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But if some property is discovered or theorised that prevents the ultimate collapse, that wouldn't kill general relativity if it doesn't affect the properties outside, would it?
No, but depending on where/how the collapse is halted, it may be more or less surprising and difficult to reconcile with general relativity. Suppose for example that the collapse is somehow halted just after forming the horizon of a supermassive black hole. This would be a serious contradiction to principles of GR, because the space-time curvature at that point is still very weak. Nothing terribly exotic is happening to the space-time there, so GR should still be perfectly valid in predicting continued collapse.
Whereas what we expect is that GR's core assumptions (like space-time being smooth on all scales) must break down near the singularity condition, since they contradict principles of quantum mechanics. It would not be too surprising at all if this turns out to be how singularities are removed.
In either case (whether something surprising happens that requires revising general relativity, or if the expected turns out and we develop GR into quantum gravitation), I would not say that GR is killed so much as extended
to a wider set of conditions. The extended theory must still reduce to GR and produce the same predictions as before for the geometry outside the hole. It must be consistent with all observations that so far agree with GR.
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Hmm... Would it be possible to find the volume for every possible observer?
Sure, although I am not sure how useful that would be for understanding a black hole's properties. In addition to the volume not being invariant, it is generally also not time-independent. In particular, you could choose to define the maximum
possible interior volume of the black hole. It turns out that this volume grows with time, even if the black hole's mass and horizon area stay the same! So you definitely cannot think of a black hole as a region that encloses a volume related to its surface area. The amount of volume it contains is a very tricky concept!
Terran wrote:Edited: I miswrote here earlier and said that it is hyperbolic, but that isn't quite right. I was mixing up the geometry with how events transform in that geometry. I've clarified this below.
The geometry of space-time is Minkowskian
. This is not in itself hyperbolic, but it does have important features of hyperbolic geometry. To see this, it is useful to start in the space-time of special relativity. Expansion of space over time is just an extension of that, where the spatial component grows with time.
The space-time of special relativity is flat (the spatial component of it is Euclidean) with a time component that does not merge with it in a Euclidean way (so it is not like Euclidean 4D space). It has the metric ds2
. That minus sign is crucial, and is what makes it Minkowskian rather than Euclidean.
The quantity ds2
has a deep physical meaning: it is the (squared) separation between events measured through space-time, or the "space-time interval". Some weirdness crops up here: because of the minus sign, it is possible for what seem to be widely separated events to actually have zero space-time separation. These correspond to events that are equally distant in space and time (like one light year apart in space and one year apart in time), and as you might guess, this exactly corresponds to paths that light can take. Light in vacuum travels a distance of zero in space-time. So if you set ds2
= 0, you construct the light cone out of the geometry.
Another key property of the space-time interval is that it is invariant. All observers agree on the space-time interval between two events. So if you measure space and time coordinates of events and build a space-time map, and then move into a different reference frame, the space and time coordinates of events will change, but they must change in a specific way in order for the space-time interval to remain the same. That transformation will be along hyperbolas. To see this, think of the curves generated by y2 - x2 = k, for k a constant.
We might ask of other types of space-times, like for the large scale expanding universe (FLRW metric), or around a spherical mass distribution (Schwarzschild metric), or a rotating black hole (Kerr metric). The global
geometry in these cases may be complicated. The rotating black hole for example has a cross relationship between time and longitude angle, because of the rotation (frame dragging). But in all cases, the geometry is at least locally
Minkowskian. If you limit your view enough, curved spaces appear flat. This is in essence a principle of special relativity. "Special" relativity means "limited" relativity, in the sense that we are limiting the size of the region being analyzed such that the effects of curvature (the tidal forces) are too small to measure.
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Finally, is there a symmetry in gravity-bound systems? Two planets orbiting for instance, if we traced out their past and future, wouldn't reflecting over the time axis not change anything?
Absolutely, provided there are no damping mechanisms (gravitational wave emission, for instance).