In the thermodynamic view of time, time’s arrow is set by Entropy- entropy must increase as time advances. In fact, entropy is nearly the only concept in physics that requires a direction of time.

In the relativistic view Time and space cannot be talked about as separate distinct entities- there is only space-time. Our human perspective of a 3 dimensional world advancing through a separate thing called time works fine for us as long as we all share the same reference frame with each-other and all that we observe. But when we move (or we observe something moving) very vast relative to us our primate intuition breaks down. We see that there is only the 4 dimensional space-time, and our relative motion, and local curvature of space can rotate us relative to the 4 axes-in our 3-dimensional perception- lengths get contracted and time becomes dilated...

Were the curvature of space-time to be positive, then the universe would eventually stop expanding and collapse back in on itself in a 'big crunch'... While that is a grim view of our fate it at least has the comforting familiarity of a beginning, middle, and end.

If space-time were flat then the expansion would stop- after an infinite amount of time…

And if the curvature is negative the universe will expand for eternity.

As for the actual curvature of space-time, it would seem that it is negative based on our observations- in fact Current observations suggest that our universe is not just expanding for eternity, but that the expansion is accelerating.

Eternal expansion, infinite time, infinite space- these are not just mind boggling concepts, they are concepts that cosmologists have found to lead to absolute madness. Madness that results from Rational, logical conclusions – but madness nonetheless.

We exist at a point in time where we can look around us and determine from observation that the universe came into being just over 13 billion years ago. Given an eternity, what are the chances that any randomly selected moment will be within 14 billion years of the start of it all? Or within 100 billion? The chances are zero. In fact, it’s infinitely more likely that you will pick a moment in time so far removed from the big bang that all traces of it have been erased- for all but an infinitely small fraction of the universe’s existence, it will be impossible to observationally determine that it even had a beginning. The acceleration of the expansion means there will be no perceivable cosmic background radiation, there will be no nearby galaxies to see receding from us… indeed there will be no solar systems or planets.

One can say ‘well, we could not possibly exist in the universe of the far future since it would be uninhabitable – there could be no one to ask these questions in the infinite future. In other words, while the universe may eventually encompass infinite time and space, sentient beings can only exist in this finite sliver of time near the beginning.

But this is where the madness starts…

We have all heard the thought experiment- If you have an infinite number of monkeys and an infinite number of typewriters and an infinite amount of time they will produce an infinite janitorial bill and infinite copies of every novel written by man (Except the novels by Dan Brown, because there are some things even monkeys won’t do).

An even more simple example: You put the parts of an unassembled watch in a box and shake it… at some point the parts will come together to assemble the watch in perfect working order. Given infinite time it is a certainty… in fact the watch will be assembled and disassembled worn down into its component atoms and those atoms recombined to reassemble the watch and infinite other watches an infinite number of times.

Life on Earth started after only a few hundred million years of ‘shaking’ and random collisions between molecules. That required no infinities at all…

Empty space is not really empty- particles are flitting in and out of existence constantly- given an infinite amount of time and an infinite volume of space, eventually a perfect copy of our galaxy will pop into existence. Of course it’s far more likely that something simpler like our solar system will arise spontaneously… popping into existence from random fluctuations in the vacuum.

And far more likely still is the creation of a single brain, pre-programmed with a lifetime of experience and memories… perhaps it even is created with the synapses firing to make it think it is typing silliness into a computer- or maybe to think it is reading silliness on a computer. In fact- given infinite time and space it is certain that both versions and infinite others will exist… There is a term for such random creations in cosmology-‘Boltzmann Brains’.

If you pick an observer at random in an infinite universe, the chances will be %100 that the observer you pick will be a Boltzmann Brain. The same odds apply to you as well. But that can’t be correct can it? We all ‘know’ we are real… right?

At this point you probably think that I have lost my mind – but I assure you this is a serious topic in cosmology- just google “The Boltzmann Brain Problem”… here are a few links for further reading:

http://www.nytimes.com/2008/01/15/science/15brain.html?pagewanted=all&_r=0

http://fqxi.org/data/essay-contest-files/Carroll_fqxitimecontest.pdf?phpMyAdmin=0c371ccdae9b5ff3071bae814fb4f9e9

https://plus.maths.org/content/dreaming-dream

http://blogs.discovermagazine.com/cosmicvariance/2008/12/29/richard-feynman-on-boltzmann-brains/#.Vm3AB3arRD9

http://www.preposterousuniverse.com/blog/2013/08/22/the-higgs-boson-vs-boltzmann-brains/