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 except the 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 is so 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 The significance of these results depends on the as ' yet unanswered question whether exterior fields of the PROBABLY the most important problems in general Kerr could or would result as the final state in a relativity today are those concerning the singulari- type dynamical treatment of the collapse of a rotating body. ties and other pathological features arising in gravita- A hint that this question may have a positive answer tional collapse. Because of the scanty nature of the comes from the recent demonstration by Israel' that experimental evidence in its favor, the acceptability or the Schwarzschild solution is unique among asymp- the unacceptability of Einstein's theory must depend totically-Rat static-vacuum solutions in being bounded largely on whether its theoretical predictions seem a simple nonsingular Killing horizon ("simple" reasonable or not. by meaning that the constant-time cross sections are topo- A great deal is now known about the gravitational logically spherical), which suggests that the family of collapse to a curvature singularity of a spherically stationary axisymmetric asymptotically Rat vacuum symmetric body. However, virtually nothing is known solutions with the same property may also be very about collapse in more general circumstances, where restricted. It can be conjectured that the low-angular- angular momentum is present for example, except that momentum Kerr fields may be the only examples. If by the results of Penrose' and Hawking' 4 singular this is the case, or even if there are other examples behavior of some sort must be expected to remain. provided that these also have pathological behavior For this reason it is interesting to examine the proper- similar to that of the Kerr fields, then grave doubt will ties of the Kerr family of gravitational fields from this have been cast on the validity of Einstein's theory in point of view since these are the only solutions of its present form. Einstein's equations known at present which could represent the exterior field of a rotating body in 1. PHYSICAL AND TOPOLOGICAL STRUCTURE asymptotically Oat space. It will be shown that the ringlike curvature singulari- A. Metric Form ties in the inner parts of the Kerr fields are comparatively The for many purposes, the most useful innocuous (they are in fact invisible except in the original and, of solutions of the source-free equatorial direction) in contrast with the all-embracing form of the Kerr family Einstein-Maxwell equations is given in terms of co- curvature singularity in the Schwarzschild solution. On can be interpreted most the other hand, there is a very complicated topological ordinates I, r, 0, and y which manifold formed taking behavior and a complete and unavoidable breakdown simply and naturally on a by of 2-plane on which and r of the causality principle. the topological product a I are Cartesian coordinates running from —oa to + oa and a 2-sphere on which 8 and p are ordinary spherical *Present address: Institute of Theoretical Astronomy, Cam- coordinates (y is periodic with period 2sr, and 0 runs bridge, England. metric tensor is ' R. Penrose, Phys. Rev. Letters 14, 57 (1965). from 0 to sr). The covariant form of the ' 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). 5 W. Israel, Phys. Rev. 164, 1776 (1967). 174 1559 i560 B RAN DON CARTE R expressed in terms of three parameters, m, e, and a, by fore are not hypersurface-orthogonal. It was by making use of these structural properties of the Acyl tensor, ds'= p'd0' 2a—sin'0drd q7+2drdu and speci6cally looking for non-hypersurface-orthogonal solutions, that the empty space metrics of the family +p s[(r'+a')' —Aa' sin'0$ sin'0dp' were derived by Kerr. ' Subsequently these metrics — '(2mr —e') sin'0d 2ap pdu were derived by Kerr and Schild, ' from a systematic —[1—p '(2mr —e') ]du', (1) study of empty solutions whose metric tensor is the sum of a Rat-space metric tensor and the and the corresponding covariant form of the electro- (locally) tensor of null vector with itself. The magnetic field tensor is product a charged solutions are also of this form, as can be seen by making F= 2ep '[(r' as coss0—)dr n du 2a—'r cos0 sin0d0 n, du the coordinate transformation —a sin'0(r' —a' cos'8)dr A dq x+iy= (r+sa)e'" sin0, z=r cos0, t=u r,—(6) +2ar(r'+a') cos0 sin0d0A dy j, (2) which the metric tensor where the abbreviations gives as p'= r'+ a' cos'0 25$t' —8 ds'= dx'+dy'+dz' dt'+ —r' 6=r' 2mr+ a'+ e—' r4+a2z2 have been used, and where the usual symbol, n, has (r(xdx+ydy) a(xdy ydx—) sds- been used for the operation of taking the antisym- xj + +dt, (7) metrized tensor product. (When e=o the electromag- r +as netic 6eld vanishes and the metric satis6es the vacuum where r is determined in terms of Einstein equations. ) implicitly x, y, s, by and axisym- These solutions are clearly stationary r4 —(x'+ y'+ s' —a') r' —a'z'= 0. metric with Killing vectors 0/0u and 0/Bq, and it is also apparent that they are invariant under the discrete However, this Kerr-Schild form of coordinate system transformation of inversion about the equatorial hyper- is rather awkward for studying global structures, be- plane tI=-,'~. Both the Inetric and the electromagnetic cause (as the price of imposing a flat-space background field forms are analytic except on the stationary ring metric on a manifold with the topology described above) —, p' r=0, 0= 'x, where vanishes. In fact, the curvature each set of values of the x, y, 2,, t coordinates corre- itself becomes singular as p' —+ 0 except in the special sponds to two different points, distinguished by the two case where e and m both vanish. In this special case different real values of r determined by (7). These there must still be a singularity of the geometry at coordinates have the further disadvantage that the p'=0, although the metric is then Qat everywhere else. axis symmetry is no longer manifest, but there is the In all cases the metric and the electromagnetic field compensating advantage that the degeneracy on the are well behaved throughout the rest of the manifold, axis itself is removed. except for the usual trivial degeneracy of spherical The generalization of the solutions to include an elec- coordinates at 0=0, 0=x. tromagnetic field was originally achieved not by a In all these spaces the Weyl tensor is of type D in systematic logical method but by an algebraic trick the Petrov-Pirani classification, the two double prin- discovered by Newman and Janis's who succeeded in cipal null vectors being given by obtaining the empty-space Kerr solutions by transfor- Schwarzschild solution which 0/Br, — mation from the (to they (5) reduce in the case when a vanishes). The charged (r'+ a') 0/Bu+ a0/0 (p+ 60/0r generalization of the empty-space Kerr metrics was By the Kundt and Trumper generalization' of the obtained by Newman, Couch, Chinnapared, Kxton, Goldberg-Sachs theorem~ these are integrable to give Prakash, and Torrence" who applied an analogous trans- two shear-free null geodesic congruences. The first of formation to the charged spherical solution of Reissner is likewise the limiting case these (which is ingoing in the sense that r decreases in and Nordstrom (which to which the solutions reduce when a vanishes).