Background: There are two broad understandings of time's existence - presentism and eternalism. Presentism suggests that only the present (the 'moving now') truly exists. Eternalism, by contrast, puts all times/moments on an equal metaphysical footing, saying that they all exist in exactly the same way. We basically just conceive of time as being another dimension, complementing the 3 spatial dimensions we are more used to. The result: a 4-dimensional space-time loaf.
The eternalist picture fits nicely with Einstein's special relativity. To extend the 'loaf' metaphor, we can understand a moment of time as being a 'slice' of the loaf, which contains all the events occuring throughout all of space at that moment. However, observers in relative motion will cut the loaf at different angles, resulting in a different set of events. That is, different events will appear to occur simultaneously to you, depending on your frame of reference. Moreover, there is no basis for judging that any frame of reference is more "right" than any other. We are left with the amazing conclusion that time is not absolute: the proposition that events E1 and E2 occur simultaneously, is not absolutely true or false. The answer will vary relative to the frame of reference.
The problem for Presentism: Presentism seems like the more 'common-sense' approach. It's the way we all intuitively understand the world. However, if special relativity is true, then presentism effectively collapses into eternalism (or at least a far more expansive conception of time's existence than it originally aimed for). In what follows, I will try to outline Greene's explanation of why this is...
First of all, let's introduce the concept of a now-list. A 'now-list', as the name suggests, is simply a list of all the events that are occuring right now. It's a sort of mental freeze-frame image of the entire universe at any given moment. This then lets us understand presentism as the claim that all and only things on the now-list currently exist.
Before we go on, there is one clarification that needs to be made to this account. (All quotes are from Greene, pp.133-139, original italics):
Nothing you see right now belongs on your now-list, because it takes time for light to reach your eyes... If you know how far away something is, you can determine when it emitted the light you see now and so you can determine on which of your time slices it belongs - on which already past now-list it should be recorded.
Here's the problem: As mentioned earlier, two observers in relative motion have different judgements of what events occur simultaneously. That is, they experience different nows, and so have different now-lists. As Greene puts it, "Observers moving relative to each other have different conceptions of what exists at a given moment, and hence have different conceptions of reality".
We never notice this in everyday life because the discrepancies are tiny. We cut the space-time loaf in almost identical slices to each other. But the differences can be amplified in two ways: either travel near light-speed (so increasing the angle of the cut), or separate the observers through a huge gulf of space (so even a tiny angle makes for a large end result).
Greene asks us to imagine a simplified scenario (ignoring motions of the planets, etc) where Chewie is sitting 10 billion light-years from Earth, and that we are at rest relative to each other. This means that we would slice up spacetime in an identical way, resulting in identical now-lists. But suppose Chewie stands up and starts to walk away from us.
This change in Chewie's state of motion means that his conception of now, his slicing up of spacetime, will rotate slightly. This tiny angular change has no noticable effect in Chewie's vicinity: the difference between his new now and that of anyone still sitting in his living room is miniscule. But over the enormous distince of 10 billion light-years, this tiny shift in Chewie's notion of now is amplified. His now and your now, which were one and the same while he was sitting still, jump apart because of his modest motion.
So which now-list provides the list of existing things? We presumably have to say "both". But then we have to conclude that a very wide range of times can all exist together. For if Chewie were to move away from us at 1000 mph, his now-list would suddenly include events on Earth that from our perspective took place 15,000 years ago! If Chewie's twin brother Dewey moved towards us at that speed, his now-list would instead include earthly events 15,000 years into our future. Chewie and Dewey exist at the same time, yet they disagree (by 30,000 years!) on what earthly events are also occuring at that moment. Both view-points are equally valid. The presentist must take both now-lists as listing existing events. Too bad that we're now stuck with the existence of events which occur thousands of years into our past and future.
[Just remember that none of the observers discussed would be able to know any of this at the time. What we really mean is that 10 billion years later, when Chewie's descendents calculate his now-lists, they will find that a bunch of cavemen belong on the same now-list as when he was zooming away from us, whereas just a moment earlier (when he was sitting still) Chewie's now-list included your reading this blog-post.]
The reasoning in the Chewie scenario can then be extended much further. If we collate all of the overlapping possible now-lists, we will end up with a substantial chunk of spacetime (all of it, in fact, if space is infinite).
So: if you buy the notion that reality consists of the things in your freeze-frame mental image right now, and if you agree that your now is no more valid that the now of someone located far away in space who can move freely, then reality encompasses all of the events in spacetime... Just as we envision all of space as really being out there, as really existing, we should also envision all of time as really being out there, as really existing, too.
Crazy stuff, physics. It's incredibly cool - and perhaps kind of surprising - how it can help to shed light on metaphysical questions such as the nature of time's existence. But there you go.