As discussed in more detail on my page Speed of Light and Theory of Relativity
, the Lorentz transformation is based on the paradox attempt to apply the usual concept of speed (which implies a velocity dependent transformation of the coordinates of the body between different reference frames) to light signals as well. In order to achieve this, any derivation of the Lorentz transformation assumes first that for light signals the usual Galilei transformation applies, and then the resultant inconsistency with the principle of the invariance of c is removed by means of illegal mathematical steps (which physically result then in the conceptually inconsistent re-scaling of the original definitions for the length and time units (length contraction and time dilation)).
In this sense, in §2 of his famous 1905 paper On the Electrodynamics of Moving Bodies
Einstein first considers the propagation of a light signal emitted by a moving light source (and subsequently reflected back to the light source by a co-moving mirror) and assumes (in violation of the principle of the invariance of c) the relative velocities c-v and c+v between the light signal and the light source, which would be in the rest frame connected to the corresponding times of the reflection at the mirror t1
and return to the light source t2
by the equations
(1) t1= x'/(c-v)
(2) t2= x'/(c-v) + x'/(c+v)
where x' is the (fixed) distance between the light source and the reflecting mirror (so this variable should not be confused here with the one used for the transformed x-coordinate in the usual notation (like used in Einstein's later algebraic derivation
As the different relative velocities c-v and c+v would obviously violate the invariance of c in the frame of the moving light source, he then postulates a functional equation intended to re-scale the time units in the moving frame such as to compensate for this, i.e.
(3) 1/2.τ(0,t2)= τ(x',t1)
where the start values have been set to zero here for simplicity, and also the y and z components not been written.
Einstein then apparently differentiates Eq.(3) with regard to x' to yield
(4) ∂τ/∂x'= (a2/2-a1).∂τ/∂t = - v/(c2-v2).∂τ/∂t
(5) a1= 1/(c-v)
(6) a2= 1/(c-v) + 1/(c+v) .
However, already at this point there are two crucial mathematical mistakes:
In Eq.(3), the arguments should be coordinates in the rest frame. However, when using the values 0 and x' for the first argument, this is a coordinate which is referred to a point moving with regard to the rest frame (with velocity v) (see part b) below how to correctly formulate the equation in terms of the rest frame coordinate x=x'+vt (as defined by Einstein) and the resultant effect on the differential equation).
On differentiating Eq.(3), Einstein is not applying the chain rule correctly: if we take τ1
) and τ2
), then the outer derivatives with regard to the second argument are ∂τ1
respectively. Now these derivatives are different, as according to Eqs.(1),(2) and (3) ∂t1
. So it is not permissible to set them both to a common derivative ∂τ/∂t .
In order to investigate this point more closely, it is necessary to first avoid the error mentioned under a) and formulate Eq.(3) in terms of the actual coordinate x rather than x'. This yields the equation
(7) 1/2.τ2= τ1
(8) τ1=τ(x1,t1) ; τ2=τ(x2,t2)
(9) x1=x'+v.t1 , t1=x'.a1 ; x2=v.t2 , t2=x'.a2 .
Differentiating (7) with regard to x' yields (using the chain rule)
(10) 1/2. (∂τ2/∂x2.v.a2 + ∂τ2/∂t2.a2) = ∂τ1/∂x1.(1+v.a1) + ∂τ1/∂t1.a1 .
Now, if we use Eqs.(9) to express ∂x1
in terms of ∂t1
respectively, Eq.(10) becomes
(11) a2.∂τ2/∂t2= 2a1.∂τ1/∂t1 .
This means that if we would set ∂τ2
= ∂τ/∂t (like Einstein did), this would require a2
, which, given Eqs.(5) and (6), is not possible (unless v=0).
The inconsistency is indeed also directly apparent if one has the Lorentz transformation
τ = γ.(t-v.x/c2)
and inserts into it (as per Einstein's definitions)
t1 = x'.a1 = x'/(c-v) ; x = x'+v.t1 ,
one gets (after a little algebra)
τ1 = γ.x'/c
Analogously one gets τ2
This is obviously inconsistent with the original assumption of the invariance of c, which requires that the time for the light signal to reach the mirror (at a distance x' from the light source) in the latters rest frame is τ1
= x'/c, and the time to return to the light source τ2
= 2x'/c. So again, Einstein's equations would only be consistent if γ=1 i.e. v=0.
In other words, there can not possibly exist a single function τ(x,t) that would satisfy the given constraint, and thus there can not be any transformation of the coordinates of a light signal between different reference frames, which of course, as explained on my page Speed of Light and Theory of Relativity
is exactly what the invariance of c implies.