Relativity: stationary vs moving In the book The Elegant Universe, Greene describes a situation in which there are two space travelers, George and Gracie, moving in relationship towards each other at a constant velocity with no other available vantage points. Both George and Gracie have an equal right to declare that they are stationary and that the other is moving.From George's perspective, he is stationary while Gracie is moving and he observes (consequently? not sure this is direct consequence) her clock to tick slower than his. However, doesn't George also have the right to claim/perceive himself as the one who is in motion, and if so wouldn't he perceive Gracie's clock to tick faster than his? This seems to contradict the first perception. How would contradicting perceptions be available to the same observer? I assume I'm off somewhere.
 A: Excellent question. And well put.
You're quite correct in saying that any obseverver can claim to be moving at a constant speed.

However, doesn't George also have the right to claim/perceive himself as the one who is in motion, and if so wouldn't he perceive Gracie's clock to tick faster than his?

No, the clock George looks at (on his wrist or whatever) is stationary with respect to himself and therefore ticks at a normal rate as far as he is concerned. Only clocks that move relative to him (move past him) will be ticking at a different rate.
A: Suppose  George and Gracie are moving toward each other, each claiming to be stationary.  
Their clocks happen to be set so they'll both chime 10PM at the moment they meet.
When Gracie's clock chimes 2PM, she says:  "I see that George has his clock set to 4PM.  He'll be here in 8 hours, with his clock chiming 10PM.  That slow clock of his will chime only 6 times in 8 hours!".
When George's clock chimes 2PM, he says:  "I see that Gracie has her clock set to 4PM.  She'll be here in 8 hours, with her clock chiming 10PM.  That slow clock of hers will chime only 6 times in 8 hours!".
Gracie believes that when her clock says 2PM, George's says 4:00, and back when George's clock said 2PM,  her own clock was saying something like 11:20AM.  
George believes that back when Gracie's clock said 2PM, his own said something like 11:20AM, and when his own clock says 2PM, her says 4PM.
In other words, they do not agree on the answer to the question "What time is George's clock showing when Gracie's is showing 2:00?".  (This disagreement is usually called the "relativity of simultaneity".)  This allows them to disagree about whose clock is running slow.
A: There is a symmetry associated with time dilation. The formula $$t=\frac{t_0}{\sqrt{1-\frac{v^2}{c^2}}}$$ will always result in time dilation for both the individuals which are in relative motion. Since v is squared, the direction (i.e. +/- ve ) or reference frame doesn't matter - as long as the time is quoted from inertial frames of reference, each observer will see the other one's clock tick slower than his/her clock.
A: No, George doesn't. Think of when you are driving or ride in a car. You don't think you are moving, you see everything else moving outside the window. Intuitively, we know we are moving, but that is not the kind of perception he is talking about.  
A: The thing about relativity is that you can't consider only one observer and still end up with 'relative' measurements. So neither George and Gracie can honestly say that "one of us is moving" or "one of us is stationary." At best, they can both say "we are both moving relative to each other." General Relativity doesn't support global frames at all, in fact, so for either observer to place themselves at the origin of a global inertial frame is a theoretical failure from the start.
That said, two observers in relative motion will each measure the other's clock to run slow, by the same proportion, because obviously the relative motion is the same both ways - it's a feature of the system as a whole.
A: The situation is symmetrical, so both will see the other one* undergo time dilation. This seems contradicting at first sight, but if any of them turned around for a meetup, they would undergo acceleration and thus change in which inertial frame of reference they are stationary in. This will result in a "speed-up" of time for the accelerating spaceship.
A: I'd just like to emphasize the point - touched on in several of the other answers - that the core of the "paradox" here is our natural, nonrelativistic tendency to implicitly assume a universal time. Such an assumption indeed is contradicted because, if true, the well orderedness of time intervals would be blatantly contradicted by the situation where both observers measured each others' ticks as slower to their own. But there is no universal time: there are two times and two spacetime co-ordinate systems, one for each observer. This situation leads to WillO's last paragraph:

In other words, they do not agree on the answer to the question "What time is George's clock showing when Gracie's is showing 2:00?". (This disagreement is usually called the "relativity of simultaneity".) This allows them to disagree about whose clock is running slow.

Although it is stated ad nauseam, time is relative and it is the implicit assumption otherwise that gives rise to apparent paradoxes like these.
As noted elsewhere, the "paradox" disappears once we think in detail about how the two observers would directly compare each others' clock readings, either by meeting up again (at least one observer must turn around) or by precise and careful signalling to each other, as discussed in the "Resolution of the paradox in special relativity" section of the Twin Paradox Wikipedia article
A: In the book The Elegant Universe, Brian Greene states on page 27, the 
"Einstein proclaimed that all objects in the universe are always traveling through space-time at one fixed speed, that of light. ........ We are presently talking about an object's combined speed through all four dimensions, three space and one time, and it is the object's speed in this generalized sense that is equal to that of light."
Thus if you have two objects which are travelling at the speed of light, but in different directions within the 4 dimensional environment known as space-time, one object will be moving across time faster that the other, due to the other moving faster across space instead. However, from either of the two objects point of view, it always appears as though it is the "other" objects clocks that are ticking slower.
The fun part is to figure out why this is the case.
To do so, you have to view the situation at a hand, in a 4 dimensional scale.
