Why are inflationary bubble universes spatially infinite from the inside but finite from the outside?

Question

In The Hidden Reality, Brian Greene describes the bubble universes that would arise from inflation as being spatially infinite as observed by a viewer on the inside of the universe, yet spatially finite when observed by a viewer on the outside of the universe. What is the intuition for this?

I'll try to paraphrase his explanation to the best of my (clearly limited) understanding. He explains that each of these observers — one inside the bubble universe, and one outside — takes their cosmic metric of time to be the value of the inflaton field, such that points in space that have the same value of the inflaton field are, for that observer, at the same age. To an outside observer, the bubble universe is expanding, and the value of the inflaton in the "middle" of the bubble is decreasing over time. My best understanding of the argument for the inside observer's perception is that if we focus on any given concentric sphere of the bubble universe (e.g., its "edge" in the language of the outside observer), the inside observer always perceives all points on this sphere as always being at the same inflaton field value and therefore at the same age. Therefore this observer perceives the bubble universe to be infinite?

Answer 1

This analogy may help: suppose you want to simulate Conway's game of life on an infinite board with unbounded, nonperiodic initial conditions (e.g., the binary digits of pi in a spiral pattern). You can't start by simulating the first time step of every cell, because it would take forever and you'd never get to the second time step (it would also take infinite storage space). But you can simulate the first time step at the origin, then the next time step of a 3×3 square centered at the origin, then the next time step of a 5×5 square, etc. This way, you will eventually simulate every particular time step, no matter how large, of every particular cell, no matter how far from the origin. The evolution of distant cells lags behind the origin, but that makes no difference to the correctness of the simulation. You also need only finite storage space at any finite simulation time.

The simulation time in this analogy corresponds to the "outside observer" time, and the number of time steps that have taken place at a particular point on the grid corresponds to the cosmological/"inside" time. You can define surfaces of constant "inside" time, all of which are infinite in size, which means that there is no particular "outside" time at which any of those surfaces are completely computed, but it doesn't matter because they are abstractions anyway; due to the speed of light limit, you'd have to wait forever to see the entirety of one of those surfaces from any vantage point.

Note that the role of the simulation in this analogy is just to provide a second time coordinate. This has absolutely nothing to do with arguments about the universe being a simulation.