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Black Holes
This is a brief series of articles on stellar evolution, neutron stars, and black holes. This series consists of 3 parts:
- Stellar Evolution - Part I of the article
- White Dwarfs and Neutron Stars - Part II of the article
- Black Holes - Part III of the article: this page
The series begins with a brief introduction to star formation, covering the birth, lifetime and death of a star. The second part describes white dwarfs and neutron stars, including special types of neutron stars, such as pulsars and magnetars. This third part describes black holes.
We'll start with a brief review of the previous sections. If you're interested, you might want to read about stellar evolution, and the process that results in a black hole. Briefly, at the end of a star's life, it has no fuel left to generate any more energy. During its lifetime, a star exists in a dynamic equilibrium between two opposing forces - pressure generated by heat of nuclear fusion, which pushes outwards to make the star expand, and gravity, which pulls inwards to make it contract. When the nuclear reactor shuts down for the final time, one of these forces is lost. There is nothing left to oppose gravity, and the star begins to rapidly collapse into itself.
The Origin of Black Holes
Massive stars fuse several elements during their lifetime. For most of their lives, they are in the main sequence, fusing hydrogen into helium. However, once the hydrogen in the core is exhausted, the star moves on to fusing progressively heavier elements, until it reaches iron. This is the end point in the active life of the star, since the fusion of iron (and elements heavier than iron) does not produce net energy. Such fusion reactions are endothermic - they absorb energy instead of producing it.
This puts a sudden brake on the activity of the star. There is nothing left to burn which would produce energy, therefore the star starts to collapse under gravity. Stellar collapse produces intense pressure and massive amounts of heat at the core. This energy comes from the gravitational potential energy of matter falling inwards. In large stars, the core temperature may reach 5 billion °K, or higher. This intense heat fuses the iron at the core. As mentioned, iron fusion is endothermic, so this drastically lowers the temperature of the core within a few seconds. The collapse accelerates, due to the loss of this residual heat, which was generating pressure. The rapid collapse is of the nature of an implosion - happening within a few seconds, and generates a massive shock wave. The rebound from this shock wave tears the star apart, throwing out most of its mass in the form of a Type II supernova, one of the most powerful and energetic events in the universe. The remaining core, which is usually a small fraction of the star's initial mass, is left behind.
The size of this remaining core determines what happens next. If the core is under 1.4 S (S is a solar mass, about 1.9891 x 1030 kilograms), it becomes a white dwarf. White dwarfs are very dense (having the mass of a Sun concentrated into an object about the size of the Earth), and are made of degenerate matter (electrons torn loose from their orbitals, so electrons and atomic nuclei exist as a fluid sometimes called the Fermi Sea). However the electron degeneracy pressure keeps them from collapsing any further.
If, however, the mass of the core is larger than 1.4 S (the Chandrashekhar Limit), the gravity is powerful enough that electron degeneracy pressure cannot prevent it from collapsing. The object then becomes a neutron star. The equations of state for neutron stars are not very well understood, so the nature of neutron star matter is a subject of debate. Neutron stars are much denser than white dwarfs, with their centers having higher density than an atomic nucleus. Neutron degeneracy pressure keeps them from collapsing further. However, if the core is even more massive, then neutron degeneracy pressure would not be sufficient to keep it from collapsing further.
How much more massive is a question for debate. The boundary is defined by the equations of state, which are uncertain for neutron stars. This boundary is represented by the Tolmann-Oppenheimer-Volkoff limit, which is somewhere between 1.5 S and 3 S. If a neutron star is heavier than this, it collapses into a black hole. This is where our knowledge fails, since no one knows what could be denser than neutron star matter. Some people posit an intermediate stage, with neutrons breaking down into quarks, to form a quark star. No quark star has ever been discovered. However, with the addition of more matter, this too collapses into a black hole.
What is a Black Hole?
In simple (but not very descriptive) terms, a black hole can be considered a region of space where the gravitational field is so powerful that not even light can escape. Black holes were predicted by general relativity, though for a long time, people disputed whether they could actually exist. Einstein himself did not believe black holes were physically possible.
However, by the 1960's, the majority of physicists were convinced that there was no known theoretical reason why black holes should not exist. Today, several black holes have been identified, both within our galaxy and outside it. It is believed that a supermassive black hole exists at the center of our own Milky Way galaxy (and at the centers of most other galaxies).
Theoretically, black holes can be created by different processes. In the early universe, when the density of matter was much higher, it's possible that local perturbations could cause material to clump together and collapse into a black hole. Some people think that high energy collisions of elementary particles (such as cosmic rays, or the Large Hadron Collider), could also produce black holes. However, the best understood process for black hole formation is through the gravitational collapse of a massive object, such as a star.
Schwarzschild Radius
Since we've defined the black hole as a region where the gravity is so intense that not even light can escape, we can calculate a boundary around the black hole where the escape velocity equals the speed of light in vacuum.
As can be seen in the equations on the left, the formula for escape velocity can be used to calculate the radius (distance from a mass M) where the escape velocity would be equal to the speed of light. Intuitively, we can see that this would require a very dense mass, so that the entire mass was contained within this radius.
This is a quick but incorrect way to calculate the Schwarzschild Radius, using Newtonian mechanics only. It's incorrect because it doesn't strictly follow the definition of the Schwarzschild Radius, which derives from mass and gravity, not directly from the speed of light. The Schwarzschild Radius is defined as the radius of a sphere of matter that was dense and massive enough so that the force of gravity inside it was intense enough that no known degeneracy pressure could prevent it from collapsing into a "point of infinite density" or gravitational singularity. Therefore if any mass could be contained within its Schwarzschild Radius, it would collapse into a black hole, i.e., an object smaller than its Schwarzschild Radius is a black hole.
In 1916, Karl Schwarzschild obtained this radius by calculating an exact solution to Einstein's field equations (from general relativity) for the gravitational field outside a spherical, non-rotating body. Note that therefore this formula only works for a spherical object, which is not rotating. Since most black holes do rotate, the formula needs to be modified to take that into account.
One significance of this radius is that it acts as a one-way barrier. This is the point at which nothing can escape from the black hole's gravity, therefore matter and energy can enter this barrier, but nothing can come out. It is not a physical barrier, since matter and energy can enter, it is simply a region of space with the property that nothing inside the enclosed volume can leave it.
The sphere defined by the Schwarzschild Radius is called the event horizon. It's a one-way boundary in space-time. It would be wrong to think of it in a Newtonian fashion (imagine light trying to leave it, and being pulled back by gravity). The real situation is that all possible paths that light can take inside the event horizon are warped in towards the singularity. Therefore the forward light cone of any particle leads to the singularity - moving forward in time means moving towards the singularity.
As we can tell from the formula, the Schwarzschild radius (SR) depends upon the mass. Every mass has some Schwarzschild radius that can be calculated for it. The Schwarzschild radius for the Sun is about 3 kilometers. If the mass of the Sun could be concentrated within a radius of 3 km, the Sun would be a black hole. The Schwarzschild radius for the Earth is about 9 mm. The SR increases linearly with mass, so for each solar mass added to the black hole, the SR increases by about 3 kilometers. At this rate, the largest black hole discovered so far - OJ287 - which has a mass of 18 billion suns, would have a SR over 7 times that of the solar system.
Another interesting effect that happens near the event horizon is time dilation. Consider two observers, Alice and Bob. Alice is approaching the black hole, while Bob sits back far away, at a safe distance from it. As Alice gets closer to the black hole, it will appear to Bob that she is slowing down. The closer she gets to the black hole, the more she will seem to slow down to Bob, until she reaches the event horizon, at which point she will appear to stop. Bob will never see her enter the event horizon - it will look to him like she's stuck at the event horizon forever.
This is a thought experiment, of course, since in reality if Alice were to approach that close, she would be torn apart into atoms and there would be nothing for Bob to observe anyway. But let's disregard that for the moment. Now, from Alice's perspective, time does not slow down. She does in fact reach the event horizon, pass through, eventually to merge with the gravitational singularity inside. For Alice, the whole trip would probably take no more than a few seconds.
Formulas used to calculate the Schwarzschild Radius (rS) and the time dilation seen by a distant observer for material entering the black hole.
The picture above shows Alice as Observer A, and Bob as Observer B. The dimensions are not to scale. The two equations shown are for the Schwarzschild radius (on top), and for the time dilation effect (below it). As can be seen, the Schwarzschild radius is a term in the time dilation equation, reflecting the fact the closer you get to the event horizon, the greater the apparent time dilation to an outside observer. At the event horizon, the term rS/r becomes 1, and the denominator on the right hand side becomes 0, making the equation not calculable ("infinite").
If you'd like to see the calculations in more detail, they are on a separate page here (will open in a new window). The page has some calculations showing how the size of the event horizon grows with the mass of the black hole, and also the effect of the gravitational field on time dilation.
Structure of Black Holes
A black hole typically shows the following regions. In the center is the gravitational singularity, where everything ends up. Surrounding it is a region (bounded by the event horizon on the outside) where space-time get horribly mangled. Outside the event horizon, a rotating black hole (and only rotating black holes) shows a region called the ergosphere, kind of like an oblate or stretched sphere. This is a region where space gets dragged around with the rotating black hole. Outside it is a region called the photon sphere, which has some peculiar properties relevant to light paths. All of these regions are discussed in detail below.
[CLICK HERE FOR LARGE VERSION] Diagram of a black hole, showing the Schwarzschild Radius, Photon Sphere, and Ergosphere.Event Horizon
As mentioned, this is a sphere of zero thickness that surrounds the black hole (the singularity) at a distance equal to the Schwarzschild radius. It bounds a region of space where space is curved so that no light paths beginning inside that region can ever leave it. Nothing leaves the black hole from inside its event horizon. Time as we know it, does not exist inside the event horizon.
From the perspective of an outside observer, matter or light falling in towards the black hole would come to a halt at the event horizon (due to time dilation). We cannot see it fall in any further, since in order to see inside it would require that something carrying information about the object should leave the black hole, and nothing of the sort does. In other words, the light cone of the event of the object's crossing of the event horizon never intersects the outside observer's world line. As the object (or light) falls towards the event horizon, all processes in the object appear to slow down. If a clock were falling towards the black hole and the outside observer could read the clock's face, the clock would appear to slow down. In the case of light, it becomes ever dimmer and more red-shifted to an outside observer, as it falls towards the event horizon. Finally as it reaches the event horizon it becomes so red shifted that it can't be seen (the wavelength of light reaching the outside observer tends to infinity).
The event horizon of a non-rotating black hole is spherical, but that of a rotating black hole is somewhat distorted from a perfectly spherical shape. Again, it's important to remember, the event horizon is not a physical barrier - it's simply a mathematical concept. Nothing actually prevents matter from falling in to a black hole.
This description of the event horizon (and of black holes in general) is based on general relativity, which is not sufficient to understand what's going on inside the black hole. It is only an approximation that holds true from the perspective of an outside observer. As you get close to the event horizon, quantum gravity effects become significant. In order to model the event horizon, we would need both relativity and quantum mechanics. While we are starting to understand some of these things (for example, quantum mechanics predicts that event horizons have a certain temperature, and therefore emit radiation - Hawking Radiation), we cannot have a complete understanding of event horizons until we have a theory of quantum gravity.
So what would an event horizon look like to an outside observer, if we were close enough to a black hole to see it? Based on what we know so far, pretty much like a black, spherical region with a very sharp boundary. Objects behind it would appear "lensed", that is, you could actually see things directly behind the black hole as some light paths from them to you follow the highly curved space-time around the black hole. They would appear as a ring, smeared around in a circle outside the event horizon. This is called gravitational lensing, giving the appearance of Einstein rings around massive objects such as black holes. You would see swirls of matter, caught in the black hole's gravity field spiraling in towards the black hole, slowing down and then seeming to disappear as it reached the event horizon. Stars nearby would have the gases sucked out of them, spinning in long spiral tracks towards the black hole. Here's an artist's impression of what it might look like.
Because of the curvature of space and the consequent bending of light rays, the apparent size of the black hole would vary from our normal experience, when approaching it. This purely has to do with the distance to the black hole, not its actual size. Even if the black hole was quite small, at a distance of about 1.5 Schwarzschild radii, the event horizon would appear to cover half the sky. At distances closer than 1.5 Schwarzschild radii, the black hole appears to wrap around our vision and extends further behind us, giving a feeling of being surrounded by it, even though it is straight ahead. This effect is real, you can turn your head around and look back, and see that the universe behind you appears to be shrinking. Very close to the black hole (but still outside it), the black hole would appear to be everywhere around you, except for a pinpoint directly behind you. The whole universe would appear shrunk into that pinpoint. You can see what this might look like visually through a computer simulation here.
Singularity
The singularity is what sits inside the black hole, where everything entering the event horizon ends up. It's a region of space where matter has infinite density, gravity is infinitely strong, and spacetime has infinite curvature. As you can tell by all the "infinites" we don't really have a clue what's going on. A non-rotating black hole's singularity is a dimensionless point. A rotating black hole's singularity is a ring situated along the plane of rotation (it has length and width, but zero thickness).
The appearance of singularities in general relativity show the limits of the theory - the places or conditions where the theory breaks down. At a gravitational singularity, quantum mechanical effects (specially quantum gravity effects) are dominant. At present, we have no quantum theory of gravity. When we do, we will no longer think of these things as singularities.
Since there's not much we can say about singularities, let's say a few things about what they're not. The space inside the event horizon is not normal space. Time inside the event horizon is not normal time. What happens to space-time inside the event horizon is unknown. Various references describe it has "mangled", "highly curved" etc., which sound somewhat descriptive but don't say much. The singularity is simply the place where this "mangledness" reaches its ultimate extent. Using words which we created to reference things and events happening in space and time don't do justice to the state of affairs inside the event horizon. Therefore, saying that a singularity is a "dimensionless point" is not strictly true. If space as we know it doesn't exist at the singularity, what does "dimensionless" mean anyway? It just goes to show that along with general relativity, our language also breaks down inside the event horizon. When we do invent language to describe it (when we have a theory of quantum gravity), we will no longer call it the "singularity". This is not something most people care about in the ordinary course of affairs, unless they are theoretical physicists. But these concepts are often used (and misused) by people philosophizing about the nature of reality and such metaphysical questions. Then it becomes extremely important to remember what we do not know, and thus remove the temptation to derive conclusions from it.
Ergosphere
Any rotating black hole "pulls" along a region of spacetime with it, due to a phenomenon called frame-dragging. As far back as 1918, the Austrian physicists Josef Lense and Hans Thirring predicted (using general relativity) that a rotating object would drag any nearby object along with it. In other words, an object close enough to a rotating black hole would start rotating around it. There is some complicated math behind it, but one way to understand it is to imagine a clock near a black hole. Because of time dilation, the clock appears to slow down, to an outside observer. If the clock starts rotating around the black hole, in the direction of rotation of the black hole, it will appear to tick faster than if it were not rotating. Time will still appear to slow down for it, to an outside observer, but it will slow down the least if the clock is rotating around the black hole in that direction. The same is true if we replace the clock with a light beam. A light beam will move around a black hole faster in the direction of rotation of the black hole.
This is simply a specific example of a more general idea of gravetomagnetism, which occurs when a mass is moving in any fashion (linearly or rotating). This terminology considers the phenomenon of frame-dragging as a fictitious force, analogous to a moving electrical charge creating a magnetic field. A moving mass creates a fictitious force, tending to pull along any nearby objects with it. There is some proof that this prediction of relativity is true, in the observation of relativistic jets emerging from black holes in quasars. Roger Penrose proved mathematically that it's possible for these relativistic jets to acquire their energy directly from the black hole, through frame dragging. There is a NASA experiment called Gravity Probe B under way, to obtain information that might corroborate frame dragging. The results should be available by 2010.
Rotating black holes would show this effect, dragging space along with them in the direction of their rotation. Because of this, any object close to a black hole would start rotating with it. In a region very close to the black hole, the speed of rotation exceeds the speed of light. An object in this region would therefore move faster than the speed of light. Just to be clear, it's not really the object moving faster than the speed of light, it's space itself moving at that speed. For this reason, such an object could never be stationary relative to anything outside the universe, because to be stationary, it would need to actively move in a direction opposite to that of the rotation, at a speed exceeding light. Which is not possible. This region where it's not possible for an object to remain stationary relative to an outside object is called the ergosphere. Its shape is that of an oblate spheroid - bulging at the equator, and flattened at the poles of the rotating black hole.
As one moves away from the black hole, the frame dragging effect decreases, and objects move more and more slowly. At a certain distance, they move at the speed of light (instead of faster than light when they are closer). At this point, it's possible for them to remain stationary with respect to an outside observer. This point marks the boundary of the ergosphere. The boundary of the ergosphere touches the event horizon at the poles, while at the equator (where it's maximum), it extends to a distance outside the event horizon equivalent to the Schwarzschild radius.
Since the ergosphere is outside the event horizon, it's possible for an object within it to escape the black hole. When it does, it will have energy accumulated from the spinning of the black hole. This is possibly what powers the relativistic jets mentioned earlier, and Roger Penrose showed that it's theoretically possible to design a system to extract power from black holes in this fashion. Of course, any power extracted will come from the energy of the black hole, and if you extract enough energy, the black hole will stop spinning.
Photon Sphere
There is a region outside the event horizon where space is still curved enough that light can travel in complete circles. This is called the photon sphere. In a non-rotating black hole, the photon sphere is 1.5 times the Schwarzschild radius, so it's half as large again as the event horizon. Think of it this way. Since the photon sphere is outside the event horizon, it's possible for light and physical objects to escape the black hole. However, the trajectories on which they can leave the black hole are limited. Light cones have to be pretty nearly perpendicular to the surface of the event horizon in order to leave the photon sphere (the exit cone).
The photon sphere has some interesting features. Light that enters the photon sphere from outside cannot leave the black hole. However, light originating from inside the photon sphere can leave the black hole, provided that its trajectory is aimed directly away from the black hole, and not at an angle. This means that if an object is orbiting the black hole inside the photon sphere, it cannot be seen from the outside through reflected light coming in from the outside. It can only be seen if it generates light of its own, or from some source of light within the photon sphere. Another interesting feature of the photon sphere is that since light in this region can orbit the black hole, if you were inside the photon sphere you could conceivably see the back of your own head, as light leaving it could circle around and reach your eyes.
The photon sphere is just the gravitational distortion of space produced by compact bodies. Any object smaller than 1.5 times its Schwarzschild radius will have a photon sphere. This includes neutron stars.
Black Hole Growth and Evaporation
Black holes grow by accumulating more mass. All black holes absorb the interstellar dust around them, as well as any cosmic radiation headed their way. This is, however, insufficient to increase their mass significantly. In order to grow, a black hole needs a source of matter relatively close by. Black holes formed in binary systems can slowly absorb the mass of the companion star. The supermassive black holes at the centers of galaxies are thought to form by a process of merger of many smaller black holes and stellar objects.
Stephen Hawking showed that black holes aren't completely black, they do in fact emit some radiation. He got this result by applying quantum field theory. Many other people have also mathematically verified this result. The radiation from a black hole is called Hawking radiation, and is in a perfect black body spectrum. The amount of radiation is proportional to surface gravity of the black hole (surface gravity being the gravity at the event horizon). Since smaller black holes have much higher surface gravity than large ones, they are expected to produce a lot more radiation than larger ones.
Since radiation is energy, and energy and mass are mutually interconvertible, black holes slowly lose mass as they emit radiation, and will therefore one day disappear. Stellar and galactic black holes are so large that it would take them a very long time to disappear, a time compared to which the present life of the universe is only a negligible fraction. However, small black holes can evaporate very fast. A black hole the mass of a car would evaporate in a nanosecond, having in that nanosecond, a luminosity 200 times brighter than the Sun. However it's possible that very very tiny black holes (with masses comparable to atomic particles) might have quantum gravity effects that make them stable.
A black hole with a mass equal to 5 Suns has a Hawking temperature of about 12 nanokelvins. This is much less than the cosmic background radiation (2.7 K). Therefore, in today's universe, such black holes are not losing mass, they are in fact gaining mass. Only when the universe ages enough that the cosmic background radiation temperature drops below their Hawking temperature, will they start to evaporate.
You may wish to read the first two parts of this article, about stellar evolution, white dwarfs and neutrons stars.