[Relativity FAQ] - [Copyright]

Original by Don Koks 21 April 1998

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-v^{2})^{(-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 And also, that the so-called "relativistic mass" of a moving object (an outdated term but still useful) also doesn't depend on its acceleration.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 *10 ^{16} 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

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 t ^{2} - delta x^{2} - delta y^{2} -
delta z^{2}*, 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

But, what is the time as shown on a *noninertial* clock connecting two widely
separated events? It isn't *delta t ^{2} - delta x^{2} -
delta y^{2} - delta z^{2}*.
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^{2}= dt^{2}- dx^{2}- dy^{2}- dz^{2}

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 *dt ^{2} - dx^{2} -
dy^{2} - dz^{2}*, 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.

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
*A ^{a}* (up to a choice of gauge). Now, we know from other considerations
that

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 *A ^{a}*
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,
*A ^{a''}*, will be identical to the field measured by our MCIF,

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).