Fizeau Relativistic Light Drag

Worked Example

Consider light passing through water (refractive index n = 1.5) moving in the same direction at v = 0.05c.

1. Calculate rest medium speed over c: 1/n = 1/1.5 ≈ 0.6666666666666666.
2. Apply relativistic velocity addition: exact light speed u/c = (1/n + v) / (1 + v/n) = (1/1.5 + 0.05) / (1 + 0.05/1.5) = (0.6666666666666666 + 0.05) / (1 + 0.03333333333333333) = 0.7166666666666667 / 1.0333333333333334 ≈ 0.6935483870967741.
3. Find Fresnel drag coefficient: 1 - 1/n² = 1 - 1/2.25 ≈ 0.5555555555555556.
4. Calculate first-order speed: 1/n + v(1 - 1/n²) = 0.6666666666666666 + 0.05 * 0.5555555555555556 = 0.6666666666666666 + 0.027777777777777776 = 0.6944444444444444.
5. The difference between exact speed and first-order speed is the signed approximation error: 0.6944444444444444 - 0.6935483870967741 = 0.0008960573476702871.

Experiment

Try the Fizeau Relativistic Light Drag module to verify the approximation error for different velocities.

1. Select the Refractive index to 1.5.
2. Change the Medium speed over c to 0.2 to see a larger divergence between exact addition and the first-order approximation.
3. Observe how the Signed approximation error increases for higher relative velocities.

Limitations

This model omits dispersion, which would modify the light drag in a physical medium. It also does not calculate Fresnel interfaces or the specific interferometer path timings used in actual experiments. Larger values of beta are used to expose the breakdown of the first-order approximation, rather than representing achievable water flow speeds.

References

Fitzpatrick, R. "The Fizeau Experiment", University of Texas at Austin. This resource discusses the exact relativistic derivation and its relation to the Fresnel drag coefficient.