Challenges in Verifying Time Dilation Using the Relativistic Doppler Formula

Time dilation is a fundamental concept in Einstein's special theory of relativity, asserting that time intervals perceived by observers in relative motion can differ. A common method to test this is through the Ives–Stilwell type experiment, which utilizes the relativistic Doppler equation. However, recent critiques suggest that the empirical verification of the relativistic Doppler equations in this experiment may not be sufficient to uniquely validate them, as other frequency shift equations could satisfy the same experimental criterion. Let's explore this in detail.

Experimental Design and Relativistic Doppler Equations

The Ives–Stilwell experiment typically involves a beam of ions moving at a relativistic speed, with two laser beams (one from the front and one from the back relative to the ions' direction of travel). In the ion's reference frame, the frequency absorbed (f_o) matches the ion's rest state resonance frequency. The absorbed frequencies in the lab frame, denoted as (f_F) and (f_B), are then measured, leading to the relativistic frequency shifts:

(f_o gamma f_B left(1 - frac{v}{c}right)) (f_{o} gamma f_F left(1 frac{v}{c}right))

From these, we derive:

(f_B gamma f_o left(1 - frac{v}{c}right)^{-1}) (f_F gamma f_{o} left(1 - frac{v}{c}right))

Consequently, the frequency ratio is:

[frac{f_B f_F}{f_o f_{o}} gamma^2 left(1 - frac{v^2}{c^2}right) 1]

Criticism of the Ives–Stilwell Experiment

The criticism lies in the empirical verification that the ratio (frac{f_B f_F}{f_o f_{o}}) equals 1. This alone does not validate the equations leading to that ratio. Importantly, the inverse relation of the relativistic red shift to blue shift factor is noted.

Algebraic manipulations reveal:

[gamma left(1 - frac{v}{c}right) left[ gamma left(1 - frac{v}{c}right) right]^{-1}]

Thus, the shift equations can be written:

(f_R f_R gamma left(1 - frac{v}{c}right)) for the receding frequency, ((f_A f_A left[ gamma left(1 - frac{v}{c}right) right]^{-1}) for the approaching frequency.

This implies that the sign change in the velocity term is not due to the velocity direction but the inversion of the shift factor, which is a hidden mathematical illusion. Therefore, any red and blue shift equations where the shift factor in one is the inverse of the other will satisfy the experimental criterion.

Further Generalizations

The general form of the frequency shift equations ((f_R f_R left(1 - frac{v^n}{c^n}right)^p)) and ((f_A f_A left(1 - frac{v^n}{c^n}right)^{-p})) for different positive numbers (n) and (p) would also be verified by the Ives–Stilwell experiment. This indicates that such an experiment does not offer a unique test for time dilation.

Conclusion

The classical frequency shift equations, while initially reflecting the source velocity direction, are sufficient for practical purposes. However, the common form of the relativistic Doppler shift equations, with a sign change in the velocity term, obscures the real implications. The Ives–Stilwell experiment should therefore not be seen as a definitive test for time dilation.

Keywords: time dilation, relativistic Doppler effect, Ives-Stilwell experiment