= Sagnac effect
{c}
{title2=$\Delta t\simeq4\boldsymbol\Omega\cdot\mathbf A/c^2$}
{wiki}
The <Sagnac effect> is a difference in return times, or an associated <interference fringe> displacement, for <light> sent in opposite directions around a closed optical path in a rotating apparatus. The <light> leaves one <beam splitter>, travels around the same loop in opposite senses, and returns to the rotating <beam splitter> and readout. For a slowly rotating apparatus in flat <spacetime>, the leading laboratory arrival-time difference is
$$
\Delta t=t_+-t_-\simeq\frac{4\boldsymbol\Omega\cdot\mathbf A}{c^2},\qquad
\mathbf A=\frac12\oint_C\mathbf r\times d\mathbf r.
$$
Here $c$ is the vacuum <speed of light>, $\boldsymbol\Omega$ is the apparatus's <angular velocity> relative to an <inertial frame>, and $\mathbf A$ is the <vector area> oriented by the chosen positive direction around the <closed curve> $C$. With positive <rotation> about the upward <normal vector>, the counterclockwise beam is labelled $+$ and arrives later. Reversing <rotation>, or reversing the orientation convention alone, changes the sign of $\Delta t$.
The surprise is that a small loop can reveal its <rotation> by comparing two beams locally, even though each local vacuum <speed of light> measurement gives $c$. A <rotating reference frame> is not a single <inertial frame>: a beam must catch the moving receiver after a full circuit. The <Sagnac area law> makes this difference a practical way to measure <angular velocity>, while the <Relativistic Sagnac delay on a circular ring> distinguishes laboratory coordinate time from the receiver's <proper time>.
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