Kerr spacetime

5.8 Kerr spacetime

Next to the Schwarzschild spacetime, the Kerr spacetime is the physically most relevant example of a spacetime in which lensing can be studied explicitly in terms of the lightlike geodesics. The Kerr metric is given in Boyer–Lindquist coordinates $(r,𝜗, φ,t)$ as

where $ϱ$ and $Δ$ are defined by

and $m$ and $a$ are two real constants. We assume $0 < a < m$ , with the Schwarzschild case $a = 0$ and the extreme Kerr case $a = m$ as limits. Then the Kerr metric describes a rotating uncharged black hole of mass $m$ and specific angular momentum $a$ . (The case $a > m$ , which describes a naked singularity, will be briefly considered at the end of this section.) The domain of outer communication is the region between the (outer) horizon at

and $r = ∞$ . It is joined to the region $r < r+$ in such a way that past-oriented ingoing lightlike geodesics cannot cross the horizon. Thus, for lensing by a Kerr black hole only the domain of outer communication is of interest unless one wants to study the case of an observer who has fallen into the black hole.

Historical notes.
The Kerr metric was found by Kerr [180 ]. The coordinate representation (118 ) is due to Boyer and Lindquist [36 ]. The literature on lightlike (and timelike) geodesics of the Kerr metric is abundant (for an overview of the pre-1979 literature, see Sharp [305 ]). Detailed accounts on Kerr geodesics can be found in the books by Chandrasekhar [54 ] and O’Neill [248 ].

Fermat geometry.
The Killing vector field $∂ t$ is not timelike on that part of the domain of outer communication where $2 ϱ(r,𝜗 ) ≤ 2mr$ . This region is known as the ergosphere. Thus, the general results of Section 4.2 on conformally stationary spacetimes apply only to the region outside the ergosphere. On this region, the Kerr metric is of the form (61 ), with redshift potential

Fermat metric

and Fermat one-form

(Equation (122

) corrects a misprint in [265 ], Equation (66), where a square is missing.) With the Fermat geometry at hand, the optical-mechanical analogy (Fermat’s principle versus Maupertuis’ principle) allows to write the equation for light rays in the form of Newtonian mechanics (cf. [7 ]). Certain embedding diagrams for the Fermat geometry (optical reference geometry) of the equatorial plane have been constructed [312 , 159 ]. However, they are less instructive than in the static case (recall Figure 11

) because they do not represent the light rays as geodesics of a Riemannian manifold.

First integrals for lightlike geodesics.
Carter [53 ] discovered that the geodesic equation in the Kerr metric admits another independent constant of motion $K$ , in addition to the constants of motion $L$ and $E$ associated with the Killing vector fields $∂φ$ and $∂t$ . This reduces the lightlike geodesic equation to the following first-order form:

( ) 2 2 2 2 ϱ(r,𝜗 )2t˙= a L − Ea sin2𝜗 + (r-+-a-)-((r--+-a-)E-−--aL), (124 ) Δ (r) L − Ea sin2 𝜗 (r2 + a2)aE − a2L ϱ(r,𝜗)2 ˙φ =-------2------+ ------------------, (125 ) sin 𝜗 Δ (r) 4 2 (L − Ea sin2 𝜗)2 ϱ(r,𝜗) ˙𝜗 = K − --------2-------, (126 ) 4 2 (sin2 𝜗 2 )2 ϱ(r,𝜗) ˙r = − K Δ + (r + a )E − aL . (127 )

Here an overdot denotes differentiation with respect to an affine parameter $s$ . This set of equations allows writing the lightlike geodesics in terms of elliptic integrals [16

]. Clearly, $˙ 𝜗$ and $r˙$ may change sign along a ray; thus, the integration of Equation (126

) and Equation (127

) must be done piecewise. The determination of the turning points where $𝜗˙$ and $˙r$ change sign is crucial for numerical evaluation of these integrals and, thus, for ray tracing in the Kerr spacetime (see, e.g., [329

, 285

, 108

]). With the help of Equations (126

, 127

) one easily verifies the following important fact. Through each point of the region

there is spherical light ray, i.e., a light ray along which $r$ is constant (see Figure 21

). These spherical light rays are unstable with respect to radial perturbations. For the spherical light ray at radius $rp$ the constants of motion $E$ , $L$ , and $K$ satisfy

L 2 2 2rpΔ (rp) aE- = rp + a − -r-−--m--, (129 ) p K-- -4r2pΔ(rp)- E2 = (rp − m )2. (130 )

The region $𝒦$ is the Kerr analogue of the “light sphere” $r = 3m$ in the Schwarzschild spacetime.

View Image

Figure 21: The region $𝒦$ , defined by Equation (128

), in the Kerr spacetime. The picture is purely spatial and shows a meridional section $φ = constant$ , with the axis of symmetry at the left-hand boundary. Through each point of $𝒦$ there is a spherical geodesic. Along each of these spherical geodesics, the coordinate $𝜗$ oscillates between extremal values, corresponding to boundary points of $𝒦$ . The region $𝒦$ meets the axis at radius $rc$ , given by $r3c − 3mr2c + a2rc + ma2 = 0$ . Its boundary intersects the equatorial plane in circles of radius $rph +$ (corotating circular light ray) and $ph r−$ (counter-rotating circular light ray). $ph r±$ are determined by $ph ph 2 2 r± (r± − 3m ) = 4ma$ and $r+ < rp+h< 3m < rp−h< 4m$ . In the Schwarzschild limit $a → 0$ the region $𝒦$ shrinks to the light sphere $r = 3m$ . In the extreme Kerr limit $a → m$ the region $𝒦$ extends to the horizon because in this limit both $rph→ m +$ and $r → m +$ ; for a caveat, as to geometric misinterpretations of this limit (see Figure 3 in [16

]).

Light cone.
With the help of Equations (124 , 125 , 126 , 127 ), the past light cone of any observation event $pO$ can be explicitly written in terms of elliptic integrals. In this representation the light rays are labeled by the constants of motion $L ∕E$ and $K ∕E2$ . In accordance with the general idea of observational coordinates (4 ), it is desirable to relabel them by spherical coordinates $(Ψ, Θ )$ on the observer’s celestial sphere. This requires choosing an orthonormal tetrad $(e0,e1,e2,e3)$ at $pO$ . It is convenient to choose $e1 ∝ ∂𝜗$ , $e2 ∝ ∂φ$ , $e3 ∝ ∂r$ and, thus, $e0$ perpendicular to the hypersurface $t = constant$ (“zero-angular-momentum observer”). For an observation event in the equatorial plane, $𝜗O = π ∕2$ , at radius $r O$ , one finds

( ) 2 2 ∘ ------ L- --(rO(rO-+-a-)sinΘ--sin-Ψ-−-arO---Δ-(rO)- E = a + r ∘ Δ-(r-) + 2ma sinΘ sin Ψ , (131 ) ( O O------ )2 r2 r2+ a2 − ar ∘ Δ (r ) sin Θ sin Ψ 3 2 2 2 2 -K- = -O----O-∘--------O------O-------------- − rO-(r∘O(rO-+-a-)-+-2ma--)cos--Θ-. (132 ) E2 rO Δ (rO) + 2ma sin Θ sin Ψ rO Δ (rO) + 2ma sinΘ sin Ψ

As in the Schwarzschild case, some light rays from $pO$ go out to infinity and some go to the horizon. In the Schwarzschild case, the borderline between the two classes corresponds to light rays that asymptotically approach the light sphere at $r = 3m$ . In the Kerr case, it corresponds to light rays that asymptotically approach a spherical light ray in the region $𝒦$ of Figure 21

. The constants of motion for such light rays are given by Equation (129

, 130

), with $rp$ varying between its extremal values $ph r+$ and $ph r−$ (see again Figure 21

). Thereupon, Equation (131

) and Equation (132

) determine the celestial coordinates $Ψ$ and $Θ$ of those light rays that approach a spherical light ray in $𝒦$ . The resulting curve on the observer’s celestial sphere gives the apparent shape of the Kerr black hole (see Figure 22

). For an observation event on the axis of rotation, $sin𝜗O = 0$ , the Kerr light cone is rotationally symmetric. The caustic consists of infinitely many spacelike curves, as in the Schwarzschild case. A light source passing through such a caustic point is seen as an Einstein ring. For observation events not on the axis, there is no rotational symmetry and the caustic structure is quite different from the Schwarzschild case. This is somewhat disguised if one restricts to light rays in the equatorial plane $𝜗 = π∕2$ (which is possible, of course, only if the observation event is in the equatorial plane). Then the resulting 2-dimensional light cone looks indeed qualitatively similar to the Schwarzschild cone of Figure 12

(cf. [146 ]), where intersections of the light cone with hypersurfaces $t = constant$ are depicted. However, in the Kerr case the transverse self-intersection of this 2-dimensional light cone does not occur on an axis of symmetry. Therefore, the caustic of the full (3-dimensional) light cone is more involved than in the Schwarzschild case. It turns out to be not a spatially straight line, as in the Schwarzschild case, but rather a tube, with astroid cross-section, that winds a certain number of times around the black hole; for $a → m$ it approaches the horizon in an infinite spiral motion. The caustic of the Kerr light cone with vertex in the equatorial plane was numerically calculated and depicted, for $a = m$ , by Rauch and Blandford [285

]. From the study of light cones one may switch to the more general study of wave fronts. (For the definition of wave fronts see Section 2.2.) Pretorius and Israel [283 ] determined all axisymmetric wave fronts in the Kerr geometry. In this class, they investigated in particular those members that are asymptotic to Minkowski light cones at infinity (“quasi-spherical light cones”) and they found, rather surprisingly, that they are free of caustics.

View Image

Figure 22: Apparent shape of a Kerr black hole for an observer at radius $rO$ in the equatorial plane. (For the Schwarzschild analogue, see Figure 14

.) The pictures show the celestial sphere of an observer whose 4-velocity is perpendicular to a hypersurface $t = constant$ . (If the observer is moving one has to correct for aberration.) The dashed circle is the celestial equator, $Θ = π∕2$ , and the crossing axes indicate the direction towards the center, $Θ = π$ . Past-oriented light rays go to the horizon if their initial direction is in the black disk and to infinity otherwise. Thus, the black disk shows the part of the sky that is not illuminated by light sources at a large radius. The boundary of this disk corresponds to light rays that asymptotically approach a spherical light ray in the region $𝒦$ of Figure 21

. For an observer in the equatorial plane at infinity, the apparent shape of a Kerr black hole was correctly calculated and depicted by Bardeen [16 ] (cf. [54

], p. 358). Earlier work by Godfrey [141 ] contains a mathematical error.

Lensing by a Kerr black hole.
For an observation event $pO$ and a light source with worldline $γS$ , both in the domain of outer communication of a Kerr black hole, several qualitative features of lensing are unchanged in comparison to the Schwarzschild case. If $γS$ is past-inextendible, bounded away from the horizon and from (past lightlike) infinity, and does not meet the caustic of the past light cone of $pO$ , the observer sees an infinite sequence of images; for this sequence, the travel time (e.g., in terms of the time coordinate $t$ ) goes to infinity. These statements can be proven with the help of Morse theory (see Section 3.3). On the observer’s sky the sequence of images approaches the apparent boundary of the black hole which is shown in Figure 22 . This follows from the fact that

the infinite sequence of images must have an accumulation point on the observer’s sky, by compactness, and
the lightlike geodesic with this initial direction cannot go to infinity or to the horizon, by assumption on $γS$ .

If $γS$ meets the caustic of $pO$ ’s past light cone, the image is not an Einstein ring, unless $pO$ is on the axis of rotation. It has only an “infinitesimal” angular extension on the observer’s sky. As always when a point source meets the caustic, the ray-optical calculation gives an infinitely bright image. Numerical studies show that in the Kerr spacetime, where the caustic is a tube with astroid cross-section, the image is very bright whenever the light source is inside the tube [285 ]. In principle, with the lightlike geodesics given in terms of elliptic integrals, image positions on the observer’s sky can be calculated explicitly. This has been worked out for several special wordlines $γ S$ . The case that $γ S$ is a circular timelike geodesic in the equatorial plane of the extreme Kerr metric, $a = m$ , was treated by Cunningham and Bardeen [67 , 17 ]. This example is of relevance in view of accretion disks. Viergutz [329 ] developed a formalism for the case that $γS$ has constant $r$ and $𝜗$ coordinates, i.e., for a light source that stays on a ring around the axis. One aim of this approach, which could easily be rewritten in terms of the exact lens map (recall Section 2.1), was to provide a basis for numerical studies. The case that $γS$ is an integral curve of $∂t$ and that $γS$ and $pO$ are at large radii is treated by Bozza [38 ] under the additional assumption that source and observer are close to the equatorial plane and by Vazquez and Esteban [328 ] without this restriction. All these articles also calculate the brightness of images. This requires determining the cross-section of infinitesimally thin bundles with vertex, e.g., in terms of the shape parameters $D+$ and $D −$ (recall Figure 3 ). For a bundle around an arbitrary light ray in the Kerr metric, all relevant equations were worked out analytically by Pineault and Roeder [277 ]. However, the equations are much more involved than for the Schwarzschild case and will not be given here. Lensing by a Kerr black hole has been visualized (i) by showing the apparent distortion of a background pattern [278 , 306 ] and (ii) by showing the visual appearence of an accretion disk [278 , 282 , 306 ]. The main difference, in comparison to the Schwarzschild case, is in the loss of the left-right symmetry. In view of observations, Kerr black holes are considered as candidates for active galactic nuclei (AGN) since many years. In particular, the X-ray variability of AGN is interpreted as coming from a “hot spot” in an accretion disk that circles around a Kerr black hole. Starting with the pioneering work in [67 , 17 ], many articles have been written on calculating the light curves and the spectrum of such “hot spots”, as seen by a distant observer (see, e.g., [75 , 12 , 174 , 168 , 108 ]). The spectrum can be calculated in terms of a transfer function that was tabulated, for some values of $a$ , in [65 ] (cf. [329 , 330 ]). A Kerr black hole is also considered as the most likely candidate for the supermassive object at the center of our own galaxy. (For background material see [106 ].) In this case, the predicted angular diameter of the black hole on our sky, in the sense of Figure 22 , is about 30 microarcseconds; this is not too far from the reach of current VLBI technologies [107 ]. Also, the fact that the radio emission from our galactic center is linearly polarized gives a good motivation for calculating polarimetric images as produced by a Kerr black hole [44 ]. The calculation is based on the geometric-optics approximation according to which the polarization vector is parallel along the light ray. In the Kerr spacetime, this parallel-transport law can be explicitly written with the help of constants of motion [61 , 277 , 316 ] (cf. [54 ], p. 358). As to the large number of numerical codes that have been written for calculating imaging properties of a Kerr black hole the reader may consult [175 , 329 , 285 , 108 ].

5.8.0.2 Notes on Kerr naked singularities and on the Kerr–Newman spacetime.

The Kerr metric with $a > m$ describes a naked singularity and is considered as unphysical by most authors. Its lightlike geodesics have been studied in [47 , 49 ] (cf. [54 ], p. 375). The Kerr–Newman spacetime (charged Kerr spacetime) is usually thought to be of little astrophysical relevance because the net charge of celestial bodies is small. For the lightlike geodesics in this spacetime the reader may consult [48 , 50 ].