Relativistic Sagnac Arrival Times

Worked Example

Let the lab radius be R = 1 meter. In the lab frame, light always travels at exactly c = 299,792,458 m/s in vacuum, regardless of the source's motion. Suppose a detector is moving along the circular path with a tangential speed of v = βc, where β = 0.1 (so v is 10% the speed of light).

Two light pulses are emitted simultaneously from the detector in opposite directions along the circle. In the lab frame, the pulses must travel a distance of 2πR, plus or minus the distance the detector has moved during the transit time. For the pulse moving in the same direction as the detector (the "positive" direction), the detector is running away, so the lab transit time T₊ is longer:

c * T_+ = 2πR + v * T_+ T_+ = 2πR / (c - v) = 2πR / (c(1 - β)) = 2π(1) / (299792458 * (1 - 0.1)) ≈ 2.3287166910574244e-08 seconds

For the counter-propagating pulse, the detector is running towards it:

c * T_- = 2πR - v * T_- T_- = 2πR / (c + v) = 2πR / (c(1 + β)) = 2π(1) / (299792458 * (1 + 0.1)) ≈ 1.9053136563197106e-08 seconds

The arrival time difference in the lab frame is:

ΔT_lab = T_+ - T_- = 2.3287166910574244e-08 - 1.9053136563197106e-08 = 4.234030347377134e-09 seconds

Because the detector is moving, its own proper clock runs slow relative to the lab clocks by a factor of γ = 1/√(1 - β²). Therefore, the arrival time difference as measured by the detector's proper clock is:

Δτ = ΔT_lab / γ = ΔT_lab * √(1 - β^2) = 4.234030347377134e-09 * √(1 - 0.01) ≈ 4.212807003968427e-09 seconds

For comparison, the classical low-speed limit is 4πRβ / c (which equates to 4AΩ/c² where A is the enclosed area and Ω is the angular velocity), giving 4.191690043903364e-09 seconds.

Experiment

You can interactively explore the Sagnac arrival times with the Relativistic Sagnac Arrival Times interactive module.

1. Change the Lab radius meters to see how the total path length affects the transit times of the light pulses.
2. Adjust the Speed fraction beta. Notice that for positive β, the positive direction lab time is larger, meaning that pulse arrives later. For negative β, the order of arrivals is reversed.
3. Observe that when β = 0, the detector is stationary in the lab frame. The pulses arrive simultaneously, resulting in a zero arrival time difference.

Limitations

This model prescribes an ideal, fixed circular light path of constant radius R in a flat Minkowski spacetime (the lab frame). It does not represent a full relativistic model of an elastic rotating ring or disk, which would involve complex considerations of stresses, strains, and Ehrenfest's paradox. The light pulses are assumed to propagate in a perfect vacuum. We also assume the detector's speed v = βc can be maintained without explicitly analyzing the forces required to sustain that rotational motion.

References

The primary reference for the exact equations and derivation of the relativistic Sagnac effect in an ideal circular path.

• Tartaglia, A. (2004). Sagnac Effect and Pure Geometry. https://arxiv.org/abs/gr-qc/0305084