This is pseudomathematics and pseudoscience. It apes the form of math and science without doing more than confirm the writer's biases against 20th century.

I. INTRODUCTION

The invariance of the speed of light in different reference frames lies at the heart of the derivation of the Lorentz transformation and thus the theory of Special Relativity. The derivation typically assumes two given events associated with the propagation of a light signal, and then looks for a linear transformation of the space and time coordinates of these events that would also correspond to the same speed of light c in a different reference frame moving with a velocity v relatively to the first. In the following it is shown that such a transformation can not exist without violating the speed of light postulate applied to an independent measurement of the speed of light in the second reference frame.

II. PROOF

Version 1

We designate the two reference frames with the indices 1 and 2 respectively and assume that in each frame we have a separate stationary array of light detectors. Let a light signal be emitted by a light source at the origin of both reference frames, the light source being at rest in frame 1 but moving with velocity v in frame 2. Assuming the detectors to be equipped with synchronized clocks and the light signal emitted at zero system time in each frame, the detector locations and associated detection times are then given by

(1) x1=c*t1

(2) x2=c*t2

where (2) follows from the speed of light postulate.

We have formally allowed different time units here, but use of the same numerical value for c implies obviously

(3) x1/x2 =t1/t2 =q

where q is some constant. As the value of q is merely a matter of convention, we can therefore, without loss of generality, choose q=1 and thus set

(4) t1 =t2 =t

and instead of (1) and (2) we have then

(5) x1=c*t

(6) x2=c*t

Note that these timings, despite being identical here due to the chosen convention, are as such completely independent of each other. Frame 2 obtains its data completely independently of the data obtained in frame 1. Equations (5) and (6) define the speed of light postulate therefore without the need of transforming the frame 1 data into frame 2. If we wanted to do the latter in addition, we would require that the coordinate transformed location of x1 coincides with x2, i.e. using (5) and (6) we would get

(7) x1'=x2=c*t=x1

It is obvious that this does not allow any velocity dependent transformation of the form

(8) x1'=a*(x1-b*t)

unless a=1 and b=0 (note that (7) requires that x1' is reflected about the origin whenever x1 is reflected about the origin.).

Version 2

In contrast to version 1, we do not take any conventions at all about the time units used in the two reference frames, so we retain the initial equations

(9) x1=c*t1

(10) x2=c*t2

Again, the corresponding detection events in both equations are completely independent of each other (the data in one frame can be obtained without having any knowledge of those obtained in the other frame). This system of equations consists thus of two independent variables (the locations of the detectors x1 and x2) and two dependent variables (the corresponding detection times t1 and t2). If we would require that the event (x2,t2) can be represented as a coordinate transformation of the event (x1,t1), we would have the additional constraints

(11) x2=x1'=f(x1,t1,v)

(12) t2=t1'=g(x1,t1,v)

where f and g are the transformation functions of the arguments.

However, (11) would turn the independent variable x2 into a dependent variable, and (12) would, via the inversion of the equation, turn the independent variable x1 into a dependent variable. There would thus be no independent variable left in the system of equations (9) and (10). This would be unacceptable both mathematically and physically.

III. DISCUSSION

As is evident from both of the above considered scenarios, any velocity dependent transformation is inconsistent with the speed of light postulate, Such a transformation is only applicable for material (massive) objects, for which the speed is not invariant in different reference frames. For those, independent measurements in the two reference frames (moving with velocity v relatively to each other) would instead of (5) and (6) yield

(13) x1=u*t

(14) x2=(u-v)*t

i.e.

(15) x2=x1-v*t

and thus a transformation equation of the form (8) follows naturally.

In case of a propagating light signal, the problem is of course not the equation (8) but (7) where x1' is assumed to be co-located with x2. If we don’t make this assumption, we can without any issues take for instance

(16) x1'=x1-v*t

as the speed of light postulate is independently satisfied by (5) and (6).

And this would of course also remove the impossible situation of having a system of equations without any independent variables (as argued in version 2 above).

The invariance of the speed of light in different reference frames lies at the heart of the derivation of the Lorentz transformation and thus the theory of Special Relativity. The derivation typically assumes two given events associated with the propagation of a light signal, and then looks for a linear transformation of the space and time coordinates of these events that would also correspond to the same speed of light c in a different reference frame moving with a velocity v relatively to the first. In the following it is shown that such a transformation can not exist without violating the speed of light postulate applied to an independent measurement of the speed of light in the second reference frame.

II. PROOF

Version 1

We designate the two reference frames with the indices 1 and 2 respectively and assume that in each frame we have a separate stationary array of light detectors. Let a light signal be emitted by a light source at the origin of both reference frames, the light source being at rest in frame 1 but moving with velocity v in frame 2. Assuming the detectors to be equipped with synchronized clocks and the light signal emitted at zero system time in each frame, the detector locations and associated detection times are then given by

(1) x1=c*t1

(2) x2=c*t2

where (2) follows from the speed of light postulate.

We have formally allowed different time units here, but use of the same numerical value for c implies obviously

(3) x1/x2 =t1/t2 =q

where q is some constant. As the value of q is merely a matter of convention, we can therefore, without loss of generality, choose q=1 and thus set

(4) t1 =t2 =t

and instead of (1) and (2) we have then

(5) x1=c*t

(6) x2=c*t

Note that these timings, despite being identical here due to the chosen convention, are as such completely independent of each other. Frame 2 obtains its data completely independently of the data obtained in frame 1. Equations (5) and (6) define the speed of light postulate therefore without the need of transforming the frame 1 data into frame 2. If we wanted to do the latter in addition, we would require that the coordinate transformed location of x1 coincides with x2, i.e. using (5) and (6) we would get

(7) x1'=x2=c*t=x1

It is obvious that this does not allow any velocity dependent transformation of the form

(8) x1'=a*(x1-b*t)

unless a=1 and b=0 (note that (7) requires that x1' is reflected about the origin whenever x1 is reflected about the origin.).

Version 2

In contrast to version 1, we do not take any conventions at all about the time units used in the two reference frames, so we retain the initial equations

(9) x1=c*t1

(10) x2=c*t2

Again, the corresponding detection events in both equations are completely independent of each other (the data in one frame can be obtained without having any knowledge of those obtained in the other frame). This system of equations consists thus of two independent variables (the locations of the detectors x1 and x2) and two dependent variables (the corresponding detection times t1 and t2). If we would require that the event (x2,t2) can be represented as a coordinate transformation of the event (x1,t1), we would have the additional constraints

(11) x2=x1'=f(x1,t1,v)

(12) t2=t1'=g(x1,t1,v)

where f and g are the transformation functions of the arguments.

However, (11) would turn the independent variable x2 into a dependent variable, and (12) would, via the inversion of the equation, turn the independent variable x1 into a dependent variable. There would thus be no independent variable left in the system of equations (9) and (10). This would be unacceptable both mathematically and physically.

III. DISCUSSION

As is evident from both of the above considered scenarios, any velocity dependent transformation is inconsistent with the speed of light postulate, Such a transformation is only applicable for material (massive) objects, for which the speed is not invariant in different reference frames. For those, independent measurements in the two reference frames (moving with velocity v relatively to each other) would instead of (5) and (6) yield

(13) x1=u*t

(14) x2=(u-v)*t

i.e.

(15) x2=x1-v*t

and thus a transformation equation of the form (8) follows naturally.

In case of a propagating light signal, the problem is of course not the equation (8) but (7) where x1' is assumed to be co-located with x2. If we don’t make this assumption, we can without any issues take for instance

(16) x1'=x1-v*t

as the speed of light postulate is independently satisfied by (5) and (6).

And this would of course also remove the impossible situation of having a system of equations without any independent variables (as argued in version 2 above).