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Is it generally accepted that the primes associated with the Lorentz Transformations signify something different to the primes associated with the relativistic effect equations?

Imagine you have Inez in an "inertial frame" with a ruler of length x -when measured in a "stationary frame". She points the ruler at Stan, who is considered to be stationary, so that is it "longitudinal". Stan wants to work out how long the ruler is according to Inez, in terms of his frame of reference. Stan considers two "events" Inez' end of the ruler (0,t) and the end she's pointing at him (x,t), therefore:

Δx'= x' - 0 = (x-vt)/sqrt(1-v^2/c^2) - (0-vt)/sqrt(1-v^2/c^2)

so

Δx' = x' = x/sqrt(1-v^2/c^2)

This is upside down ... why?

Note that when you do the same thing with the temporal Lorentz equations, you don't run into the same problem.

Imagine that Stan now considers two other events, the first tick of Inez' clock (x,0) and the second tick of the same clock (x,t), so:

Δt'= t' - 0 = (t-vx/c^2)/sqrt(1-v^2/c^2) - (0-vx/c^2)/sqrt(1-v^2/c^2)

and

Δt'= t' = t/sqrt(1-v^2/c^2)

This is the correct equation for "time dilation". What's going on?

PS - my suspicion is that our concept of time dilation is screwy, or rather that time dilation happens on part of the stationary observer in relation to the "inertial" observer and that the primes are not used consistently, but I am willing to listen and learn if anyone has a better explanation.

Imagine you have Inez in an "inertial frame" with a ruler of length x -when measured in a "stationary frame". She points the ruler at Stan, who is considered to be stationary, so that is it "longitudinal". Stan wants to work out how long the ruler is according to Inez, in terms of his frame of reference. Stan considers two "events" Inez' end of the ruler (0,t) and the end she's pointing at him (x,t), therefore:

Δx'= x' - 0 = (x-vt)/sqrt(1-v^2/c^2) - (0-vt)/sqrt(1-v^2/c^2)

so

Δx' = x' = x/sqrt(1-v^2/c^2)

This is upside down ... why?

Note that when you do the same thing with the temporal Lorentz equations, you don't run into the same problem.

Imagine that Stan now considers two other events, the first tick of Inez' clock (x,0) and the second tick of the same clock (x,t), so:

Δt'= t' - 0 = (t-vx/c^2)/sqrt(1-v^2/c^2) - (0-vx/c^2)/sqrt(1-v^2/c^2)

and

Δt'= t' = t/sqrt(1-v^2/c^2)

This is the correct equation for "time dilation". What's going on?

PS - my suspicion is that our concept of time dilation is screwy, or rather that time dilation happens on part of the stationary observer in relation to the "inertial" observer and that the primes are not used consistently, but I am willing to listen and learn if anyone has a better explanation.

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