Let two inertial co-ordinate systems (ICR's),
S and ,
be in standard configuration. This means that
is moving with constant velocity
relative to S in the x-direction as in the following figure
and that the spatial origins, O and , and the three spatial axes, and , co-incide at time .
We shall convert units of time (eg. seconds) to a length (eg. meters) by multiplying time by the speed of light c - thus our measure of time will be ct which has units of length (this assumes implicitly that c is the same in both reference frames - one of the postulates of relativity). We wish to determine Cartesian co-ordinates in , , as functions of Cartesian co-ordinates in S, (ct,x,y,z), using reasonable assumptions. In other words will be a function of ct,x,y and z, i.e. , etc. It will sometimes be convenient to adopt an index notation where the four co-ordinates (ct,x,y,z) are labelled by an index a taking on four possible values 0,1,2,3 with x0=ct, x1=x, x2=y and x3=z so that (ct,x,y,z)=(xa). Similarly for primed co-ordinates an index notation is sometimes useful, , where the prime is placed on the index so that the index itself can be used to distinguish between the two co-ordinates systems. The change from (ct,x,y,z)=(xa) to is called a co-ordinate transformation.
The derivation of the explicit form of the co-ordinate transformation
proceeds in four steps:
Step 1): The transformations are linear
Consider a clock moving with constant velocity, showing a time . The path of the clock in S can be described by four functions. Since the clock is moving with constant velocity in Sequal increments of must correspond to equal increments of the co-ordinates (xa) labeling the position of the clock in S. Thus is constant and . Since is moving with constant velocity relative to S, the clock must aslo be moving with constant velocity in hence the same argument implies that is constant and . Now treating as functions of xa the chain rule for differentiation implies
Thus and can only be true if , in other words the transformations must be linear in xa. In mathematical symbols this means
where are constants and are sixteen functions, independent of xa but possibly depending on v - the velocity of S relative to . If S is in standard configuration relative to then for all four values of a=0,1,2,3.
These conditions can be summarised in the single formula
which can be thought of as a matrix formula with a matrix and and xa column vectors,
The matrix with components can be inverted to give xa in terms of ,
where L-1(v) is the inverse matrix to L(v), i.e. with the identity matrix. Since the (xa) co-ordinate system is moving in the negativedirection relative to the with speed v it is clear that bear the same relation to (xa) as (xa)do to , except that the sign of v is reversed. Mathematically this means that L-1(v)=L(-v).
We shall now determine the sixteen functions .Step 2):
At time the two planes co-incide for all y and z, as in the following figure
At time the two planes co-incide. Since the relative motion is in the x-direction and there is no rotation (by assumption), the planes co-incide as in the following figure
We can apply the same argument with S and interchanged, which requires that we replace L(v) by L-1(v)=L(-v), to deduce that . Hence .
Now if we reflect , without changing the other co-ordinates in S, it should be clear that z and do not change (since we have just proven that is independent of x). But changing the sign of xchanges the sign of v, since the relative motion is in the x-direction. Hence , thus , so . The sign can be determined by the trivial observation that v=0 should give the identity transformation, thus .
A similar argument applied to the two planes allows us to conclude that
In summary, we have now that the transformation matrix must be of the form
Up until now we have only really used the postulates of relativity to streamline the notation. Now it will be used to full effect. First suppose a flash of light is emitted from the origin O=(0,0,0) of S at t=0 (and so also from the origin of at ). The flash expands with the speed of light, c which is the same in both reference frames, as a spherical shell whose radius at time t is given by x2+y2+z2=c2t2 in S and by in . Now we already know that and so
Demanding that this hold true for all t and any x with |x|<ct gives three conditions
on four unkowns. We can express these as four functions of a single parameter by using the identity for any real to write
where is a function of v which is yet to be determined (the minus sign is for later convenience). We have now arrived at the following form for the transformation matrix:
The spatial origin of is determined by . In S the point moves with speed v in the x-direction, i.e. it has x co-ordinate x=vt. Thus
so can be written as an inverse hyper-trigonometric function
Note that the properties of the function now imply that -c<v<c (see the figure below). is called the rapidity of the transformation.
Using we have
It is conventional to define
and then the transformation matrix can be written as
Thus we have finally arrived at the following form for the transformation
These are called Lorentz Transformations or sometimes Lorentz Boosts, to distinguish them from rotations - the name ``boost'' is unfortunate as there is no acceleration involved.
In matrix notation the Lorentz Transformations can be represented as
The rapidity has the useful property that it is additive under successive transformations in the same direction with and . This is most easily established using matrix multiplication to show that
(use the hyperbolic trigonometric identities
). Thus these two transformations are equivalent to a single transformation with rapidity .