Black Holes: #3 The Second Horizon


Today in the continuation of our series on Black Holes, you can find the previous two instalments (here and here) we go beyond the event horizon and discover that there exists a second, inner horizon whose nature has drastic implications on our understanding of space-time.

First I want to outline two hypotheses that will be central to our discussion. The weak and strong cosmic censorship hypotheses introduced to constrain the nature of black holes. Remember before when we spoke of black holes, it was explained that the point in the very center is a place of such infinite density that all our theories and frameworks for describing space and time breakdown here. The mathematics simply cannot model the extremity that exists at this point. This point is called the singularity and it shrouded from our observation due to the fact that black holes have event horizons, this is surface a certain radius from the center of the black hole from which, when information passes through it cannot then escape. If you fall past this radius there is no coming back. This is why we cannot fully understand the inner nature of a black hole and why of course it appears black. (For a longer explanation of these phenomena please see the posts linked above).

Now, the weak cosmic censorship hypothesis reiterates this point and firmly states that all singularities must come with this surrounding horizon, meaning they cannot be directly observed and must stay hidden to observers who have not crossed the threshold of no return. This is nicely summed up by the phrase that there can be no naked singularities. General Relativity is strictly PG-13. The strong cosmic censorship hypothesis is of a different ilk. This hypothesis states that the general relativity is a deterministic theory, deterministic meaning that given the initial conditions of a system, its future state is entirely predictable. This is the clear, stable world we know and love. A bit more of a discussion (with some philosophical consequences) of deterministic theories, along with their antithesis, probabilistic theories, can be found here.

Now to the crux of today’s post. In the 1960s mathematicians discovered a solution of Einstein’s field equations which described a system that was no longer deterministic, i.e. contradicting the strong cosmic censorship hypothesis. This came in the form of a rotating black hole. Far outside a black hole we can use classical mechanics to describe the universe, which as we’ve said is a deterministic theory, it gives a clear forecast of how the system we’re looking at will evolve, given its current information. Near a black hole we must move from classical mechanics to general relativity yet we can still think of the world as deterministic, GR and Einstein’s equation still manage provide a single forecast for how space-time will evolve. In fact this continues to work clearly and without ambiguity even when we cross the event horizon, there is still one clear future, the snag comes when we reach the second horizon.


This second horizon is known as the Cauchy Horizon, predicted by mathematicians as lying beyond the event horizon, deeper inside the black hole. Here when analysed, Einstein’s equations become erratic, spewing out multiple solutions, telling us that many different configurations of space-time can then occur, none of which are more justified than another. The mathematics of general relativity there suggests the universe is inherently unpredictable beyond this point, that there is no single path that will be followed given the initial conditions. The path at the Cauchy horizon branches off from one into many, like the frayed end of a rope.

So how to deal with this unsettling fact? Mathematician Roger Penrose offered the first argument. His angle was that sticking to the strong cosmic censorship conjecture the universe must be inherently deterministic and as such there must be some error in our understanding of the nature of the Cauchy horizon and the effect it has on Einstein’s equations. Penrose suggested that the Cauchy horizon is unstable and it cannot exist in the physical sense in which it had been postulated. In fact any matter passing through the black hole and hitting the Cauchy horizon would cause it to instantaneously collapse to a singularity itself. Penrose claims that Cauchy horizons can only exist in an idealised universe where nothing else exists except a single black hole in question. However if you introduce any other matter into the universe, it will eventually fall into the black hole, hit the Cauchy horizon and cause its dramatic collapse. Therefore Penrose attempts to save the strong cosmic censorship conjecture by claiming the Cauchy horizon cannot split the paths in the first place because it has been postulated in error and is only an idealised mathematical solution to General Relativity.

However recently a mathematician called Dafermos at Stanford University has offered a different and insightful alternative. He postulated that the Cauchy horizon does form a singularity but not of the extreme kind as suggested by Penrose or of the kind we know exist at the center of black holes. The horizon pulls on the surrounding spacetime and matter but does not cause it to collapse entirely and as such there is a continuation of spacetime past its border. However (and here is the key part) this continuation is not smooth. Einstein’s equations evaluate the changes in spacetime over infinitesimal increments but these infinitesimal increments need to be joint up smoothly for the mathematics to work. The Cauchy horizon causes a discrete break in the spacetime so much so that Einstein’s equations are not longer satisfied or even applicable..

So because they can no longer be applied correctly, they no longer give us meaningful solutions and we are not faced with having to take the multiple conflicting outputs seriously. Determinism preserved. However the theory is rather unsatisfactory in the sense that we’ve managed to retain what the mathematics has told us about the genuine existence of a Cauchy horizon and do away with the conflicting multiple futures but the price to pay is that we cannot extend our mathematical analysis beyond the second horizon because the best theory we have, embodied by Einstein’s equations, breaks down. An alternative theory needs to step in whose mathematics can smoothly transition across the Cauchy horizon, deeper into the Black Hole. Border control at the Cauchy horizon requires documentation in the form of mathematics we unfortunately do not yet possess.






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