- #1

ANvH

- 54

- 0

Given

##Δ\acute{t}=\gamma(Δt-\beta c^{-1}Δx)## (1)

##Δ\acute{x}=\gamma(Δx-\beta c Δt)## (2)

and the inverse transformations,##Δt=\gamma(Δ\acute{t}+\beta c^{-1}Δ\acute{x})## (3)

##Δx=\gamma(Δ\acute{x}+\beta c Δ\acute{t})## (4),

where the primes refer to the emitter frame (the moving frame of reference) and the non-primes refer to the receiver frame (the stationary frame or rest frame of reference).Length contraction is defined by measuring the object length ##Δ\acute{x}## in the moving frame while held stationary from the rest frame; moving the line of sight of the receiver accomplishes this and justifies setting ##Δt## to zero. This circumstance can be met by using Equation (2). The resulting equation is ##Δ\acute{x}=\gammaΔx## and it is then rearranged to ##Δx=Δ\acute{x}/\gamma##. Since ##\gamma > 1## it suggests a length contraction.

For time dilation the object in the moving frame needs to be stationary, while applying a transform of the starting and ending times in the stationary frame. In this case ##Δ\acute{x}## has to be set to zero, and this can be met by Equation (3) and it leads to ##Δt=\gammaΔ\acute{t}##. While this reasoning seams to be straightforward there is a huge inconsistency. To obtain a length contraction, Equation (2) is utilized and the final result is rearranged, contrary to the time dilation derived from Equation (3). Somehow the symmetry of the Lorentz and inverse Lorentz transformations is violated, because Equation (2) belongs to the first set (the Lorentz transformations), and Equation (3) belongs to the second set (the inverse Lorentz transformations).

To exemplify this ambiguity, Equations (1-4) can be reduced to

##Δ\acute{t}=\gammaΔt## (5)

##Δ\acute{x}=\gammaΔx## (6)

and the inverse transformations,##Δt=\gammaΔ\acute{t}## (7)

##Δx=\gammaΔ\acute{x}## (8).

What should one pick: (5) and (6) or (7) and (8)?