Original by Don Koks 21 April 1998

# Does a clock's acceleration affect its timing rate?

It's often said that special relativity is based on two postulates: that all inertial frames are of equal validity, and that light travels at the same speed in all inertial frames. But in real world scenarios, objects almost never travel at constant velocity, and so we might never find an inertial frame in which such an object is at rest. To allow us to make predictions about how accelerating objects behave, we need to introduce a third postulate.

This is often called the "clock postulate", but it applies to much more than just clocks, and in fact it underpins much of advanced relativity, both special and general, as well as the notion of covariance in general (that is, writing the equations of physics in a frame-independent way).

In one form, the clock postulate is this: the Lorentz factor

```                    gamma = (1-v2)(-1/2)
```

only depends on v, and does not depend on any derivatives of v, such as acceleration. So this says that an accelerating clock will count out its time in such a way that at any one moment, its timing has slowed by a factor (gamma) which only depends on its current speed; its acceleration has no effect at all. In other words, our accelerated clock's rate is identical to the clock rate in a " momentarily comoving inertial frame" (MCIF), which we can imagine is holding an inertial clock that for a brief moment is moving alongside of us, so that our relative velocity is momentarily zero.

This also means that moving rods are shortened by the same factor; that is, that the amount of shortening is independent of their acceleration. And also, that the so-called "relativistic mass" of a moving object (an outdated term but still useful) also doesn't depend on its acceleration.

Although the clock postulate is just that, a postulate, it has been verified experimentally up to extraordinarily high accelerations, as much as 1016 g in fact (in the 1960's by Pound and Rebka). Of course, the postulate also speaks of more than acceleration, it speaks of all derivatives of v with respect to time. But still it can be shown to be a reasonable thing to assume, because it leads to something which has been tested in other ways.

## The spacetime metric, or interval

This something is the idea of a spacetime metric. Typically, when learning special relativity, at some stage we note that the "interval" between two events, delta t2 - delta x2 - delta y2 - delta z2, is independent of the (inertial) frame in which we make our measurements. So although relativity has taught us to throw away our ideas of the absoluteness of space and time, even so this idea of something else which is observer-independent leads to a new absolute thing called spacetime. And since the interval is frame independent, we can also say immediately that the time delta T elapsed on an inertial clock connecting two events is just delta T2 = delta t2 - delta x2 - delta y2 - delta z2, since in its own inertial frame, the clock doesn't move.

But, what is the time as shown on a noninertial clock connecting two widely separated events? It isn't delta t2 - delta x2 - delta y2 - delta z2. In fact, what we do is this: we assume the clock postulate holds. So, when an accelerating clock moves from one event to another that is infinitesimally close, we can say that the infinitesimal time dT elapsed is given by

```              dT2 = dt2 - dx2 - dy2 - dz2
```

since this is the time which elapses on a clock in the MCIF. And we can now integrate this dT along the clock's worldline to get the actual elapsed time that it shows. This is why we can give some structure to spacetime, because it's possible to talk about the "length" of a curved worldline as being given by the time shown on a clock moving along that world line, even though the clock itself is accelerating (since the worldline is curved---we are still only talking about flat space here). If the "length" between two infinitesimally separated spacetime points was something that depended on how a clock connecting them moved, then it wouldn't be an intrinsically geometric thing. The clock postulate geometrises relativity by saying that we don't need to consider how the clock moves; the time shown by any clock connecting two closely separated events is just dt2 - dx2 - dy2 - dz2, and it doesn't matter how the clock moves.

This is why the interval is written using infinitesimals; it's not necessarily because we have an idea at the back of our mind that we might want to deal with curved spacetime; rather it's because in this form the interval embodies the clock postulate. But now the grand thing is that this idea of spacetime structure allows us to make the transition to general relativity, and the great success of that theory lends plausibility to the parts that make it up---and one of these is the clock postulate.

## Generalising the clock postulate

The clock postulate can be generalised to say something about measurements we make in a noninertial frame. First, it tells us that noninertial objects only age and contract by the same factor as that of their MCIF. So, any measurements we make in a noninertial frame that use clocks and rods, will be identical to measurements made in our MCIF. But we now choose to extend the clock postulate to include all measurements (though perhaps it can be argued that all measurements only ever use clocks and rods anyway). This idea leads onto "covariance", which is a way of using tensors to write the language of physics in a way that applies to all frames, noninertial as well as inertial. How is this done?

Take, for example, the electromagnetic field. Suppose some measurements were made in the (inertial) lab frame, and it was determined to have components Aa (up to a choice of gauge). Now, we know from other considerations that Aa is a four-vector. That means, it transforms between inertial frames just as do (dt, dx, dy, dz). In the language of tensors, we can say that from one inertial frame (t x y z) to another (t' x' y' z'), the Aa transforms like

```
a'
a'         del x        b
A    =    ----------   A        ...(1)
b
del x
```

This much can be shown without the clock postulate, because it deals only with inertial frames. The question is, in this language, how does Aa look in a noninertial frame?

Suppose we are in a noninertial frame, and wish to know what values the field will take on for us. The clock postulate tells us that the field in our frame, Aa'', will be identical to the field measured by our MCIF, Aa'. But that frame is inertial, so we know how to calculate Aa': it's just given by equation (1) above. Next step: how can we relate our co-ordinates to the way that Aa transforms?---after all, the MCIF always exists, but we may not want to consider it explicitly.

Again, we make use of the clock postulate. It says that our measurements of distance and time are unaffected by any pseudo forces we feel, so that they are identical to those of our MCIF:

```
a''       a'
dx    =   dx
```

This is just another way of saying that

```
a''               a'
del x             del x
---------    =    ----------
b                 b
del x             del x
```

so that we can put it all together to write

```
a''     a'
A    =  A     (clock postulate)

a'
del x      b
=   ---------   A  (Lorentz transform)
b
del x

a''
del x      b
=   ---------   A  (clock postulate)
b
del x
```

or finally

```
a''
a''      del x       b
A    =   ---------   A
b
del x
```

which is just like (1) above, except now it holds in a noninertial frame! So, we've managed to write our original transformation law (1), which only held between two inertial frames, in a way which looks just the same as (1), except that it's using different co-ordinates. And this is the idea behind covariance, that it's possible to write the laws of physics for arbitrary frames, even noninertial ones, in one way: the way of eqn (1).