Chapter 3 Characterizing black holes

  In Chapter 1, we introduced the concept of a mass singularity,forming in gravitational collapse, and surrounded by an event horizon. Examples of such objects that are not spinning are called Schwarzschild black holes and this term specifically denotes black holes that are not rotating: in the jargon, they have no spin.Simply put, the only characteristic that distinguishes one Schwarzschild black hole from another (other than location) is how massive it is. In Chapter 7 we will learn how black holes grow but for now, it will suffice to know that collapse under gravity is the key ingredient. If there is any rotation whatsoever in the pre-collapsed matter, however gentle, then as the collapse occurs the rotation rate will increase (unless something acts to stop that happening). This arises due to a remarkable physical law known as the conservation of angular momentum. This law is illustrated by a pirouetting skater: as she pulls her arms in she spins faster. In the same way, if the star that gives rise to the black hole is gently rotating then the black hole that it ultimately forms will be spinning significantly and is termed a Kerr black hole. Most stars are in fact rotating, because they themselves are formed from the gravitational collapse of slowly rotating massive gas clouds. (If such a gas cloud had even a minute amount of net rotation then the collapsing cloud will have non-zero angular momentum, and as the matter occupies an increasingly smaller volume the final rotation of the collapsed object may well be rather rapid.) Thus we see that rotation, more commonly called spin, is likely to be a prevalent, if not actually a ubiquitous, characteristic for black holes that have just formed from the collapse of matter.We now believe that spin is as inevitable in real astrophysical black holes as it is in current-day politics (though in the latter case it arises from something other than the conservation of angular momentum!).

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  We have now stated that a second physical parameter, that of spin or angular momentum, is a characteristic that distinguishes one black hole from another just as mass does. Thus, there are two properties of black holes that are important to keep in mind as we study the behaviour of black holes: mass and spin. In principle,there is a third characteristic of black holes that might be relevant to their behaviour: electrical charge. This is also a conserved quantity in physics, and the forces between electric charges,known as electrostatic forces, have a number of resemblances to gravitational force. A key similarity is that both are (on large scales) examples of inverse-square laws meaning that, in the case of two massive objects, as you double the distance that separates them from one another the gravitational force they experience reduces to a quarter of the original value. A key difference is that while gravity is always attractive, electrostatic charges are only sometimes attractive (when the two bodies are oppositely charged,i.e. one is positive and the other is negative). They are at other times repulsive (when the bodies have charge of the same sign,either both positive or both negative, they repel each other). If two charged bodies have the same type of charge, then electrostatic repulsion will tend to prevent them coalescing, even if gravity is tending to attract them. So while charge could in principle be a third property of black holes that one might hope to measure,in reality a charged black hole would be rapidly neutralized by the surrounding matter. It is therefore a good operational assumption that there are only two relevant properties of black holes that distinguish one from another: mass and spin.That's all!

  Now, you might wonder whether black holes could be distinguished by their composition. One might have been formed from a hydrogen gas cloud, another from a helium gas cloud.Why should it be that the provenance of the collapsed matter that gave rise to the black hole isn't manifested in the measurable properties of the black hole subsequently formed? That's because information can't get out of the event horizon! Light is the means by which information might be transmitted, but we have already seen in Chapter 1 that it cannot escape from inside the event horizon of a black hole. Thus the chemical composition of the matter that fell into the black hole can have no effect on the properties of the black hole as determined from the outside. It would not be correct to think of gravity as something that needs to `get out of ' the black hole. The continued existence of a gravitational field external to the black hole is something that is laid down in the formation of the black hole as spacetime becomes distorted. No influence from inside the black hole could change the external field after the event horizon has formed.

  Black holes have no hair

  When we are asked to describe another person, a distinguishing characteristic that is often included is their hair (for example,strawberry blonde, or silver grey or chocolate brown). There are sometimes clues in the nature of people's hair as to their age or their nationality. Information about further physical characteristics such as `Body Mass Index' might provide information on their diet. In contrast to humans, black holes are entities that have absolutely no distinguishing characteristics other than their mass and their spin (neglecting charge for the reasons noted above). This is captured in the breviloquent phrase`Black holes have no hair', coined by JohnWheeler to emphasize that there is nothing about a black hole that bears any evidence of the nature of its progenitor star. Not its shape, not its lumpiness,not its landscape, not its magnetism, not its chemical composition.Nothing. Calculations done by, amongst others, the Belarusian physicist Yakov Zel'dovich demonstrated that if a non-spherical star with a lumpy surface collapsed to form a black hole, its event horizon would ultimately settle down to a smooth equilibrium shape having no lumps or bumps of any kind. So, a black hole never has a bad hair day! The only things you can know about it are its mass and spin.

  Spin changes reality

  Perhaps the most remarkable feature of a spinning black hole is that the gravitational field pulls objects around the black hole's axis of rotation, not merely in towards its centre. This effect is called frame dragging. A particle dropped radially onto aKerr black hole will acquire non-radial (i.e. rotating) components of motion as it falls freely in the black hole's gravitational field.

  What this means for a test particle having spin (such as a small gyroscope) is that if it falls freely towards a rotating massive body,such as a Kerr black hole, it will acquire a change to its spin axis.It is as though its local frame of reference was dragged by the rotation of the central massive body. This phenomenon,discovered in 1918, called the Lense-Thirring effect actually occurs not just around black holes, but to some extent around any spinning object. If you put a very precise gyroscope in orbit around the Earth, the frame dragging causes the gyroscope to precess.

  It is Einstein's field equations that describe the mathematics of black holes and, as also mentioned in Chapter 1, Karl Schwarzschild solved these equations for the case of the stationary(non-rotating) black hole, a remarkable achievement given that he did this in 1915, the same year that Einstein introduced his general theory of relativity. The case of the spinning black hole was treated much later by New Zealander Roy Kerr in 1965. A few years after this, the Australian Brandon Carter explored Kerr's solution further still. Carter carried out an in-depth investigation into the consequences of the Kerr metric. He established that a spinning black hole causes a dramatic swirling vortex in the spacetime that surrounds it which arises because of the dragging of the reference frame. An example of a vortex is a whirlwind-close to the centre of the whirlwind the air swirls rapidly, carrying with it anything in its path, be it hay in a hay field or sand in a desert. Further from the whirlwind the air (and hence hay or sand) rotates much more slowly. So it is too, with spacetime surrounding a spinning black hole: far away from the event horizon, the speed at which spacetime itself rotates is slow, but at the horizon, spacetime itself spins with the same speed that the horizon spins.

  The event horizon for the spinning (Kerr) black hole is much the same as for a non-spinning (Schwarzschild) black hole, except that the faster the black hole is spinning, the deeper the gravitational potential well: a Kerr black hole forms a deeper gravitational potential well than a Schwarzschild black hole of the same mass,and therefore a Kerr black hole can be a more powerful energy source than a non-spinning one, a point to which we return in Chapter 7. In the meantime, it is helpful to summarize this behaviour by saying the event horizon of a Schwarzschild black hole depends only on mass, but that of a Kerr black hole depends on both mass and spin.

  An outstanding question is whether there could be, even in principle, any spacetime singularities that are not enclosed within and hidden by event horizons-a so-called `naked singularity'. By definition, all black hole solutions to the Einstein field equations do have event horizons and, as shown in Chapter 1, no light and therefore no information can escape from within such horizons.All black hole singularities are believed to be enclosed within event horizons and therefore not `naked', so that direct information about the singularity is inaccessible from the rest of the Universe. The so-called cosmic censorship conjecture was formulated by the British mathematician Roger Penrose and states that all spacetime singularities formed from regular initial conditions are hidden by event horizons and that there are no naked singularities out in space.

  How much spin is too much?

  There is a limit to how much angular momentum a black hole can have. This limit depends on the mass of the black hole, so that a more massive black hole can spin faster than a less massive black hole. A black hole that is rotating close to this maximum limit is known as an extreme Kerr black hole. It is possible to show that if you try to spin up a black hole, to make an extreme Kerr black hole, by firing rapidly rotating matter into it (i.e. giving it a stir)then centrifugal forces prevent the matter from even entering the event horizon.

  Somewhat further out from the event horizon of a rotating black hole is another significantmathematical surface which is known as the static limit. The dragging of inertial framesmeans that if the spin of the massive body is non-zero then there are no stationary observers inside of this surface: every physically realizable reference frame inside the static limit must rotate.Within this surface, space is spinning so fast that light itself has to rotate with the black hole, i.e. it is impossible to remain motionless. The region between the static limit and the event horizon is known as the ergosphere, which rather confusingly is not spherical, as shown in Figure 10. In equatorial directions the ergosphere is much larger than the event horizon, but in the polar directions the radius of the ergosphere is the same as the radius of the event horizon. The resulting shape of the ergosphere is an oblate spheroid, resembling the shape of a Jarrahdale pumpkin(without the stalk). The first two syllables of ergosphere, however,come from the Greek noun 俽gon relating to `work' or `energy' (as in `ergonomics') from which the old unit of energy, the erg, is also derived. It is intriguing to note that in addition there is a Greek verb ergo which means to enclose and keep away, appropriately for the nature of the ergosphere. Perhaps this may have been in the minds of Roger Penrose and Demetrios Christodoulou who coined and championed the name of this region around a spinning black hole. The importance of the ergosphere is that it is the region within which energy can be extracted away from the black hole.

  10. The different surfaces around a Schwarzschild (stationary)black hole and around a Kerr (spinning) black hole (in the frequentlyused representation of `Boyer-Lindquist' coordinates).

  Since inside the ergosphere space is spinning, particles of matter within that space also get swept up into a rotational motion.Considerable rotational energy is therefore stored in this rotation of space, a very important point to which we return in Chapter 8.

  White holes and wormholes

  Einstein's equations of General Relativity are particularly rich and allow many different solutions describing alternative versions of curved spacetime. This provides an almost inexhaustible source of possible universes for cosmologists to describe and think about.Which type of universe we actually live in is a matter that can only be decided by observation (if at all!). But that doesn't stop mathematical physicists playing around with Einstein's equations to find all kinds of interesting solutions.

  One intriguing object that can be dreamt up by mathematical physicists is what is called a white hole. A white hole behaves just like a black hole but with the direction of time reversed (imagine a movie played backwards). Instead of matter being sucked in, it is spewed out. Instead of the event horizon marking out the region from which you can never escape, it stakes out the region into which nothing could ever enter. Once matter exits from a white hole, it can never return there; its entire future is outside. As we see in Chapter 6, a black hole is formed from a collapsing star and must eventually evaporate by the laws of quantum mechanics into Hawking radiation (see Chapter 5). A white hole, on the other hand, could only result from radiation that for some reason spontaneously assembles into a black hole. It is not easy to understand how this could happen in practice, and moreover Douglas Eardley has demonstrated that white holes are inherently unstable.

  When Einstein and his student Nathan Rosen were playing around with Einstein's equations in the 1930s, they found an interesting solution. If a region of spacetime could be strongly curved, it might be possible for it to become sufficiently folded that two parts of spacetime which had previously been separated by a large distance could become connected by a small bridge,or wormhole, as shown in Figure 11. The enormous distances between the stars and galaxies have always been unfavourable for those writers who wish to set human dramas on a cosmic stage,and wormholes (also known as Einstein-Rosen bridges) have provided the perfect plotting device for writers to transport their heroes and villains about. This mathematical invention has been an absolute boon to the writers of science fiction, because it provides a ready means for traversing enormous distances through space and thereby to sustain various highly artificial and unbelievable plot devices. Yet again, we have no observational evidence that wormholes actually exist in our Universe. In addition, there is considerable theoretical evidence that a wormhole, once formed, would not be stable for very long. It seems that to keep a wormhole propped open, one needs a large amount of negative energy matter, and all normal matter has positive energy (this is connected with the fact that gravity is normally always attractive). Normal matter passing through a wormhole may be enough to destabilize and destroy it, causing it to turn into a black hole singularity.

  11. A wormhole connecting two otherwise separate regions ofspacetime.

  If wormholes did exist, and could be maintained for any reasonable length of time, they would have some surprising and bizarre properties. Not only would they provide a means for taking an enormous shortcut across a vast expanse of space, but they would also allow a traveller to journey back in time. One would then be able to construct closed time-like curves, loops in spacetime in which the light cones form a ring (see Figure 12) so that, like in the movie Groundhog Day, a person travelling along a closed time-like curve would simply repeat their same experiences over and over again.

  In fact, there are a number of solutions to Einstein's equations in addition to wormholes which have this alarming and counterintuitive property. In 1949, the mathematician Kurt G?el found a solution that described a spinning universe, and this contains exactly the same sort of closed time-like curves which pass through events again and again in an endless Groundhog Day cycle. (Evidently `free will' is not part of the field equations!)The part of the Kerr solution thought to have genuine physical significance in the real world is that which describes the spacetime outside of the event horizon. However, it is unclear whether the part of the Kerr solution inside the event horizon, while mathematically sound, has any physical relevance. In this part of the Kerr solution, the singularity is not a point (as it is for the non-rotating black hole) but has the form of a rapidly rotating ring(however, the physical validity is very speculative). This ring-like singularity is surrounded by closed time-like curves. On such a curve, your future is also in your past and you have the theoretical possibility of murdering one of your own grandparents before they had produced your parents! Thus the existence of closed time-like curves seems to create the possibility of all kinds of paradoxes relating to time travel. One possible solution to this is to admit that we do not have a theory that links quantum mechanics (which describes the very small) and general relativity (which describes the very massive), in other words a theory of quantum gravity.We don't know the physics of extremely massive but very small objects.Most physicists think we need this to fully understand the behaviour of spacetime very close to singularities. Thus it may be that these strange solutions to Einstein's equations do not actually occur in the Universe because they are prohibited by its fundamental quantum mechanical nature. Quantum effects may,for example, destabilize wormholes. Stephen Hawking believes this to be the case and has called this principle the `Chronology Protection Conjecture'. He has quipped that this is the underlying principle that keeps the Universe safe for historians.

  12. A closed time-like loop, on which your future becomes your past.

  There is much about the interior of rotating black holes that pushes our understanding of fundamental physics to the limits and therefore to where much of our description is highly speculative. By contrast, the rotation of black holes and their effect on their surroundings is something that has enormous practical significance for understanding what we can see with our telescopes. Thus our next step is to consider in more detail what happens to matter when it falls into a black hole.