Rotating Frames, Sagnac Timing, and Light Cylinders

A rotating ring cannot carry a globally Einstein-synchronized clock network because local simultaneity fails to close around the loop. The same obstruction appears experimentally as unequal return times for oppositely directed light pulses—the Sagnac effect. Relativity also imposes a local admissibility condition: a point fixed in uniformly rotating coordinates represents an observer only inside the light cylinder.

Required background. Special Relativity provides Minkowski causal geometry, four-vectors, and mass-shell kinematics; Relativistic Observers and Measurement provides observer rest spaces and invariant decompositions of measured quantities. Helpful background. Calculus and Linear Algebra provides multivariable differentiation, linear maps, inner products, and basis changes; Ordinary Differential Equations provides initial-value problems, linearization, and asymptotic solution methods.

Rotating coordinate lines and physical observers

Start from inertial cylindrical coordinates:

$$ds^2=-dT^2+dR^2+R^2d\Phi^2+dz^2.$$

Define rotating coordinates by

$$t=T,\qquad r=R,\qquad \phi=\Phi-\Omega T.$$

Since $d\Phi=d\phi+\Omega dt$,

$$\boxed{ds^2=-(1-\Omega^2r^2)dt^2 +2\Omega r^2dt\,d\phi+dr^2+r^2d\phi^2+dz^2.}$$

The cross term is essential: time is not orthogonal to the rotating spatial coordinate lines.

A fixed $(r,\phi,z)$ curve has tangent $\partial_t$ and norm

$$g_{tt}=-(1-\Omega^2r^2).$$

It is timelike only when

$$\boxed{|\Omega|r\lt1.}$$

At $r_{\rm L}=1/|\Omega|$, called the light cylinder, the curve is null; beyond it, a fixed rotating position would require superluminal tangential speed. The coordinate transformation itself remains invertible there: its metric determinant does not vanish. What fails is the interpretation of the fixed-coordinate curves as a physical observer congruence. The corresponding rigidly rotating map, metric, and limiting radius are given in Pauri and Vallisneri 2000, §4, printed pp. 7–8, Eqs. (13)–(15).

Inside the cylinder, a rotating observer has

$$u=\gamma\partial_t,\qquad \gamma=(1-\Omega^2r^2)^{-1/2},$$

where the normalization uses coordinate time $t=T$. Its accelerometer reads the inward magnitude

$$\alpha=\gamma^2\Omega^2r.$$

Thus “at rest” in the rotating chart does not mean force-free.

Sagnac timing on a ring

Restrict light to an ideal evacuated circular guide at $r=R$ with $|\Omega|R\lt1$, neglecting reflection delays. The guide redirects the rays: a circular light path is not a freely propagating null geodesic in flat spacetime. To label the two directions without ambiguity, take $\Omega\gt0$ and let increasing $\phi$ point in the same sense as the apparatus’s rotation. Setting $ds^2=dr=dz=0$ gives

$$0=-(1-\Omega^2R^2) +2\Omega R^2\frac{d\phi}{dt} +R^2\left(\frac{d\phi}{dt}\right)^2.$$

The two angular coordinate velocities are

$$\frac{d\phi}{dt}=-\Omega\pm\frac1R.$$

The pulse moving in the positive inertial angular direction has $d\phi/dt=1/R-\Omega$ and must gain $2\pi$ on the emitter, so its return time is $t_+=2\pi R/(1-\Omega R)$. The oppositely directed pulse has return time $t_-=2\pi R/(1+\Omega R)$. Hence

$$\Delta t=t_+-t_- =\frac{4\pi\Omega R^2}{1-\Omega^2R^2}.$$

The emitting clock has $d\tau=\sqrt{1-\Omega^2R^2}\,dt$, so it measures

$$\boxed{\Delta\tau =\frac{4\pi\Omega R^2}{\sqrt{1-\Omega^2R^2}}.}$$

The sign reverses with the chosen orientation or with the sense of rotation; the magnitude is the observable delay. With the $\pm$ labels tied to fixed angular directions, reversing $\Omega$ interchanges which pulse returns first. The relation between Sagnac delay and the synchronization gap is given in Gourgoulhon 2010, PDF slides 37–40.

Why synchronization does not close

Einstein synchronization is local: neighboring comoving clocks choose events simultaneous in the instantaneous rest space of the rotating observer. Along the ring that condition is $u_\mu dx^\mu=0$. Since $u_\mu$ is proportional to $g_{t\mu}$,

$$g_{tt}\,dt+g_{t\phi}\,d\phi=0 \quad\Longrightarrow\quad dt=\frac{\Omega R^2}{1-\Omega^2R^2}\,d\phi.$$

Integrating once around gives the nonzero gap

$$\Delta t_{\rm sync} =\frac{2\pi\Omega R^2}{1-\Omega^2R^2}.$$

The starting clock converts this coordinate-time mismatch to

$$\Delta\tau_{\rm sync} =\frac{2\pi\Omega R^2}{\sqrt{1-\Omega^2R^2}} =\frac12\Delta\tau.$$

Returning to the starting clock therefore assigns it a different time. One may adopt a single-valued rotating time coordinate, or enforce local Einstein synchronization on successive pairs, but not both globally around the ring. The factor-of-two relation to the two-beam delay is displayed in Gourgoulhon 2010, PDF slides 38–39.

Limits and common failures

• For $|\Omega|R\ll1$, $\Delta\tau\simeq4\Omega A$, where $A=\pi R^2$. Restoring units gives $4\Omega A/c^2$, and the exact admissibility condition is $|\Omega|R\lt c$.
• The delay compares two beams on the same apparatus and does not depend on an arbitrary remote synchronization convention.
• The light cylinder is not a material wall, a curvature singularity, or a black-hole event horizon. It is the boundary where the chosen helical coordinate flow becomes null.
• The rotating chart is a coordinate system; the rotating ring is a physical congruence with stresses and an admissible radial domain. Conflating them hides the superluminal-worldline problem.
• A globally rigid disk cannot be spun up from rest while preserving all proper distances. Uniformly rotating motion can be studied once established, but its preparation is a separate dynamical question.

Transfer task: recover the area law

Expand the exact proper-time delay through the first relativistic correction.

Solution

With $v=\Omega R$,

$$\Delta\tau =4\Omega A(1-v^2)^{-1/2} =4\Omega A\left(1+\frac12v^2+O(v^4)\right).$$

The leading term depends only on the oriented area for a circular loop. The correction combines two effects: the laboratory return-time difference contributes a factor $(1-v^2)^{-1}=1+v^2+\cdots$, while conversion to the detector’s proper time contributes $\sqrt{1-v^2}=1-v^2/2+\cdots$. Their product gives the net $+v^2/2$ correction; time dilation alone would have the opposite sign. Both displayed terms reverse sign under $\Omega\to-\Omega$. This series is valid only for $|v|\ll1$; the divergence as $|v|\to1$ belongs to the exact expression and cannot be inferred from a finite truncation.

Lapse, Shift, and Measurements in Noninertial Coordinates rewrites this rotating metric so that the cross term, observer choice, and light-cylinder condition become parts of one general decomposition.

References

• Éric Gourgoulhon, Special relativity from an accelerated observer perspective (PDF), printed slides 35–40, Observatoire de Paris, 2010.
• M. Pauri and M. Vallisneri, “Märzke–Wheeler Coordinates for Accelerated Observers in Special Relativity,” Foundations of Physics Letters 13(5), 401–425, 2000, doi:10.1023/A:1007861914639; Open PDF, §4, printed pp. 7–8, Eqs. (13)–(15).

Further reading

• Éric Gourgoulhon, Special Relativity in General Frames: From Particles to Astrophysics, Chapter 13, Graduate Texts in Physics, Springer, 2013; consult the author’s errata.