While reading Barrett O’Neill’s book The Geometry of Kerr Black Holes, it struck me that the text would be much more informative if the illustrations were live and fully interactive. This applies particularly to the depictions of basic parts of the metric structure in the second chapter. This page provides the interactive visualizations that can be presented in a modern browser via a suitable library, here MathCell. Consider this a prototype for textbooks of the future.

The Kerr metric generalizes the Schwarzschild metric to describe a rotating black hole. It can be written rather simply in Boyer-Lindquist coordinates as

$ds2 =Δρ2 [dt-a sin2θdφ ]2 -sin2θ ρ2 [(r2 +a2)dφ -adt ]2 -ρ2Δ dr2 -ρ2 dθ2$

with the variable combinations

$ρ2=r2 +a2 cos2θ Δ=r2 -2mr+a2$

Expanding the squared quantities, the metric can also be written

$ds2 =[1 -2mr ρ2] dt2 -ρ2Δ dr2 -ρ2 dθ2 -[r2 +a2 +2mra2 ρ2 sin2θ] sin2θ dφ2 +4mra ρ2 sin2θ dφdt$

The parameter m measures the mass of the black hole and the parameter a its angular momentum per unit mass. Setting $a=0$ recovers the Schwarzschild metric for a black hole without rotation.

There are three different rotation types of Kerr black hole, determined by the relative values of the two parameters:

 $0<{a}^{2}<{m}^{2}$ slowly rotating Kerr black hole (slow Kerr) ${a}^{2}={m}^{2}$ extreme Kerr black hole ${a}^{2}>{m}^{2}$ rapidly rotating Kerr black hole (fast Kerr)

In the interactive visualizations that follow, the mass parameter will be set equal to one for simplicity. The different rotation types will then be distinguished by their behavior above and below $a=1$ as well, with a range from zero to two covering essential differences.

The horizons of a rotating black hole, its points of no return for infalling matter and energy, are given by the radial values where the combination Δ is equal to zero. The quadratic behavior of this combination is easily visualized:

The slow Kerr has two horizons, an inner and an outer. In the Schwarzschild limit, the inner horizon coincides with its singularity at the origin. The extreme Kerr has a double root for $a=m=1$ and so only one horizon. The fast Kerr has no horizons, leaving the inner structure of the black hole exposed.

One significant way the Kerr metric differs from the Schwarzschild is that it does not fail for $r=0$ unless one also has $θ=π2$ . That means that one can in principle continue through the origin to another spatial region where $r<0$ . It also implies that the singularity of a Kerr black hole can be described as a ring rather than a point.

These features can be visualized relative to one another by displacing all radial variables outward by a unit, but scaling the interior parts of the black hole by the parameter a to emphasize that the singularity is not passable in the Schwarzschild limit. The two horizons, along with the sphere at the origin and the region beyond it, look like this:

The sphere in red represents the origin $r=0$ which is mathematically traversable: real-world travel through it is another matter entirely. The inner horizon collapses into the singularity in the Schwarzschild limit and converges with the outer horizon for the extreme Kerr. Since the fast Kerr has no horizons, the ring singularity is naked.

The spacetime at the horizons actually has intrinsic curvature as discussed in Section 9 of this reference, so that the horizons should perhaps more properly be visualized as oblate ellipses. That will be left as a possible future addition.

The Kerr metric displays interesting behavior when the coefficient ${g}_{tt}$ changes sign. The surface

$ρ2-2mr =r(r-2m) +a2cos2θ$

defines what is known as the ergosphere, inside of which the formerly temporal variable becomes spacelike. This implies that physical objects cannot remain stationary relative to a distant observer and must rotate with the black hole. Since the nature of a geodesic does not change along its course, a physical object can follow a timelike geodesic into and out of the ergosphere, taking the enforced rotation into account. The interval along the geodesic remains positive (for this choice of metric signature) due to a contribution from the off-diagonal component ${g}_{t\phi }$ .

Visualized in direct coordinates with the singularity unscaled at the origin, the ergosphere looks like this:

A cross section is used to display both the inner and outer surfaces of the ergosphere. In the Schwarzschild limit the ergosphere coincides with the outer horizon.

An interesting circumstance not discussed in O’Neill’s book is that the ergosphere remains for a fast Kerr black hole. The inner and outer surfaces of the ergosphere merge into one surface for this case, and there are clearly regions above and below the singularity that are outside of the ergosphere. This indicates one would be able hover above the singularity of a fast Kerr black hole without being forced to rotate, if a naked singularity is possible.

Since negative values are allowed for the radial variable, the metric function ${g}_{\phi \phi }$ can also become negative, changing the nature of the metric over a small region. It can be shown that causality violations are possible within this region, so that it is called the time machine. The isosurface defining the time machine is

$ρ2(r2 +a2) +2mra2 sin2θ =r4 +a2(1 +cos2θ) r2 +2ma2 sin2θ r +a4 cos2θ$

One way to visualize it is displacing the radial variable by an amount larger than the angular momentum parameter. The ring singularity is then a circle at the same displacement and the time machine sits just inside it:

In this interactive graphic the ring singularity is not scaled to vanish in the Schwarzschild limit, so that should be kept in mind. The cross section of the time machine becomes thinner as the angular momentum decreases, finally becoming a thin spherical shell as the ring singularity shrinks into the Schwarschild point singularity.

O’Neill curiously does not include any depiction of the time machine, even though it is relatively simple to produce. Moving on, consider the graphs and plots in the fourth chapter characterizing geodesics.

The first-order geodesic equations can be written relatively cleanly as

$ρ2 t· =[(r2 +a2 )2Δ -a2 sin2θ ]E -2maLr Δ ρ2 φ· =2maEr Δ +[1 sin2θ -a2Δ ]L ρ4 r·2 =R(r) =[(r2 +a2)E -aL]2 -Δ[Q +(L-aE )2 +qr2] ρ4 θ·2 =Θ(θ) =Q-[a2 (q-E2) +L2 sin2θ ]cos2θ$

The new constants along geodesics appearing here are energy E, angular momentum L and the Carter constant Q. The parameter q determines the nature of the geodesic: for the signature being used, $q=1$ for timelike geodesics, $q=0$ for null geodesics and $q=-1$ for spacelike geodesics. This parameter has an opposite sign to that used by O’Neill.

A slight rearrangement of the last geodesic equation

$ρ4 θ·2 +[a2 (q-E2) +L2 sin2θ ]cos2θ =Q$

produces what can be interpreted as an energy equation for the angular behavior of geodesics. The first ‘kinetic’ part of the equation is not strictly the same as classical kinetic energy, since ${\rho }^{2}$ is not constant like a mass parameter, but the behavior of this term is similar to kinetic energy. As long as ${\rho }^{2}$ remains nonzero, then ${\rho }^{4}{\stackrel{·}{\theta }}^{2}\to 0$ implies $\stackrel{·}{\theta }\to 0$ , which is equivalent to turning points in this variable.

For nonzero geodesic angular momentum, the shape of the angular potential is controller by the relative values of ${L}^{2}$ and the combination

$E¯ =a2 (E2 -q)$

When $\overline{E}\le {L}^{2}$ the potential has a single minimum at $π2$ and geodesics traverse angular values centered symmetrically about the equator. When $\overline{E}>{L}^{2}$ the single minimum splits into two located at the angles

$sin4θ =L2 E¯ → θ=sin −1 (L2 E¯ )14 , π-sin −1 (L2 E¯ )14$

with the corresponding minimum value

$Qmin =−E¯ +2L E¯ -L2$

The behavior is easily visualized:

Input parameters have been kept separate to allow more variation, with the controlling value $\overline{E}$ indicated as well. The level of Q can be adjusted to demonstrate the limits on angular motion imposed by the turning points of the potential, just as for classical motion in a potential well.

For $Q\ge 0$ geodesics again traverse angular values centered symmetrically about the equator. When input parameters allow $Q<0$ , geodesics are restricted in angular range to one hemisphere or the other and do not cross the equator: they are then called vortical geodesics. They are a distinctive feature of the relativistic rotation of the Kerr black hole.

The angular potential for zero intrinsic geodesic angular momentum, $L=0$ , no longer has vertical asymptotes at zeros of the sine function. It looks like this:

For all values of $\overline{E}$ there are now positive values for the Carter constant for which the angle θ increases continuously. There are also two distinct sets of potential well restrictions in angular range: about the equator $θ=π2$ when $\overline{E}<0$ , and around the poles $\theta =0$ and $\theta =\pi$ when $\overline{E}<0$ . Curiouser and curiouser...

Rearranging the third geodesic equation to exhibit an energy equation for radial behavior of geodesics is a bit problematic. Expanding the right-hand side of the equation,

$ρ4 r·2 =(E2 -q)r4 +2mqr3 +[a2 (E2-q) -L2 -Q]r2 +2m[Q +(L-aE )2] -a2Q$

ones sees that the constant term is not independent of the remainder of the ‘potential.’ It is simpler in this case to assume an energy of zero and inspect where the horizontal r-axis crosses the potential to determine limits on radial movement.

O’Neill plots the right-hand side as is and describes limits based on intervals of the horizontal axis beneath the potential. This is counterintuitive and confusing, so the visualization here will plot the negative of the right-hand side to allow the usual interpretation of potential turning points. With the notation for the equation

$ρ4 r·2 +∑ k=04 ckrk =0$

and mass set to unity as usual, the potential and a couple of its critical coefficients are

The coefficient c3 is only zero for null geodesics. Manipulation of the inputs can display both bound geodesics and flyby geodesics, as well as transits of the ring singularity.

Another device used by O’Neill is an “r-L” plot, an isoplot of $R\left(r,L\right)$ for various values of the other input parameters. As description of all the applications made of this type of plot might become tedious, let a single example suffice for the time being:

Manipulation of the inputs indicates that there are a variety of output configurations to be explored.

Numerical integration of the geodesic equations does not figure highly in O’Neill but would be quite interesting in an interactive context. That will be left for a separate presentation.