Lorentz Inversion

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In the Lorentz Transformation we see the centerpiece of the theory of Special Relativity. We deduced it from Einstein's postulates and we can use it to deduce new facts that come out of the relationships among inertial frames in space and time.

But does the Lorentz Transformation actually conform to the first postulate or have we, perhaps, deceived ourselves? Those equations for converting one observer's measurements into the equivalent measurements that another, moving, observer would make look too complicated to reflect any proposition that the laws of physics must look the same to any observers regardless of any relative motion between them. However, we have a way to test that suspicion. We start with the equations in question:

(Eq'n 1)

(Eq'n 2)

Lower-case Louis wants to take Upper-case Ulric's measurements of the time elapsed and distance spanned between two given events and convert them into the equivalent time and distance that he would have measured and he would use Equations 1 and 2 to do so. But if Lower-case Louis had made the measurements and Upper-case Ulric had to calculate the equivalent measurements in his frame, what equations would Ulric use? We can start to find out by multiplying both Equations 1 and 2 by the square root and then adding to both sides of each the velocity term. We thus obtain

(Eq'ns 3a & 3b)

We can solve those two equations by the standard methods of algebra to obtain:

(Eq'ns 4a & 4b)

Equations 4 differ from Equations 1 & 2 by the interchange of the upper-case and lower-case variables and by the replacement of the subtraction sign in the numerator by the addition sign. Replacement of the algebraic sign within the transformation formulae merely reflects the fact that Louis and Ulric have agreed on which direction they take as the positive x-direction and the fact that one observer's frame moves in the negative x-direction for the other observer while that other observer's frame moves in the positive x-direction for the first observer. Those differences have no impact upon the laws of physics, so we can say that as far as the laws of physics are concerned Equations 4 are identical to Equations 1 & 2. Thus we can say that the Lorentz Transformation does, indeed, conform to Einstein's first postulate of Relativity, as it should.

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