Original by Don Koks, 2009.
A commonly heard phrase in the realm of special relativity is "Moving clocks run slowly". But—even in the context of special relativity—is it always true? The answer is no. It's only true when a clock's ageing is measured in an inertial frame. This assumption of inertiality might not always be stated explicitly in textbooks, but it's always there.
Neglecting the complications introduced by gravity, the frame you occupy qualifies as inertial if, when you hold a mass in front of you and let it go, it hovers there indefinitely. In such a frame, you can indeed say that all moving clocks run slowly. But if your frame is not inertial, the situation becomes much more complicated.
One way to see how moving clocks might not run slowly is to consider an inertial clock being orbited by another clock that is so close as to be almost touching. This close separation means we can reduce almost to zero any complications that would be caused by having to account for the finite speed of signals passed from one clock to the other. The inertial clock measures the orbiting clock to age slowly (see the FAQ entry Does a clock's acceleration affect its timing rate? for a discussion of this). This can only mean that the orbiting clock measures the inertial clock to be ageing quickly. The frame of the orbiting clock is accelerated, and the (inertial) clock that moves within this frame ages quickly, not slowly.
This complication with noninertial frames lies at the heart of the so-called Twin Paradox. (See The Twin Paradox FAQ entry for a discussion of the scenario.) Twins Terence and Stella are initially together on Earth, and Stella blasts off in a rocket to visit a distant star. Terence is almost inertial—he becomes fully inertial if Earth's gravity is neglected, which it can be for the purpose of the discussion. He knows that, being inertial, he is entitled to say that all moving clocks run slowly, and that includes Stella's. So when Stella returns home, Terence is not surprised to find that she has aged less than he has.
What about Stella? If she could claim to have been inertial for the whole trip, then she would maintain, correctly, that Terence should be younger than herself (because moving clocks run slowly in an inertial frame!), and there would then be a real problem. But she simply cannot claim to have been inertial for the whole trip. She might have been inertial for some or most of her trip, but she cannot have been inertial for all of it. When she reached the distant star and fired her return rockets to come back home, she most definitely was not inertial. If she held a ball in front of her while her rocket braked, the ball would suddenly zoom off to one side until it hit a wall. This is not a subjective observation, somehow dependent on frame; if the ball hits the wall, then it hits the wall, and Stella's frame fails the test for being inertial. Her inertiality, or lack of it, is an absolute thing. For this time of braking, however brief, Stella inhabits an accelerated frame, and clocks in such a frame can indeed run faster than her own. They can also run slower or even backwards! The details depend not only on their motions, but also on their positions relative to Stella.
The analysis of events in an accelerated frame is actually quite complicated. While she is no longer inertial, Stella maintains that not only can a clock run slowly or quickly, but that its speed of ageing is also dependent on how far away it is from her. If she travels at constant velocity for almost the whole distance to and from the star, only braking for a short time at the star itself, then she will maintain that most of Terence's ageing happened while she braked. The farther away the star is, the more Terence must have aged while she braked.
If you wish to imagine a more gentle accelerated frame than what astronaut Stella has to deal with, consider sitting in a rocket that accelerates constantly at 10 m/s2. You will feel the force of the rocket motor as one Earth gravity, so you'll probably naturally orientate yourself so that "down" is what you're used to on Earth. In fact, your experience will be as if you were sitting in the room you are in right now. Just what it means to synchronise clocks in such a frame does require some thought, but it can be done. When everything is running smoothly, you'll be able to measure that time just "above" you runs a little quickly, while time farther "up" runs more quickly still, and so on farther up, without limit. Time "below" you runs a little slowly, while time farther "down" runs more slowly still. On a plane below you at a distance of very close to one light year (coincidentally; the actual figure is c2 divided by your acceleration), time stops altogether. Light signals from events closer and closer to this plane will take longer and longer to reach you, and light from the plane itself will take forever to reach you. (You won't actually see events almost frozen very close to the plane, because their light will be redshifted out of visibility.) Light from events below the plane can never reach you at all, and this prompts the plane to be called a "horizon". In fact, although you cannot know what is happening below this plane, it turns out that you can infer time below it is going backwards.
Because the distance to the horizon lessens as your acceleration increases, you can imagine an extreme scenario in which your acceleration is so high that the horizon is arbitrarily close to you. So even though Stella might brake for an arbitrarily short time, and we might think that we can ignore those few short moments when analysing the Twin Paradox from her frame, we must remember that the effects of this high acceleration on Stella's measurements are extreme. In particular, she'll measure Terence to be ageing extremely quickly in these few moments, because he's very high "above" her, where Stella maintains that time runs extremely quickly during the short interval of her braking.
This dependence of the flow of time on position makes accelerated-frame calculations complicated, but that only serves to enrich relativity theory. In fact, this is just how Einstein progressed from special relativity to his theory of gravity, general relativity. He postulated that uniform acceleration is indistinguishable from a uniform gravitational field. (It's true that uniform gravitational fields don't actually exist, but that doesn't stop thought experiments from being done with them.) In this way, he was able to use accelerated-frame ideas to step into the realm of gravity. In our current age of global positioning technology, the satellites that orbit Earth have amply demonstrated that the speed of a clock really does depend on its distance from us in a noninertial frame precisely as predicted by general relativity, for these satellites must make heavy use of relativity in their calculations that yield positions on Earth to such high accuracy.
Some people argue that this idea of clocks running slowly or quickly in Stella's frame is nonsense; after all, they say, how can a decision made by Stella to brake cause Terence's ageing rate to change? But this is not what relativity is saying; of course as far as he is concerned, Terence is unaffected by Stella's braking! While an analysis of cause and effect can readily be made in the context of relativity, such an analysis is not required to describe the ordering of events in Stella's frame. And statements such as "this clock runs slowly" and "that clock runs quickly" are just that: a description of the order of events in Stella's frame.
For an in-depth analysis of ageing and clock rates in an accelerated frame, see Chapter 7 of "Explorations in Mathematical Physics" by D. Koks (Springer, 2006).