Black Holes

Theoretical physicist Mahdi Godazgar on Einstein’s equations, the event horizon, and spaghettification

videos | June 3, 2016

It’s just over one hundred years now since Einstein came up with his theory of general relativity, which is a theory of gravitation. And in contrast to the previous theory of gravitation, which was due to Newton, this is a geometric theory, it doesn’t rely on mysterious actions at a distance the way the Newton’s theory did. And basically what it’s saying is that gravitational attraction is due to curvature of the spacetime around us, and the curvature is created by matter, and this interplay between geometry on the one hand and matter on the other hand is set through the Einstein equations, which is basically that the curvature of the spacetime is equal to the energy of matter. This is at the heart of Einstein’s theory of general relativity.

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If we assume that we now turn off the matter, we have no matter prevent, one would assume that Einstein’s equation will tell us that our spacetime is flat. But this isn’t true in fact, it’s just that Einstein’s equations with curvature of spacetime equals zero. It is itself a very complicated equation, there remain many different solutions, and probably the most important solution of this equation is a black hole solution.

So black hole solutions are solutions to this equations where the region, the black hole region is such that an observer that goes into that region can’t communicate with the outside world, with the distant observer. The way a black hole is defined is via the surface, which is called ‘the event horizon’. And outside this event horizon, it’s like orbiting a very dense star – there isn’t anything special about it. But as soon as one force thwarts this surface – the so-called event horizon – strange things start to happen. For example, if one is sitting very far away as a distant observer from this surface and watching something fall in, then from that person’s perspective it feels as if it’s taking an infinite time for that person to approach the surface. Whereas for the person themselves they reach that surface and go beyond it within a finite time. For them that surface has no special features or characteristics.

Of course, if it was you or I who were falling into the black hole then because we have a length and the difference in the gravitational forces, say, between our legs and our heads if we are dropping with our heads first then it would mean that we’d basically be broken up by the tidal forces due to the strong gravitational field. This is called spaghettification whereby we just get streched, and stretched, and stretched until we completely break up. If we assume that we were just some tiny particle then – nothing special happens.

What’s more strange is that once we are inside this surface time and space exchange. So the notion of time becomes space and the notion of space becomes time.

And just as outside the black hole in everyday life we are forced to follow time forward, inside the black hole we are forced to follow space forward. And what this means in that everything is moving towards one thing, and what Penrose and Hawking showed in the 1960s is that under very generic circumstances, if you have a strong gravitational field such as you do in a black hole then singularities are always going to appear, so that everything inside the black hole is together moving into a singularity. The theorem of theirs is called the singularity theorem and it is one of the most important results in general relativity. It appears, say, in cosmology – we can apply this notion to cosmology by saying if we have, if we trace back our evolution of the universe, again, we will always end up in a singularity.

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So how are we to view these singularities? They are very generic and from an astrophysical perspective, for example, black hole is expected to appear. We know for sure that at the center of most galaxies there are black holes. And because of it Penrose and Hawking said that there are singularities inside them. Or recently because of the new gravitational wave detection where they found a signal of two black holes colliding with one another and coalescing to form one big black hole. That’s given as an indicationg that the black holes are even more ubiquitous than we’ve imagined. So clearly, an appearance of a singularity is not a very good thing for a theory and it indicates the breakdown of a theory. And in some sense many people argue that this is one of the greatest features of G. Not only it is a very successful theory in explaining, say, the discrepancies in Newton’s encountered with say the orbit of Mercury around the Sun. It has been incredibly successful in explaining many phenomena in nature, and it also demonstrates very clearly that something is wrong with the theory, that we need to move beyong Einstein’s theory of general relativity, and black holes really are only the window into this theory.

One of the biggest challenges in fundamental gravitational research is to come up with a theory that resolved these singularities, such that the theory doesn’t break down and we clearly know that black holes will exist and if one goes through the horizon then space and time will reverse, everything will move towards a singular point, but then just very very close to that singular point – what happens?

That is the main question and that, because of the very small dinstances, is related to quantum theory because we know: as you go to smaller and smaller distances, then quantum effects become important. So we are not surprised in facts that G. will break down on those scales, and that leads to, say, a search for a quantum theory of gravity, and there are proposals along those lines. Say, string theory, where one tries to formate a theory in which black holes are not singular.

Within the context of string theory, for example, there is one proposal which is called ‘the fuzzball proposal’, which says that in fact, what we as a black hole, classically saying in Einstein’s theory, is in fact an averaging out actually smooth surfaces. There are solutions to this string theory – a theory beyond Einstein’s theory, whereby you have solutions which are all smooth, they have no singularities, but the way you put these solutions together and average them out – it appears as though there is a singularity inside. Even though this proposal is very difficult to realise in four dimensions – in the world that we live in (three spacetime dimensions and one time dimension). If one goes to higher [number of] dimensions, say, five dimensions – then, certainly, one can find solutions which are smooth where there appears to be an event horizon which one cannot escape. But inside there is no singularity – and this is very nice, because it’s some kind of a resolution of that problem, although there are problems that need to be addressed and understood.

PhD; Research fellow, King's College and DAMTP, University of Cambridge
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