Chapter 4 Falling into a black hole . . .

  How close is too close?

  

  Before we can consider in detail what would happen if you or your belongings had the misfortune to fall into a black hole, it is important to understand the effect of an observer's particular perspective, or frame of reference. This means that different observers see very different things. Exactly what your perspective is on an object falling into a black hole depends on how far away you are from that object (and indeed whether you are that object!). Consider a particle of light, a photon, that is outside the event horizon of a black hole: since it is outside the horizon, it can in principle escape. Inside the event horizon it would be a different story-the photon could not escape the gravitational field of the black hole. But even outside the event horizon, a photon that is travelling away from the black hole will not escape completely unscathed. The photon suffers a loss in its energy due to the work it has to do against gravity. This is an example of a gravitational potential well; just as energy would be needed to haul yourself upwards out of a deep well, so the photon needs to expend energy to pull itself away from the region near a massive object. The effect has even been measured for photons moving in the Earth's gravity.The energy of a photon is inversely related to its wavelength: a high-energy photon has a short wavelength whereas a low-energy photon has a long wavelength. The photon loses energy as it retreats away from the black hole, so its wavelength increases.This changes the colour of the light, moving from the blue (short wavelength) towards the red (long wavelength) end of the spectrum (this effect is called redshift). This sort of redshift,known as gravitational redshift, arises where spacetime itself stretches out, or is curved, for example by the effect of a massive body such as a black hole. Note that John Michell, despite having significant original thoughts about dark stars, was incorrect in thinking that the velocity of light decreases as it climbs out of the potential well.We now know that it is the wavelength(hence frequency) of light that is affected by the presence of a massive star.

  What happens to time near a black hole?

  In Chapters 1 and 2 I described how spacetime is distorted by the presence of a mass (i.e. something which produces its own gravitational field) and this means that not just space, but also time is affected close to a black hole.

  Imagine you want to keep a safe distance from a Schwarzschild black hole but you want to learn more about how time behaves nearby. Thus you have arranged for twenty-six fixed observers to be stationed close to the black hole's event horizon but definitely safely outside it. These observers are named A to Z, and are arranged in a line with A closest to the event horizon and with Z being nearest to you, safely far away. Each observer from A to Z has a good clock with which to measure their local time, at their particular location. As part of the deal to persuade A to Z to participate in this experiment, you had offered them each an inducement in the form of a gift of an additional, unusual clock that had been adjusted so that the time on it would read the same as the time on your clock at your safe distance. Participant Z,closest to you, would find that the two clocks in his possession read slightly different times because his own clock, which measures local time (`proper time' in the jargon),would be running slightly more slowly than the gift clock which matches the time you measure at your rather safer and more remote distance.The collated results of participants Z to A would display a remarkable effect: closer to a black hole, a clock measuring time`runs more slowly' compared with the distant time as reported on the participants' specially adjusted gift clocks. This effect,described by Einstein's theory of general relativity, is known as time dilation. The effectwould be greater and greater for the observers nearer the start of the alphabet who are nearer to the black hole. The greater the proximity to a black hole, the more slowly a local clock (of any kind: atomic, biochemical) will run compared to a clock used by a distant observer.

  Suppose you were multi-tasking your experiments with a different set of twenty-six observers at the same distances from a different black hole. They are arranged in just the same way as their namesakes near the first black hole. However, in this second case,the black hole has twice the mass of the black hole in your first experiment. The unusual clocks you had prepared as gifts for this second set of observers would need to be radically altered as for your original experiment, but the rate at which each unusual clock has to be adjusted is exactly double that of the rate needed for the corresponding clock in the first set of gift clocks at the exact same distance from the centre of the first black hole which has half the mass of the second. These time dilation effects are larger if the black hole mass is larger, and also become more extreme the closer you get to the event horizon.

  Note that this time dilation is not a consequence of some additional light-travel time for a clock closer to the black hole and hence further from you, the safely-distant observer: there is not merely a time offset for an observer further away from the black hole. The closer a clock is to a black hole, the slower is the rate at which time is measured to flow on that clock, no matter what reputable means you use to measure that flow of time. Time itself is stretched (or, indeed, dilated).

  What is the corollary of time dilation near a black hole? This causes effects that happen in the frame of an observer very local to the black hole to be measured to be very different from those in the frame of an observer who is very distant, worlds apart in fact.

  Let's now consider what happens if in your first experiment,observer A became a little careless and dropped his first clock(the one with which he could measure proper time at his location)so that it fell towards the black hole. Despite this disaster, he would be nonetheless safely gripping onto the gift clock with which you had enticed him to participate in the experiment. Both you and A would see his first clock move towards the hole. The clock would find itself moving into the black hole, more and more rapidly. You and A would gradually notice that the time you read on the plummeting clock becomes even more discrepant with the time on A's other clock (namely the clock that was adjusted to run faster than the local clock in order that it would read the same time as the one corresponding to your time). After a while both you and A would begin to notice that time stops for the plummeting clock. A photon emitted at the event horizon towards a distant observer appears to stay there indefinitely.What happens to anything that falls into a black hole after it has passed within the critical radius of the event horizon is unknowable to an external observer. So the event horizon may be regarded as a hole in spacetime. No light will emerge from within the event horizon,as we saw in Chapter 1. That is why it is black. However, in the reference frame of the dropped clock plummeting through the event horizon, life is very far from unchanging. From the clock's perspective, it would travel to the singularity in a mere one ten-thousandth of a second, assuming that the black hole had a mass of ten times that of our Sun. If the clock had the misfortune to fall into a supermassive black hole with a mass one billion times that of our Sun (such as we meet when we study quasars in Chapter 8), its journey time inwards between the vastly larger event horizon and the singularity would be a more leisurely few hours.

  Tidal forces near a black hole

  Suppose in a weak moment, person A wonders about jumping,feet first towards the black hole, in hopes of being reunited with the clock he dropped. What would happen? Such a leap would prove to be a big mistake, as the survival outcome would be zero.The difference between the gravitational force on his feet and the force on his head would become extreme. This is a feature of any inverse-square force field, such as gravity from a massive body.The Earth is rather a long way from the moon, yet even the small differences in gravitational force due to the moon experienced on opposite sides of the Earth, known as tidal forces, are at the root of why the tides come and go about twice per day. In general, these forces resulting from differences in gravity in different places are called tidal forces. There are additional factors that enrich the details of the rising and falling of tides such as the gravitational force due to the relative angle of the moon, and the detailed shapes of continental masses. But even if the surface of Earth were entirely covered by ocean without land, there would still be tides with the amplitude of the sea level varying by about 20 cm twice per day, simply because of the differential gravitational force experienced by points on the planet at different distances from the Sun.

  Let's now consider the smaller distance between me and the centre of the Earth. As I sit typing this chapter, my head is somewhat over a metre higher than my feet which are on the floor of my study. My feet are thus closer to the centre of the Earth than my head is. Because the gravitational force follows an inverse-square law behaviour as though all the mass of the Earth were located at the very centre of the Earth, and because my feet have a smaller distance to this centre they feel a stronger force, or pull, to the centre of the Earth than my head does. But actually,the difference is rather slender: for a height difference of one metre the difference in gravitational force is three parts in ten million. This is such a slight difference because I am about6,400 km from the centre of the Earth. Much closer to a point mass such as a black hole, the difference in gravitational force experienced at points just a metre apart in the direction towards the black hole would be vastly more extreme. So extreme that close to the singularity A's feet would be stretched away from his knees and the rest of his body beyond what his tendons and muscles could hold together, and he would be elongated into something resembling long spaghetti. Best not to jump.

  Dynamic spacetime

  The rotation of a black hole makes an important difference regarding how close matter can orbit around it, and this relates to how much energy can be extracted from it. From the work of Roy Kerr and his solution to the Einstein field equations, we know that the smallest orbit that a particle can have around a black hole without falling in depends on just how fast the hole is spinning.The faster a black hole is spinning, the closer the matter can get before the hole swallows it, as illustrated in Figure 13. If you drop something straight down into a spinning black hole, it will start orbiting the hole even though there is nothing but empty spacetime outside the hole. Outside the ergosphere, it is possible to overcome this frame dragging using rockets, but not inside it. In the region inside the rotating black hole's ergosphere, just outside its event horizon, nothing can stand still. The spinning hole actually drags the spacetime and hence its contents around with it.A further aspect of this frame-dragging is that even if light itself is going against the direction the black hole is rotating, it will be carried in the reverse direction around the hole.

  13. Gas can orbit closer to a spinning black hole than to a non-rotatingone.

  Orbiting around a black hole

  It is interesting to ponder what would be the sequence of events if our Sun were to spontaneously metamorphose into a black hole right now. The first that you or I could know about it would be eight minutes later; the beautiful Spring sunlight by which I am writing would come to an abrupt halt. Although the luminosity of the single star we call our Sun is tiny by comparison with the quasars and microquasars discussed in Chapter 8, it is sufficiently close to the Earth that it provides on average about a kilowatt per square metre of power to our planet. Remarkably, this has been enough to sustain all life on the planet, allowing plants to grow and then be eaten by animals that are then eaten by other animals.The Sun has been the engine behind it all. But if fusion ceased in the Sun and it were (contrary to all expectation) to collapse into a black hole, then it would go very dark and we would all eventually die. (This is a bit of a gloomy outlook, but I encourage the reader to hold fast until Chapter 7, where we learn that our Sun is not the kind of star to form a black hole-it's too lightweight for that.)However, dynamically speaking, as far as planet Earth and the whole Solar System of planets, dwarf planets, and asteroids are concerned, nothing will change at all. All massive bodies in orbit around the Sun will continue in pretty much the same orbits. The way that gravity works is that whether the Sun has the same extent that it has now, or whether it collapses to a singularity within an event horizon of 3 km, the gravitational attraction outside the Sun would remain unchanged. The spherical collapse under gravity to a black hole would not change the angularmomentumof the orbiting bodies at all, so the patterns and progressions and tides within the Solar System would continue utterly unaltered by the lack of sunshine.

  Some new orbits would be possible however, much closer to the black-hole Sun than were possible previously when the solar plasma was in the way. However, these orbits could not get too close to the event horizon. The details of the warping of spacetime by a mass singularity mean that it is not possible to orbit just outside the event horizon itself. Attempting a circular orbit there would require corrective action by rockets in order to maintain the orbit. In fact, the mathematics shows that the closest that we or any other mass particle could exist on a stable circular orbit near a stationary black hole would be at a distance three times that of the Schwarzschild radius away. You have been warned.

  Actually, unstable circular orbits are possible up to half this distance away from a Schwarzschild (non-spinning) black hole.This distance defines a spherical surface that is sometimes called the photon sphere. Even for a photon, these orbits are unstable,and before too long an orbiting photon would either slither in towards the black hole, never to return, or indeed away into space.

  For a Kerr black hole though, one that has spin, the situation is different for the orbits near the black hole. In particular, there are two photon spheres, in contrast with the one photon sphere around a stationary Schwarzschild black hole. The outermost sphere is for photons that are orbiting oppositely to the direction of rotation of the black hole (the ones we say are on retrograde orbits). Inside this is the photon sphere for photons travelling in the same sense around the black hole as it is rotating (on prograde orbits). For a very slowly rotating black hole that isn't so very different from a Schwarzschild black hole, these two photon spheres are very nearly co-spatial. For black holes of increasing spin, these surfaces are increasingly further apart.

  Moving closer in towards a rotating black hole, there is another important surface (discussed in Chapter 3), called the static limit.This is the surface at which nothing can remain static with respect to a distant observer: it is just impossible to sit still this close to a rotating black hole, no matter how powerful the rockets you might be equipped with. At this surface, even retrograde light rays are dragged along in the direction of rotation. It is still possible to escape from this close to a rotating black hole, with sufficient propulsion, but it's just not possible for anything to remain stationary and non-rotating here. Moving inwards still further,the next surface of significance is the event horizon we met in Chapter 1, the one-way membrane that we met originally in the context of Schwarzschild black holes. Crossing this outwards isn't possible and crossing it inwards has an ineluctable destiny, just as for the static black hole.

  An orbit around a Kerr black hole is not generally confined to a plane. The only orbits confined to a plane are those in the plane that contains the equator (i.e. the plane of mirror symmetry of the spinning black hole). Orbits out of this equatorial plane move in three dimensions. These orbits are confined to a volume that is limited by a maximum and minimum radius and by a maximum angle away from the equatorial plane.

  The details of the spin of a black hole have a dramatic effect on how close particles may encounter the black hole, which itself depends on their direction of travel relative to the spin. For a maximally spinning black hole, the photon sphere for light rays orbiting in the same sense (prograde) as the black hole spin has a radius that is half of what the Schwarzschild radius would be. For light rays on retrograde orbits, the radius of their photon sphere is twice the Schwarzschild radius. For particles with mass that are on prograde orbits, the innermost stable circular orbit on which they can move is again at half of the Schwarzschild radius. For those on retrograde orbits, such a close distance would be unstable: their innermost stable circular orbit is at 4.5 times the Schwarzschild radius. Thus, a rotating black hole enables particles on prograde orbits to orbit more closely without reaching the point of no return at the event horizon, more closely than if the black hole were non-rotating. In Chapter 7, we consider the importance of just how close matter can orbit before falling onto a black hole and how much energy may be consequently leveraged.