= Relativistic Sagnac delay on a circular ring
{c}
{title2=$\Delta\tau=4\Omega A/(c^2\sqrt{1-\Omega^2R^2/c^2})$}
Consider an ideal circular vacuum path of laboratory radius $R$, a limit of closely spaced <mirrors> constraining <light> around the circle at laboratory <speed> $c$. The source and detector rotate uniformly together with signed <angular velocity> $\Omega$, where $|\Omega|R<c$, in flat <spacetime>. Let $L=2\pi R$ be the laboratory circumference and $A=\pi R^2$ the positive <area>. The $+$ pulse travels counterclockwise and the $-$ pulse clockwise, independently of the sign of $\Omega$. Both depart from the detector at $t=0$.
Unwrap the circle into a laboratory arc coordinate. The two first-return conditions are
$$
ct_+=L+\Omega Rt_+,\qquad ct_-=L-\Omega Rt_-.
$$
Consequently the exact laboratory arrival times and their difference are
$$
t_+=\frac{L}{c-\Omega R},\qquad
t_-=\frac{L}{c+\Omega R},\qquad
\Delta t=\frac{4\Omega A}{c^2-\Omega^2R^2}.
$$
The denominators $c\mp\Omega R$ are closing rates in these laboratory equations; they are not local <speed of light> measurements by a rotating observer. The two arrival <spacetime events> lie on the same detector <world line>. Its constant <speed> is $|\Omega|R$, so <proper time> along that <world line> gives
$$
\Delta\tau=\sqrt{1-\frac{\Omega^2R^2}{c^2}}\,\Delta t
=\frac{4\Omega A}{c^2\sqrt{1-\Omega^2R^2/c^2}}.
$$
Thus laboratory coordinate-time and rotating-detector <proper time> delays agree only to leading order at small $|\Omega|R/c$. The radius and circumference here are laboratory quantities, not lengths obtained by laying comoving rods around the rim. Finite mirror polygons, transparent media, and <optical dispersion> require their own path and propagation assumptions before using an exact formula.
An exact dimensionless check takes $\Omega R/c=1/5$. Relative to $L/c$, the laboratory arrival times are $5/4$ and $5/6$, giving $\Delta t/(L/c)=5/12$. The detector instead records $\Delta\tau/(L/c)=\sqrt6/6\simeq0.408248$. The leading <Sagnac area law> would give $\Delta t/(L/c)\simeq2/5$. These are model values chosen to make the relativistic distinction visible, not attainable parameters of the historical apparatus.
The rotating loop does not constitute a universal measurement of the <one-way speed of light>. It compares completed circuits at a single readout; assigning a <velocity> to each remote leg additionally requires a clock synchronization convention. Local <Einstein synchronisation> around a rotating rim does not close into a single globally synchronized clock network: the accumulated mismatch is another expression of the rotating-path asymmetry. Local comoving vacuum measurements still give $c$ in both directions. <Special relativity> predicts a nonzero closed-loop delay because the receiver moves during propagation, while no single comoving <inertial frame> covers the whole circuit.
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