In his book 'Relativity: The Special and General Theory', Einstein gives a seemingly simple and elegant derivation of the Lorentz transformation which goes as follows:
he considers the 'equations of motion' of a light signal in the unprimed and primed frame
(1) x-ct=0
(2) x'-ct'=0
and correspondingly for a light signal travelling along the negative x-axis:
(1a) x+ct=0
(2a) x'+ct'=0
(note that Eqs.(1a) and (2a) are not written explicitly in Einstein's book (see
here)).
From these equations, he concludes then
(3) x'-ct'= λ.(x-ct)
(4) x'+ct'= μ.(x+ct)
and by adding and subtracting Eqs.(3) and (4) he gets the formal Lorentz transformation
(5a) x' = ax -bct
(5b) ct' = act -bx
where
(6) a= (λ+μ)/2
(7) b= (λ-μ)/2 .
Now it is obvious that the equations (1),(2) on the one hand and (1a),(2a) on the other are algebraically inconsistent unless the first set of equations is restricted to x,x'>0 in the first case and x,x'<0 in the second. So the generalized Eqs.(3) and (4) must then be restricted in the same sense in order not to violate the original constraint. But Einstein simply ignores this, and assumes that (3) and (4) hold both for positive and negative x,x' in order to be able to add and subtract the equations.
The resultant algebraic inconsistency is best evident if one uses actually different variables for the light signals travelling along the positive and negative x-axis, i.e. if we define
(8) x1=ct ; x1'=ct'
(9) x2=-ct ; x2'=-ct' ,
we have evidently the identities
(10) x2= -x1
(11) x2'=-x1'.
With these definitions, Einstein would obviously not have been able to proceed beyond his Eqs.(3) and (4), as the former would depend on x
1,x
1' but the latter on x
2,x
2'. It is only by violating Eqs.(10) and (11) and setting instead x
1=x
2 and x
1'=x
2' that Einstein is able to add and subtract the equations, and not surprisingly, the algebraic inconsistency reappears then indeed again in the resultant Lorentz transformation, because we obtain
(12) x1' = ax1 -bct
(13) x2' = ax2 -bct ,
and if we now insert (10) and (11) into (13), we get
(14) -x1' = -ax1 -bct
i.e.
(15) x1' = ax1 +bct
and by comparison with (12) we can thus conclude
(16) b=0
(unless t=0 (and hence x
1=x
1'=x
2=x
2'=0)).
With Einstein's interpretation of the constants, this means that the Lorentz transformation only applies to all values of x and t if the relative velocity of the reference frames is zero, which obviously would be a pointless result.
As an even simpler illustration for what Einstein has actually done here, consider the equation
(I) y(x)=ax + b
with the constraints
(C1) y(1) = 1
(C2) y(-1) = -1
The task is to determine the coefficients 'a' and 'b' by applying the constraints to (I). Now since (C1) results in 1=a+b and (C2) in 1=a-b, it is obvious that this requires b=0. But since b=0 is not what Einstein likes, he decides to modify the constraints such that (I) is valid for all b, i.e. he changes (C1) and (C2) to
(C1') y(1) = a +b
(C2') y(-1) = -a +b .
"Fine" Einstein says, "now I have a system of equations that is consistent for all 'b' (and 'a' at that)", but unfortunately it has nothing to do with the problem anymore. The task was not to find a set of constraints that are consistent with (1) irrespective of the value of the coefficients, but to apply the constraints (C1) and (C2) to (I) and thus to find the coefficients.
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As indicated above already, this sign inconsistency is due to the restriction that equations (1),(2) hold only for x>0, but (1a),(2a) only for x<0. Einstein was obviously implying this initially, but, aided by his notational inconsistencies, he suddenly dropped this restriction when he got to Eqs.(3),(4) and assumed that x and x' can be positive as well as negative for both equations (because this is the only way the equations can be added and subtracted). This however is obviously inconsistent with the initial constraints. The latter clearly require that in Eq.(5a) the reversal of the sign of x results in the reversal of the sign of x', but this can only be the case if b=0.
This sign error is actually also instrumental for the well known
'light sphere' derivation of the Lorentz transformation (where the quadratic form produces the same effect).
The correct interpretation of the invariance of the speed of light is of course that there can not possibly be any velocity dependent coordinate transformation as it would be inconsistent with the invariance principle (see the page regarding the
Speed of Light for more (which addresses the inconsistencies from a less mathematical viewpoint)).
Any other events not related to the propagation of a light signal are of course anyhow completely unaffected by the constancy of the speed of light (i.e. the constraints Eq.(1)-(2a) do not apply), that is the usual Galilei transformation holds.
For a discussion of the arguments given on this page see the
Relativity Discussion Page.
See also the analysis of
Einstein's original 1905 derivation.