The main theme of the 2020 Nobel Prize in Physics is black holes, and it has been awarded to Roger Penrose, Andrea Ghez, and Reinhard Genzel. The Nobel committee awarded half of it to Ghez and Genzel for their pioneering work in showing that there is a supermassive black hole at the centre of our galaxy, the Milky Way; the other half has been awarded to Roger Penrose for showing that ‘black hole formation is a robust prediction of the general theory of relativity.’ This three-part series of articles focuses on Penrose’s contribution.
Part I of the series can be read here.
I promise you that we won’t do any maths after this, but if you can bear with me, you will gain very incredible insight and appreciation for some of the deepest mysteries in the history of cosmology. All it requires is some minor dealings with fractions, and we will only look at the first two terms.
Let’s look at what happens to our equation when the radius r is the Schwarzschild radius i.e. r=2GM
The Schwarzschild radius divided by itself is 1 and 1-1 is zero, so the first term (next to dt) vanishes, which is all well and good. Let’s see what happens when we plug it in the second term. I should point out for those who are not familiar: if you look at the second term, you’ll see it’s raised to the power of -1. In this case, this means that once we solve whatever is inside the bracket, that term then becomes the denominator i.e. we have to divide dr by our answer.
The inside term is the same as before, so we already know the answer is zero. So now we have to divide by zero, and anything divided by zero gives us infinity, and hence the whole equation becomes infinite. Our equation crashes.
Now, what exactly does this mean? Well, we know that this means that something weird is happening at the event horizon where r=2GM. We say there is a singularity there which you can’t pass. A singularity can have multiple meanings (more on this later), but for our intents and purposes, this means that either our theory breaks down and we need a better one, or that singularity is something real in which case we have something like an infinite curvature.
This puzzled Einstein deeply, and he initially thought that it implies that if an object was falling towards the black hole, it would bounce back outwards as soon as it reached the event horizon (his reasoning was not exactly clear). He later went on to conclude that this singularity was an artefact of mathematics and not something real in the physical universe.
Was Einstein, right? Well yes, he was (mostly). It turns out that the singularity at the event horizon is a ‘coordinate singularity’. This means that the singularity is not actually in space-time but is instead a rudiment of the coordinate system being used. A coordinate system can be vaguely thought of as a measuring system; you can represent the same thing in different coordinate systems. For instance, you can draw a right-angled triangle on a Cartesian plane and represent it using either Cartesian system i.e. (x,y) or polar coordinates (θ, r). It is possible to convert from one system to another while maintaining the overall geometry (in this case at least) e.g. the length of the hypotenuse of the triangle will be the same whether you measure it in polar or Cartesian coordinates. A less accurate but perhaps a more straightforward intuition might be to think of coordinate systems as different measuring systems e.g. I can say something like ‘The nearest coffee shop to my house is 1 mile away,’ or I can say ‘The nearest coffee shop to my house is 1.6 km away’; they are both describing the same amount of distance in different units of measurement. Building on this example, it should be noted that some measuring units are better than others, depending on what you are trying to measure. For example, it makes more sense to use centimetre if you measure some line on a small piece of paper than using miles or lightyears. In a similar vein, some coordinate systems are better than others, depending on the geometry you are working in. As it turns out, by introducing a new coordinate system such as the Eddington–Finkelstein coordinates (we won’t go into the details here) and rewriting the metric in this coordinate system, the singularity at r=2GM vanishes and it, in fact, becomes possible to traverse the event horizon. But then, at this point, our story becomes even stranger.
Referring back to the Schwarzschild solution above, knowing that the singularity at r=2GM is an artefact of a bad coordinate system rather than an actual singularity, we can turn our attention to the other problem that arises that some of you might have already noticed. This time, we will push beyond the event horizon and towards the ‘centre’ of the black hole where r=0. When we plug in that value, we find the equation blows up to infinity once again, another singularity! But this time, it can be shown after doing some maths (for the avid reader, this is done by calculating Kretschmann scalar for the Schwarzschild geometry and then plugging in the value of r=0) the singularity at r=0 is not a coordinate singularity but rather a “curvature singularity”. A curvature singularity persists in all coordinate systems, and this implies that there is an infinite density or curvature at least in some sense if the general theory of relativity actually holds true at the centre of a black hole.
An important bit of note here regarding the singularity at r=0 is the fact that singularity is not a location in space that you travel to but rather a moment in time. This is because the causal structure of space-time (or perhaps, more appropriately, the light cone structure) ‘inside’ the event horizon is different from the one outside it. As an example, imagine a photon travelling towards the event horizon from outside the black hole along a radial path. As soon as the photon reaches the horizon and starts to cross it (in reality the photons would be feeling the effects of gravity before it crosses the horizon), the spatial component (in this case the radial path of the photon) becomes a time coordinate i.e. (+t=-r) and the time coordinate becomes the spatial or radial coordinate. This means that when the photon is in the region where r is smaller than 2GM and it moves towards the ‘centre’ where r=0, the photon is actually moving in time where r=0 exists as a temporal point in the future. Hence, it’s approaching the singularity. The photon can’t escape and travel back out of the event horizon because it would mean that the photon has to travel towards r=2GM which, in the new causal structure, means that the photon has to travel back in time. Since it is not possible for the photon to travel back in time, it inevitably has to go towards the future, towards the singularity where time will end for the photon. The photon’s fate was sealed as soon as it ‘crossed’ the event horizon or the Schwarzschild radius and in a sense, it was captured by the singularity.
So far Schwarzschild has shown that it is possible to solve Einstein’s equation for a spherically symmetric space-time, but the charge was that it is just one mathematical solution and that too an idealised one (assumes spherical symmetry) and had problems such as coordinate singularity. Robert Oppenheimer, along with others (as alluded to before), showed that gravitational collapse actually continues beyond R=2GM thus not a real singularity (plus a plethora of mathematical arguments that we already discussed also provide us with reasons for believing R=2GM is not a real singularity, even including some later work of Einstein himself). However, even Oppenheimer’s collapse assumed complete spherical symmetry and hence was thought to be an idealisation that did not exist among actual astrophysical black holes.
Kerr Geometry and Spinning Black Hole
Just about two years before Roger Penrose’s famous contribution that we are going to get to, Roy Kerr, a mathematician from New Zealand, provided a more general solution to Einstein field equations for rotating black holes. Kerr did not assume spherical symmetry; instead, he modelled the space-time around a massive rotating object as stationary ( roughly speaking, spinning in the same direction) and he also used the vacuum equations along with some minor additional assumptions to build the equation that describes Kerr geometry.
Kerr black hole has a Kerr geometry (just as was the case with Schwarzschild geometry). However, unlike the Schwarzschild case where you only needed mass,you need both mass and angular momentum to define the geometry in Kerr geometry. Kerr black holes are considered to be more like real astrophysical black holes since they are rotating and they form an accretion disk around them, but it’s the internal structure of them that is extremely exotic.
According to Kerr geometry, these black holes have two event horizons: an internal one and an external one. There is also a curvature infinity at r=0 (technically misleading) ‘inside’ the internal horizon, but this singularity is very different from the one we encounter in Schwarzschild geometry because rather than being a point, this singularity is a ring.
This gives rise to some mind-boggling implications. For example, if we take the case of a photon travelling towards the centre of the black hole then, like Schwarzschild black hole, as soon as the photon crosses the external event horizon, the temporal coordinates shift with the spatial ones, and as you travel towards the inner horizon, you are actually travelling towards a future time rather than a location in space. When you hit the inner horizon and pass it, weirdly enough, the coordinates shift again, and once again you are travelling in space, where the ring singularity lies towards the centre. There are many questions you can ask about the possibility of time travel into the past (via generation of closed timelike curves): Does it mean it’s possible to escape this kind of black hole? (it isn’t); Is it possible to traverse the ring singularity? I won’t address these questions here because they are not extremely important for our story and because Kerr geometry deserves an article treatment in its own right. Suffice to say for now, not all implications of Kerr geometry are realised in actual astrophysical black holes (fun fact: if you extend Schwarzschild geometry then you get descriptions of white holes and wormholes which are thought not to be realised either).
We will discuss a special case for Kerr geometry that has some relation with Penrose’s work. This is a special case of the Kerr black hole when the ‘angular velocity’ is approximately greater than the mass contribution of the black hole. Under such a scenario, it is possible for a photon to start at ring singularity and travel infinitely further away from it; this, in turn, means that for such a Kerr black hole, the event horizons have vanished and the singularity is exposed to an observer outside the black hole. Such a singularity is known as a ‘naked singularity’.
At this point, one can wonder if people had started taking the existence of black hole seriously since the main charge against Schwarzschild and Oppenheimer was the fact that their solution and models both assumed an exact spherical symmetry which is not actually realised in nature, so they are mathematical fictions. Kerr, however, did not assume spherical symmetry yet his solution for rotating black holes (geometrical) still had a curvature singularity like Schwarzschild. So maybe these black holes actually exist with a singularity in them. But the fact of the matter is that Kerr’s solution is still very symmetrical. In fact, rather than being spherically symmetric, Kerr’s solution is axisymmetric (looks the same from all directions when the object rotates around its axis). This charge was levelled against Kerr by Roger Penrose in his famous 1965 paper (not adversely). Since if we were pedantic then Kerr’s solution is still an idealisation and might not actually hold true in nature.
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