# How can a finite amount of matter be uniformly distributed in a flat, infinite space?

There are some properties of the Universe I find in the (mostly popular) literature which are often described as "the most probable in case of our Universe". I can't put them together in a way that doesn't confuse me. Here they are:

• the Universe curvature is flat
• the space is infinite
• the amount of matter is finite
• the matter distribution is uniform

The first three assumptions seems to contradict the last, because from what I understand, if you run in one direction long (and fast) enough through a flat, infinite space with a finite amount of matter, you will eventually run out of galaxies and go into an empty space. Is that true, or I just don't understand some of the above concepts?

• Does this have to do with the schwarzschild-metric or schwarzschild space-time? – raul Oct 17 '15 at 7:50
• @raul Your comment lead me to start reading about the Schwarzschild metric and made me realize, I have to do a lot more of it, so to be able to answer Your question. Thanks. – kuonirat Oct 17 '15 at 8:34
• I just say it because maybe you were confusing things like "flat" with "asymptotically flat", and it seems to be this way because you're considering the presence of mass in order to describe the "universe", but this is always curve or asymptotically flat. – raul Oct 17 '15 at 8:37

We don't know whether the universe and the amount of matter in it are finite or infinite, so in the absence of certain knowledge we use a mathematical model. For the universe this model is the solution to the equations of general relativity called the FLRW metric. This has a number of possible solutions depending on the density of the matter, and we can divide the solutions into two classes:

• finite universes containing a finite amount of matter$^1$

• infinite universes containing an infinite amount of matter

Plus there is one special case of an infinite universe containing no matter at all.

There are no solutions that have an infinite universe with a finite amount of matter. So your list of most probable properties of the universe is wrong. There's nothing to stop you hypothesising an infinite universe with a finite amount of matter, but there is no support for such an idea in physics.

$^1$ in this context matter means anything that appears in the stress-energy tensor plus a cosmological constant if applicable

• "There are no solutions that have an infinite universe with a finite amount of matter." - this is the key sentence which makes my confusion go away. So it seems like the uniformity is the most widely believed property of our Universe, if all the others are more debatable. Thank you! – kuonirat Oct 17 '15 at 11:33