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**New Theories / Re: In regards to the 1st Postulate of STR**

« **on:**19/02/2015 09:45:48 »

What about a round-trip race? Well, let's walk the dog on that one shall we...

We will utilize the same photon/electron race scenario as before, only now we add the requirement for the photon and electron to be reflected (after traveling an identical distance along the positive x-axis) and travel an identical distance back along the negative x-axis. The new round-trip times will be reflected as rτ for the photon and rt for the electron.

When the race occurs within the stationary laboratory frame (v = 0), no special calculation is needed (Lorentz transformations reduce to Galilean format) and the ratio of the photon time span to the electron time span WRT the stationary laboratory frame is (1/2 the trip is in the positive x-axis direction, the other 1/2 of the trip is in the negative x-axis direction, where both photon and electron end up returning to their perspective points of origin);

Photon: rτ = τ'+τ'

rτ = 2τ'

Electron: rt = t'+t'

rt = 2t'

Ratio #1: rτ/rt = 2τ'/2t'

rτ/rt = τ'/t'

When the race occurs within the inertial frame at velocity (v), the calculation of the ratio of the photon time span to the electron time span WRT the stationary laboratory frame IAW the Lorentz transformations is;

Photon: rτ = (τ'+vx'/c^2)γ + (τ'+v(-x')/c^2)γ

rτ = (τ'+vx'/c^2)γ + (τ'-vx'/c^2)γ

Since it is a photon, the second postulate gives x' = cτ'

rτ = (τ'+v(cτ')/c^2)γ + (τ'-v(cτ')/c^2)γ

rτ = (τ'+v(τ')/c)γ + (τ'+v(τ')/c)γ

rτ = τ'(1+v/c)γ + τ'(1-v/c)γ

rτ = τ'γ+τ'(v/c)γ+τ'γ-τ'(v/c)γ

rτ = 2τ'γ

*NOTE: (2τ'γ) was expected as this is the same round-trip time that would occur for the photon if the race occurred along the y-axis (reference "http://en.wikipedia.org/wiki/Michelson–Morley_experiment", 'Length contraction and Lorentz transformation' section). See also "http://www.people.fas.harvard.edu/~djmorin/chap11.pdf".

Electron: rt = (t'+vx'/c^2)γ + (t'+v(-x')/c^2)γ

rt = (t'+vx'/c^2)γ + (t'-vx'/c^2)γ

rt = t'γ+t'(vx'/c^2)γ+t'γ-t'(vx'/c^2)γ

rt = 2t'γ

*NOTE: (2t'γ) was expected as the same relativistic delay (γ factor) encountered by the photon would also be expected to affect the electron in order for the measured race results to appear the same to the inertial frame at velocity (v) as when the race occurred within and was measured by the stationary laboratory frame.

Ratio #2: rτ/rt = (2τ'γ)/(2t'γ)

rτ/rt = τ'/t'

The ratios are identical as expected since "The laws of physics are the same in all inertial frames of reference." This means that the race results measured by the stationary laboratory frame when the race occurs within the stationary laboratory frame as well as the race results measured by the inertial frame at velocity (v) when the race occurs within the inertial frame at velocity (v) will be identical. This is Einstein's first postulate in action!

Why are the ratios equal for the round-trip races, yet unequal for the one-way races?

We will utilize the same photon/electron race scenario as before, only now we add the requirement for the photon and electron to be reflected (after traveling an identical distance along the positive x-axis) and travel an identical distance back along the negative x-axis. The new round-trip times will be reflected as rτ for the photon and rt for the electron.

When the race occurs within the stationary laboratory frame (v = 0), no special calculation is needed (Lorentz transformations reduce to Galilean format) and the ratio of the photon time span to the electron time span WRT the stationary laboratory frame is (1/2 the trip is in the positive x-axis direction, the other 1/2 of the trip is in the negative x-axis direction, where both photon and electron end up returning to their perspective points of origin);

Photon: rτ = τ'+τ'

rτ = 2τ'

Electron: rt = t'+t'

rt = 2t'

Ratio #1: rτ/rt = 2τ'/2t'

rτ/rt = τ'/t'

When the race occurs within the inertial frame at velocity (v), the calculation of the ratio of the photon time span to the electron time span WRT the stationary laboratory frame IAW the Lorentz transformations is;

Photon: rτ = (τ'+vx'/c^2)γ + (τ'+v(-x')/c^2)γ

rτ = (τ'+vx'/c^2)γ + (τ'-vx'/c^2)γ

Since it is a photon, the second postulate gives x' = cτ'

rτ = (τ'+v(cτ')/c^2)γ + (τ'-v(cτ')/c^2)γ

rτ = (τ'+v(τ')/c)γ + (τ'+v(τ')/c)γ

rτ = τ'(1+v/c)γ + τ'(1-v/c)γ

rτ = τ'γ+τ'(v/c)γ+τ'γ-τ'(v/c)γ

rτ = 2τ'γ

*NOTE: (2τ'γ) was expected as this is the same round-trip time that would occur for the photon if the race occurred along the y-axis (reference "http://en.wikipedia.org/wiki/Michelson–Morley_experiment", 'Length contraction and Lorentz transformation' section). See also "http://www.people.fas.harvard.edu/~djmorin/chap11.pdf".

Electron: rt = (t'+vx'/c^2)γ + (t'+v(-x')/c^2)γ

rt = (t'+vx'/c^2)γ + (t'-vx'/c^2)γ

rt = t'γ+t'(vx'/c^2)γ+t'γ-t'(vx'/c^2)γ

rt = 2t'γ

*NOTE: (2t'γ) was expected as the same relativistic delay (γ factor) encountered by the photon would also be expected to affect the electron in order for the measured race results to appear the same to the inertial frame at velocity (v) as when the race occurred within and was measured by the stationary laboratory frame.

Ratio #2: rτ/rt = (2τ'γ)/(2t'γ)

rτ/rt = τ'/t'

The ratios are identical as expected since "The laws of physics are the same in all inertial frames of reference." This means that the race results measured by the stationary laboratory frame when the race occurs within the stationary laboratory frame as well as the race results measured by the inertial frame at velocity (v) when the race occurs within the inertial frame at velocity (v) will be identical. This is Einstein's first postulate in action!

Why are the ratios equal for the round-trip races, yet unequal for the one-way races?