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If the universe is undergoing inflation, and there is a minimum scale that things can exist at (the Planck length), does that mean that new Planck-sized domains have to be continuously popping into existence? If not, does that mean that the Planck length is constantly changing?

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This is the problem of the Big Snap as explained by Tegmark. –  user10442 Jul 10 '12 at 8:41
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This question is very puzzling. Why can volumes of space keep growing without limit, when we think there is a graininess somewhere down there?

To understand it best, you should go to other places where volume space keeps popping up out of nowhere like this. The best example is near a black hole horizon. In this case, if you trace back outgoing light rays, you will find that they all emerge from an infinitesimally thin skin right next to the horizon. This is obvious, because light rays on the horizon are stopped, so two light rays that peel off the horizon at two different late times were squished very close together earlier.

So the entire exterior volume of a black hole is peeling off a thin skin next to the horizon. This property, when it is considered as a property of outgoing Hawking radiation is called the "transplanckian problem". Outgoing modes, which have normal wavelengths by the time they get away from the black hole, are traced back to impossibly compressed wavelengths near the horizon.

This is not a problem classically, but if you imagine some sort of graininess to space, you run into trouble. How do you get the grains to appear as you have this expansion going on?

The resolution to this, as to all problems of quantum gravity, is the holographic principle. You need to change point of view regarding the domain of quantum gravity. The gravity doesn't live in the bulk of spacetime, but on holographic surfaces, and the bulk is reconstructed. Any graininess is boundary graininess not bulk.

This reconstruction means that new space can be created, it is not paradoxical, so long as the boundary degrees of freedom that describe the new space are the same as they were previously. For the case of the black hole, the geodesics peeling off of the horizon may keep on making new space-volume, but any physical matter which they contain loses it's identity and merges with the black hole degrees of freedom once you trace it back to close to the horizon.

Similarly, in deSitter space, there is a cosmological horizon all around you. As the universe inflates, geodesics separate exponentially and go to join the cosmological horizon. But this horizon is not doing anything, it is just sitting there, so any putative holographic description of the space doesn't involve producing new degrees of freedom, it just means that the old degrees of freedom have a finite time before they reach the horizon and, from the point of view of the holographic description, they thermalize.

This is one of the properties of classical GR which really cries out for holographic interpretation. If you take a naive space-time graining, without holography, the constant creation of space from horizons and inflation is very mysterious.

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This Wikipedia article on the Metric expansion of space might provide a better answer, but here goes my understanding, good or bad.

I don't think it makes sense to think that new Plank domains come into existence. After all, the Plank distance is just a unit of measure. The distance between objects in the Universe does grow, so expressing the distance between them will require an ever-increasing number of Plank distances, but nothing new has been created. The Plank length, as a unit of measure, is the same as it was. There was an early time when everything fit in a Plank-distance-sized Universe.

Easier: you measure the diameter of a baloon with a long ruler. As you inflate it, it will take more and more centimeters to express its diameter, but no new space is created.

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This analogy is not great--- the question is more fundamental than this. –  Ron Maimon Sep 8 '12 at 2:25
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