This page presents a number of videos and still images of Kerr black holes which were computed using a program I wrote. For the impatient, jump straight to the videos; or read on for the explanations, including a (hopefully accessible) course on black holes.

- Contents
- Introduction
- A not-so-short course on black holes
- Conventions for all images and videos
- Still images
- Videos
- The
`kerr-image`

program

A black hole in general is a region of space from which nothing, not even light, can escape the gravitational pull (or, to put things in a different way, clearer though perhaps not quite accurate, space-time itself is “falling” inward faster than the speed of light, so one would need to go faster than light merely to stand still something which is impossible for a material object). See further below for much more detailed explanations, and/or see the Wikipedia article on the subject for a start. But for the moment, suffice it to say that of the not-so-many different kinds of black holes in (theoretical) existence, the Schwarzschild black hole is the simplest, non-rotating one, and the Kerr black hole is one which adds angular momentum, i.e., an axis of rotation. While the Schwarzschild black hole is (somewhat deceptively) simple, having as only feature a spherical event horizon which acts as a limit of no-return beyond which all things are inexorably attracted to the central singularity where they are crushed, the Kerr black hole exhibits a tantalizingly complex geometry with two nested horizons and a ring-shaped singularity which act as gates to even more surprising features (travel to different universes, a so-called “negative space”, and a kind of time machine).

These features (although largely theoretical, and presumably absent
in real black holes), along with the catchy name wormhole

, have
made the black hole a popular theme in popular imagination or
science-fiction.

Strangely enough, however, it appears that **nobody has,
hitherto, computed any actual pictures or videos** of what one
would see, when traveling through such features. (I know
that Alain
Riazuelo has computed similar videos, e.g., close views of
Schwarzschild black holes, traveling through a Reissner-Nordström
black hole, and some images of Kerr black holes, but none like the
ones presented here.) So the point of this page is to demonstrate
what the Kerr black hole and its intriguing features *look
like*. This is a question that had been obsessing me ever since I
haad learned about Kerr black holes.

But before I get to the videos themselves, there are a number of things I have to explain if any of this is to make scientific sense (of course, feel free to skip ahead if you just want to see things).

A black hole in general is a region of space from which nothing, not even light, can escape the gravitational pull (or, to put things in a different way, clearer though perhaps not quite accurate, space-time itself is “falling” inward faster than the speed of light, so one would need to go faster than light merely to stand still something which is impossible for a material object). The limiting surface beyond which no return is possible is known as an event horizon.

The properties of black holes, to be discussed in more detail below, are described in the general framework of general relativity: mathematically idealized black holes (meaning, in particular, that they are eternal and stationary) are represented by a number of exact solutions of Einstein's equations (i.e., the equations of general relativity) such as the Schwarzschild metric solution and the Kerr metric solution. The Schwarzschild metric is the general relativistic description of any static and spherically symmetric gravitational field outside of the matter creating it, and, in particular, when extended up to and beyond the event horizon it describes a static and spherically symmetric black hole. The Kerr metric adds one parameter of a handful that a black hole can possess, namely angular momentum, and therefore describes a stationary rotating black hole (but with no electric charge, let alone any of the more exotic things like the NUT parameter). This is physically relevant to the geometry of space-time, because the black hole's rotation drags space-time along with it through a process known as the Lense-Thirring effect, so that not only particles coming close to the black hole are gravitationally attracted by it (and once through the event horizon can only fall inward) but they are also made to rotate in the black hole's direction with respect to distant stars (and beyond a surface known as the static limit, they can no longer remain at a fixed angle). Furthermore, the Kerr black hole, in its mathematicall idealized version, exhibits a number of distinctive geometric features, having two concentric horizons, a ring-shaped singularity beyond which lies a “negative space”, and more remarkably, connects an infinite number of universes together (acting as a so-called wormhole).

**The Schwarzschild radius.** The Schwarzschild black
hole may seem dull. Its main feature is a single event horizon in the
shape of a sphere. The radius `r` of that sphere, known as
the Schwarzschild radius, is proportional to the black
hole's mass `M`, and is given by
`r`=2`G`·`M`/`c`²,
where `G`
is Newton's
constant and `c` is
the speed of
light: this is usually written `r`=2`M` because
relativists use a system of units with `c`=1 and
even `G`=1 (these constants can be inferred by dimensional
analysis), and the proportionality constant
2`G`/`c`² translates to 2.953250077km per solar
mass, meaning that one would have to concentrate the (nonrotating)
mass of the sun in a sphere just under 3km in radius (the
Schwarzschild radius of the sun) in order to turn it into a black
hole. For this reason, black hole masses are often expressed in
kilometers (though there is then an ambiguity as to whether the value
given is the Schwarzschild radius 2`M` or the mass itself
which is one half that). For larger black holes, it can even be
expressed in seconds, meaning light-seconds, in other words, by
dividing once more by the speed of light
(using `G`/`c`³ instead
of `G`/`c`²): the Sun then has a Schwarzschild
radius of 9.850981898µs (that's just under 10 microseconds);
this value is also the order of magnitude of the typical time,
experienced by the falling observer, to reach the central singularity
once it has crossed the horizon (the maximal possible time is π/2
times the Schwarzschild radius-as-time, and for an observer with zero
velocity at infinity and zero angular momentum it is 2/3 times the
Schwarzschild radius-as-time).

The typical masses of actual black holes in the Universe are thought to fall in two distinct ranges: stellar black holes, resulting of the collapse of a massive star, are in the range of a few solar masses (typically 4 to 15), while galactic black holes, found in the nuclei of nearly all galaxies, are in the range of hundreds of thousands to billions of solar masses. The mass of the black hole used in all the videos in this page is about one million solar masses, which determines the speed of the videos (to double the mass, simply play the videos two times slower, etc.).

Inside the horizon of a Schwarzschild black hole, there is no
feature of interest: space and time are, one might be said,
interchanged, and all matter is inexorably drained inward (just like
all matter, outside the black hole, must progress toward the future,
inside it must progress toward the center, and it would be just as
futile to try to stop this progress as to try to stop the flow of
time), and reaches the central singularity in a finite (and
absolutely bounded, see above) amount of proper time. This central
singularity is a point (a point in *space* for outside
observers, but a point in *time* for falling observers once
they have crossed the horizon) where the curvature of space-time
becomes infinite, and general relativity therefore breaks down: what
actually happens there must be described by another (presently
unknown) theory such as string theory or loop quantum gravity or
something else, but one thing is certain, is that whatever matter goes
there is violently torn to pieces by tidal forces.

**A side note about tidal forces, the equivalence principle,
and the size of black holes.** There are three effects
naturally associated with gravity: (1) the gravitational
potential, which is the total sum of the gravitational forces to be
overcome in order to escape to infinity, (2) the gravitational
force itself, and (3) the tidal effect, which is
the *difference* in gravitational force from one place to
another. These three effects, caused by a mass `M` at a
distance `r`, are roughly given
as `G`·`M`/`r`, `G`·`M`/`r`²
and `G`·`M`/`r`³ respectively:
in other words, they are proportional to the mass `M` causing
them, but inversely proportional (1) to the distance, (2) to
the *square* of the distance (Newton's law), and (3) to
the *cube* of the distance to that mass. Black holes are
formed when the effect (1) becomes too large
(the escape
velocity, which is one form of effect (1), becomes larger
than the speed of light), which is the reason why the Schwarzschild
radius is proportional to the mass of the object (not its square root
or cube root). Effect number (2), the gravitational force
itself, can always be neutralized by freely falling (this is basis of
the theoretical axiom in general relativity known as
the equivalence
principle, and in practice this is used on Earth in the form
of parabolic
flight, or unwittingly by people in an elevator that breaks down):
this is also true around a black hole: an observer *freely
falling* toward a black hole, even within the event
horizon, *experiences no gravitational force* and can only
detect the gravitational effect by observing distant stars. Effect
number (3), however, is real, and this is the actual physical
expression of the *curvature* of space-time (in mathematical
terms,
the Riemann
tensor): this is the effect (the difference of gravitational
attraction between the head and feet of a falling observer) which
would destroy someone coming too close to the singularity of a black
hole. This effect decreases as `M`/`r`³,
however, so *at the horizon*,
where `r`∝`M` [is proportional to

], we
the effect decreases as 1/`M`²: in other words, the
tidal effects felt at the horizon (or at any
given `r`/`M` ratio) are all the weaker when the
black hole is large. Interestingly, the point at which tidal effects
become too large for a human to withstand (say, of the order of
10`g` between head and feet, i.e., ∼60 in SI
units of 1/s²) is independent of the size of the black hole: it
occurs, for an observer freely falling in a Schwarzschild black hole
approximately 0.1s before the singularity is reached. Now of course
for a stellar black hole with a typical Schwarzschild radius of
fractions of milliseconds, a human observer will be torn apart well
before the horizon is crossed; but for a galactic black hole, the
journey can be survived well beyond the horizon. A similar note is
that the black hole “density”, although that concept is
not very meaningful, is not necessarily large: just like tidal forces,
it is in `M`/`r`³ and therefore decreases as
1/`M`² if we choose to limit the black hole at the
horizon.

**On time and the gravitational redshift.** Even
though the journey from horizon to central singularity takes a finite
amount of time for the falling observer experiencing it, distant
observers never get to see this: the distant observer never sees
anything falling inside the event horizon, because this would mean
that photons have managed to escape this region for the object to be
seen; so what the distant observer sees is that the infalling object's
fall slows down as it nears the horizon, and becomes infinitely slow
instead of crossing it: this is because photons emitted just before
the crossing of the horizon have a harder and harder time reaching the
distant observer, and therefore take a longer and longer time to do
so—they are said to
be gravitationally
redshifted to infinity. In fact, from the distant observer's
point of view, nothing beyond the horizon really exists, and a
physical black hole's mass could be said to consist of the mass of the
collapsing star, ever frozen in the last stages of its collapse.

Note that, from the infalling observer's perspective, on the other
hand, nothing remarkable happens when the horizon is crossed: the
distant observer does *not* appear infinitely sped up or
blueshifted, the view of the outside world does *not* appear
compressed to a point, and the black hole, in fact, appears to be
still ahead (as the videos on this page will show). (Also note that
the infalling observer can still see his feet when they are ahead
toward the black hole's center: even though light rays cannot go up
from his feet to his eyes, his eyes are falling faster toward the feet
than some photons coming from his feet, and this makes no difference
in virtue of the equivalence principle.)

What is true is that observers who somehow manages to *stand
still* (e.g., by firing their rockets) near the black hole's
horizon will see the black hole occupying more and more of their
visual field, until at the horizon (when it is no longer possible to
remain still) it occupies all of it but for a single point behind.
This is part of a general phenomenon in relativity, a form of
the aberration
of light: objects *toward* which one is going at
relativistic speeds appear *smaller*, whereas objects *away
from which* one is going appear *larger*; and just to stand
still near a black hole one is, in effect, moving away from it at
relativistic speeds (as the red queen explained to Alice, it takes a
lot of running just to stand still).

**On the photon
sphere.** In this respect, there is one interesting distance
around a Schwarzschild black hole, which is larger than the
Schwarzschild radius: it is located at `r`=3`M`,
i.e., one-and-a-half Schwarzschild radii, and it is known as
the photon sphere. This is the distance at which, for an
observer standing still, the black hole occupies precisely one half of
the visual field. This is because it is the distance at which photons
themselves will orbit the black hole circularly (this orbit is
unstable, however). In other words, the horizon is the distance at
which photons emitted *outward* from the black hole are
standing still, whereas the photon sphere is the distance at which
photons emitted *orthogonally* from the black hole remain at
this constant distance and circle around the black hole in an orbit:
but since light rays always *appear* to be straight, to an
observer standing still on the photon sphere, the photon sphere seems
like an infinite plane, with the black hole occupying half of space
beyond it, and the outside world occupying the other half of space.
Coming closer to the black hole, between the photon sphere and the
horizon, the observer standing still will see an inversion of
convexity: the outside world seems like a sphere whereas the black
hole occupies the rest of the visual field. This phenomenon also
manifests itself in the centrifugal force: at a distance greater than
the photon sphere, the centrifugal force felt by an observer rotating
around the black hole is directed outward, as expected, and can
therefore be made to balance the gravitational force which is directed
inward, resulting in a circular orbit. Inside the photon sphere, it
is no longer possible to orbit the black hole: the centrifugal force
is directed *inward*, and the faster one goes the more one
feels pulled toward the black hole (however, it is still possible to
stand still by exerting a force, e.g., by firing one's rockets; but
the closer to the horizon, the more inefficient this force, and just
like no force can cause a material object to go faster than light, no
force can keep material particles still once they have crossed the
horizon).

A special note should also be made about the last stable
orbit which, for a Schwarzschild black hole, is situated at a
distance of `r`=6`M`, or three times the
Schwarzschild radius (twice further than the photon sphere). This is
the distance at which a circular orbit has the smallest total energy.
Closer than this (but still outside the photon sphere), orbits are
still possible, but they are no longer stable: any perturbation on the
orbit will either send the orbiting particle falling into the black
hole or escaping away from it (either to a more distant bound orbit,
or even to infinity if the initial circular orbit was tight
enough).

Now introduce rotation and we get a new object: the Kerr black hole.

The Kerr black hole is described by two numbers: its
mass `M`, just like a Schwarzschild black hole (and we will
still call Schwarzschild radius the quantity 2`M`,
even though the horizon is smaller than that), and secondly its
angular momentum `J` or, as is generally given, its angular
momentum per unit mass `a`=`J`/`M`.
Actually, it is the
ratio `a`/`M`=`J`/`M`², or
angular momentum per mass *square*, which will play the major
role: this is sometimes called the fraction of maximality,
and a black hole can only form when `a`/`M` is less
than 1 in natural units (which means `G`/`c` =
2.23×10^{−19} in SI units of
kg^{−1}·m^{2}·s^{−1};
or, if you will, the fraction of maximality is
(`c`·`J`)/(`G`·`M`²)).
For a fraction of maximality greater than 1, the Kerr metric no longer
describes a black hole, but a “naked” singularity, one
which is not surrounded by horizons (they have been, in intuitive
terms, ripped apart by the centrifugal force),
and it
is thought that such things (at least the one described by the
Kerr metric) do not form in reality (it is nevertheless interesting
from a mathematical perspective): presumably a collapsing star
evacuates its excess angular momentum in the form of gravitational
waves so that it can actually collapse to a black hole.

It should be noted that typical astronomical bodies, say, the Earth, have a fraction of maximality well above 1 (in the case of the Earth, it is around 740). This does not contradict the above conjecture, it merely says that the Earth is far from being a black hole, and if it were to collapse to a black hole, it would probably lose much of its angular momentum before doing so (or else, its angular momentum would prevent it from so collapsing).

Unlike the Schwarzschild black hole, the Kerr black hole no longer has spherical symmetry: it has a well-defined axis of rotation, and an equatorial plane perpendicular to this axis. The horizon is no longer a sphere, as will be explained below.

At what speed is the black hole rotating? There is no easily
defined exact answer to this question, but one can at least give an
order of magnitude: the parameter `a` divided by the square
of the Schwarzschild radius,
i.e., `a`/(4`M`²)=`J`/(4`M`³),
is a ballpark value for “the angular velocity of the black
hole”, at least near the (outer) horizon. With this in mind,
the black hole in the videos on this page, which has a fraction of
maximality of 80% and a Schwarzschild radius 2`M` of
10 (light-)seconds, can be said to be rotating at 0.04rad/s, or 1
revolution every 160 seconds approximately (but keep in mind that this
is only an order of magnitude: for example, the singularity might
rather be said to rotate at 1 revolution every 25 seconds).

**Frame-dragging and precession.** What are the
effects of this rotation? The frame-dragging,
or Lense-Thirring, effect states that, in essence, any
rotating and gravitating mass “drags” space-time along
with it. This dragging can be expressed in various ways: one way is
as a force (sometimes known as a gravitomagnetic force,
because it is the gravitational analog of magnetic forces if the
ordinary Newton law is compared to the electrostatic force) which
applies to particles approaching the rotating mass (in its equatorial
plane, say) and tends to drag them along the rotation (if my
computations are correct, the value of this force, in the lowest
approximation, is
2`G`·`J`·`m`·`v`/(`c`²·`r`³)
where `J` is the angular momentum causing the gravitomagnetic
force, `m` is the mass of the test particle, `v` is
its velocity and `r` is the distance; a more general form for
a particle not approaching directly the rotating mass in its
equatorial plane should be analogous to the force exerted on a test
particle by
a magnetic dipole). Another way to express this is that, because
this force exerts a torque on a gyroscope, it causes a gyroscopic
precession, which is analogous to
the Larmor
precession of electrically charged gyroscopes in a magnetic field,
except that gravitomagnetic precession applies equally to *all*
gyroscopes: the angular velocity of precession
is `G`·`J`/(`c`²·`r`³).
Now since a gyroscope is the canonical way to determine what a fixed
direction is, one can say that the rotating mass actually drags
space-time along with it, at this rate of precession: an observer
might be *inertially non-rotating*, i.e., might believe from
its own measurements that it is not rotating (as witnessed by a
gyroscope or any other means), but still observe the distant stars
rotate at the frame-dragging rate indicated above.

This matter of precession is made more confused by
the fact that there is (at least!) another cause of gyroscopic
precession in general relativity, known
as de Sitter
precession or the geodetic effect. This one does
depends not on the fact that the gravitating mass rotates (i.e., it
takes place even around a Schwarzschild black hole), but on the fact
that the *observer* is rotating around the gravitating mass.
To the lowest order, the rate of de Sitter precession
is: `G`·`M`·`ω`/(`c`²·`r`),
where `ω` is the angular velocity of the observer
around the gravitating mass `M`. The intuitive way to
imagine de Sitter precession is that the gravitating mass warps
the curvature of trajectories around it; in fact, we have already
mentioned the case of the photon sphere around a Schwarzschild black
hole, which appears to be flat (=an infinite plane) whether one looks
at it with light rays or whether one measures the (non-existing)
centrifugal force when going along it—another aspect of the fact
that the photon sphere appears flat is that a gyroscope pointing, say,
toward the black hole, will continue to do so when transported along
the photon sphere: so from the point of view of distant observers,
this gyroscope is precessing so as to always point toward the black
hole, and this is an aspect of de Sitter precession. The
electromagnetic analog of de Sitter precession
is Thomas
precession. Both the Lense-Thirring and the de Sitter
effects have been confirmed experimentally around the Earth by
the Gravity Probe B NASA mission,
and this
page does a good job at clarifying how the two relate. Around a
Kerr black hole, however, lower-order approximations are no longer
sufficient, and the Lense-Thirring and the de Sitter precession
effects are combined together in a way that makes it hardly meaningful
to separate the two.

**On the structure of the Kerr black
hole.** What about the influence of rotation on the structure
of the black hole? As the fraction of
maximality `a`/`M` increases, the (outer)
horizon ceases to be a sphere (it becomes oblate) and shrinks,
whereas *another* horizon, known as the inner
horizon, appears and grows. The outer and inner horizons are shown,
for a maximality fraction of 80%, in red and green on the diagram to
the right, which represents a cut of a Kerr black hole along a
meridian. Outside of the outer horizon, tangent to it at the poles
and bulging at the equator, there is another surface, known as
the static limit (shown in dashed red on the diagram to the
right): this is the limit past which the Lense-Thirring effect becomes
so large that it is no longer possible to remain motionless with
respect to distant stars. (There is also an inner static limit,
though its meaning is not quite as clear.) When the fraction of
maximality is greater than 1, there are no longer any horizons, and
the outer and inner static limits join at the poles to form a somewhat
torus-shaped region (actually, its section looks a bit like a pacman)
of non-staticity.

The standard terminology is to number I, II and III the regions of space which are respectively: (I) outside the outer horizon, (II) between the outer and inner horizons, and (III) inside the inner horizon (including “negative space”, see below). In region II, just as within the horizon of a Schwarzschild black hole, time and space coordinate exchange their roles, and any material particle is inescapably drawn forward. Within region III the traveler is again free to travel around (although strange things await us there, as will be explained).

Also interesting is what happens to the singularity: in the
Schwarzschild case, it was a single point at the center of the black
hole. In the Kerr case, it takes the shape of a *ring* on the
equatorial plane (represented by the fat yellowish dot on the cut
diagram to the right). Rather unexpectedly, it turns out to be
repulsive (although in a complicated way), and point particles falling
toward it will *not* hit it except if they are exactly on the
equator (and even then, there are conditions on angular momentum).
Furthermore, this ring is not just a singularity of infinite
curvature, it is also a *cut discontinuity* (as symbolized by
the purple segment on the cut diagram): by going *through* the
ring, one enters a region of space-time different from the region
which would be reached by going around it (typically in the space
between the ring singularity and the equator of the inner horizon).
This region (reached by going through the ring) is
called negative space,
because the `r` coordinate becomes negative there.

**What is the r coordinate anyway?** Here I
must digress on the meaning of the coordinates. The

distance to the black hole(not to the black hole's center, but to some loosely defined abstraction of the black hole, perhaps the ring singularity). For example, when we write that the observer is at, say, a distance of 1.5 Schwarzschild radii from the black hole, we really mean that

distanceis an abuse of language. The surfaces

[#] One striking phenomenon is that
these `r`=constant “spheres”, for sufficiently
small `r` (and in particular, the inner horizon
always) *have negative
(Gaussian)
curvature* at the pole, something which is obviously impossible
(at the axis) for a surface of revolution in Euclidean space.

**A note about the other coordinates:**
The `r` coordinate is one of the so-called Boyer-Lindquist
coordinates which are the ones in which the Kerr metric takes its
simplest form (but not the form in which it was originally discovered
by Kerr). The `θ` coordinate is a form of latitude
(well, colatitude, because as is usual in spherical coordinate
systems, the angle is measured down from the north pole); except that
it isn't really the angle for much the same reason that `r`
isn't really a distance: `θ` should rougly be
considered as a kind
of reduced
(co)latitude or the colatitude angle of
a spheroidal
coordinate system (π/2−`ν` in Wikipedia's
notation, whereas `r` would be analogous
to `a`·sinh(`μ`), where `a` has,
fortunately, the same meaning). The coordinates `t`
and `φ` also present their own difficulties: `t`
is supposed to be the time measured by distant observers
and `φ` the longitude measured by distant observers, but
while the general direction of these coordinates is clear (moving
in `t` or `φ` while keeping `r`
and `θ` fixed corresponds to the symmetries of the
black hole,
or killing
vectors), it is much less obvious how to synchronize `t`
and `φ` across different values of `r`
and `θ`. This is where Boyer-Lindquist coordinates,
Kerr ingoing coordinates and Kerr outgoing coordinates differ (they
have the same `r` and `θ` coordinates):
Boyer-Lindquist coordinates synchronize `t`
and `φ` by declaring that surfaces of equal `t`
and `φ` should be orthogonal to surfaces of
equal `r` and `θ`, whereas Kerr ingoing and
Kerr outgoing coordinates synchronize them by ingoing and outgoing
light rays respectively. Boyer-Lindquist coordinates are much more
symmetric, but they are ill-defined at the horizons: the `t`
and `φ` coordinates have a logarithmic singularity there,
essentially because of the fact that distant observers never see any
object cross the horizon to fall inside the black hole.

The
ring singularity acts a little like the magical door of fantasy
worlds: by going through it we end up in a different place than if we
go around it. If we go around it, nothing remarkable happens. But if
we go through it (I mean through the disk bounded by the ring, of
course: it is best to avoid the ring itself, where curvature is
infinite, if we do not wish to be crushed to something unnatural),
the `r` coordinate becomes negative, and starts measuring the
opposite of the distance to the black hole (which, from that side, no
longer looks like a black hole): this region of space-time is known
as negative space.

This negative space region is infinite, and, if we go far away in
it in any direction (for very negative values of `r`, that
is, away from the black hole), then space becomes flat again: so there
is, in effect, an infinite world tucked beyond the black hole's ring
singularity. This can be thought of as the black hole's flip side, or
evil twin brother; but it should be emphasized that there is nothing
strange about negative space in itself (deep negative space is just a
flat region of space, and, of course, from the point of view of
negative space, it is positive space which is lies beyond the black
hole's ring singularity). It is the negative side of the black hole
(or negative black hole—*not* to be confused with
the white hole which will be
described later), and the region near it, which is strange, not the
deep negative space.

One question I am not addressing is whether we
reach the *same* negative space by crossing the ring
singularity from north side or from the south side (or, in a somewhat
similar line of thought, if we enter negative space from the north, go
around the ring in negative space, then cross it again from the north,
do we re-enter the same positive space as we left). Mathematically,
the most natural answer to this question is yes

, because for
the Kerr manifold to be an algebraic variety in a certain sense
demands it; but general relativity, and the Kerr solution, is agnostic
about this, becaues it only makes prescriptions about the local
geometric properties of space-time, not about its global topology.
(In a certain sense, the question is meaningless, because the Kerr
metric is a mathematical abstraction and is *empty*, and it is
meaningless to ask whether two empty and identical regions of
space-time are actually *the same* or not. Real life black
holes might not have a negative space anyway.)

A first
surprise is that, in the negative side, the black hole is repulsive:
there are no orbits around it with negative `r`, and it takes
a considerable amount of energy to enter deep negative space
(essentially, the particle's rest mass energy divided by the black
hole's fraction of maximality, so the particle must be relativistic).
Another surprise is that that there are no horizons on the negative
side: seen from the deep negative space, the black hole is
a *naked* singularity. This is one reason why it is believed
that there are no negative black holes in our universe (i.e., black
holes for which *we* would be in the deep negative sapce). A
third surprising characteristic of the negative side of the black hole
is that it contains the Carter time machine: there is a
region, rougly in the shape of a torus having the ring singularity as
its inner equator, in which a material particle can travel
at *infinite* speed along the ring, thus returning to initial
position in space *and time* after a finite amount of proper
time, or even go back in time. It takes a tremendous amount of energy
and calibration to do this (I know it because I've tried to produce a
geodesic that does so, and failed), but in principle it is possible.
And, of course, this is one of the points where it should be
emphasized that the interior region of the Kerr metric is a
mathematical abstraction which probably does not describe real life
black holes accurately (more about
this later).

Suppose we have fallen inside a Kerr black hole, and do not wish to cross the singularity into negative space, because the purple color there does not go well with our complexion. Is there another way out? Or are we stuck forever inside the black hole (or inside region III, as one calls the region inside the inner horizon)?

The answer, perhaps surprisingly, is: yes, we can leave the region just as we entered it, and there is no difficulty in crossing the horizons. How can this be when I defined a black hole as a region of space which it is impossible to escape? To explain this, I have to admit that I have been lying the whole time about the very nature of the black hole: in the mathematical idealization represented by the Kerr metric, the exact same region of space (but not time) contains not only a black hole, but also its Tao counterpart, the white hole.

This should, really, have been obvious, for a number of reasons.
Here's one: the laws of physics (or, at least, general relativity) are
invariant under time-reversal. So if it makes sense for a particle to
fall toward a gravitating mass, crossing its event horizons, it must
make physical sense for it to go the other way. Or, to put things
differently: imagine a test particle dropped into a black hole with
zero initial velocity, and falling inside its horizons (its course is
more complicated, but not entirely unlike, the parabolic fall of a
pebble to Earth, taken from the summit of the parabola); now any free
fall trajectory has a well-defined *past* as well as a future,
and that past cannot be a bound orbit, so it *must* be coming
from the inside of the black hole (just as the past part of the
pebble's parabolic trajectory must have been going up before it could
be going down). But then what we have is not a black hole, it is a
white hole: the two are (roughly speaking) in the same region in
space, but in a different region in time.

Or consider this: what do we see when we look at a black hole? In
a real black hole, the answer is simply (and rather
logically): black

. The reason for this is that what we are
seeing are the last remnants of the collapsing star that formed the
black hole, ever frozen in time in the very last stages before the
horizon was formed, and near infinitely redshifted to the point where
they are undetectable—if we try to project light rays back in
the past, to determine what we could be seeing in that direction,
these rays extend far to the past as they approach the horizon, and
eventually trace back to the point where the black hole was formed.
But what about the sort of mathematically idealized black hole that I
have been talking about all along? By backtracing the light rays from
the direction of the black hole, we eventually hit the horizon, but we
do so infinitely back in time, and the horizon we hit is not a black
hole horizon: it is a white hole horizon, or antihorizon,
since light is leaving it. The black hole horizon itself, on the
other hand, is undetectable until we cross it.

So the scenario is now roughly this: when we look at the black
hole, we are actually not seeing the black hole, we are seeing the
white hole that resides in the same place. We see the white hole's
outer horizon and, behind that, its inner horizon (actually, for
reasons to be described later on, there are *two* white hole
inner horizons visible behind the outer horizon), taking up exactly
the same space as the outer horizon because any light ray crossing one
must necessarily cross the other; even beyond this, we can glimpse the
inner region (region III), including the negative space and,
bounding that, the singularity. What happens if we fall into the
black hole? The white hole horizon does not suddenly get closer, it
still appears ahead of us, and we will never actually reach it,
because it is impossible to cross a white hole horizon while going
in. However, the black hole horizon suddenly becomes visible: it
takes the form of a “bubble”, springing from the white
hole horizon and suddenly engulfing us, and seemingly growing faster
than the speed of light. (Of course, this is only a manner of
speaking: in reality, horizons are not visible at all, so nothing
remarkable happens when we cross the horizon.) The place where this
bubble seems to appear is the place where we have crossed the horizon,
but at the point in time where white hole becomes black hole, the
birthpoint of the black hole.

The white hole should not be confused with the repulsive side of
the black/white hole found in negative space. The terminology is
disastrous, and things are made even worse by the fact that some
people do (either by mistake or by an unfortunate change in an already
bad terminology) use the term white hole

to describe the
negative space side… In the sense in which I use it (and which
seems standard), the white hole is attractive, not repulsive: this
does not contradict the fact that things are (or can be) coming out of
it, just like a pebble thrown in the air is attracted toward the Earth
but still moving away from it.

I'm afraid things have become a bit confused
at this point: the cure for that is to introduce the Carter-Penrose diagram,
which is what is shown on the right. A Carter-Penrose diagram displays a
two-dimensional surface of space-time, here with `θ`
and `φ` (angular coordinates) kept constant (here it is
probably best, for various reasons, to imagine that the diagram
represents the black hole's axis in space and time). But rather than
using the `t` coordinate as one axis and the `r`
coordinate as the other (we have already noted that `r` plays
the role of time and `t` the role of space in region II,
i.e., inside the horizons), the Carter-Penrose diagram systematically
arranges so that time will be vertical (flowing upward, say), and
space will be horizontal. This is done by rectifying
the light cones
so that rays of light emitted by a point on the Carter-Penrose diagram will
always go at angles of 45° (toward the upper left and the upper
right), and, of course, rays of light received by a point on the
Carter-Penrose diagram will always come from angles of 45°, from the
lower left and the lower right. So, in essence, the rules of the
Carter-Penrose diagram are this: any given point can view things which are
situated below it, between the two 45° lines crossing at that
point (the past light cone), and can travel to any point
situated above it, between the same two lines extended in the other
direction (the future light cone). Furthermore, the
Carter-Penrose diagram arranges so as to bring infinite space down to a
finite region.

(It should be said, however, that the Carter-Penrose diagram is not uniquely defined by these constraints: there is still a lot of freedom left to determine it. For example, the point in the center of a region of the penrose diagram does not have any special significance, and can be chosen arbitrarily.)

The Carter-Penrose diagram for the Kerr metric looks like the figure on
the right. Suppose we are in the region marked I

: then we can
see the region marked II (WH)

(the white
hole, i.e. the region between the two horizons in the white hole), and
we can navigate to the region marked II (BH)

(the black hole) and from then to the region marked III

(the
inside of the black hole, from which one can leave to another white
hole region, and so on). The point where the regions
marked II (WH)

and II (BH)

meet is the birthpoint

of
the black hole, it is the point from which we see the horizon bubble
appear when we enter the black hole. Generally speaking, squares
colored blueish are regions I (outside the horizons), squares colored
pinkish are regions III (inside the horizons, including negative
space), and squares colored light gray are the regions II of a white
hole whereas squares colored darker gray are regions II of a black
hole.

This diagram illustrates the following surprise: the
(mathematically ideal Kerr) black hole is actually not just a black
hole, but an infinite sequence of black hole and white hole
alternating in time, and connecting an infinite sequence of universes.
This is sometimes described as a wormhole. If we enter the
black hole's region III, we are free to leave through the white hole,
but the region I in which we arrive is a *different* region I
from the one we started in, i.e., a parallel universe. (Or, again,
it *could* be the same as the one we started from: general
relativity and the Kerr metric are agnostic about that.) If we
navigate from one region I (say, the one marked I

on the
diagram) to another (say, the one two squares immediately above it),
distant observers in the region I of origin will see us take a
seemingly infinite amount of time to cross the black hole's horizon,
so from their point of view we are entering the black hole beyond the
end of time; distant observers in the region I of destination can view
our entire journey, but if they try to compute the `t`
coordinate at which we left the white hole, they will find it to be
infinitely far back in the time. So, in a certain sense, traveling
through the wormhole from one region I to another means traveling
beyond infinity in time.

Now pause to consider the `r` coordinate: there is a point
in space and time (or rather, one for each `θ`
and `φ`), which I have called the birthpoint

of
the black hole, where outer horizons of black hole and white hole meet
(where the horizon bubble appears), and, in fact, meet transversely.
How is this possible if they are all at the same `r`
coordinate? The answer is that the `r` coordinate is
degenerate at the horizons: it is a bit like the
function `x`·`y` (product of both
coordinates) on the Euclidean plane: the curves
with `x`·`y`=constant are hyperbolæ
when the constant is nonzero, but when it is zero, the curve
degenerates to two intersecting lines; the outer horizons of black
hole and white hole are roughly similar to this situation, and so are
the inner horizons; indeed, lines of constant `r` on the
Carter-Penrose diagram will look like hyperbolæ in the vicinity of a
horizon crossing, lying entirely in region I, II or III according as
the constant is greater than the radius of the outer horizon, between
the two, or less than the inner.

Now there is one more feature
about this diagram which I should address: every region I and III has
a sibling on the other side of the central vertical axis. Is this
just a sign that we have eliminated two coordinates? (One might
believe, for example, that two symmetric regions I are simply the same
region on either side of the black hole, and although it is not
possible to pass from one to the other by going through the black
hole, one could go around it; but this is not the case.) No, this
duplicity corresponds to something real: if a traveler starts in the
region marked I

on the diagram, its sibling on the other side
of the diagram is a region which the traveler will see while in region
II (BH) (it appears as a kind of window past the
horizon bubble, in blue on the previous image), but which one can
never travel to; however, one might meet travelers from that region
which have also fallen into the black hole. (Interestingly, this
sibling region I also exists for the Schwarzschild black hole; the
Carter-Penrose diagram for the Schwarzschild black hole is rather simple,
with just two sibling regions I, a black hole region which is bounded
above by a singularity in the future, and a white hole region which is
bounded below by a singularity in the past.) Similarly, a traveler
entering the black hole from the region marked I

on the diagram
above has the choice, once in the region
marked II (BH)

, of which region III to enter,
either the one marked III

or its sibling: these regions are
genuinely different, and if two travelers enter one each, they will
not find each other (however, they are not permanently separated: they
can be reunited if they travel to a common region I, as they can).

How does one choose

which region III to
enter? I must admit that I do not completely understand this. If one
is traveling along a geodeisc (i.e., freely falling), then the
condition is simple: it is the ratio of angular momentum (as measured
along the black hole's axis: this is a conserved quantity) to energy
that determines whether the particle enters one or the other region
III. In general, what matters is whether the Boyer-Lindquist
coordinate `t` tends to +∞ or −∞
when `r` tends to the radius of the inner horizon, but that
doesn't enlighten us very much, because the
Boyer-Lindquist `t` isn't a natural coordinate for the
traveler in region II (although it does tell us that in each one of
the two regions III under consideration one can witness the entire
future history of the region I directly below it in the Carter-Penrose
diagram). When comparing two of the videos I made, it appears that
the region III one enters is determined by the fact that the
“bubble” of the inner horizon opens on one side or the
other (corotating or counterrotating side) of the window to the
sibling region I; however, this can't be the whole story, because one
can enter either region III even by following trajectory purely on the
black hole's north polar axis (though in this case only one is
accessible through a geodesic: for the other, one will have to
accelerate *away* from the black hole).

I have already said it, but I really must insist on the following
fact: the Kerr black hole, and specifically its interior, is a
mathematical idealizations representing an eternal and unchanging
black hole that has existed forever (and, in a certain sense, beyond
forever) and will exist forever. *Real black holes are not
believed to behave that way.* Now I am a mathematician and not a
physicist, so this is really not my domain of expertise, but I should
still say something about real black holes.

The Kerr solution is known by the No-Hair Theorem to reflect reality in region I (outside the outer horizon) and I believe it is thought to be accurate also for the black hole region II (between the inner and outer horizons). On the other hand, the white hole region II component in the past (i.e., the region of the Carter-Penrose diagram which is below and to the center of the region I one lives in) definitely does not exist in a real black hole: that place is occupied by the collapsing star whose mass gave existence to the black hole; and even if a white hole somehow managed to exist, it would be unstable. The situation is similar to the following: a light bulb falling on the floor and breaking to a thousand shards of glass spread over the place, or a thousand shards of glass coming together and jumping in the air to form a light bulb, are both equally possible from the point of view of newtonian physics since they are time-reversible; however, the former is a commonplace occurrence and the latter would rightly be held a miracle; black hole horizons and white hole (anti)horizons are somewhat analogous. More generally, the whole interior region (III) of the Kerr solution, along with its ring singularity and negative space, is thought to be unstable (see, e.g., this; the fact that the Kerr singularity is repulsive is probably a good hint), and there is no reason to believe that the interior of a real black hole should look like it.

What the interior of real black holes actually looks like, as far
as I know, remains an open question. It is *conceivable* that
it might give birth to baby universes

, for example, in which
case some form of the wormhole properties of the idealized Kerr black
holes might, in fact, be true (but nothing says the journey would be
pleasant).

See this page for a more knowledgeable account of what happens (or at least, what reasonably might happen) inside real black holes, and on the instability of region III.

The black hole represented in all the images and videos on this
page (except for the picture of the photon sphere of a Schwarzschild
black hole) is always the same: it has a maximality fraction of 80%
and, in the videos, a mass of about one million, and precisely
1015127, solar masses. This odd figure is simply there because I
chose a Schwarzschild radius 2`M` of exactly 10
(light-)seconds, or a mass `M` equivalent of 5 seconds;
actually, the mass is arbitrarily set to 1 unit and the videos are
computed in steps of 0.01 units of proper time, and there are 20
frames per second. Only the speed of the videos reveals the mass of
the black hole (e.g., play them two times slower and the black hole
now has a mass two times larger). The still images, of course, do not
come with an indication of scale, so the black hole mass could be
anything (but I still like to think that it's the same).

This gives a radius `r` of the external horizon, here
1.6`M`, of approximatively 2.4 million kilometers
(gigameters), or 0.016 astronomical units, or about 6 times the
earth-moon distance.

The maximality fraction of 80% translates to a rotation
parameter `a`=`J`/`M` of
3.6×10^{17}·m²/s, and an angular
momentum `J` of
7.3×10^{53}·kg·m²/s (this is
something like twenty billion times the total angular momentum of the
Solar system).

All the videos I computed are *inertial*, in the following
sense: (A) they follow a geodesic, which means the observer
is *freely falling* at all times (and, by virtue of the
equivalence principle, would experience weightlessness in the
spaceship), and (B) the direction of view, and the orientation of
view are parallel-transported, which means that they are stabilized
(e.g., with a gyroscope) and the observer does not experience any
rotation (centrifugal force).

I thought it would have broken the elegance of the
whole concept to have accelerating, or even rotating, observers.
(This is why, for example, I did not make a video showing a closed
time loop as can be found in the Carter time machine: I was unable to
compute a *geodesic* which crosses itself in space-time, and I
am not entirely sure whether it exists.) But there is also an element
of realism in the postulate (A): for an acceleration to make a
noticeable difference in the trajectory, it would have to be
comparable with one million gees, so if we think of the observers as
being human, they had better not be accelerating.

The images show a number of spheres (where by sphere

here I
mean a surface with `r`=constant; they are not really
spheres), including all horizons. All spheres are checkered in an
identical way, with twenty-four longitudinal stripes and twelve
latitudinal stripes (that is, actually, eleven latitudinal stripes
plus two patches at the poles counting as half-stripes), consistent
with the black hole's axis. The longitudinal stripes on the horizons
rotate with the black hole at the natural rate for the horizons
(which, for the particular black hole under consideration, is
0.05rad/s for the outer horizon and 0.2rad/s for the inner horizon).
The color scheme for the horizons is identical with that of the
Carter-Penrose diagram shown above:
red/orange/brown colors are outer horizons, green colors are inner
horizons; lighter colors are white hole horizons, darker colors are
black hole horizons. The blue (celestial) sphere is placed outside
the horizon at some distance (it is supposed to be at infinity; in
reality, it is not, but it is still pretty far), and it does not
rotate; actually, there are two different colors of blue to
distinguish regions I which are placed symmetrically in the Carter-Penrose
diagram. The purple sphere is placed at some distance in negative
space, and it also does not rotate.

To avoid things getting too crowded, a convention is made that only
four consecutive horizons are drawn: anything behind that (any light
ray which, when traced back, crosses more than four consecutive
horizons) is cut off and shown in black. Also shown in black are rays
which did not hit a sphere in a certain amount of time (time

meaning, here, integration steps *or* the affine parameter used
to parametrize light rays): this explains some black regions sometimes
appearing in the videos (see on
limitations below).

The images do not display any redshift or blueshift. This would not make any sense since the colors are simply conventional.

It goes without saying (in fact, the contrary would not even make any sense), but the images take into account the finite speed of light (and, even more obviously, the bending of light by the black hole: that's the whole point, isn't it?).

The `kerr-image`

program doesn't
do any anti-aliasing (to get that, one would have to compute a larger
image and scale down: this was done on some videos and on all the
still images): this explains why the videos are terribly jittery and
show unpleasant moiré effects.

The iteration limit has probably been set too low. This explains a number of black regions appearing on the videos.

The regions where two horizons cross is not handled correctly, because it requires the use of a special coordinate system that is very difficult to convert to. This explains why, wherever two horizons cross, a kind of snow pattern appears around the intersection: this is a numerical artefact. Also numerical artefacts are various kinds of traces that look like a kind of powder, typically near the singularity, or in the alignment of the black hole's axis, or sometimes in other places (I sometimes don't understand why they appear), and generally speaking whenever the edges seem indistinct.

I have provided all the necessary data files to reproduce the
videos using the `kerr-image`

program. For videos, the recipe for production is this: simply
cut the provided data file at every blank line, feed each fragment
to `kerr-image`

on the standard input and it will produce
the corresponding frame (as a PPM file) on the standard
output. Play these at 20 frames per second (there is one data block
per 0.01 black hole mass-as-time, so playing at 20fps
will give `M`=5s, or Schwarzschild radius of 10s, as
announced). You might wish to fiddle with the compilation parameters
of `kerr-image`

, though (e.g.,
increase `MAX_NBSTEPS`

and `MAX_PROPERTIME`

to
eliminate black regions, possibly decrease `BASIC_STEP`

to
improve image quality, increase `FARSPHERE_RADIUS`

in the
last stable orbit video (I used a value of 30); and, of course,
set `NBCOLS`

and `NBLINS`

to the desired image
size). I am not providing the rest of the video generation chain
(e.g., the way I combined the images and the video encoding command
lines and all that) because it's a mess of command lines and Perl and
shell scripts with paths or numeric values hardcoded all over the
place. (If someone really insists, I can send him something, but I
don't think it's worth it.) I'm also not providing the program used
to *compute* the data files in the first place (it is easy to
extract from the `kerr-image`

program a program which will
compute the trajectory of a material geodesic, however, that will not
solve the problem of parallel transport of the direction vectors),
because it too is a mess (and the lazy geodesic and last stable orbit
videos were, in fact, computed in a wholly different way).

This section reproduces images already shown elsewhere on this
page, so as to comment them a little and provide the input
to `kerr-image`

which can be used to reproduce them.

Here, the black (well, white) hole is observed
from a good distance (`r`=25`M`, i.e. 12.5
Schwarzschild radii away) and at a latitude of 11.4°, i.e., near
the equator. (The second image is the same, zoomed in.) The observer
is sitting motionless with respect to distant stars and is looking
straight down into the black hole. The data file
for `kerr-image`

is as follows:

0 2 25 0.19801980198019801980198019801980198 0 0 0 0 1 0 -1 0 0 0 0 0 0 1 0 1 0 0

(It's up to anybody's guess why I chose
cos(`θ`)=20/101.)

The red sphere is the outer horizon. Note that both its poles are
visible simultaneously (north pole is up and south pole is down, by
convention). Inside, we can see both inner horizons (on
the Carter-Penrose diagram, they are under the
region marked II (WH)

): the yellowish-green
is the “normal” one, corresponding to photons escaping the
white hole with small (or negative) angular momentum with respect to
the black hole's axis, and behind it lies the white hole's
“normal” interior from the point of view of our universe
(analogous to the black hole interior in which the observer falls in
my first video), whereas the
blueish-green interior horizon, the “other” one, to the
left, is its sibling. The separation between them is a clear line,
because it's a matter of angular momentum to energy ratio which
horizon gets crossed
(see around here for
more explanations); however, there are a number of image artefacts
around this line, because my program does not handle horizon crossings
correctly. Note that the inner horizons, together, take exactly the
same space as the outer horizon, because any photon (or material
particle) crossing one must necessarily cross the other.

The blue lines represent the celestial sphere around the black
hole. Note the following fact: the pole of the celestial sphere which
is visible above the black hole's north pole (about center of the top
part of the first/unzoomed image) is the *south* celestial
pole. This is because the photons coming from that direction have
traveled half a turn around the black hole.

Negative space is visible if you look very closely, but it will be more obvious on the next image:

This is a relatively similar image. This time, the observer is
at a distance of 10`M` (5 Schwarzschild radii) only, again
motionless with respect to distant stars, and at a latitude of
30°:

0 2 10 0.5 0 0 0 0 1 0 -1 0 0 0 0 0 0 1 0 1 0 0

The next image has been used to illustrate the moments immediately after crossing the horizon:

This is actually frame 185 of the first video (but rendered here with better quality settings). Its parameters are:

1.85000000e+00 0 1.39529962e+00 7.45506508e-01 2.87508661e+00 -1.98655344e-01 -1.38706110e+00 -6.88084298e-02 1.73274639e+00 -2.64201189e-01 -1.33164883e+00 -6.39149322e-02 8.62844619e-01 -3.57546145e-01 2.49176857e-01 5.68609750e-02 -1.74800070e-01 9.49499870e-01 -8.79981423e-03 4.36280541e-01 -1.24093549e-01 -1.34461090e-01

We are in region II here (marked II (BH)

on our usual Carter-Penrose diagram): the black
hole horizon is now visible in brown, forming a bubble around us,
whereas the *white* hole horizon is still visible in bright red
(and seems to be still ahead). It is futile to try to escape the
black hole, now, because the horizon bubble expands faster than light.
Of course, a real observer, at this stage, would not see anything
remarkable, because the grids represented on the horizon are purely
imaginary: a real black hole does not have any markings on its
horizon! And note that the celestial sphere is not much more deformed
than in the previous images taken from region I.

One interesting thing visible at this point, however, is a
“window” opening up to the sibling region of the region I
we are coming from. The blue disk around the center of the image is
the sibling region in question; the dark red grid barring the way is
the outer horizon of the black hole reachable from *that*
region, and the bright orange is the outer horizon of
the *white* hole in that sibling region I (again, see the
Carter-Penrose diagram, which uses the same color code, to make sense of
this). Notice how the window has opened precisely at the limit where
the two inner horizons of the white hole met. As we continue falling
toward the black hole (see the video), the window gets larger at
first, and then contracts again: it is impossible for a material
particle to reach our sibling universe (but we can meet particles
coming from there if the fall in the black hole).

Naturally, the inner region of the black hole (or even the inner
horizon) is *not* visible since it lies ahead of us. The inner
horizon will also appear as a bubble opening up to engulf us. (At
this point, we can choose which region III to enter, although by
freely falling this particular observer is bound to enter the
“normal” region III, i.e., the one marked III

on
the Carter-Penrose diagram on this page.)

The next image was used to illustrate what negative
space and the ring singularity look like, when seen from inside region
III. Its input for `kerr-image`

is:

0 2 0.2 0.8 0 0 0 0 1 0 0 0 0 -1 -1 0 0 0 0 1 0 0

The black hole inner and outer horizons are in green and brown, and
the outside is visible behind them. The ring singularity is the edge
of window to negative space visible as the purple blob. The observer
is motionless with respect to distant stars (not a very likely option
at this place, but physically possible),
at `r`=0.2`M` and a latitude of 53°; but in this
region, it is perhaps better to describe position using the distance
to the axis `ℓ`=0.49`M` and the distance
above the equatorial plane `z`=0.16`M` (knowing that
the ring singularity is
at `ℓ`=`a`=0.8`M`
and `z`=0). We are actually surprisingly close to negative
space, and in fact, even a view from the close negative space would
appear quite similar: it is only once we have overcome the strong
potential barrier guarding the entry to negative space that we will
really seem to be inside it, as in the next image:

0 2 -1.8 0.8 0 0 0 0 1 0 1 0 0 0 0 0 0 -1 0 1 0 0

This is what the black hole looks like from the flip side (for a process leading up to something like this, see the second video below).

**Note:** Every video is viewable on YouTube, and also
downloadable as a WebM (VP8) or AVI (H.264) file through
BitTorrent: recent versions of many standard players such
as VLC
or MPlayer should be able to
read both. The links in what follows are the links to
the *torrent* files to be given to some BitTorrent client
program (e.g., Vuze) which will
then download the video itself. If you cannot get BitTorrent to work,
you can also download the video directly through HTTP by
removing the `.torrent`

extension at the end of
the URL (it will then end in `.webm`

or `.avi`

) but please do this only if you cannot download
through BitTorrent.

In this video (43.3″ long), which is the first I computed, the observer falls into the black hole (crosses the outer horizon at 8.5″ and the inner horizon at 12.2″), grazes on negative space (at 13.2″), and emerges again from what is now a white hole (at 14.9″ for the inner horizon and 18.6″ for the outer).

The observer's energy is 1.2 times his rest mass (meaning he is
mildly relativistic), and his angular momentum along the black hole
axis is zero (so his only rotation can be said to be caused by the
Lense-Thirring effect). The journey begins at a distance `r`
of 1.75 Schwarzschild radii (3.5`M`) and a latitude
(`π`/2−`θ`) of 53° (north), and
ends at a distance `r` of 3.25 Schwarzschild radii
(6.5`M`) and a latitude very
close[#] to the starting one
(51°). The ~~black~~ white hole appears very much larger
in the emerging part of the journey than in the infalling part,
because the observer is moving away from it at relativistic speed.

[#] Technically,
the geodesic is said to be vortical (its Carter
constant `Q` is negative) in that its latitude remains purely
on one side of the equator (the geodesic does cross the equator, but
only by entering negative space).

I made two different versions of this video: a “combined view version”, here on YouTube, simultaneously shows the observer's front view (in the upper left quadrant), his rear view (in the upper right quadrant), a longitudinal cut and Carter-Penrose diagram (in the lower left quadrant) and the values of the Boyer-Lindquist coordinates and their derivatives (in the lower right quadrant). The YouTube version also contains a number of annotations explaining what is going on at various points in the video. The “front view version”, here on YouTube, only shows the observer's front view.

Video links:

Combined view (960×720) | Front view (640×480) | ||
---|---|---|---|

On YouTube: | Combined view | Front view | |

Through BitTorrent: | WebM format (VP8 codec) | Combined view (13MB) | Front view (11MB) |

AVI format (H.264 codec) | Combined view (17MB) | Front view (15MB) | |

Data file: | Data file |

In this video (47.4″ long), the observer agains falls into the black hole (crosses the outer horizon at 8.1″ and the inner horizon at 11.7″), but then remains in region III by crossing the ring singularity and entering deep negative space.

The observer's energy is equal to 1.574 times his rest mass (he is even more relativistic than in the previous video): this was chosen so as to be barely sufficient to cross the threshold into deep negative space, and this is the reason of the slowing down around 25″. His angular momentum along the black hole axis is 0.243 (in units of observer mass times black hole mass). This was chosen to conserve a latitude of about 60° throughout the journey (the geodesic in question is, once again, vortical).

As previously, the “combined view
version”, here
on YouTube, simultaneously shows the observer's front view (upper
left) and rear view (lower right), a polar cut and horizontal
projection (lower left, left and right) and the values of the
Boyer-Lindquist variables and their derivatives (upper right). The
top view is shown over (partially obscuring) the bottom view up to
16″ in the video, after which the bottom view pops up because it
seemed more interesting from then on. I didn't provide a Carter-Penrose
diagram because it's not as interesting (only two horizons are
crossed). The horizontal projection (seen from above) is given in the
perspective of Boyer-Lindquist coordinates (i.e., the angle is the
angle `φ` of Boyer-Lindquist coordinates, and twists
infinitely when a horizon is crossed); the edges of the horizons are
shown where they cross the equatorial plane, but the orbit is not in
the equatorial plane: this may be a bit confusing, and I'm sorry I did
that.

Video links:

Combined view (960×720) | ||
---|---|---|

On YouTube: | Combined view | |

Through BitTorrent: | WebM format (VP8 codec) | Combined view (15MB) |

AVI format (H.264 codec) | Combined view (23MB) | |

Data file: | Data file |

In this video (40.2″ long), just
like in the first, the observer falls
into the black hole (crosses the outer horizon at 12.8″ and the
inner horizon at 19.3″), and emerges again from what is now a
white hole (at 20.9″ for the inner horizon and 27.3″ for
the outer). The important difference with respect to the previous two
videos is that the interior region (III) penetrated is
the *other* one
(recall that every
region I and III in the Carter-Penrose diagram has a “sibling”,
and that the traveler entering the black hole can choose to enter one
or the other of the two sibling interiors). The stay in region III
itself is brief and violent (and the computed images during this part
of the video are unfortunately ripe with numerical artifacts:
specifically, the big black blob briefly seen around 20″
is *not* a real phenomenon), but the point is not so much the
stay itself as the moments leading up to it (note that the inner
horizon “bubble” forms on the eastern side of the white
hole, and note that the window to the sibling region I is much larger
than in the previous videos).

The observer's energy is equal to his rest mass (meaning he has
zero velocity at infinity), and his angular momentum along the black
hole axis is 1.3 (in units of observer mass times black hole mass).
The journey begins at a distance `r` of 1.75 Schwarzschild
radii (3.5`M`) and a latitude
(`π`/2−`θ`) of 4.3° north, and
ends at the same distance but this time with a latitude of 10.9°
south. In other words, this video is much more equatorial
than the first: this is because a
vortical geodesic cannot enter this “other” interior.

As previously, the “combined view version”, here on YouTube, simultaneously shows the observer's front view (left), a longitudinal cut (right, top) and Carter-Penrose diagram (right, middle), and the values of the Boyer-Lindquist coordinates and their derivatives (right, bottom). I didn't compute the rear view, because I don't think it will show anything significantly new or interesting.

Video links:

Combined view (800×480) | ||
---|---|---|

On YouTube: | Combined view | |

Through BitTorrent: | WebM format (VP8 codec) | Combined view (14MB) |

AVI format (H.264 codec) | Combined view (25MB) | |

Data file: | Data file |

In this video (2′35.7″
long), the observer orbits the black hole, on the equatorial plane, at
a constant radius `r`, namely 2.907`M` (1.453
Schwarzschild radii, or 4.36 million kilometers) which, for this black
hole, is the position of the last stable direct orbit
(here direct

means rotating in the same direction as the
black hole itself

). It may not be as intense as the previous
videos, but it does show a number of interesting phenomena such
as Einstein
rings.

The observer orbits the black hole once every
3′0.8″ *as seen by distant observers*, but in
1′23.9″ as measured by his own clock. So after
1′23.9″ in the video, we are back to our starting point
(from the point of view of distant observers, who think that
3′0.8″ have elapsed), but not looking in the same
direction because our gyroscope has precessed. It takes
2′35.7″ for the gyroscope to be pointing at the same
direction again (with respect to the black hole), and that is how long
the video lasts (distant observers, of course, would say that
5′35.3″ have elapsed).

The observer's energy is equal to 87.8% of his rest mass (so he is bound and cannot escape to infinity without extra energy), and his angular momentum along the black hole axis is 2.38 (in units of observer mass times black hole mass).

I should also note that the prominent blue cross which is visible,
say, around 2′ in the video, is part of an optical phenomenon of
the Einstein ring kind: it is the *equator* of the distant
sphere (aligned with the equator of the black hole) which is visible
as a circle around the black hole.

There is a unique version of this video, here on YouTube, because it didn't seem interesting to add anything else.

Video links:

Unique version (640×480) | ||
---|---|---|

On YouTube: | Unique version | |

Through BitTorrent: | WebM format (VP8 codec) | Unique version (44MB) |

AVI format (H.264 codec) | Unique version (66MB) | |

Data file: | Data file |

This video (31.4″ long), shows a very peculiar trajectory in the Kerr space-time, sometimes known as the lazy geodesic: this is a geodesic which remains on the equatorial plane and is forever locked between the horizons of the black hole: it bounces back and forth between outer and inner horizon, and between black hole and white hole regions II while never actually crossing fully into region I or region III (though it sees a lot of them, corresponding to a succession of different universes in pairs). On the Carter-Penrose diagram, it can be identified with the vertical line in the middle: whereby one can see that it actually does cross horizons, but it crosses two of them simultaneously (re-entering a black hole exactly as it leaves a white hole, and vice versa). The whole phenomenon is periodic, with a period equal to π times the Schwarzschild radius (as time) of the black hole, i.e., four our particular black hole, 10π=31.4 seconds. Here we start with the observer halfway between outer and inner horizons, and going down: the inner horizon is touched at 4.9″ (after which the observer is in a white hole region II), and the outer horizon at 20.6″ (after which the observer is again in a black hole region II).

The lazy geodesic corresponds to one with exactly zero energy, zero angular momentum along the black hole's axis (and zero Carter constant).

The whole thing is not easy to interpret and I'm not sure I understand exactly what it means what we're seeing. But in a certain sense, it must be said that the lazy geodesic does not move at all: the black hole is moving and changing periodically around it.

There is a unique version of this video, here on YouTube, because it didn't seem interesting to add anything else.

Video links:

Unique version (640×480) | ||
---|---|---|

On YouTube: | Unique version | |

Through BitTorrent: | WebM format (VP8 codec) | Unique version (32MB) |

AVI format (H.264 codec) | Unique version (58MB) | |

Data file: | Data file |

`kerr-image`

programThe program I used to compute all the images can be downloaded from here. It is in the Public Domain, so you can play with it all you want.

From the comments of the program:

This program reads 22 numbers from standard input, separated by whitespace. Their meaning is as follows:

- first number is a float, and is ignored (typically used to store observer's proper time when making animations),
- second number is the integer 0, 1 or 2, to indicate whether all
the following coordinates are given in ingoing Kerr (0), outgoing
Kerr (1) or Boyer-Lindquist (2) coordinates, (note that for ingoing
and outgoing Kerr coordinates, the
`t`coordinate is`V`−`r`or`V`+`r`, not the null coordinate`V`), - next four numbers are the coordinates of the observer's current
position, in the order:
`r`, cos(`θ`),`t`and`φ`(note the order, and note that it's cos(`θ`) and not`θ`itself which is given), - next four numbers are the coordinates of the observer's current
velocity, in the same order and convention (i.e.,
d
`r`/d`s`, −sin(`θ`)·d`θ`/d`s`, d`t`/d`s`and d`φ`/d`s`), - next four numbers are the components of the vector indicating the direction in which the observer is looking (forward-pointing vector), again with the same order and convention,
- next four numbers are the components of the vector indicating the direction which the observer thinks is “right” (right-pointing vector), again with the same order and convention,
- final four numbers are the components of the vector indicating the direction which the observer thinks is “up” (up-pointing vector), again with the same order and convention.

The four vectors (velocity and forward-, right- and up- pointing) will be orthonormalized by the Gram-Schmidt algorithm prior to any processing by the program (and their values before and after this orthonormalization will be printed out to standard error). Of course, the first vector should be timelike and the next three should be spacelike (or you will get no meaningful output).

Here is an example possible set of input values, producing a view which is not too uninteresting:

0 2 10 0.5 0 0 0 0 1 0 -1 0 0 0 0 0 0 1 0 1 0 0

This will compute the view of the black hole by an observer sitting
motionless with respect to distant stars at a distance of 10 units (5
Schwarzschild radii) at a latitude of 30 degrees
(`θ`=π/3), and looking straight down into the black
hole.

The program then produces a ppm file on standard
output, whose number of lines and columns is given
by `NBLINS`

and `NBCOLS`

defined at compile
time. By default, the viewing angle is set to about 100 degrees
vertically and even more horizontally (it is assumed that pixels are
square), but a zoom factor can be introduced using the
variable `ZOOM_FACTOR`

at compilation. To produce partial
images, specify a range of lines on the command line (e.g. ```
kerr
image 120 240
```

to compute lines from 120 to 239 inclusive), and
concatenate the resulting output files (only the concatenation will be
a well-formed ppm file, not each individual output); the default is to
produce a full image of the size specified at compile time.

The black hole parameters are specied
by `BLACKHOLE_MASS`

and `BLACKHOLE_ANGMOM_PER_MASS`

at compile time (by default
1 and 0.8). It is advised to leave `BLACKHOLE_MASS`

untouched, since everything would only change by a scale factor
anyway.

The way this works is that geodesics for each light ray arriving at
the observer are integrated, partially using the first integrals of
motion, for every pixel in the image. (Partially

here means
that the equations on `r` and cos(`θ`) are
kept as second-order because this avoids a degeneracy when the
derivative tends to zero, but the value of the derivative is regularly
recomputed.) The geodesic integration part of this program
does *not* assume that the geodesic is lightlike (as it is
here), so a trivial modification of it can be used to compute the path
of timelike or spacelike geodesics.

Note: this program does not handle correctly the region where two horizons cross (where neither ingoing nor outgoing Kerr coordinates are adequate). This will result in some “snow” around such regions. This is a numerical artefact. Also, rays coming very close to the axis (or, a fortiori, interesting it) will cause problems.

Note: this program does not attempt to do any anti-aliasing on the resulting image. Do do this, compute a larger image, and downscale it.