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:

                ds = dx + dy + dz − c dt .

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:

                ds = cdt − Rdθ .

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 − /c θ)         where         γ = (1 − /c)−½

to obtain the corresponding metric by transforming ds. This yields

                ds = cdt' − Rdθ' .

The invariance of the line element yields that the circular travelling θ'(t') = + ct'/R of the two light beams travelling along x + y = R with velocity c is seen from S' as two light beams travelling along x' + y' = R 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−v2πR / c+v = 2πR (1 / c−v1 / c+v) = 4πRv / c−v .

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, Birkhuser 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/EckardtsWeb-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


Links

(08.01.2008) An Editorial Note by G. 't Hooft in Found. Phys.

(29.01.2008) Remarks on Evans' Web Note #100-Section 7: The Sagnac Effect

(25.01.2008) Remarks on Evans' Web Note #100-Section 6: SCR

(16.01.2008) Remarks on Evans' Web Note #100-Section 5: EM field

(08.01.2008) Remarks on Evans' Web Note #100-Section 4: The Aharonov Bohm effect

(05.01.2008) Remarks on Evans' Web Note #100-Section 3: Field and Wave equation

(01.01.2008) Remarks on Evans' Web Note #100-Section 2: Torsion and Bianchi identity

(27.12.2007) Remarks on Evans' Web Note #103

(19.12.2007) Myron now completely confused

(14.12.2007) Evans' Central Claim in his Paper #100

(10.12.2007) How Dr. Evans refutes the whole EH Theory




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