Is Our Universe "Finite"?

Is Our Universe Finite?

A while ago I drew the pictures above to try to understand current ideas about the size of the universe. The diagrams are based on some pictures I saw in the book "Our Mathematical Universe".

The diagrams show two dimensional slices of four dimensional spacetime. The blue stuff is "inflationary material" which expands at an enormous rate. The current theory of inflation states that universes like ours form as bubbles in the inflationary material as some of the inflationary material changes phase and "evaporates" out as non-inflationary material. An important point is that the sides of this bubble are moving away from each other way too fast for anything - even light - to travel from one side to the other.  The 1st diagram illustrates the point that in this model there is room for more, far more, than one universe.

The yellow region in the 2nd and 3rd diagrams is what is known as a light cone. The point in the middle is an event (e.g. you turning on a lamp); the yellow region above consists of all the points in spacetime (aka "events") which that can affect; and the yellow region below are the spacetime events which can plausibly be a cause (e.g. someone sending you a message to turn on the lamp). The boundary of the light cone represents things travelling towards you or away from you at the speed of light. Nothing goes faster than the speed of light, and so nothing outside of the light cone can either cause or be caused by the event in the middle.

One of the features of relativity is that if something has no causal relationship to you/now (i.e. it's outside your present light cone) then there is no absolute meaning in saying it has already occurred, or that it is yet to occur. You are perfectly free to choose whichever coordinate system you want - in some of these a specific event outside your light cone will have an earlier "time coordinate" than "now" and in others it will have a later "time coordinate". The equations don't care what coordinate system you choose and it doesn't have any physical significance either.

Diagrams 2 and 3 show the same universe and the same light cone, but with two different coordinate systems both of which work perfectly well. But if we ask the question "is the universe finite?" we get different answers in each. In the first, if you follow a "surface of constant 'time coordinate'" (i.e. a horizontal line from the centre of the light cone) it hits the edges of the universe after a finite amount of "space coordinate", suggesting that the universe is indeed finite. However, in the second diagram, if you follow a "surface of constant 'time coordinate'" you never hit the edge of the universe, suggesting that the universe is in fact infinite.

Now note that the coordinate system chosen does not make any difference to any actual predictions we might make about any given experiment. In both your timeline is constrained by your present light cone and you can therefore never actually reach the walls of the universe. And so it turns out the question of whether "the universe is finite" is not actually a question about the physical nature of the universe but instead one about the coordinate system we - arbitrarily - choose to apply. A more physically meaningful question is "can you reach an edge of the universe" and the answer to that is: no (probably).

I mentioned that surfaces of constant time in relativity are just arbitrary coordinates and not in themselves physically meaningful - the time coordinate is just one of four coordinates which can be used to give an address to every event. Having said that there is a thing called proper time which is physically meaningful and this is the amount of time experienced by an object between two events on its trajectory (usually called its "timeline")$^\dagger$. What this means is that it is physically meaningful to ask how much proper time has elapsed for an object that has reached us from the beginning of the universe (i.e. from the bottom of the light cone to the centre). The answer is around 13.8 billion years$^{\dagger_2}$. Thus it is meaningful to ask the age of the universe - from the point of view of this location in spacetime - but not really meaningful to ask the 3D volume of universe, or even whether that volume is finite.

More Mathsy Answer

Four dimensional volume is another thing though. The following integral is an invariant, meaning that it evaluates to the same number whatever coordinate system $x^\mu$ is used:

$$
\int_\Omega \sqrt{-g}\ d^4x
$$

In the above $g$ is the determinant of the metric $g_{\mu\nu}$, which is a directly measurable thing given an existing coordinate system, and $\sqrt{-g}\space d^4x$ can be reasonably interpreted to signify a volume$^{\dagger_3}$.  So the 4D volume is genuinely either finite or infinite.  The question of which it is, is equivalent to the - still unanswered - question of the fate of the universe.

FOOTNOTES

$\dagger$: Note that the timeline is important since the proper time between two events depends not just on the two events but also on the path taken between them. For example, if you launch a satellite, let it orbit for a year, and then return to you on Earth, its clock will show about 20 milliseconds more time to have elapsed than your watch.

$\dagger_2$: I say "around" because it depends on the object's timeline, and an object that travels at the speed of light from the beginning of the universe - e.g. from the bottom right of the light cone to the centre - will experience exactly zero seconds of proper time. But for most massive objects it's around 13.8 billion years.

$\dagger_3$: The minus sign appears because $g_{\mu\nu}$ has one negative eigenvalue and 3 positive instead of 4 positive eigenvalues like 4D euclidean space has.