Fair enough, but what the fuck does "geodesically complete / incomplete" then mean at all? I'm a layman, not a mathematician or cosmologist.

Teuton wrote:What can cause a great deal of misunderstanding is that "eternal" is used in physics as a technical term with a specific meaning. Prof. Paul Steinhardt told me personally via email that in the physical discussion "eternal" does not mean "infinitely old" but "geodesically complete". Correspondingly, "isn't eternal" does not mean "has a finite age" but "is geodesically incomplete".

Interesting, hmmmmmmmmmm

Rumraket wrote:Fair enough, but what the fuck does "geodesically complete / incomplete" then mean at all? I'm a layman, not a mathematician or cosmologist.

I wanted to ask the same question as well.

Suppose you look at what happens to a point particle as it moves around. It traces out a curve - its worldline in spacetime. If the point particle is free, i.e. not acted upon by any force other than gravity, the curve is a special type of curve called a geodesic. A curve in spacetime is mathematically a map from a subset of the real line into spacetime. The number in this subset that you're mapping is called the curve's parameter. For point particles like electrons etc, you could use the particle's proper time as a parameter. This is the time that would be measured by a hypothetical clock attached to the particle. For particles which travel at the speed of light, however, you can't do that (because there is no proper time for these particles), so you have to use some other paramterization for the curve (an affine parameter).

Anyway, we have this map from a parameter into the spacetime. The spacetime is geodesically complete in the past direction if we can keep on extending timelike or null geodesics to arbitrarily large negative values of their parameters. The reason you might NOT be able to do this is if the spacetime has a singularity which these geodesics would encounter when winding back their parameters in the past direction.

There is a small technicality here - if I take Minkowski space and whip out a point, I have a perfectly good spacetime, but it's not geodesically complete. Geodesics approaching the point I've whipped out are inextendible, but there's no singularity there. To avoid such pathologies, they usually restrict discussion to maximally extended spacetimes (maximal extension would put back any points that somebody's artificially removed).

So past incompleteness is used to characterize the existence of a singularity in the past.

If we go strictly by our present theories,

QM: λ ~ 1/m
Relativity: Schwarzschild radius ~ m

At planck's length, the mass would equal to its Schwarzschild radius, and therefore automatically collapse into a black hole.

All this to say is that we know absolutely zilch of what is going on at Planck's length, and anything said is pure speculation. Anyone talking about singularity, which comes from relativity, is neglecting what QM is saying. And until we develop a theory that circumvent what QM + R is saying, we should keep our mouth tightly shut.

Rumraket wrote:Fair enough, but what the fuck does "geodesically complete / incomplete" then mean at all? I'm a layman, not a mathematician or cosmologist.

"BGV [= Borde, Guth, Vilenkin] show that following an observer into the past along its worldline, there is a bound on the integral of H [= Hubble constant] over the observer's proper time. Therefore if H always exceeds some positive number, then the past-proper time of the observer must be finite, hence the worldline geodesically incomplete."
— Anthony Aguirre ("Eternal Inflation, Past and Future", p. 20)

Some more information from Guth:
Eternal inflation and its implications‡
Alan H. Guth

There is of course no conclusion that an eternally inflating model must have a
unique beginning, and no conclusion that there is an upper bound on the length of
all backwards-going geodesics from a given point. There may be models with regions
of contraction embedded within the expanding region that could evade our theorem.
Aguirre and [49, 50] have proposed a model that evades our theorem, in which the
arrow of time reverses at the t = −∞ hypersurface, so the universe “expands” in both
halves of the full de Sitter space.
The theorem does show, however, that an eternally inflating model of the type
usually assumed, which would lead to Hav > 0 for past-directed geodesics, cannot
be complete. Some new physics (i.e., not inflation) would be needed to describe the
past boundary of the inflating region. One possibility would be some kind of quantum
creation event.