Einstein's Derivation of the Lorentz Transformation

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

From these equations, he concludes then_{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
_{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 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.

(1) x-ct=0

(2) x'-ct'=0

(1a) x+ct=0

(2a) x'+ct'=0

From these equations, he concludes then

(3) x'-ct'= λ^{.}(x-ct)

(4) x'+ct'= μ^{.}(x+ct)

(5a) x' = ax -bct

(5b) ct' = act -bx

(6) a= (λ+μ)/2

(7) b= (λ-μ)/2 .

(8) x_{1}=ct ; x_{1}'=ct'

(9) x_{2}=-ct ; x_{2}'=-ct' ,

(10) x_{2}= -x_{1}

(11) x_{2}'=-x_{1}'.

(12) x_{1}' = ax_{1} -bct

(13) x_{2}' = ax_{2} -bct ,

(14) -x_{1}' = -ax_{1} -bct

(15) x_{1}' = ax_{1} +bct

(16) b=0

(unless t=0 (and hence xWith 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. |

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.

Thomas Smid (M.Sc. Physics, Ph.D. Astronomy)

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See also my sister site https://www.plasmaphysics.org.uk