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The Sagnac Effect and the Chirality of Space Time
Prof. R. M. Kiehn, Emeritus Physics, Univ. of
Houston www.cartan.pair.com rkiehn2352_at_aol.com D
imdim December 5 2009
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This presentation consists of several parts 1.
Fringes vs. Beats
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This presentation consists of several parts 1.
Fringes vs. Beats 2. The Sagnac effect
and the dual Polarized Ring Laser
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This presentation consists of several parts 1.
Fringes vs. Beats 2. The Sagnac effect
and the dual Polarized Ring Laser 3. The
Chirality of the Cosmos
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(And if there is time a bit of heresy) 4.
Compact domains of Constitutive
properties that lead to non-radiating
Electromagnetic Molecules

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(And if there is time a bit of heresy) 4.
Compact domains of Constitutive
properties that lead to non-radiating
Electromagnetic Molecules
with infinite Radiation Impedance ?!
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(And if there is time a bit of heresy) 4.
Compact domains of Constitutive
properties that lead to non-radiating
Electromagnetic Molecules
with infinite Radiation Impedance ?! Or
why an orbiting electron does not radiate
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1a. Fringes vs. Beats
?1 e i(k1 r - ?1 t) ?2 e i(k2 r - ?2
t)
Superpose two outbound waves k1 ? k2, ?1 ? ?2

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1a. Fringes vs. Beats
?1 e i(k1 r - ?1 t) ?2 e i(k2 r - ?2
t)
Two outbound waves superposed ?k k1 - k2
?? ?1 - ?2
?1 ?2 exp (?kr/2 - ??t/2) ?1
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Fringes vs. Beats
?1 e i(k1 r - ?1 t) ?2 e i(k2 r - ?2
t)
Two outbound waves superposed ?k k1 - k2
?? ?1 - ?2
?1 ?2 2 cos(?kr/2 - ??t/2) ?1
Fringes are measurements of wave vector
variations ?k (t constant, r varies)
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Fringes vs. Beats
?1 e i(k1 r - ?1 t) ?2 e i(k1 r - ?2
t)
Two outbound waves superposed ?k k1 - k2
?? ?1 - ?2
?1 ?2 2 cos(?kr/2 - ??t/2) ?1
Fringes are measurements of wave vector
variations ?k (t constant, r
varies) Beats are measurements of frequency
variations ?? (r constant, t varies)
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Phase vs. Group velocity
Phase Velocity ?/k C/n C Vacuum Speed n
index of refraction
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Phase vs. Group velocity
Phase Velocity ?/k C/n C Lorentz Speed n
index of refraction Group Velocity /?//k
??/?k Phase Velocity C/n ? ??/?k Group
Velocity
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4 Propagation Modes
Outbound Phase ?1 e i(k1
r - ?1 t) ?2 e i(- k2 r ?2 t)
k
k

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4 Propagation Modes
Outbound Phase ?1 e i(k1
r - ?1 t) ?2 e i(- k2 r ?2 t)
k
k

Note opposite orientations of Wave and phase
vectors
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4 Propagation Modes
Outbound Phase ?1 e i(k1
r - ?1 t) ?2 e i(- k2 r ?2 t)
k
k
Inbound Phase ?3 e i(k3 r
?3 t) ?4 e i(- k4 r - ?4 t)
k
k
Note opposite orientations of wave and phase
vectors
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4 Propagation Modes
Mix Outbound phase pairs or Inbound phase pairs
for Fringes and Beats.
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4 Propagation Modes
Mix Outbound phase pairs or Inbound phase pairs
for Fringes and Beats.
Mix Outbound with Inbound phase pairs to produce
Standing Waves.
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4 Propagation Modes
Mix all 4 modes for Phase Entanglement
Each of the phase modes has a 4 component
isotropic spinor representation!
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1b. The Michelson Morley interferometer.
The measurement of Fringes
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Most people with training in Optics know about
the Michelson-Morley interferometer.
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Viewing Fringes.
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The fringes require that the optical paths are
equal to within a coherence length of the
photons. L C decay time 3 meters for Na
light
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Many are not familiar with the use of multiple
path optics (1887).
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1c. The Sagnac interferometer.
With the measurement of fringes (old)
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The Sagnac interferometer encloses a finite
area, The M-M interferometer encloses zero area.
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The Sagnac interferometer responds to
rotation The M-M interferometer does not.
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1d. The Sagnac Ring Laser interferometer.
With the measurement of Beats (modern)
Has any one measured beats in a M M
interferometer ??
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Two beam (CW and CCW linearly polarized) Sagnac
Ring with internal laser light source
Linear Polarized Ring Laser Polarization fixed
by Brewster windows
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4 Polarized beams CWLH, CCWLH, CWRH, CCWRH
Sagnac Ring with internal laser light source
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Ring laser - Early design Brewster windows for
single linear polarization state
Rotation rate of the earth produces a beat signal
of about 2-10 kHz depending on enclosed area.
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More modern design of Ring Laser
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Hogged out Quartz monolithic design
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Ring Laser gyro built from 2-beam Ring lasers on
3 axes
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These aircraft use (or will use) Ring Laser gyros
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These missiles use Ring Laser gyros
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Aerospace devices use ring Laser gyros
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Under water devices use ring Laser gyros
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2. The Sagnac effect and Dual polarized Ring
lasers.
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Dual Polarized Ring Lasers
Non-reciprocal measurements with a Q
1018 Better than Mossbauer
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Dual Polarized Ring Lasers
Non-reciprocal measurements with a Q
1018 Better than Mossbauer
This technology has had little exploitation !!!
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Non-Reciprocal Media.
As this is a meeting of those who like a bit of
heresy, and Optical Engineers, who know that the
speed of light in media can be different for
different states of polarization, let me start
out with the first, little appreciated, heretical
statement
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Non-Reciprocal Media.
As this is a meeting of those who like a bit of
heresy, and Optical Engineers, who know that the
speed of light in media can be different for
different states of polarization, let me start
out with the first, little appreciated, heretical
statement
In Non-Reciprocal media, the Speed of light not
only depends upon polarization, but also depends
upon the direction of propagation.
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Non-reciprocal Media Faraday rotation or
Fresnel-Fizeau
Consider Linearly polarized light passing through
Faraday or Optical Active media
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Non-reciprocal Media Faraday rotation or
Fresnel-Fizeau
Consider Linearly polarized light passing through
Faraday or Optical Active media
Exact Solutions given by E. J. Post 1962
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These concepts stimulated a search for apparatus
which could measure the effects of gravity on the
polarization of an EM wave,
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These concepts stimulated a search for apparatus
which could measure the effects of gravity on the
polarization of an EM wave, and ultimately to
practical applications of a Sagnac dual polarized
ring laser.
Every one should read E. J. Post The Formal
Structure of Electromagnetics North Holland 1962
or Dover 1997
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The Faraday Ratchet can accumulate tiny phase
shifts from multiple to-fro reflections. The
hope was that such a device could capture the
tiny effect of gravity on the polarization of the
PHOTON.
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The Faraday Ratchet can accumulate tiny phase
shifts from multiple to-fro reflections. The
hope was that such a device could capture the
tiny effect of gravity on the polarization of the
PHOTON.
It was soon determined that classical EM theory
would not give an answer to EM - gravity
polarization interactions.
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More modern design of dual polarized Ring Laser
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Technique
Tune to a single mode. If no intra Optical Cavity
effects, then get a single beat frequency due to
Sagnac Rotation.
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Tune to a single mode. If no intra Optical Cavity
effects, then get a single beat frequency due to
Sagnac Rotation.
If A.O. and Faraday effects are combined in the
Optical Cavity, then get 4 beat frequencies.
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Conclusion
The 4 different beams have 4 different phase
velocities, dependent upon polarization and
propagation direction.
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Experiments conducted by V. Sanders and R. M.
Kiehn in 1977, using dual polarized ring lasers
verified that the speed of light can have a 4
different phase velocities depending upon
direction and polarization. The 4-fold Lorentz
degeneracy can be broken. Such solutions to the
Fresnel Maxwell theory, subject to a gauge
constraint, were published first in 1979. After
patents were secured, the full theory of singular
solutions to Maxwells equations without gauge
constraints was released for publication in
Physical Review in 1991. R. M. Kiehn, G. P.
Kiehn, and B. Roberds, Parity and time-reversal
symmetry breaking, singular solutions and Fresnel
surfaces, Phys. Rev A 43, pp. 5165-5671,
1991. Examples of the theory are presented in
the next slides, which shows the exact solution
for the Fresnel Kummer singular wave surface for
combined Optical Activity and Faraday Rotation.
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Generalized Fresnel Analysis of Singular
Solutions to Maxwells Equations (propagating
photons)
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Generalized Fresnel Analysis of Singular
Solutions to Maxwells Equations (propagating
photons)
Theoretical existence of 4-modes of photon
propagation as measured in the dual polarized
Ring Laser.
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The 4 modes correspond to 1. Outbound LH
polarization 2. Outbound RH polarization 3.
Inbound LH polarization 4. Inbound RH
polarization
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Fundamental PDEs of Electromagnetism A review
Maxwell Faraday PDEs
Maxwell Ampere PDEs
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Lorentz Constitutive Equations -- The Lorentz
vacuum
Substitute into PDE,s get vector wave equation
Phase velocity
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EM from a Topological Viewpoint.
USE EXTERIOR DIFFERENTIAL FORMS, NOT
TENSORS Exterior differential forms, A, F and G,
carry topological information. They are not
restricted by tensor diffeomorphisms For any 4D
system of base variables
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EM from a Topological Viewpoint.
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EM from a Topological Viewpoint.
USE EXTERIOR DIFFERENTIAL FORMS, NOT
TENSORS Exterior differential forms, A, F and G,
carry topological information. They are not
restricted by tensor diffeomorphisms F is an
exact and closed 2-Form, A is a 1-form of
Potentials. G is closed but not exact, 2-Form. J
dG, is exact and closed.
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EM from a Topological Viewpoint.
USE EXTERIOR DIFFERENTIAL FORMS, NOT
TENSORS Exterior differential forms, A, F and G,
carry topological information. They are not
restricted by tensor diffeomorphisms/ F is an
exact and closed 2-Form, A is a 1-form of
Potentials. G is closed but not exact, 2-Form. J
dG, is exact and closed. Topological limit
points are determined by exterior
differentiation dF 0 generates Maxwell
Faraday PDEs dG J generates Maxwell Ampere
PDEs For any 4D system of base variables
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EM from a Topological Viewpoint.
dF 0 generates Maxwell Faraday PDEs dG J
generates Maxwell Ampere PDEs A
differential ideal (if J0) for any 4D system of
base variables
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EM from a Topological Viewpoint.
dF 0 generates Maxwell Faraday PDEs dG J
generates Maxwell Ampere PDEs A
differential ideal (if J0) for any 4D system of
base variables
Find a phase function 1-form ? ?kmdxm ?
?dt Such that the intersections of the 1-form, ?,
and the 2-forms vanish
?F 0 ?G 0
Also require that J 0.
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?F 0 ?G 0 In Engineering Format
become
k E - ?B 0, k B 0, k H ?D
0, k D 0,
Six equations in 12 unknowns. !! Need 6 more
equations The Constitutive Equations
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Constitutive Equation examples
Lorentz vacuum is NOT chiral, ? 0
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Constitutive Equation examples
Generalized Complex Constitutive Matrix
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Constitutive Equation examples
Generalized Complex Constitutive Matrix
Generalized Complex Constitutive Equation
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Chiral Constitutive Equation Examples
Generalized Chiral Constitutive Equation ? ? 0
? Gamma is a complex matrix.
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Chiral Constitutive Equation Examples
Diagonal Chiral Constitutive Equation
? Gamma is complex
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Chiral Constitutive Equation Examples
Diagonal Chiral Constitutive Equation
? Gamma is complex The real part of Gamma
represents Fresnel-Fizeau effects. The Imaginary
part of Gamma represents Optical Activity
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Chiral Constitutive Equation Examples
Diagonal Chiral Constitutive Equation
The Wave Speed does not depend upon Fresnel
Fizeau ? Expansions (the real diagonal
part). The Wave Speed depends upon OA
expansions, (the imaginary diagonal part). The
Radiation Impedance depends upon both
expansions.
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Chiral Constitutive Equation Examples
Fresnel-Fizeau ? rotation diagonal Chiral ?
expansions
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Chiral Constitutive Equation Examples
Fresnel-Fizeau ? rotation diagonal Chiral ?
expansions
Combination of Fresnel-Fizeau rotation, ?,
about z-axis and Diagonal Optical Activity
Fresnel-Fizeau expansion, ?.
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Chiral Constitutive Equation Examples
Fresnel-Fizeau ? rotation diagonal Chiral ?
expansions
Combination of Fresnel-Fizeau rotation, ?,
about z-axis and Diagonal Optical Activity
Fresnel-Fizeau expansion, ?. WILL PRODUCE 4
PHASE VELOCITIES depending on POLARIZATION and K
vector
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This Chiral Constitutive Equation
Explains the Dual Polarized Sagnac ring laser
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Sagnac Effect Fresnel Surface The index of
refraction has 4 distinct values depending upon
direction and polarization.
Z axis Index of refraction 4 roots ?1/3 ? -
1/2
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3. The Chirality of the Cosmos
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3. The Chirality of the Cosmos
Definition of a chiral space A chiral space is an
electromagnetic system of fields E, B, D,
H constrained by a complex 6x6 Constitutive
Matrix which admits solubility for a real phase
function that satisfies both the Eikonal and the
Wave equation.
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3. The Chirality of the Cosmos
Definition of a chiral space A chiral space is an
electromagnetic system of fields E, B, D,
H constrained by a complex 6x6 Constitutive
Matrix which admits solubility for a real phase
function that satisfies both the Eikonal and the
Wave equation. Hence any function of the phase
function is a solution to the wave equation.
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3. The Chirality of the Cosmos
Definition of a chiral Vacuum The chiral Vacuum
is a chiral space which is free from charge and
current densities. J 0, ? 0
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3. The Chirality of the Cosmos
Definition of a chiral Vacuum The chiral Vacuum
is a chiral space which is free from charge and
current densities. Can the Cosmological Vacuum
be Chiral ?
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3. The Chirality of the Cosmos
Definition of a chiral Vacuum The chiral Vacuum
is a chiral space which is free from charge and
current densities. Can the Cosmological Vacuum
be Chiral ? Can the chirality be measured ?
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The Lorentz Vacuum
For the Lorentz vacuum, it is straight forward to
show that there is no Charge-Current density and
the fields satisfy the vector Wave Equation.
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The Simple Chiral Vacuum
For the Lorentz vacuum, it is straight forward to
show that there is no Charge-Current density and
the fields satisfy the vector Wave Equation.
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Use Maple to solve more complicated cases
Six equations 12 unknowns k x E - ? B 0, k
x H ? D 0 Use Constitutive Equation to yield
6 more equations
Define
Technique Use constitutive equations to
eliminate, say, D and B This yields a 6 x 6
Homogenous matrix in 6 unknowns. The determinant
of the Homogeneous matrix must vanish
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The determinant can be evaluated in terms of the
3 x 3 sub matrices of the 6 x 6 complex
constitutive matrix and the anti-symmetric 3 x 3
matrix, n x composed of the vector, n k
/?. The determinant formula is
The general constitutive matrix can lead to
tedious computations. A Maple program takes away
the drudgery.
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Conformal off-diagonal chiral matrices
Simplified (diagonal ?) Constitutive matrix for
a chiral Vacuum

• ?? i ??
• ? ? 1 ? ? 1

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Conformal Rotation chiral matrices
Simplified (diagonal ? Fresnel rotation ?)
Constitutive matrix for a chiral Vacuum
Leads to Sagnac 4 phase velocities
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Semi-Simplified Constitutive Matrix with
Conformal Rotation chiral submatrices

• f Fresnel Fizeau diagonal real part
  (conformal expansion)
• ? Fresnel Fizeau antisymmetric real part
  (rotation)
• ? Optical Activity antisymmetric imaginary
  part (rotation)
• ? Optical Activity diagonal imaginary part
  (conformal expansion)

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The Wave Phase Velocity and the Reciprocal
Radiation Impedance depend upon the
anti-symmetric rotations, and the conformal
factors of the complex chiral (off diagonal)
part of the Constitutive Matrix.
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The Wave Phase Velocity and the Reciprocal
Radiation Impedance depend upon the
anti-symmetric rotations, and the conformal
factors of the complex chiral (off diagonal)
part of the Constitutive Matrix.
(All isotropic conformal rotation chiral
matrices have a center of symmetry, unless the
Fresnel rotation, ?, is not zero)
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As an example of the algebraic complexity, the
HAMILTONIAN and ADMittance determinants are shown
above for the semi-simplified case.
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Fresnel Fizeau Conformal f does not effect
phase velocity AO Conformal ?
modifies phase velocity Fresnel Fizeau Rotation
? modifies phase velocity AO rotation
? modifies phase velocity
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Fresnel Fizeau Conformal f does not effect
phase velocity AO Conformal ?
modifies phase velocity Fresnel Fizeau Rotation
? modifies phase velocity AO rotation
? modifies phase velocity
All factors give an effect on chiral admittance
(cubed)
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Fresnel Fizeau Conformal f does not effect
phase velocity AO Conformal ?
modifies phase velocity Fresnel Fizeau Rotation
? modifies phase velocity AO rotation
? modifies phase velocity
All factors give an effect on chiral admittance
(cubed)
IN fact it is possible for the admittance ADM to
be ZERO, But this implies the radiation
impedance Z goes to infinity (not 376.73 ohms) !!!
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The idea that chiral effects could cause the
Admittance to go to Zero is startling to me. Zero
Admittance ? infinite Radiation Impedance, Z !
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The idea that chiral effects could cause the
Admittance to go to Zero is startling to me. Zero
Admittance ? infinite Radiation Impedance, Z
! Can this idea impact antenna design?
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And now some heresy
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Zero Admittance ? infinite impedance
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Zero Admittance ? infinite impedance What would
be the effects of a chiral universe on Cosmology
??? ?
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Zero Admittance ? infinite impedance What would
be the effects of a chiral universe on Cosmology
??? Is the Universe Rotating as well as Expanding
?
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Zero Admittance ? infinite impedance What would
be the effects of a chiral universe on Cosmology
??? Is the Universe Rotating as well as Expanding
? Could the chiral effect be tied to dark matter
-- where increased radiation impedance causes
compact composites to bind together more than
would be expected ??
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Zero Admittance ? infinite impedance What would
be the effects of a chiral universe on Cosmology
??? Is the Universe Rotating as well as Expanding
? Could the chiral effect be tied to dark matter
-- where increased radiation impedance causes
compact composites to bind together more than
would be expected ?? -- Could the infinite
radiation impedance be tied to compact composites
such as molecules and atoms which do not Radiate
?
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Hopefully these questions will be addressed on
Cartans Corner
Optical Black Holes in a swimming pool
http//www.cartan.pair.com
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Some Examples from Maple
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Real f 0, ? 0 Imag ? 1/3, ? 0
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Real f 0, ? 0 Imag ? 0, ? 2
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Real f 0, ? 0 Imag ? 1/3, ? 1/3
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The 4-mode Sagnac Effect - with No center of
symmetry
Real ? 1/3, f 0, Imag ? 1/6, ? 0
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Ebooks Paperback, or Free pdf
http//www.lulu.com/kiehn or
http//www.cartan.pair.com email
rkiehn2352_at_aol.com
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