Quasi-Spherical Light Cones of the Kerr Geometry

Frans Pretorius and Werner Israel

Canadian Institute for Advanced Research Cosmology Program

Department of Physics and Astronomy

University of Victoria

P.O. Box 3055 STN CSC

Victoria B.C., Canada V8W 3P6

Quasi-spherical light cones are lightlike hypersurfaces of the Kerr geometry that are asymptotic to Minkowski light cones at infinity. We develop the equations of these surfaces and examine their properties. In particular, we show that they are free of caustics for all positive values of the Kerr radial coordinate . Useful applications include the propagation of high-frequency waves, the definition of Kruskal-like coordinates for a spinning black hole and the characteristic initial-value problem.

PACS numbers: 0240, 0420

1 Introduction

The Kerr geometry is a vacuum spacetime with a Weyl tensor of Petrov type D. According to the Goldberg-Sachs theorem [1], it therefore possesses two congruences (ingoing and outgoing) of shear-free lightlike geodesics. Historically, these congruences played an essential role in the discovery of the Kerr solution [2], because the metric takes a simple explicit form in coordinates adapted to them.

In the original [2],[3] (Eddington-Kerr) form of the metric, the ingoing congruence consists of parametric curves of the Eddington- Kerr coordinate , often referred to as an “advanced time”. This nomenclature can be misleading, because hypersurfaces of constant are actually spacelike, not lightlike, if the Kerr rotational parameter is nonzero. (The lightlike congruences are “twisting”, i.e., not orthogonal to any hypersurfaces, and hence not tangent to lightlike hypersurfaces if ).

Thus, the Eddington-Kerr form of the metric, and the Boyer-Lindquist form [3] derived from it by an elementary coordinate transformation, correspond to a “threading” [4] (i.e., a one-dimensional foliation) of the manifold by twisting lightlike geodesics. However, a “slicing” (three dimensional foliation), in particular a light-like slicing, is for many purposes more advantageous and often corresponds more closely to the physics.

Light cones of the Kerr geometry have not previously been a subject of systematic study to our knowledge. Perhaps this stems in part from a feeling that such hypersurfaces would quickly develop caustics because of the twist inherent in the metric. Yet as characteristics these surfaces obviously play a key role in the physics of Kerr black holes, e.g in the propagation of wave fronts and high-frequency waves, and in characteristic initial value problems.

In this paper we first derive the general solution for axisymmetric lightlike hypersurfaces of the Kerr geometry (Sec. 2). We then focus on a particular foliation (invariant under time-displacement) by (either ingoing or outgoing) “quasi-spherical” light cones, defined as degenerate 3-spaces whose spatial sections are asymptotically spherical at infinity. These hypersurfaces become the standard Minkowski light cones when the Kerr mass parameter vanishes; if , , they are the familiar spherical light cones, , of Schwarzschild spacetime (where is the “tortoise” coordinate). Quite generally, for arbitrary and , they have the remarkable property of being free of caustics for all positive values of the Kerr radial coordinate .

These appealing features suggest that coordinates adapted to quasi- spherical surfaces should be especially convenient and useful for the description of the Kerr geometry. The drawback (and it is considerable) is that the metric is no longer expressible in an explicit elementary form, because the new coordinates are elliptic functions of the Boyer-Lindquist coordinates.

We now briefly outline the contents of the paper. We begin in Sec. 2 by solving the eikonal equation for axisymmetric lightlike hypersufaces of the Kerr geometry. The general solution is expressed in terms of integrals of elliptic type, and involves an arbitrary function of one variable, to be fixed when the “initial” (e.g. asymptotic) shape of the surface is given. (If the geometry becomes flat and the integrals can be reduced to elementary form; in Sec. 3 we digress briefly to illustrate this.) Some general properties of axisymmetric lightlike surfaces are developed in Sec. 4. In Sec. 5, guided by the results of Sec. 3 for Minkowski light cones, we make the special choice of appropriate for asymptotically spherical light cones in Kerr space. The key result, that these are free of caustics for , is demonstrated in Sec. 6. In Secs. 7 and 8 we show how the “quasi-spherical” coordinates , defined by these light cones can be used to construct high-frequency solutions to the wave equation and Kruskal-like coordinates for Kerr black holes. Sec. 9 summarizes the limiting cases that can provide serviceable approximations for the elliptic functions which appear in the quasi-spherical form of the metric. Finally, in Sec. 10 we present some results of numerical integrations for the inward evolution of the light cone generators.

2 Axisymmetric lightlike hypersurfaces

The equation of an arbitrary axially symmetric lightlike hypersurface of the Kerr geometry is expressible in terms of elliptic integrals as we now proceed to show.

We write the Kerr metric in its standard (Boyer-Lindquist) form




and we note the useful identities


The equation


represents an axisymmetric lightlike hypersurface (ingoing for , outgoing for ) if , i.e., if


It is easy to obtain a particular (separable) solution of (6) depending on two arbitrary constants (a “complete integral”) by adding and subtracting an arbitrary separation constant on the right -hand side. Let us define


(Note the useful identity


which follows from (3).) A complete integral


of (6) is then obtained by integrating the exact differential


at fixed . When (10) is integrated, a second, additive integration constant appears which we shall denote as , where is an arbitrary function.

We next proceed in the usual way to promote this complete integral, depending upon the arbitrary constants and , to a general solution involving an arbitrary function.

In (9), is a function of three independent variables , and (not counting and ), and a more complete expression for its differential is


where is the partial derivative


Its explicit form may be taken to be


Up to now, we have taken to be a constant, i.e., in (11). But we achieve the same effect (i.e., (11) still reduces to (10) ) even when provided we require that . In other words: the function , given by (9) with now a function , remains a solution of (6) provided its extra dependence on through does not change the algebraic form of the differential (10). This will indeed be so if we impose the constraint


This condition fixes the dependence of on for any given choice of . Thus we now have a general solution, depending upon an arbitrary function .

The explicit form of the general solution (9) is then


where the radial dependence has been arranged to ensure convergence of the definite integral at its upper limit. When performing the integrations in (13) and (15), we are allowed to treat as merely a passive constant parameter – it is in fact a function of the limits of integration ,, not of the integration variables – with the constraint (14) imposed a posteriori. It is possible, though not especially illuminating or useful, to express each of these integrals in terms of standard elliptic integrals.

3 The case : light cones in Minkowski space

As a simple illustrative example (and because we shall need some of the results later), we specialize in this section to the case . The Kerr line-element reduces to the metric of flat spacetime expressed in terms of oblate spheroidal coordinates – related to Cartesians by


The second integral in (13) can be reduced to the same general form as the first integral when . Assume that we are in a domain where . Setting


and making the substution


we find


Thus the two integrals in (13) can be combined into a single integral, to give




and the form of is given by the “unprimed” version of (18); equivalently,


The function is arbitrary. As an example, let us consider the simplest choice, . The functional dependence corresponding to this choice is determined by the constraint , which requires that by inspection of (20). Thus (22) gives


by (16), showing that is the spherical polar angle. With known from (23), it is straightforward to integrate (10) to obtain


Thus is the usual spherical radius, and our solutions of (6) are in this case the Minkowski light cones.

4 Axisymmetric lightlike hypersurfaces: general properties

Returning to the general case, we shall now derive a number of results applicable to all axisymmetric lightlike hypersurfaces of Kerr.

Since the function is determined by the constraint , the partial derivatives of can be read off from (13):


It follows from (10) and (25) that , i.e. that and are orthogonal with respect to the intrinsic 2-metric


of the spatial sections of the Kerr geometry. Since is independent of and , it follows further that is constant along the lightlike generators, i.e.,


If and are adopted as coordinates in place of and , the 2-metric (26) becomes


where we have made use of (10), (25) and (8). The identity (4) can be re-arranged in the form




is the Bardeen or ZAMO angular velocity which characterizes orbits having zero angular momentum. This enables us to recast the Kerr metric (1) in the form




from which the null character of the 3-spaces is directly evident. The degenerate intrinsic metric of these 3-spaces is


showing that the generators rotate with angular velocity relative to stationary observers at infinity. (This is directly obvious from (27), which shows that for an axisymmetric hypersurface). From (33) we see that caustics will develop when the degenerate volume element tends to zero, i.e. when (recalling that has a fixed value along each generator).

The integrability of (25) for the exact differential provides a condition on the integrating factor . This takes the form of an evolution equation for along the generators. A straightforward calculation, which calls upon (7), (8), (13) and (25) itself, yields


where the subscript indicates that the partial derivative is being taken at fixed .

Inversion of the differential relations (10) and (25) gives


which will be of use in subsequent sections.

5 Quasi-spherical light cones

From here on we confine attention to a special class, “quasi-spherical light cones”, defined as lightlike hypersurfaces which asymptotically approach Minkowski spherical light cones at infinity. Define an angle by


In Minkowski space, the generators of light cones are radial straight lines, suggesting (as we confirmed in Sec. 3) that is the spherical polar angle. For , the oblate coordinates become indistinguishable from spherical coordinates, so we have the condition


for spherical light cones in Minkowski space. This asymptotic condition must therefore also hold for quasi-spherical surfaces in Kerr space. This fixes the arbitrary function in (13). We conclude that


generates the solution for quasi-spherical surfaces. The radial function is given by (15) with (compare (21) with )


The equation of the hypersurface is then , with and the function determined by the constraint .

6 No caustics for positive

We now prove the rather remarkable result that quasi-spherical light cones are free of caustics for all positive values of the Kerr radial coordinate .

This is trivially true if , when these surfaces are simply light cones in Minkowski space with vertices at the spatial origin, represented by , or in oblate spheroidal coordinates according to (16). We shall prove that it is true a fortiori for by effectively showing that when is larger, the generators converge less rapidly as .

As noted in Sec. 4, formation of a caustic along a generator is signalled by


We consider in turn the behaviour of each factor in (40) along an ingoing generator, so that is positive and decreasing, with fixed. Because of the equatorial symmetry we need only consider “northern” generators, i.e., we may assume (which specifies the initial, asymptotic value of when ) to be acute, and positive, at least initially. (Note from (10) that equatorial symmetry of implies .)

Since decreases with at fixed according to (25), the factor must increase and remain positive. Any possible caustic in the Kerr positive- sheet cannot arise from the behaviour of .

We proceed to consider the other factors in (40). From the definition (7),


We shall further show that


Thus, none of the three factors , and can reach zero sooner for positive than they do in flat space, and hence they do not reach zero for any positive value of .

To establish (42), we note that the function is determined by the vanishing of as given by (38). Taking the differential at fixed , and using (7) gives


which is manifestly positive. Turning to , this is defined as a function of by according to (25). Inverting the order of partial differentiation and using (38),






Each term in (46) is manifestly positive. This completes the proof.

The fact that adding mass to the Kerr field should reduce the focusing of ingoing radial light rays is at first sight paradoxical, but can be explained by the circumstance that the material source is not at the origin, but located on the singular equatorial ring , (see (16)). The source has a peculiar, “demi-pole” structure [5], made possible by the double-sheeted structure of the Kerr manifold: a ring of positive mass in the sheet is bonded to a ring of negative mass in the sheet . Light rays heading inwards toward the origin in the sheet are deflected outward by the ring and defocused. Repulsive effects have become dominant by the time they pass through the disk the rays are then refocused and form a caustic in the sheet . This description is more than just hand-waving: the Keres Newtonian analogue model of the Kerr field [6] has a source structure and equations of motion closely resembling Kerr (apart from frame dragging), and displays precisely this behaviour.

Figure 1 (a result of numerical integrations described in Sec. 10) shows the ingoing generators mapped onto the flat background of the Kerr-Schild decomposition, using the rectangular coordinates defined in (16). (Effects of frame dragging are not shown). According to the Keres-Kerr model, the gravitational “force” should become repulsive for , and, indeed, the generators have inflection points near this radius.

Null generators of the quasi-spherical light cones
projected onto the Cartesian plane for
Figure 1: Null generators of the quasi-spherical light cones projected onto the Cartesian plane for , . The azimuthal twist (not shown in this two-dimensional representation) depends on the choice of azimuthal coordinate; it is zero for the coordinate introduced in Sec. 8. In order to follow the generators through the equatorial disk and into the negative- sheet, the “southern half” () of the positive- sheet has been cut away, and replaced by the “northern” half () of the negative- sheet. Note that only well within the outer horizon does the mass of the spacetime significantly affect the behaviour of the generators (which would continue as straight lines intersecting the origin if were zero). The caustic emanates from the singularity , (if ) and creeps inwards toward the axis and “downwards” toward increasingly negative values of .

7 Eikonal solution of the wave equation

Since the quasi-spherical light cones are characteristics of the wave operator, the coordinates are well adapted for representing asymptotically spherical high-frequency modes.

The wave equation for on the Kerr background (31) is


where gives the area element (degenerate volume element) on the light cone (33).

Introducing the ansatz


into (48), we find




has no explicit -dependence. We can re-express in (50) as a partial derivative with respect to . Defining an angular function


and recalling (35),we find


Therefore (50) can be rewritten


If is large, we can apply an iterative procedure to (54) and (51) to develop the solution for in inverse powers of . In the lowest approximation, we simply equate the coefficient of in (54) to zero. This yields the “eikonal approximation”


with an arbitrary function . By linear superposition we can build from this an arbitrary high-frequency solution


where is an arbitrary function of its three arguments varying rapidly with time.

From (55) and (53) we see that crests of high-frequency azimuthal (-dependent) waves twist about the axis with angular frequency , the ZAMO angular velocity. Because and depend on , propagation of azimuthal waves produces latitude-dependent phase shifts. The latitude-dependence is, however, always bounded, even when at horizons.

In particular, wave-tails propagating inwards from the event to the Cauchy horizon of a spinning black hole experience a blueshift which is modulated by an oscillatory, latitude-dependent factor when – an effect first noted by Ori [7]. However, since is no larger than the multipole order , this effect is more than offset by the natural power- law decay of the higher multipole wave-modes with advanced time , . It does not affect the uniformity of the leading- order divergence of blueshift at the Cauchy horizon, where (the inner-horizon surface gravity) is a constant. These features, which are expected to extend at least qualitatively to waves having (initially) lower frequencies, are important when considering the back-reaction of blueshifted wave-tails on the geometry near the Cauchy horizon [8].

8 Kruskal coordinates

It is straightforward to transform the Kerr metric (31) into a Kruskal-like form. We introduce retarded and advanced times and , and associated Kruskal coordinates and , by the definitions


Here, is the surface gravity of the horizon under consideration, defined for the outer () and inner () horizons by


Then (31) becomes




The first () term is manifestly regular at the horizon sheets and . The last term, involving the Boyer-Lindquist coordinates and , is not. We therefore define the advanced and retarded angular coordinates , by


where was defined in (52). It is straightforward to show that


with defined as in (43). Thus the last term in (59) can be expressed in a manifestly regular way as


with , . In (63) we can choose either sign for , depending on which sheet of the horizon is of interest. For example, and are regular on the “upward” (future) sheets of both outer and inner horizons, with the nice bonus feature that is constant over each ingoing light cone and is constant along each ingoing generator.

Is there a single azimuthal coordinate which regularizes the metric simultaneously on both past and future sheets of (say) the outer horizon, including the bifurcation surface? Since takes a constant value over this horizon, a possible choice for the desired coordinate is , and in fact agrees with on the future sheet and on the past sheet. But develops the undesirable features characteristic of rigidly rotating axes in the outer regions of the space.

We briefly mention that following standard methods one could construct a Penrose diagram of the Kerr spacetime using the Kruskal-like coordinates (60), valid up to formation of the caustic surface. Such a diagram would not look any different from the usual textbook examples [9] except that a single 2-dimensional diagram is all that would be needed to illustrate the causal structure of the spacetime. This is because the ingoing () and outgoing () lightlike congruences intersect at surfaces of constant , represented by a single point on a diagram where the compactified coordinate system is derived from (60). The causal future of observers on the surface of intersection is entirely contained within the future-directed wedge of , (as is evident from (31) by noting that when the line-element is spacelike).

9 Limiting cases and approximations

The quasi-spherical coordinates and , and the coefficients in the quasi-spherical forms (31) and (59) of the Kerr metric are complicated elliptic functions of the Kerr-Boyer-Lindquist coordinates ,. It may therefore be useful to record here the simple forms these functions reduce to in the two opposite limiting cases, and .

(i) , : This was studied in Sec. 3, and we simply list the results.


(ii) , : This is just Schwarzschild spacetime, and one readily finds


Comparison with numerical results described in the next section suggest that the , case functions can provide serviceable analytic approximations outside of the outer horizon for broad ranges of and . For example, with and , all of the quantities except in (64), (65) and (66) differ by no more than roughly 1 to 3 percent from the true (numerically integrated) values, outside of . ( converges only logarithmically to the Minkowski value in the limit , and of course diverges at the horizon).

10 Numerical evolution of the generators

In Figure 1 we illustrated the behaviour of the generators and the nature of the caustic that forms in the negative sheet of the Kerr manifold. The curves were obtained by numerically integrating the evolution equations of the null generators of the hypersurface:


with affine parameter . In particular


To detect the formation of a caustic (see 40) we track the evolution of , and to reconstruct surfaces of constant in the plane we calculate the change of along the generators. If we treat as the independent variable, from (70), (15), (25) and (34) we find



At the horizons and its derivative with respect to diverge. These divergences are coordinate singularities which can be avoided numerically by subtracting off the infinities that occur there. Thus define as


where [] is the surface gravity at the outer()[inner()] horizon as defined in (58). So we actually integrate along the generators and retrieve from the result using (73).

For asymptotic intitial conditions for and , we use the relations listed in (64) and (65) for a sufficiently large initial . Using (43) and (46) one can show that and for large along a given generator. The initial condition for is arbitrary ( is still a solution to (6)).

Figure 2 below is a close-up within the inner horizon of the case illustrated in Figure 1, showing the projections of surfaces of constant onto the plane. This is the only region where surfaces start showing significant deviations from the spheres of the massless scenario. Note that the caustic surface does not coincide with a hypersurface slice : following the generators inward a caustic first develops where the hypersurface meets the ring singularity, after which it quickly “unravels”. We only present plots for and ; except for variations in scale there are no qualitative differences in the shapes of the curves and surfaces for arbitrary non-zero and positive .

Null generators and surfaces of constant
Figure 2: Null generators and surfaces of constant projected onto the Cartesian plane (). This is a close-up of Figure 1 focusing on the region between the inner horizon and caustic. As in Figure 1 the northern half of the negative- sheet replaces the southern half of the positive- sheet to illustrate the evolution of the generators through the equatorial disk.

11 Concluding Remarks

Despite the complexity of the coordinate transformations linking them to the familiar Boyer-Lindquist coordinates, the quasi-spherical coordinates , introduced in this paper and the light cones associated with them provide new and useful insights into the structure of the Kerr geometry and wave propagation in Kerr spacetime. We anticipate that they will find increasing use as their special advantages become apparent.

Acknowledgements This work was supported by NSERC of Canada and by the Canadian Institute for Advanced Research.


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