| Note that the title of this post is just a personal opinion of Kerr. But he did show, as mentioned in the article, that the typical argument for the necessity of singularities within horizons (or more generally trapped surfaces) was faulty. Note that this is not about any Quantum Gravity stuff, these arguments concern what happens in pure general relativity. Also, since this is not explained in the article (and mostly assumed as prerequisite knowledge in the paper by Kerr), some words on the difference between affine and geodesic length: In general relativity, we consider so called geodesics: These are paths observers would take while moving only under the influence of the curvature of spacetime / under the influence of gravity, i.e. without any other forces acting on them. There exist different kinds of geodesics depending on their speed: We distinguish between - space-like geodesics (not relevant for this discussion), - time-like geodesics (speed of geodesic is less than speed of light c), which are the paths all observers with mass will take and - null geodesics (speed of geodesic = c), which are exactly the possible paths for light rays to take. For all of these we can define the geodesic length. With the right sign conventions, this is equal to the length as measured in proper time, i.e. the amount of time that will pass for observers moving along the geodesic as they themselves will measure on clocks they bring with them. But importantly, for null geodesics this geodesic length is 0: For observers moving at the speed of light, no time passes! So in this sense, all light rays have the same length. But there is a more general notion: Affine length. This affine length in general is not very well defined (it depends on an extra choice, the affine parametrization ([1]), that you have to make for each geodesic), but when considering a geodesic it is either finite for all of these parametrizations or infinite for all of them. Thus the concept of finite affine length makes sense to talk about, and this is defined for all three kinds of geodesics. Note also that a time-like geodesic has finite affine length if and only if it has finite geodesic length. Now back to the question considered in the article: I'm pretty sure that if we knew that there existed time-like maximal geodesics (i.e. geodesics that you can't continue any further) with finite geodesic length, i.e. finite proper time passing along them, then they would necessarily end in some kind of singularity. So the question becomes: If you have a black hole (of which we can - from the outside - observe the existence of some kind of horizon boundary), then do there necessarily exist time-like geodesics of finite length inside it? AFAIK we don't know the answer to this. But there is a slightly related result: We know that for any horizon boundary, there exist light rays of finite affine length (also denoted FALLs, which I think stands for finite affine length light rays) within it. This feels very similar to the result we wanted for time-like geodesics, but what Kerr's paper shows is that the intuitive belief that these FALLs must necessarily end in a singularity is wrong. There exist null geodesics inside a specific solution for the inside of a horizon boundary (the Kerr metric), that at both ends asymptotically approach a horizon boundary tangentially (never intersecting these horizon boundaries), but which have in total a finite affine length. In particular these geodesics don't intersect or end in singularities. [1]: For time-like geodesics, we normally parametrize them by proper time, i.e. we distinguish the points along the path by the proper time that has passed since some starting time. We can describe the geodesics in this parametrization using the geodesic equation plus the additional condition that we normalize the four-velocity (you don't need to understand what this last condition means, just note that this is not necessarily a part of the geodesic equation). Affine parametrizations are a slight generalization of this, where we drop the condition on the four-velocity, i.e. we consider any solutions to the geodesic equation. We thus label the points on our curve by some more general number, called e.g. a, which does not necessarily have to be the same as the proper time, but which does increase as we move along the path. For time-like observers, the geodesic equation ensures that this affine parameter a will just be some scaled and shifted version of the proper time, i.e. one can imagine going along the path with a clock that is going too fast or too slow (the constant shift doesn't really matter). As an equation we have a = lambda * tau + C,
where tau is the proper time and lambda≠0 and C are some constants. In particular if we measure the length of the path with respect to the affine parameter a, we will just get lambda times the geodesic length of the path (the length as measured with respect to the proper time tau).For null geodesics the proper time is not an affine parameter, but still any two affine parameters a and a' will be related by some scaling and constant shift, i.e. a = lambda * a + C.
In particular if we consider the length of the path as measured with respect to an affine parameters (by just considering a at the endpoint minus a at the starting point of our geodesic), the lengths will either be finite for all affine parametrizations or infinite for all of them. |