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
as
where
and
are defined by
and
and
are two real constants. We assume
, with the Schwarzschild case
and the extreme Kerr case
as limits. Then the Kerr metric describes a rotating
uncharged black hole of mass
and specific angular momentum
. (The case
, 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
. It is joined to the region
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 [181]. 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 [306]). Detailed
accounts on Kerr geodesics can be found in the books by
Chandrasekhar [54
] and
O’Neill [248].
Fermat
geometry.
The Killing vector field
is not timelike on that part
of the domain of outer communication where
. 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 [313, 160]. 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
, in addition to the
constants of motion
and
associated with the Killing
vector fields
and
. This reduces the
lightlike geodesic equation to the following first-order form:
Here an overdot denotes differentiation with respect to an affine
parameter
. This set of equations allows writing
the lightlike geodesics in terms of elliptic integrals [16
]. Clearly,
and
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
and
change sign is crucial for numerical
evaluation of these integrals and, thus, for ray tracing in the
Kerr spacetime (see, e.g., [330
, 285
, 109
]). 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
is constant (see Figure 21). These spherical
light rays are unstable with respect to radial perturbations. For
the spherical light ray at radius
the constants of
motion
,
, and
satisfy
The region
is the Kerr analogue of the “light
sphere”
in the Schwarzschild spacetime.
Light cone.
With the help of Equations (124, 125, 126, 127), the past light cone
of any observation event
can be explicitly written in
terms of elliptic integrals. In this representation the light rays
are labeled by the constants of motion
and
. In accordance with the general idea of
observational coordinates (4), it is desirable to
relabel them by spherical coordinates
on the
observer’s celestial sphere. This requires choosing an orthonormal
tetrad
at
. It is convenient to choose
,
,
and, thus,
perpendicular to the hypersurface
(“zero-angular-momentum observer”). For an
observation event in the equatorial plane,
, at radius
, one finds
As in the Schwarzschild case, some light rays from
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
. In the Kerr case, it corresponds to
light rays that asymptotically approach a spherical light ray in
the region
of Figure 21. The constants of
motion for such light rays are given by Equation (129, 130), with
varying between its extremal values
and
(see again Figure 21). Thereupon,
Equation (131) and Equation (132) determine the
celestial coordinates
and
of those light rays that approach a spherical light
ray in
. 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,
, 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
(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. [147]), where
intersections of the light cone with hypersurfaces
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
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
, 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 [282] 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.
Lensing by a Kerr black hole.
For an observation event
and a light source with
worldline
, 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
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
, the observer sees an
infinite sequence of images; for this sequence, the travel time
(e.g., in terms of the time coordinate
) 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
.
If
meets the caustic of
’s past light cone, the image is not an Einstein
ring, unless
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
. The case that
is a circular timelike
geodesic in the equatorial plane of the extreme Kerr metric,
, was treated by Cunningham and Bardeen [67
, 17
]. This example is of
relevance in view of accretion disks. Viergutz [330
] developed a
formalism for the case that
has constant
and
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
is an integral
curve of
and that
and
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 [329] 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
and
(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 [276
]. 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 [277
, 307
] and (ii) by showing
the visual appearence of an accretion disk [277, 281, 307]. 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, 175, 169, 109
]). The spectrum can
be calculated in terms of a transfer
function that was tabulated, for some
values of
, in [65]
(cf. [330
, 331]). 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 [107].) 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 [108]. 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, 276, 317]
(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 [176, 330, 285, 109].
5.8.0.2 Notes on Kerr naked singularities and on the
Kerr-Newman spacetime.
The Kerr metric with
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].