove) thatcausality breakdown may be expected to result fromthe collapse of a rotating body. The theorems ofPeDrose' and Hawking' ' indicate that somethingpathological must be expected to occur in a situation ofgravitational collapse of a rotating body, but differentopinions may be held about the nature of the break-down. From a physical point of view the least seriouskind of breakdown would be the local development ofdensity or curvature singularities, and this is also thekind of breakdown which has been considered mostwidely in the past. The reason why this would not bevery serious is that og.e would expect in any case thatgeneral relativity would need to be modified in condi-tions of extreme curvature, in order to accommodatequantum theory, and one might be sanguine enoughto hope that the necessary modifications would curethe trouble. However, it is also conceivable, as has beensuggested by Lifshitz and Khalatnikov, " that thecurvature singularities which are familiar in highlysymmetric solutions do not exist in more general cases.

The Kerr solutions have a lower symmetry groupthan any other solutions in which (as far as the author

"K.M. Lifshitz and I. M. Khalatnikov, Zh. Eksperim i Teor.Fiz. 89, 149 (1960) LEnglish transl. : Soviet Phys. —JETP 12, 108(1961)j.

knows) an analytic study of a curvature singularity hasbeen made (although the separability which has madethis possible is itself a symmetry of a kind), and theresults of the previous section seem to lend a certainamount of support to this last idea. Thus as the sym-metry is progressively reduced, starting from theSchwarzschild solution, the extent of the class of geo-desics reaching the singularity is steadily reduced like-wise, until in the case with both charge and rotationthere are almost none at all, which suggests that afterfurther reduction of the symmetry, incomplete geodesicsmight cease to exist altogether. Even if a few incompletegeodesics remain in the fully general case, their im-portance is overshadowed by the causal pathology,which seems to increase as the symmetry is reduced.(However, in spite of this apparent effect the existenceor lack of symmetry may not be as important as itappears, for the Taub-N. U.T. space which has beendiscussed by Misner" is much more highly symmetricthan the Kerr solution, aod yet its global behavior is insome ways worse: It has geodesics which are incompletein a region which is locally nonsingular, and it also hasclosed timelike lines confined within any neighborhood,no matter how thin, of the causality horizon of the well-behaved part, whereas in the Kerr solutions, in thecases when there are causality horizons (at r= r ), anyclosed timelike line must at some stage penetrate deeplyinto the bad part beyond these horizons. )

All these things suggest that the breakdown ingeneral relativity may be of a global rather than (or aswell as) of a local nature, in which case it is veryserious indeed. If this turns out to be the case then onewill not be able to expect to cure the trouble by minormodifications significant only in regions of high curva-ture, so that the whole theory might have to beabandoned, or at least drastically reformulated.

ACKNOWLEDGMENTS

The author would like to thank Dr. S. W. Hawking,Dr. R. W. Lindquist, Professor C. W. Misner, Dr.R. Penrose, and Dr. D. W. Sciama for many usefuldiscussions and ideas. He is also grateful for the en-couragement and suggestions of the late Dr. R. H.Boyer.

"C. W. Misner, in ReLativity Theory and Astrophysics: Rela-tivity und Cosmology, edited by J. Ehlers (American MathematicalSociety, Providence, 1967), Vol. 8, p. 160.