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Gamma= 1/sqrt(1-v^2/c^2) = 2 (Assumed)

We have L=L’/Gamma…………………………………….1

And dT= dT’ * Gamma ………………………………2

Let primed frame is that of Bob and unprimed that of Dave.

Let us define identity as an equation in which values on both sides of the equal sign are equal.

And a formula in which given values at right side are calculated and assigned to a symbol at left side of the equal sign.

Equation 1 is treated as a formula. Proper length L’ is worked on and assigned to L. Therefore, L is the length in the Bob’s frame as calculated by Dave. If L’ is 100 meters, Dave disagrees and says that the length is only 50 meters. This is proper length contraction.

This is not the case with equation 2. It is treated as an identity.

Proper time of Bob, say 5 years, is multiplied by Gamma. To retain the identity, dT is increased to 10. In this case Bob’s clock doesn’t run slow. It is Dave’s clock that is running fast.

If equation 2 is used as a formula then according to Dave, Bob’s clock is running fast. That is even if Bob measures 5 years in his clock, according to Dave, this same time is 10 years.

But as soon as we use the word ‘measure’ we must treat the equations as formulae and not as identities, because we are finding values.

So equation 1 is correct but equation 2 should be dT= dT’ / Gamma. Now Bob’s time as measured by Dave is less, IOW according to Dave, Bob’s clock runs slow.

This scenario can be viewed from another angle.

We keep rod and clock in the frame of which the measurements are to be made.

So we keep a clock/s and a rod of length L’ in Bob’s frame. For length we have,

x = (x’+vt’)*Gamma

x2– x1= ((x’2 – x’1 ) + v(t’2–t’1)) * Gamma

or L= L’*Gamma

In the above we take help of the space-time diagram. We keep the rod on the line parallel to x’-axis. On this axis t’2 = t’1

Similarly

t2 – t1 = ((t’2–t’1)+(x’2–x’1)*v/c^2)*Gamma

or dT = dT’ * Gamma

In the above we keep two synchronized clocks on the line parallel to the time axis of primed frame.

In my opinion, once we derive Lorentz equations, we should show some respect to these basic equation. We can have dispute with the concepts behind the derivation but once derived, they cannot be manipulated further as we like. Present equations 1 and 2 are convenient but mathematically inconsistent.

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Any institution in which voices are gagged is autocratic and harmful to society

–Vilas Tamhane