We have seen that there is a black hole at the centre of our own galaxy. And indeed, one quickly realised that black holes are ubiquitous in all the Universe. Now, probably your first reaction is to think that if there’s a black hole somewhere, it is going to swallow, to suck all the matter and we will be sucked into the black hole itself. Well, this is a little more complicated and you can safely travel away from the black hole if you’re at a sufficient distance. So let me explain this to you with the example of the waterfall. So here is a waterfall and you certainly have already swam in a river, not checking whether there was a waterfall further downstream.
And indeed, if the waterfall was far enough, you could safely swim in the river. Now, at some point, you might have crossed a line, which I will call the horizon of the waterfall. And as soon as you have crossed this line, you could be the best swimmer in the world, you’ll be attracted to the waterfall and you fall into what I will call the black hole. So you see this is a sort of a model of a black hole and this line is the horizon and similarly, you have an horizon for a black hole. That means that this is a line of no return. If you cross it, then you’re attracted to the central singularity of the black hole.
On the other hand, if you are away, if you are at a distance larger than the size of the horizon, then you can safely either swim in the river or just travel in space time. You might be orbiting around the black hole, but you will not be falling. You will not be sucked into the black hole singularity.
So as long as we are outside the horizon of the black hole, which is the spherical surface around the black hole, we are protected from all the strong effects of the matter, which is falling into the singularity of the black hole. Now still, very close to this horizon, the gravitational field of a black hole is important and so there are important gravitational effects. Let me illustrate this type of effects by having here a model of the horizon, so this red line here sort of imitates the surface of the horizon and just imagine that I’m an astronaut who is going to cross the horizon.
Now, I am very close to the horizon as you see and so the gravitational effects are important. And we’ve seen that one effect of gravitation is to change the way time is running. So this means that the time that is shown on my watch is different from the time that is shown on the space station that is orbiting some distance away from the horizon and is still different from the time shown in the clocks on Earth. And precisely because the time is running differently, when I cross the horizon, the different people in different reference frames will see different things.
So in my reference frame, I’m falling into the horizon and I’m in free fall like the elevator in free fall and so I should see nothing special because this is as if space time was not curved. I’m in free fall. On the other hand, on the space station, which is orbiting at some distance, the space station, in order to counteract the gravitational attraction of the black hole has to be in an accelerated motion. They have to start their engines in order to prevent the fall into the black hole. And so time is running differently in the accelerated reference frame of the space station and similarly, with the Earth.
And so that means that whereas in my case, I’m just freely falling into the black hole, what they will see on the TV screens of the space station is a situation completely differently. Time will be running much more slowly and so they will never see me crossing the horizon. It will take them an infinite amount of time to see me crossing the horizon. On top of that, the photons that I’m sending for information to be on the TV screen are red shifted which is the same time of red shift that we have seen in the case of the expansion of the Universe.
And so little by little, they will not be visible and so on the TV screens, they will see an image which is blurred more and more, so eventually, my motion will be slowing down, the picture will be blurring more and more, and so in the end they won’t see anything, but they will never see me reaching the horizon of the black hole. So now comes the moment let me cross the horizon.
Let me cross the horizon.
Now that I’m on the other side of the horizon, let me take the opportunity to describe a particular feature of time and space inside the horizon. Well, it turns out that the role of time and space is interchanged. So that means that whereas on the other side, time had an arrow, from past to future, on this side, it’s space, which has an arrow. Because I’m attracted towards the centre, towards the singularity, so I’m inexorably attracted towards the centre. And so you see that in this way space has an arrow. On the other hand, time can be perceived at different stages.
So that means that in the same way as on the other side, I could see different places at the same time, now, I could see simultaneously different parts of my own history. So me now, a couple of seconds earlier, five seconds earlier, and so on, and even in the future.
If you scratch your hair trying to understand this inversion between time and space, it’s about time that we discuss the no-hair theorem. Well the name of that theorem is due to John Archibald Wheeler and the reason for this name is the fact that Wheeler considers that black holes are very simple objects, especially the horizon of a black hole is as simple as a bald head. Now, I don’t have a bald head here, so let me produce an egg, which is a fairly good model.
And so what he tells you is that the black hole horizon is an extremely simple sphere in a sense and moreover, seen from the outside, the black hole is only characterised by three numbers, the mass of the black hole, the charge of the black hole, because black holes could have electric charge, and also what we call the angular momentum which characterises the rotational motion of the black hole because the horizon could be in rotational motion. And so strictly speaking, it doesn’t have no hair, it has got three hair, the charge, the mass, and the angular momentum.