The Lorentz Transformation and The Postulates of Special Relativity
Einstein postulated that
the laws of physics are the same in all inertial systems- no preferred inertial systems,
the speed of light in free space has the same value C in all inertial systems.
To accomodate the above assumptions, we must
obtain equations of transformation between two inertial systems moving uniformly with respect to each other which will keep the speed of light invariant. (=> Lorentz transformation)
examine the laws of physics to check whether or not they keep the same form (i.e. invariant) under this new transformation. Those laws that are not invariant will need to be generalized so as to obey the principle of relativity. We expect the generalization to be such that the new laws will reduce to the old ones when v/c << 1.
Derivation of the Lorentz transformation equations
Let's consider two inertial frames S and S' with a common x (x')- axis and y', z' axes parallel to y, z axes respectively. To simplify the algebra we choose the relative velocity to be along the x (x')-axis (without loss of generality).
The space-time coordinates of an event are t,x,y,z in S and t',x',y',z' in S'.
The differential form of the above equations is
Now if we accept the assumption of the space-time homogeneity, the
coefficients
cannot depend on coordinates t,x,y,z. Therefore we can integrate
the above equations to obtain
We simplify the situation once more by setting t=t'=0 at the origin at the instant the origins O and O' coincide. Then b0 = b1 = b2 = b3 = 0. Since the x-axis coincides continuously with the x'-axis, it follows that y'=z'=0 for y=z=0, x, t arbitrary. =>
Also the x-y plane should transform over to the x'-y' plane. Then it follows that z'=0, y arbitrary. => a32 = 0 => z' = a33z. Likewise y' = a22y.
Now consider a rod lying along the y-axis extending from y=0 to y=1. The length of the rod measured by an S-observer is 1, and to an S'-observer the length is a22. If we fix the same rod along the y'-axis, then an S'-observer should measure the length to be 1 and an S-observer should measure it to be 1/a22. The symmetry between the two inertial frames requires that
Likewise a33-1. Also from the symmetry (or isotropy) argument we conclude that t' and x' cannot depend on y and z.
For the origin O', x'=0 and x=vt
Summarizing the results obtained so far,
Now we require that the speed of light is the same C in all directions in both S and S'. Then for a light wave originated at the origin at t=t'=0,
The inverse transformation equations are
Note that the above two sets of transformation equations
are identical in form except that v changes to -v. For the
more general case of an arbitrary diretcion of
,
