Original by Don Koks, 2009.
A commonly heard phrase in the realm of special relativity is "Moving clocks runslowly". But—even in the context of special relativity—is it alwaystrue? The answer is no. It's only true when a clock's ageing is measured in aninertial frame. This assumption of inertiality might not always be statedexplicitly in textbooks, but it's always there.
Neglecting the complications introduced by gravity, the frame you occupy qualifies asinertial if, when you hold a mass in front of you and let it go, it hovers thereindefinitely. In such a frame, you can indeed say that all moving clocks runslowly. But if your frame is not inertial, the situation becomes much morecomplicated.
One way to see how moving clocks might not run slowly is to consider an inertial clockbeing orbited by another clock that is so close as to be almost touching. This closeseparation means we can reduce almost to zero any complications that would be caused byhaving to account for the finite speed of signals passed from one clock to theother. The inertial clock measures the orbiting clock to age slowly (see the FAQentryDoes a clock's acceleration affect its timing rate? for adiscussion of this). This can only mean that the orbiting clock measures theinertial 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. (SeeThe Twin Paradox FAQ entry for a discussion of thescenario.) Twins Terence and Stella are initially together on Earth, and Stella blasts off ina rocket to visit a distant star. Terence is almost inertial—he becomes fully inertialif Earth's gravity is neglected, which it can be for the purpose of the discussion. He knowsthat, being inertial, he is entitled to say that all moving clocks run slowly, and that includesStella's. So when Stella returns home, Terence is not surprised to find that she has aged lessthan he has.
What about Stella? If she could claim to have been inertial for the whole trip, then shewould maintain, correctly, that Terence should be younger than herself (because moving clocks runslowly in an inertial frame!), and there would then be a real problem. But she simply cannotclaim to have been inertial for the whole trip. She might have been inertial for some or mostof her trip, but she cannot have been inertial forall of it. When she reached thedistant star and fired her return rockets to come back home, she most definitely was notinertial. If she held a ball in front of her while her rocket braked, the ball would suddenlyzoom off to one side until it hit a wall. This is not a subjective observation, somehowdependent on frame; if the ball hits the wall, then it hits the wall, and Stella's frame fails thetest for being inertial. Her inertiality, or lack of it, is anabsolute thing. For this time of braking, however brief, Stella inhabits anaccelerated frame, and clocksin such a frame can indeed run faster than her own. They can also run slower or evenbackwards! The details depend not only on their motions, but also on their positions relativeto Stella.
The analysis of events in an accelerated frame is actually quite complicated. While she isno longer inertial, Stella maintains that not only can a clock runslowly or quickly, butthat its speed of ageing is also dependent on how far away it is from her. If she travels atconstant velocity for almost the whole distance to and from the star, only braking for a short timeat the star itself, then she will maintain that most of Terence's ageing happened while shebraked. 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 forceof the rocket motor as one Earth gravity, so you'll probably naturally orientate yourself so that "down" iswhat you're used to on Earth. In fact, your experience will be as if you were sitting in the room youare 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 slowlystill. On a plane below you at a distance of very close to one light year (coincidentally; the actualfigure isc2 divided by your acceleration), time stops altogether. Light signalsfrom events closer and closer to this plane will take longer and longer to reach you, and light from the planeitself will take forever to reach you. (You won't actuallysee events almost frozen very closeto the plane, because their light will be redshifted out of visibility.) Light from events below theplane can never reach you at all, and this prompts the plane to be called a "horizon". In fact, althoughyou cannot know what is happening below this plane, it turns out that you can infer time below it isgoingbackwards.
Because the distance to the horizon lessens as your acceleration increases, you can imagine anextreme scenario in which your acceleration is so high that the horizon is arbitrarily close toyou. So even though Stella might brake for an arbitrarily short time, and we might think thatwe can ignore those few short moments when analysing the Twin Paradox from her frame, we mustremember that the effects of this high acceleration on Stella's measurements are extreme. Inparticular, she'll measure Terence to be ageing extremely quickly in these few moments, because he'svery high "above" her, where Stella maintains that time runs extremely quickly during the shortinterval 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 Einsteinprogressed from special relativity to his theory of gravity, general relativity. He postulatedthat uniform acceleration is indistinguishable from a uniform gravitational field. (It's truethat uniform gravitational fields don't actually exist, but that doesn't stop thought experimentsfrom being done with them.) In this way, he was able to use accelerated-frame ideas to stepinto the realm of gravity. In our current age of global positioning technology, the satellitesthat orbit Earth have amply demonstrated that the speed of a clock really does depend on itsdistance from us in a noninertial frame precisely as predicted by general relativity, for thesesatellites must make heavy use of relativity in their calculations that yield positions on Earth tosuch high accuracy.
Some people argue that this idea of clocks running slowly or quickly in Stella's frame isnonsense; after all, they say, how can a decision made by Stella to brake cause Terence's ageingrate to change? But this is not what relativity is saying; of course as far as he isconcerned, Terence is unaffected by Stella's braking! While an analysis of cause and effectcan readily be made in the context of relativity, such an analysis is not required to describe theordering 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'sframe.
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).