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The Sagnac Effect

David Morgan-Marr has embarked on a fun project of writing up the various proofs we have that the Earth is a sphere. One of these is the Sagnac effect, which he introduces with a simple explanation that plainly depends on the angular velocity of the apparatus; yet, later, he claims that the effect depends on the rotational velocity and clearly (given how he uses it) means this is the linear velocity along the path of the rotation, rather than the angular velocity. This left me baffled, so I set about reading up on the matter, starting with his links to sources. The first two of these deal with an experiment done at Clearing, Illinois (now subsumed into Chicago) by Michelson. This definitely fits with the effect being proportional to angular velocity, not the linear speed of movement induced by it (and Michelson's Δ is a phase change, with λ as wave-length, so that his λ.Δ is the c.Δt difference in path-lengths below). Unfortunately, (surprise!) his third source was behind a pay-wall. So I searched for papers about it on arxiv.org and found these:

• c.Δt ≈ 4.A·w/c, in which the A·w is surely a scalar triple-product of the two edge-displacements of the square with the angular velocity vector, surely divided by radian. This paper notes some controversies about the effect. It also gives a general-relativistic formula (which they then go on to critique and refine, since the light's paths in the two routes round the system aren't quite the same): c.Δt = −2.∮dxⁱg_i 0/g_0 0, integrating round the 3-space path traversed by the light (and, I suspect, assuming a time-constant metric g); the difference in proper time between the two paths then scales this Δt by the square root of the magnitude of g_0 0 measured at the detector. Section 3 shows that distance from the centre of rotation is relevant to the relativistic corrections; but not to the leading order term, which only depends on the angular speed of rotation, albeit analysing a Mach-Zehnder apparatus, which rejoins Sagnac's beams half-way round the loop, rather than coming full circle.
• c.Δt ≈ 2.L.v/c where L is the length of the light's path and v is the speed of the light-emitter and observer with respect to the rest-frame of the loop the light traverses (IIUC). No rotation is required, although the practical matter of ensuring the emitter and observer (assumed comoving) remain on the light's trajectory is nicely addressed by making the emitter and observer traverse that trajectory. None the less, if the trajectory has a straight segment, into which the light is emitted and from which it is retrieved after one cycle, the observer and emitter can measure the difference while making a single pass along that straight segment. Observer advances h at speed v while backward light travels L−h, so L−h = h.c/v, and h = L/(c/v +1); observer advances k at speed v while forward light travels L+k, so k = L/(c/v −1); the difference is c.Δt = k−h = 2.L.c/v/(c.c/v/v −1) = 2.L.v/c/(1 −v.v/c/c). For v/c small, this is nice and close to 2.L.v/c; for larger v/c, we'll get a factor of √(1 −v.v/c/c) from either the observer's and emitter's proper time being slowed relative to that of the loop, half cancelling the denominator.

The first is doing general relativity and fine-tuning the higher order terms; the second is looking at it all as geometry, without worrying about anything beyond the principal order. Ho hum, so how about I work this out for myself. Michelson's rectangle was 2010' by 1113', i.e. 612.648 metres (east to west) by 339.2424 metres (north to south), at latitude 41°46' N (0.729 radians or 0.116 turns north of the equator), which has sine 0.6661 and cosine 0.74586; so the 2010' east-west span is 0.4427 minutes of arc (variation in longitude) and the 1113' north-south span is 0.1828 arc minutes (variation in lattitude). Our predicted c.Δt is of order 33.582 nm, 17.64e−12 of the circuit the light traverses, so we need to keep track of errors small compared to the variation in longitude and latitude, including the difference between distances along Earth's surface and along straight lines; but not of general-relativistic time-dilation due to altitude differences (1 part in 1e16 per metre), as long as they're only of order a few metres. (This last would get strained by a similar-sized loop built on the (North or South) face of a large building, e.g. to measure the effect near the equator.)

Simple geometry

So let's have a longitude-latitude rectangle with longitude range 2.r and latitude ranging from N to N+2.n; N may be anywhere from left turn to right turn, but n and r may be assumed relatively small; chose the edge nearer the equator as the one with latitude N, so that n and N have the same sign and Sin(N+2.n) > Sin(N) but Cos(N+2.n) < Cos(N). Assume the corners lie on a surface of constant distance R from Earth's centre (implying small variations in effective gravitational potential, arising from the difference in distance from the spin axis). Send the light in at the west (i.e. back) corner at latitude N (i.e. on the longer of the east-west edges, nearer the equator). My illustration treats the sign of N and n as upwards with east to the right; as to the chirality of the resulting system of co-ordinates, note that you can view this illustration as depicting the scene from below or from above. Let w be the angular speed of Earth's spin.

Connect these corners with straight pipes (rather than following Earth's surface) full of vacuuum, with suitable mirrors and windows in relevant corners. The north-south sides have length 2.R.Sin(n); the east-west ones have lengths 2.R.Sin(r) times Cos(N) and times Cos(N+2.n). Note that Cos(N) +Cos(N+2.n) = 2.Cos(N +n).Cos(n) by combining the sum and difference formulae for Cos(N +n ±n); and Cos(n) is close to 1, while Sin(n) and Sin(r) are, respectively, close to n/radian and r/radian. Our path's corners won't be exact right angles, as the east-west edges aren't quite equal; we'll tweak our mirrors to be (interior for the half-silvered one, the rest exterior) bisectors of the quadrilateral's corners. The circumference of our loop is then 4.R.Sin(n) +4.R.Cos(n).Cos(N +n).Sin(r). Dividing that by c gives us a rough approximation to the transit time for light traversing our loop, albeit we expect to deviate slightly above and below that due to Earth's rotation. The area enclosed by our loop is 2.R.R.Cos(N +n).Sin(2.n).Sin(r); the inner product of its normal (nominally placed at its centre, so latitude N+n) with Earth's spin will throw in a factor of w.Sin(N +n). to make w.R.R.Sin(2.N +2.n).Sin(2.n).Sin(r)

Measure time from the moment that light entering the apparatus hits and passes through the half-silvered mirror for the first time; take the point on Earth's spin axis closest to this mirror as spatial origin of co-ordinates, with the Earth's spin axis as z-axis and a y-axis pointing, at t = 0, at the mirror; have the x axis point in the direction of the mirror's instantaneous movement at that moment. Our mirrors, starting with the half-silvered one and proceeding around clockwise, follow [t, x, y, z] trajectories

• [t, R.Cos(N).Sin(w.t), R.Cos(N).Cos(w.t), R.Sin(N)]
• [t, R.Cos(N+2.n).Sin(w.t), R.Cos(N+2.n).Cos(w.t), R.Sin(N+2.n)]
• [t, R.Cos(N+2.n).Sin(w.t+2.r), R.Cos(N+2.n).Cos(w.t+2.r), R.Sin(N+2.n)]
• [t, R.Cos(N).Sin(w.t+2.r), R.Cos(N).Cos(w.t+2.r), R.Sin(N)]

in this system of co-ordinates (which isn't quite locally inertial, it's formally (in the region of interest) accelerating outwards at the rate of Earth's gravitational field; but my rough estimate is that the consequences of this are small compared to the Sagnac effect).

Written by Eddy.