Kerr black holes images and videos

[View of a Kerr black hole]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 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 not-so-short course on black holes

What is a black hole?

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).

A crash-course on Schwarzschild black holes

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=2G·M/c², where G is Newton's constant and c is the speed of light: this is usually written r=2M 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 2G/c² 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 2M 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/c³ instead of G/c²): 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/r² and G·M/r³ 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/r³, however, so at the horizon, where rM [is proportional to], we the effect decreases as 1/M²: 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 10g 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/r³ and therefore decreases as 1/M² 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).

[View of a Schwarzschild black hole from the photon sphere]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=3M, 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=6M, 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).

The effects of rotation: the Kerr black hole

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 2M, 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/M², 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·m2·s−1; or, if you will, the fraction of maximality is (c·J)/(G·M²)). 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/(4M²)=J/(4M³), 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 2M 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 2G·J·m·v/(c²·r³) 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/(c²·r³). 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/(c²·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.

[Cut of a Kerr black hole]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 r coordinate is supposed to measure the distance from the black hole, but how does one measure the distance from a black hole anyway? One cannot put a ruler in space, because inside the horizon there is no way to keep the ruler fixed, so what does it mean to say that the horizon is at this or that distance from the center of the black hole? In the Schwarzschild case, there is a simple answer: since the black hole has spherical symmetry, we have well-defined non-rotating spheres of constant distance around it, these spheres are ordinary Euclidean spheres, and to define the radius of such a sphere, measure its area, divide by 4π and take the square root. This defines the r coordinate. In the Kerr case, there is no such simple procedure: defining what a “sphere” means is difficult, and keeping it non-rotating is also difficult (or impossible, inside the static limit). There is, however, a standard coordinate called r. It does not meaningfully measure the distance from center of the the black hole (for example, the ring singularity is given by r=0 on the equatorial plane but it is still a ring and not a point; also, when the mass of the black hole is made to zero, so space becomes flat despite the remaining a parameter, the r coordinate does not tend to the distance from the origin); but does tend to such a distance for large values of r, and it is a very important and useful coordinate, and this is what is usually meant when, by abuse of language, one speaks of 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 r=1.5×2M, and calling it the distance is an abuse of language. The surfaces r=constant are not spheres, although they are sometimes called that way (for example, the statement is often found that the Kerr black hole horizons remain spherical); they can better be pictured as something like oblate spheroids (although even this is inaccurate in many ways[#]). One sign of their importance is that, for example, they are covered by orbits around the black hole (i.e., it is possible, when starting at a given r not too close to the horizon, to remain in orbit at that same value of r and densely cover the whole surface with that r value). The horizons are also surfaces with r constant (namely r=M±√(M²−a²), where of course the + sign gives the outer horizon and the − sign the inner). To measure the distance to the center of the black hole, although that is ill-defined, a more accurate quantity might be √(r²+a²sin²(θ)) (equal to r at the poles and √(r²+a²) at the equator), and it is this quantity which has been used as distance from the origin in the cut diagram above, or when we say that the radius of the ring singularity is a (in natural units, of course: this actually means J/(c·M)).

[#] 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.

Beyond the singularity: negative space

[View from inside a Kerr 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.)

[View from negative space]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).

White holes and antihorizons

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. [Crossing the horizon of a Kerr black hole]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.

Tying many universes together with the Carter-Penrose diagram

[Carter-Penrose diagram of a Kerr black hole]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).

Some remarks about real black holes

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.

Conventions for all images and videos

The black hole used

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 2M 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 1.6M, 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 3.6×1017·m²/s, and an angular momentum J of 7.3×1053·kg·m²/s (this is something like twenty billion times the total angular momentum of the Solar system).

All videos are inertial

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.

Color scheme

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).

More remarks

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?).

Limitations

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.

Reproducing the images

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).

Still images

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.

[View of a Kerr black hole][Same image, zoomed]Here, the black (well, white) hole is observed from a good distance (r=25M, 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:

[View of a Kerr black hole]This is a relatively similar image. This time, the observer is at a distance of 10M (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:

[Crossing the horizon of a Kerr black hole]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.)

[View from inside a Kerr black hole]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=0.2M and a latitude of 53°; but in this region, it is perhaps better to describe position using the distance to the axis =0.49M and the distance above the equatorial plane z=0.16M (knowing that the ring singularity is at =a=0.8M 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:

[View from negative space]

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).

Videos

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.

Visiting the black hole and emerging in a different universe

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 (3.5M) and a latitude (π/2−θ) of 53° (north), and ends at a distance r of 3.25 Schwarzschild radii (6.5M) 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 viewFront 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

Diving through the ring singularity and into deep negative space

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

Visiting the “other” interior of the black hole

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 (3.5M) 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

Orbiting the black hole at the last stable orbit

In this video (2′35.7″ long), the observer orbits the black hole, on the equatorial plane, at a constant radius r, namely 2.907M (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

The lazy geodesic, locked between the horizons

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

The kerr-image program

The 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:

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).

[View of a Kerr black hole]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.