On Tue, 21 Feb 2006 22:58:16 +0100, "Paul B. Andersen"
<paul.b.andersen@hiadeletethis.no> wrote:

>
> The prediction of the ballistic theory
> for the four-mirror Sagnac ring.
> ----------------------------------
>
> I will assume the reader knows what a four mirror
> Sagnac ring is.
>
> Let the transit time for the light to go from the centre
> of one mirror to the centre of the next be t.
>
> Let the distance from the centre of the ring to
> the centre of each mirror be r.
>
> Let the peripheral speed of the centre of the mirror be v.
>
> We will calculate the transit time t.
> The calculations will be done from the inertial
> frame of reference in which the centre of the ring
> is stationary.
>
> The second mirror will, when it is hit by the light pulse,
> be in the position theta = pi/2 + v*t/r
>
> The length of the light path will be the chord:
>    d = 2*r*sin(pi/4 + v*t/(2*r))
>
> Measured in the stationary frame, the light will
> have to be emitted at an angle theta/2  referred
> to the mirror frame.
>
> The speed of the light between the mirrors will
> according to the ballistic theory then be:
>    c' = sqrt(c^2 - v^2*sin^2(pi/4 + v*t/(2*r))) + v cos(pi/4 + v*t/(2*r))
>
> The transit time can then be found by solving the equation
>    c'*t = d
>
> These equations are exact, and are valid for both
> beams, v being positive for the beam going with
> the rotation, negative for the beam going in the opposite
> direction.
>
> It is not a trivial task to solve this equation exact,
> so let us make some approximations.
>
> We have the Taylor series:
>    sin(pi/4 + x) = (1 + x - x^2/2 - x^3/6 + x^4/20 ..)/sqrt(2)
>    cos(pi/4 + x) = (1 - x - x^2/2 + x^3/6 + x^4/20 ..)/sqrt(2)
>    sin^2(pi/4 + x) = 0.5 + x - 2x^3/3 + ..
>    sqrt(1 + x) = 1 + x/2 - x^2/8 + x^3/16 ..
>
> First order approximations:
> ---------------------------
> Using these equation give the following first order
> approximations (that is first order in v/c and vt/r)
> for c' and d:
>
>    d ~= sqrt(2)*r*(1 + v*t/(2*r))
>    c'~= c + v/sqrt(2)
>
> Solving the equation c'*t = d then yields:
>    t = sqrt(2)*r/c
>
> The transit time has no first order dependency of the speed.
> That means that the difference in the time for the two beams
> to go around the ring will to a first order approximation be:
>   delta_t = 4 (t(v) - t(-v)) ~= 0
>
> Second order approximations:
> ----------------------------
> A second order Taylor expansion of the equations:
>    d = 2*r*sin(pi/4 + v*t/2*r)
>    c' = sqrt(c^2 - v^2*sin^2(pi/4 + v*t/(2*r))) + v cos(pi/4 + v*t/(2*r))
> yields:
>    d ~= sqrt(2)*r + v*t/sqrt(2) - v^2*t^2/(4*sqrt(2)*r)
>    c'~= c + v/sqrt(2) - v^2/(4*c) - v^2*t/(2*sqrt(2)*r)
>
> The equation c'*t = d will then be:
>    (v^2/(4*sqrt(2)*r))*t^2 - (c - v^2/(4*c))*t + sqrt(2)*r = 0
>
> Solving this, ignoring fourth order terms and higher, yields:
>    t = sqrt(2)*r/c + r*v^2/(sqrt(2)*c^3)
>
> Note that this is actually a third order approximation,
>    ((r/c)*(v^2/c^2)). A second order term would be ((r/c)*(v/c)).
>
> Since there is no term with (v/c), the time for both beams
> will have the same dependency on the square of the speed.
> That means that the difference in the time for the two beams
> to go around the ring will to a third order approximation
> (second order in v) be:
>    delta_t = 4*(t(v) - t(-v)) ~= 0
>
> The reason for why the time for the beam to go around
> the ring in either direction will increase slightly
> with the speed can be understood if we observe the beams
> in the frame of reference rotating with the ring.
> In this frame, the speed of light will be c, and
> the distance between the mirrors will be sqrt(2)*r.
> (That's why the first order approximation is t = sqrt(2)*r/c)
> However, observed in this frame the path of the beam
> going with the rotation will be slightly concave
> while the path of the beam going in the opposite
> direction will be slightly convex. To a third
> order approximation, both path lengths will increase
> equally much.
>
> We can thus conclude that the ballistic theory
> predicts that the time difference for the beams
> to go around the ring has neither a first nor
> a second order dependency on the angular velocity.
>
> Sagnac falsifies the ballistic theory.
>
> Numeric analysis.
> -----------------
>
> As mentioned above, it is not easy to solve the equation
>    c'(v,t)*t = d(v,t) exact.
> It can however be solved numerically.
> I have done so, and the results are shown below.
>
> The predictions of SR are calculated as well by solving
> the equation c*t = d(v,t).
>
> The results are compared to the experimentally verified
> Sagnac equation delta_t = 4Aw/(c^2 - v^2)
>
> The radius r = 1m.
>
> The precision of the calculation is in the order of
> 10^-25 seconds. Numbers smaller than this are set to zero.
>
> "error" below is the deviation from the 'correct' Sagnac value
> divided by the Sagnac value. error = 1 means that no effect is predicted.
>
>          |     ballistic           |        SR                |  Sagnac
>
>   v [m/s]      d_t [s]     error         d_t [s]     error         d_t [s]
>
>  1.0e-01  0.00000000e+00 1.00e+00    8.90120043e-18 1.12e-10    8.90120043e-18
>  1.0e+00  0.00000000e+00 1.00e+00    8.90120043e-17 8.73e-11    8.90120043e-17
>  1.0e+01  0.00000000e+00 1.00e+00    8.90120043e-16 1.67e-12    8.90120043e-16
>  1.0e+02  0.00000000e+00 1.00e+00    8.90120043e-15 8.02e-13    8.90120043e-15
>  1.0e+03  6.46234854e-25 1.00e+00    8.90120043e-14 1.48e-11    8.90120043e-14
>  1.0e+04  6.60241994e-22 1.00e+00    8.90120043e-13 1.48e-09    8.90120044e-13
>  1.0e+05  6.60261538e-19 1.00e+00    8.90120010e-12 1.48e-07    8.90120142e-12
>  1.0e+06  6.60274633e-16 1.00e+00    8.90116742e-11 1.48e-05    8.90129947e-11
>  1.0e+07  6.61585739e-13 9.99e-01    8.89789216e-10 1.48e-03    8.91111538e-10
>
> The predictions of SR are correct with high precision,
> while the predictions of the ballistic theory are way off.
> (The error is equal to the Sagnac value.)
>
> The ballistic theory predicts no Sagnac effect.
> Sagnac falsifies the ballistic theory.

Unfortunately, you omitted all the relevant material.

If you run my program www.users.bigpond.com/hewn/sagnac.exe you will observe
that rays that start out 90 apart do not reunite at the same point on the
detector.
The two beams are displaced sideways by different amounts.

A second omission is an analysis of what actually happens at each reflection.  
A third involves the 'axis' of photons.

You standard 'srian' analysis is exactly that which would apply to a 'sound
sagnac' in still air.
It basically assumes that light moves in a medium.

>
> Paul

HW.
www.users.bigpond.com/hewn/index.htm