Remarks on Evans' paper #100 - Section 7.

G.W. Bruhn, Darmstadt University of Technology

29.01.2008

(Quotations from Evans' papers are displayed in black.)

With the introduction to his paper #100 [1.1]. Evans announces a review of major themes of his ECE theory. Evans Section 7 [1.7] deals with the Sagnac effect. In contrast to Evans' opinion that effect is best understood for a long time in the scope of Relativity. The reader is kindly requested to have a look at the literature [12],[13],[14].

Sketch of the calculation

We consider a light fibre placed in the x,y-plane along a circle of radius R around the centre.

Let (t,x,y,z) be the coordinates of a Minkowskian inertial-frame with its origin centered at the axis of rotation of the fibre-device with rotation around the z-axis. Then the metric of the frame is given by:

ds² = dx² + dy² + dz² − c² dt² .

We introduce cylindrical coordinates (r,θ) instead of (x,y) by x = r cos θ, y = r sin θ, and restrict our consideration to the submanifold r=R, z=0 (given by the light fibre) to obtain the metric of the submanifold:

ds² = c²dt² − R²dθ² .

Relative to this coordinate frame S the points of the fibre are travelling with the (constant) velocity

v = R dθ/dt = R ω .

We introduce a new coordinate frame S'(t',θ') where the points of the fibre are at rest:

θ' = γ(θ − ωt) , t' = γ(t − ^R²ω/_c² θ) where γ = (1 − ^R²ω²/_c²)^−½

to obtain the corresponding metric by transforming ds². This yields

ds² = c²dt'² − R²dθ'² .

The invariance of the line element yields that the circular travelling θ'(t') = + ct'/R of the two light beams travelling along x² + y² = R² with velocity c is seen from S' as two light beams travelling along x'² + y'² = R² with the same velocities c as well in clockwise or counter-clockwise sense respectively. However, the distances to be covered until the beams meet the world-lines of their initial point are different:

Let L' denote the distance measured in S' once around the circular light fibre. Then the time T_-' of the clockwise running signal to return to its initial point is given by

L' = c T_-' + v T_-' , hence T_-' = ^L' / _c+v ,

while the counter-clockwise signal needs the time T₊' given by

L' = c T₊' − v T₊' , hence T₊' = ^L' / _c−v .

The time difference is (with v=Rω, L'=2πR)

Δt = ^2πR / _c−v − ^2πR / _c+v = 2πR (¹ / _c−v − ¹ / _c+v) = ^4πRv / _c²−v² .

The result of that measurement in S' is transmitted to the non-rotating inertial system S and gives information about the angular velocity ω of S' relative to S.

That's the whole story in the simple case of a rotating circular light fibre. The reader will find more general configurations considered in the literature, e.g. in [12], [13], [14].

References

[1.1] M.W. Evans, A Review of Einstein-Cartan-Evans (ECE) Field Theory (Introduction of Paper #100),
http://www.atomicprecision.com/blog/2007/12/27/introduction-to-paper-100/wp-filez/a100thpaperintroduction.pdf .

[1.2] M.W. Evans, Geometrical Principles (Section 2 of Paper #100:
A Review of Einstein-Cartan-Evans (ECE) Field Theory,
http://www.atomicprecision.com/blog/wp-filez/a100thpapersection2.pdf .

[1.3] M.W. Evans, The Field (Section 3 of Paper #100:
A Review of Einstein-Cartan-Evans (ECE) Field Theory,
http://www.atomicprecision.com/blog/wp-filez/a100thpapersection3.pdf .

[1.4] M.W. Evans, Aharonov Bohm and Phase Effects in ECE Theory (Section 4 of Paper #100:
A Review of Einstein-Cartan-Evans (ECE) Field Theory,
http://www.atomicprecision.com/blog/wp-filez/a100thpapersection4.pdf .

[1.5] M.W. Evans, Tensor and Vector Laws of Classical Dynamics and Electrodynamics (Section 5 of Paper #100) ,
http://www.atomicprecision.com/blog/wp-filez/a100thpapersection5.pdf .

[1.6] M.W. Evans, Spin Connection Resonance (Section 6 of Paper #100) ,
http://www.atomicprecision.com/blog/wp-filez/a100thpapersection6.pdf .

[1a] M.W. Evans, Development of the Einstein Hilbert Field Equation . . .,
http://www.aias.us/documents/uft/a103rdpaper.pdf .

[1b] M.W. Evans, Proof of the Hodge Dual Relation,
http://www.atomicprecision.com/blog/wp-filez/a100thpapernotes16.pdf .

[1c] M.W. Evans, Some Proofs of the Lemma,
http://www.atomicprecision.com/blog/wp-filez/acheckpriortocoding5.pdf .

[1d] M.W. Evans, Geodesics and the Aharonov Bohm Effects in ECE Theory,
http://www.aias.us/documents/uft/a56thpaper.pdf .

[2] S.M. Carroll, Lecture Notes on General Relativity,
http://xxx.lanl.gov/PS_cache/gr-qc/pdf/9712/9712019v1.pdf, 1997.

[3] S.M. Carroll, Spacetime and Geometry,
http://xxx.lanl.gov/PS_cache/gr-qc/pdf/9712/9712019v1.pdf, 1997.

[4] F.W. Hehl and Y.N. Obukhov, Foundations of Classical Electrodynamics, Birkhäuser 2003

[5] G.W. Bruhn, Consequences of Evans' Torsion Hypothesis,
ECEcontradictions.html .

[6] G.W. Bruhn, Remarks on Evans' paper #100 - Section 2,
onMwesPaper100-2.html .

[7] M.R. Spiegel, Vector Analysis,
in Schaum's Outline Series, McGraw-Hill.

[8] G.W. Bruhn, Evans' "3-index Î-tensor" ,
Evans3indEtensor.html .

[9] G.W. Bruhn, Comments on Evans' Duality,
EvansDuality.html .

[10] G.W. Bruhn, F.W. Hehl, A. Jadczyk , Comments on ``Spin Connection Resonance
in Gravitational General Relativity'' , ACTA PHYSICA POLONICA B Vol. 39/1 (2008)
pdf . html

[11] G.W. Bruhn, Remarks on Evans/Eckardt’sWeb-Note on Coulomb Resonance,,
RemarkEvans61.html .

[12] WIKIPEDIA, Sagnac effect / Calculations
http://en.wikipedia.org/wiki/Sagnac_effect#Calculations

[13] N.N. , Reflections on Relativity, Sect. 7.2: The Sagnac Effect,
http://www.mathpages.com/rr/s2-07/2-07.htm

[14] O. Wuckwitz , Sagnac effect, twin paradox and space-time topology ...,
arXiv 2004