Teleparallel Treatment of theEhrenfest, Sagnac and Field

Rotation Paradoxes

William M. Pezzaglia

Department of PhysicsSanta Clara UniversitySanta Clara, CA 95053

Email: wpezzag@clifford.org

• Rotating Frames are Equivalent to Torsion

• Torsion Explains Sagnac & Ehrenfest

• Possible solution to Schiff field rotation paradox?

21st Pacific Coast Gravity Meeting, Univ Oregon 2005Mar25-26

http://www.clifford.org/∼wpezzag/talks.html

INDEX

Talk 4:45 pm, Friday 2005Mar25

Title 1

Index 2

I. Rotational Relativity 3

II. Harres-Sagnac Effect 4

III. Ehrenfest Paradox 5

IV. Field Rotation Paradox 6

V. Simultaneity and Spacetime Split 7

VI. Anholonomic Transformation 8

VII. Teleparallel Coordinates 9

VIII. Principles and Paradoxes 10

IX. Barnett/Feynman Paradox 11

X. Schiff Paradox 12

XI. Partial Solution 13

XII. Charge and Invariance 14

XIII. Special EM Rotational Relativity 15

XIV. Some Points 16

XV. References 17

III. Ehrenfest Paradox

A). Born’s Off-Diagonal Metric (1909)

Galilean Coordinate

Transformation

t′ = t

θ′ = θ − ωt

ds2 = [c2 − ω2r2]dt′2︸ ︷︷ ︸Centrifugal

− 2ωr2dθ′dt′︸ ︷︷ ︸Coriolis

− r2dθ′2 − dr2

• Cross term describes Coriolis and Sagnac

• Circumference (dt′ = 0): C′ = 2πris Born rigid (unchanged)

B). Ehrenfest Paradox (1909)

•Lab Frame: Circumference must be Lorentzcontracted while radius is not: C < 2πr

•Rot Frame: Einstein argues setting disc intorotation dynamically stretches circumferenceby curving space (i.e. not Born rigid):

C′ = γ2πr

C = C′/γ = 2πr

γ ≡ 1√1−ω2r2

c2

•But Born metric is FLAT. Contradiction!

V. Simultaneity: Spacetime Split

A). Orthogonalize Time

• Observers require: time ⊥ space

• Born metric has: g4θ 6= 0

• Langevin(1935), Adler, Bazin, Schiffer(1975) redefine time: dt?2 = dt′ − γ2ωr2

c2dθ′

Metric: ds2 = (c2 − ω2r2)dt?2 − γ2r2dθ?2 − dr2

B). Ehrenfest Paradox Explained

• Length measurement require: dt? = 0

• Rotating Frame Circumference: C′ = γ2πr

• Lab Frame: C′ = C′/γ = 2πr

C). Fix one thing, break something else

• Sagnac Effect now unexplainable

• Coriolis Missing?{

θr4

}=

{rθ4

}= 0

• Curvature induced (but how?)

• Rotation Addition Formula Bad

VI. Anholonomic Transformation

A). How did we induce curvature?

• Adler’s coordinate transformation is nonholonomic(dt?

dθ?

)=

[γ2 −γ2r2ω/c2

−ω 1

] (dtdθ

)

• dt? non-integrable (path-dependent)

• Curvature and Torsion induced (Kleinert 1997)

B). Sagnac Effect from Torsion

• τ4rθ = Γ4

rθ − Γ4θr =

[∂

∂r?,∂

∂θ?

]t = 2γ4rω

c2

• Sagnac as non-closure (Corum 1977)

∆T =∮

dt =∫

drdθ τ4rθ = 2ωπR2

c2−ω2R2

C). Adler’s is Riemann-Cartan

• Connection: Γσµν =

{σµν

}+ Kσ

µν

• Metrical Connection gives Centrifugal:{

r44

}

• Contorsion gives Coriolis: Kθr4 , Kr

θ4

VII. Teleparallel Coordinates

A). Want Relativistic Theory

• Observers require Local Lorentz Frame with fixed “c”

• Born metric has non-orthogonal time: g4θ 6= 0

• Adler/Langevin has non-orthogonal space: g12 6= 0

B). Anholonomic Frame

• Franklin(1922), Trocheres(1949), Takeno(1952), Co-

rum(1977, 1980) suggest angular Lorentz Transf.,

dt′ = γ(dt− ωr2/c2 dθ)

dθ′ = γ(dθ − ωdt)

• Flat Metric: ds2 = c2dt′2 − r2dθ′2 − dr2

C). All Phenomena from Torsion

• Sagnac from: τ4rθ = 2γ2rω/c2

• Ehrenfest/Thomas Precession from τθrθ = γ2rω2/c2

• Clock Asynchronization from: τ4r4 = γ2rω2/c2

• Contorsions give kinematics,

Kθr4 , Kr

θ4 (Coriolis)

Krθθ (Centripetal), Kr

44 (Centrifugal)

VIII. Principles and Paradoxes

A). Special Rotational Relativity

• Carmeli(1986): “The laws of physics are the same in

all rotationally unaccelerated systems (bodies) having

constant angular velocities relative to each other”

• Paradoxes show there is no special theory of

rotational relativity. Carmeli’s idea didn’t work.

B). General Rotational Relativity

• Einstein: Accelerated reference frame is equivalent to

a frame at rest with curvature

• Propose: Rotating reference frame is equivalent

to a frame at rest with torsion

C). Rotational Electrodynamics

• Corum(1980) has shown that rotationallyconsistant electrodynamics requires torsion

Gauge Dependent: Fµν = ∂µAν − ∂νAµ − τσµνAσ

Constitutive Eqn1√g∂µ(

√gFµσ) = jσ − 1

2τσµνFµν

︸ ︷︷ ︸Machian Current?

• Quantities transformed from Lab → Rotating Frame

do obey these, but it does not provide insight to source

of interior fields in Barnett experiment where jσ = 0

X. Schiff Paradox (1939)

A). Lab Frame

• Net Charge:

Qa + Qb = 0

• Outside: ~E = 0

• Non-zero Dipole:

m = 12ωQ(a2 − b2)

Bz(r) = µ04π

ωQr3

(b2 − a2)

B). Lorentz Transf. by v = ωr

• B′z = γBz

• E′r = γωrBz

γ ≡ 1√1−ω2r2

c2

C). Rotating Frame

• No Current =⇒ B′z = 0 ?

• No net Charge =⇒ E′r = 0 ?

• Yet a + particle at rest at point r will expe-rience outward force in rotating frame?

XI. Partial Solution

A). Charges Don’t Cancel in Σ′

Charge in rotating frame: Q′a = 2πLσ′a

From Transformation: σ′a = σa

√1− ω2a2/c2

Lab Frame Charge Density: σa = Q/(2πaL)

Thus: Q′a = Q√

1− ω2a2/c2

Net Charge: ∆Q′ = Q′z −Q′b ' Qω2

2c2(b2 − a2)

B). Yields Nonzero Electric Field

Gauss Equation: 1r∂r(rE′r) + τ4

r4Er = ρ

Non Zero Field: E′r ' γ∆Q2πεorL = γQω2(b2−a2)

4πεoc2rL

Ouch, expected: E′r = γωrBz = µoγQω2(b2−a2)8πc2r2

So we are off by a factor of 2r/L ?

XII. Charge and Invariance

Did we violate charge conservation?

NO! Conservation applies IN a framenot BETWEEN frames

A). Spinning Mass Analogy

Dixon’s Invariant: p2 − S2/λ2 = (moc)2

Dynamic Mass: m2 = m2o

[1 + S2

(moλc)2

]

Radius of gyration λ, m ' mo + S2

2moλ2

B). New Charged Particle Analogy

• Current: j = em p

• Dipole Moment: U = e2m S

• Propose Invariant: j2 − 4U2

c2λ2 = (eo c)2

• Dynamic Charge: Q2 = Q2o

[1 + 4U2

(Qoλc)2

]

XIII. EM Rotational Relativity

A). Special Rotational Relativity

Carmelli (1986) proposes rotating transformation

m′ = γ(m− ω · S/c2)

S′ = γ(S −mωr2)

γ ≡ 1√1−ω2r2

c2

This leaves (m2 − S2/r2) invariant

Fundamental particles: want invariant mass and spin!

B). EM Analogy

Q′ = γ(Q− 2ω · U/c2)

U ′ = γ(U − 12Qωr2)

γ ≡ 1√1−ω2r2

c2

This leaves (Q2 − 4U2/r2) invariant. For fundamental

particles: want charge and dipole moment invariant!

C). Application

Consistent with our Schiff Paradox treatment,

U ′ = 0 , Q′a = Qa

√1− ω2a2/c2

XIV. Summary

• Propose: A Rotating Frame is equiv-

alent to a Rest Frame with Torsion

• For consistency, Electromagnetism

MUST couple to torsion

• Rotational EM Paradoxes are the

most problematic for attempting a

“theory of rotational relativity”

XV. References

1. Feynman, Lectures on Physics, (1975) p. 14-7.

2. Einstein, The Principle of Relativity, Chapter VII.

3. Adler, Bazin and Schiffer, Introduction to General Rela-

tivity, (1975) pp. 120-131.

4. J.F. Corum, “Relativistic Rotation and the Anholonomic Ob-

ject”, J.Math.Phys. 18, 770 (1977); “Relativistic Covariance and

Rotational Electrodynamics,” J.Math.Phys. 21, 2360 (1980).

5. P. Langevin, Comptes rendus 200, 48 (1935).

6. A. L. Kholmetskii, “One century later: Remarks on the Bar-

nett experiment,” Am. J. Phys. 71, 558-561 (2003).

7. L. Schiff, “A question in general relativity,” Proc. Natl. Acad.

Sci. USA 25, 391-395 (1939).

8. M. Carmeli, “Rotational Relativity Theory,” Int. J. Theor.

Phys., 25, No. 1, 89 (1986).

8. P. Franklin, “The Meaning of Rotation in the Special Theory

of Relativity,” Proc Natl. Acad. Sci. USA 8(9), 265-268 (1922).

9. H. Takeno, “On Relativistic Theory of Rotating Disk,” Prog.

Theor. Phys. 7(4), 367-376 (1952).

10. M.G. Trocheries, “Electrodynamics in a Rotating Frame of

Reference,” Phil. Mag. 40(310), 1143-1154 (1949).

11. B. Kursunoglu, “Spacetime on The Rotating Disk,” Proc.

Cambr. Phil. Soc. 47, 177 (1951).

12. H. Kleinert, “Nonabelian Bosonization as a Nonholonomic

Transformation from Flat to Curved Field Space,” Annals. Phys.

253, 121-176 (1997); “Nonholonomic Mapping Principle for Clas-

sical and Quantum Mechanics in Spaces with Curvature and Tor-

sion,” Gen. Rel. Grav. 32, 769 (2000).