Certainly not in any kind of Cartesian deformation. But it might be interesting and informative to try to discover a consistent, non-terminal system for graphing the surface of a sphere, or conclusively proving that it is not possible.
I think it follows from the no-hair theorem. If there were a nonsingular set of coordinates, you could pick one of them, and comb the hair in the coordinate direction.
I think you mean the Brouwer fixed-point theorem (with its colourful metaphors about porcupines and salon products).
This is why we love you around here. And why we’re glad you’re asymptotic and not exponential.
And this is why we love you.
But this is the thread winner:
Nitpick: The BBT doesn’t insist that the universe was a single point during the singularity. If it’s infinite now, then it was infinite then too (according to Brian Greene, anyway). The requirement is that density was infinite, so if the universe is finite, then it would also have been infinitesimally small (a point).
Of course, the “observable universe” is finite and would have been a single point at that time.
You can almost always count on news articles about cosmology to gloss over that point and assume the universe is finite, just to simplify the text. I don’t blame them, but we just have to be careful about extrapolating from what we learn from news articles, or even books by Brian Greene.
Well crap. I’m going to have to get in the habit of using Preview, because I totally goofed up a quote above. Lemme try that again.
This is why we love you around here. And why we’re glad you’re asymptotic and not exponential.
Think of it this way: for any coordinates covering the entire sphere there must be two basis vector fields, however by the hairy ball theorem each basis vector field must be discontinuous or be 0 at some point (and thus not be a basis at that point), meaning the coordinates will always have some form of bad behaviour in the coordinate system.
The expansion of the observable Universe is not solely due to the expansion of space. If we think of the big bang singularity as being a moment in time, then the observable Universe is like the inside of the light cone of an event at the singularity* and the light cones of the event always shrinks to a point at the event even when space is not expanding.
Now each particle horizon is unique to each observer at rest relative to the CMBR, so from this point of view it seems like the big bang singularity must be made up of multiple events and hence as I said we are specifically viewing it as a moment in time, this suggests it was a moment of time made up of different points in space. This is what I guess Brian Greene means when he says the big bang singularity was not a single point in space.
However some care (as always!) is needed. Firstly the above is true wherever there are particle horizons which can exist in both infinite and finite Universes (and will exist for all FLRW cosmologies with realistic parameters), so by the above “argument”, even finite Universes don’t shrink to single points in space at the big bang. However the real problem is that spatial volume(even to the point of being finite/infinite) can’t be really meaningfully defined at the big bang singularity so whether it is finite or infinite in size is moot. In fact it is a little dangerous to think of singularities as being events or sets of events in spacetime at all as at the very least they are places where they don’t obey the rules that we have set down for spacetime.
*This contradicts what I go onto say in the last paragraph about it being dangerous to think of singularities being sets of events. In order not to get bogged down in detail about what I actually mean here, I’ll skip over this point.
I should’ve made clearer: particle horizon = boundary of the observable Universe.
Sometimes an eternal Universe gets bored.
Has any stellar object ever been positively identified as one?
That’s the first thing that jumped out at me, too: what they essentially do is take the paths particles follow in classical general relativits (geodesics) and replace them by the ‘trajectories’ the hidden variables (particle positions) in Bohmian mechanics follow. This is something whose significance is not at all clear to me—basically, that seems to me to be a kind of semiclassical approach (quantum matter on classical spacetime), which I don’t see as being very well founded in regions of size/curvature at or around the big bang. It seems to me that this should be exactly where you’d have to think about bona fide quantum gravity effects.
Also, the singularity is avoided basically by dint of the fact that Bohmian trajectories can never cross, and hence, can’t have been ‘all in one place’, so to speak; but that, to me, seems to address a different issue from the physical singularity in big bang models. I’m not sure it’s necessarily the case that even if you can get all matter to avoid one another, the metric doesn’t blow up anywhere. (But maybe the attendant relativists can elaborate on this point…)
The known object that is most likely to be a black hole is Sagittarius A*, which is generally accepted to be the black hole at the center of the Milky Way galaxy. Even then, we haven’t actually “seen” the black hole or its accretion disc; the evidence largely stems from the fact that there’s nothing else it could be given its incredibly large mass (4.1 million times that of the sun) and its relatively small size (only 17 times larger than the size of the Sun, if not smaller.) Astronomers have proposed a large-scale project to actually get images of the event horizon; check back on ten years.
…And I just realized that I earlier typed “no-hair theorem” when I meant “hairy ball theorem”. And Asymptotically Fat either made the same mistake I did when he read it, or he recognized my mistake and managed to figure out what I meant anyway.
For reference, the “hairy ball theorem” is that you can’t comb a hairy ball smooth (there must be some point that has a whorl or a cowlick or a bald spot), while the “no-hair theorem” basically says that black holes are really boring, and can’t have most of the properties that other objects have.
This is a point I must say struck me too: approximating quantum gravity as a correction to classical physics will surely have a limit and that limit would seemingly be reached in the very early Universe.
It’s a little complicated, but an essential element to Hawking’s singularity theorem(s) is that two arbitrarily-close past-directed timelike geodesics originating at some event A will intersect at some finite proper time. This can be shown to happen when certain energy conditions are obeyed. The other element is that when certain causality conditions are obeyed this can be shown to lead to geodesic incompleteness and, depending on how strict your conditions are, that all past-directed timelike curves originating on some spatial slice have a proper time which is less than or equal to some finite maximum.
These ideas are in many ways stronger than whether the metric blows up anywhere and indeed the actual proofs offer no clues as to where the metric might blow up if at all it does blow up.
If the congruences of geodesics that intersect and form the essential element of the proof of Hawking’s singularity theorem are replaced by congruences of geodesics that don’t cross, then the singualrity theorem doesn’t apply (that of course doesn’t prove there aren’t singularities of any type, but I don’t think that would necessary). However to me saying that for particle trajectories, the classical geodesics are replaced by Bohmian trajectories doesn’t mean that necessarily the singularity theorem would still not apply to the background spacetime.
I actually thought of the point I made before I read you post, so it was very clear what you meant, but I did notice your mistake, but I didn’t bother pointing it out:D
So that must explain the black holes.