Harry I. Ringermacher

KRONOTRAN Enterprises, LLC, Delanson, New York

Mark S. Conradi

Washington University, St. Louis, Missouri

Brice N. Cassenti

United Technologies Research Center, East Hartford, Connecticut

Search for Effects of an Electrostatic Potential

on Clocks in the Frame of Reference of a

Charged Particle

NASA/CR—2005-213999

November 2005

AIAA–2001–3906
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Harry I. Ringermacher

KRONOTRAN Enterprises, LLC, Delanson, New York

Mark S. Conradi

Washington University, St. Louis, Missouri

Brice N. Cassenti

United Technologies Research Center, East Hartford, Connecticut

Search for Effects of an Electrostatic Potential

on Clocks in the Frame of Reference of a

Charged Particle

NASA/CR—2005-213999

November 2005

National Aeronautics and

Space Administration

Glenn Research Center

Prepared under Contract NAS3–00094

Prepared for the

37th Joint Propulsion Conference and Exhibit

cosponsored by the AIAA, SAE, AIChE, and ASME

Salt Lake City, Utah, July 8–11, 2001

AIAA–2001–3906
Acknowledgments

Thank you to Marc Millis for having the daring and fortitude to envision and create the BPP program and support

this effort as part of it. It is an honor to be included in the first trials. Caleb Browning, Mark Conradi’s graduate

student, assembled the probe and took the very thorough NMR measurements supporting the conclusions of this

work. Mark Conradi led as well as participated in the Washington University effort and contributed the excellent

and novel NMR probe designs permitting such precise measurements. Brice Cassenti expended considerable effort

developing the classical and quantum theory supporting the work, the detail of which is largely relegated to the

Appendices (half the report). We thank Judy Keating for her assistance in organizing the final report and providing

insightful “clock suggestions.” We also thank Larry Mead for supportive discussions revolving around this new

view of General Relativity.

Available from

NASA Center for Aerospace Information

7121 Standard Drive

Hanover, MD 21076

National Technical Information Service

5285 Port Royal Road

Springfield, VA 22100

Available electronically at http://gltrs.grc.nasa.gov
Search for Effects of an Electric Potential on Clocks in the Frame of Reference of Charged Particles

Executive Summary

A series of Nuclear Magnetic Resonance (NMR) experiments were performed to search for a change

in the precession rate of the proton magnetic moment in a hydrogen atom subjected to an intense

electrostatic potential, as predicted by a new theory coupling space-time to matter and charge. The theory

self-consistently imbeds classical electrodynamics within the framework of non-Riemannian space-time by

way of introduction of an electrodynamic Torsion tensor into Einstein's equations. It predicts that an

intense, external electrostatic potential should measurably shift the internal clock of a charged particle

analogous to the gravitational red shift. The internal clock refers to time as seen by a particle of charge-tomass ratio, e/m, in its rest frame.

The precession rate (Larmor frequency) of the proton was set to 354 MHz in an 8.4T magnetic field of a

superconducting magnet. The three experiments included:

(1)

(2)

(3)

A proton subject to a spatially and temporally constant 5kV electrostatic potential -voltage on

versus voltage off and voltage reversed NMR signals.

A proton subject to a spatially constant but time varying 6kV electrostatic potential.

A proton in a hydrogen atom physically moved fllrough a spatially non-uniform, constant 5kV

electric field-voltage on versus voltage off and dummy voltage NMR signals

No change in the proton's clock (the 354 MHz proton NMR resonance in a hydrogen atom) was

observed in any of the three experiments. The experiments were done very carefully so that shifts in

frequency of as little as 0.01ppm - that is about 3 Hz out of 354 MHz - could be observed. 5,000V was

typically used resulting in a predicted shift of approximately 5ppm, more than enough to see with the stated

sensitivity. The null results of the first two experiments were consistent with the predictions of the new

theory and thus were not definitive. The third experiment was particularly important. The null result could

not be explained by the present theory. This theory is, like the theory of gravitation, a single particle

theory. However, a hydrogen atom also has an electron that interacts with the proton. In order to maintain

the single particle character of a metric theory while simultaneously including the effects of the electron, a

new conjecture, the "Clock Principle" (CP) is proposed. This principle states that the work done on the

proton's clock changes its time. This automatically includes the work done by all the forces acting on the

proton, including the external electric field and the electron. The net work done on the proton by the

external electric field is cancelled against the work done by electron's force on the proton. We take this to

be the reason for no clock effect. This is not the usual picture envisioned by General Relativity (GR), a

theory depending on geometry alone.

CP may be false for the present theory and only coincidentally true for GR due to the equivalence

principle, leaving only potential rather than potential energy (and therefore work) responsible for time

changes.

If CP is true, it follows directly that negative rest mass cannot exist. Its

existence violates the equivalence principle. This does not preclude the existence

of negative energy since the general energy-momentum relation has both positive

and negative energy roots (see "Clock Conclusions").

If CP is shown to be false then it follows that no metric theory incorporating

electromagnetism is possible, thus formally ending Einstein's dream through a real experiment.

We demonstrate CP for flame clocks: the NMR clock, the atomic clock and the pendulum clock. These and

all known clocks can be shown to satisfy the conjecture.

NASA/CR—2005-213999 iii

Table of Contents

Executive Summary .............................................................................................. iii3

I. Introduction ..................................................................................................664

II. Theory and Approach

Ideal Experiment ............................................................................................ 5

Present Approach ............................................................................................ 5

III. Experiments

Experiment 1- Constant Potential ......................................................................... 6

Experiment 2- Time Varying Potential ................................................................... 9

Experiment 3- Physical Displacement of Hydrogen Atom Through High Electric Field ......... 11

IV. Conclusions

Experimental Conclusions ................................................................................. 17

Implied Conclusions ........................................................................................ 17

Relation of the Metric to the Lagrangian and Work Done ..................................... 17

Application to the Hydrogen atom ................................................................ 18

The Clock Principle

Potentials, potential energy and work ............................................................ 20

Einstein Rocket--Equivalence Principle for clock changes ................................... 20

Einstein rocket replaced by negative mass ....................................................... 21

Inconsistency between General Relativity and a neutral mass dipole ........................ 22

Clock conclusions ................................................................................... 23

V. Future Directions

ESR Test of the Single Particle Lagrangian Formalism ................................................ 23

A Classical Measurement of the Quantum Aharonov-Bohm Effect .................................. 25

VI. Appendices

Appendix A.- On the Compatibility of Gravitation and Electrodynamics ........................... 25

Appendix B.- Metric for Constant Fields ................................................................ 30

Appendix C.- Potential Due to Two Oppositely Charged Circular Loops and Plates .............. 32

Appendix D.- Potential for Hydrogen in Electric Fields ............................................... 35

Appendix E.- Relation Between Metric, Lagrangian and Work on a Clock ......................... 38

Appendix F.- Time Dilation and Work Done for a Hydrogen atom .................................. 41

Appendix G.- Quantum Aharonov-Bolma Effect Consistent with New Theory ..................... 43

Acknowledgements ............................................................................................... 45

References ......................................................................................................... 45

NASA/CR—2005-213999 v

iii 

1 

1 

2 

3 

6 

8 

14 

14 

14 

15 

17 

17 

18 

19 

20 

Microwave Cavity Test of the Single Particle Lagrangian Formulation .....................................20 

21 

23 

29 

31 

35 

39 

43 

45 

References ..........................................................................................................................................47

I. Introduction

The goal of the present work is to investigate an electromagnetic alternative to exotic physics for the

purpose of coupling matter to space-time. Electromagnetic forces have distinct advantages. They are 1040

times stronger than gravity. They can be manipulated at will. Resources to create intense fields of virtually

any geometry are readily available. However, there is currently no accepted theory linking electrodynamics

directly with the geometry of space-time other than to curve it via extremely high energy densities. The

mainstream approach taken is "bottom up", attempting to unite all forces in the context of quantum gauge

field theory which has to-date been successful in unifying the weak and electromagnetic forces and

describing the strong force in what is known as "the standard model". Gravity and therefore space-time

geometry remains isolated from the internal geometry of gauge theory.

The proposed experiment is a measurable, predicted consequence of a completed and solved theory

[1], linking space-time geometry and electrodynamics, describing effects of intense electrostatic potentials

on time as seen in charged particle reference frames. Tiffs is grounded upon E. Schr6dinger's later works on

gravitation theory [2]. In Iris work, Schr6dinger attempted to link electromagnetism to geometry through a

non-symmetric affine connection (Torsion tensor). He failed at the attempt, primarily because of an error

of oversight. The theory upon which the present work is based corrects this error [3], resulting immediately

in the definition of a new type of affine connection - an electrodynamic connection - that precisely

matches Schr6dinger's concepts and furthermore satisfies the classical equations of motion for both charge

and spin, thus finding a common denominator for both external space-time coordinates and internal spin

coordinates. Following Schr6dinger, once the connection is identified, it is merely a formality to extract

the Einstein field equations and find solutions subject to the appropriate boundary conditions for

electromagnetic fields.

II. Theory and Approach

Ideal Experiment

The theory [summarized in Appendix A] predicts that a particle of charge e, and mass m, immersed in

a suitable electric field but nnshielded and supported will see, in its rest frame, a time differing from the

proper time of an external observer arising from the electrostatic potential at its location. In general, from

the theory, the time shift in a clock interval is related to the potential difference between the two points;

dr2 - 1+ 2K'((p2 - (Pl) (1)

dq

where K" = -e/mc 2. As an example, let us suppose that we have a distribution of test charges, say free

protons, between two concentric cylindrical electrodes, an inner cylinder of radius R and an outer cylinder

1

of radius R. We further suppose we are able to create a situation in which the protons are momentarily at

2

rest while experiencing an intense electric field. For example, they may be at the extreme of an oscillation

in a vacuum. The relation between proper time elements at any two positions r 1and r within the medium,

for the case of a cylindrical potential was found as an exact solution of the theory and is given by

61"2 -- 1+ 2KAln r2 , (2)

d/"1 I"1

where A is the line charge density. This equation is exactly analogous to that for the gravitational red

shift. The negative electrode is the zero potential reference for the free proton.

One possible clock for such a test is Nuclear Magnetic Resonance. A proton placed in an intense

electric field within a radio-frequency transverse field " H " coil aligned orthogonally to a nniform

1

magnetic field, H 0, is resonant at the Larmor frequency, co = 7 H 0, where the gyromagnetic ratio, 7, for the

proton spin is proportional to e/m. We thus have a natural clock. From eq.(2) we expect the proton's

clock frequency to depend on its position with respect to the zero potential electrode at g I:

NASA/CR—2005-213999 1
og(r) : og(R 1)/1 + 21¢A In Rlr

The Larmor field distribution is then given by:

(3)

H(r)=H(R_)II+ 2_kIn R_ )-r (4)

From this it is straightforward to calculate the NMR lineshape and shift that will result when the electric

field is turned on as compared to the field off. Tiffs is accomplished by performing a convolution of eq.(4)

with the sharp Lorentzian line (seen in Fig. 1). Figure 1 shows the result of tiffs calculation for R / R =0.50.

1 2

With the E-field off, all the protons resonate simultaneously at the Larmor frequency and a sharp sweep

field line will result, it's width broadened by the H -field inhomogeneity and local fields. With the

0

electric field on, protons at different potentials will contribute to the signal at different times in the sweep

thus further broadening and shifting the line while reducing the line intensity. We assume the natural width

is much less than the electrostatic broadening. Under ideal circumstances, for a supported proton in a 8T

magnetic field with a 5kV/cm electric field, a line shift and broadening of approximately five parts per

million is expected. Pulsed NMR with FFT evaluation can be used in place of line sweeps.

g(H)

0.5

0

--40 --20 0 20 40

H-- Ho

AH

Figure 1. Normalized NMR lineshape as a function of magnetic field for electric field off

(sharp line) and on (broadened). AH is the line width.

Present Approach

In practice it is experimentally difficult to "support" a charged particle. Generally, this can be

accomplished electromagnetically, but then, by definition, the electric field and force at the particle location

must vanish since the charge does not accelerate. Nonelectromagnetic support leaves only gravitational

balance, which is very weak, and the nuclear force, which is very strong. At first sight, it would appear

from the field equations that the presence of an electric field at the charged particle site is essential. But

this may in fact not be the case, since the metric solutions to the theory depend on the potentials rather than

on the electric fields. Thus it would appear that only potential differences are necessary to produce

observable effects. The approach we have chosen uses the proton in a hydrogen atom. It is supported

electromagnetically. The consequences of this approach will be the main subject of the conclusions of this

work. A detailed description of the three experiments performed follows.

NASA/CR—2005-213999 2
III. Experiments

Three experiments were performed. The first experiment, a constant potential applied to the proton in

the hydrogen atom, was expected to produce null results both classically and within tiffs theory. It was

considered a "warm-up" experiment to verify the operating system. A constant potential is classically

arbitrary to within a gradient of a scalar. This is a manifestation of the "Coulomb gauge condition" for the

field and thus has no physical effects. Thus, a null result is expected.

The second experiment, a time-varying potential applied to the proton in the hydrogen atom, was

initially expected to produce an effect. However, it was found during the course of tiffs program that tiffs

theory was also invariant under a pure time-varying potential [4], a result consistent with the classical

"Lorentz gauge condition" which is the relativistic generalization of the Coulomb gauge condition. That is,

it states that the potential is also arbitrary to an additive time-derivative of a scalar. Thus, a null result is

also expected.

The third experiment involves the physical displacement of a proton in a hydrogen atom through an

electric field while NMR is performed on the proton. An observable shift in the proton NMR line is

predicted. Care was taken to use a non-uniform electric field since Cassenti has shown [Appendix B] that a

uniform field, such as that between two large, parallel, metal plates can lead to a possible null effect or, at

the very least, confusion in interpretation of the data. Cassenti has calculated the electric field distribution

for the chosen electrode geometry thus ensuring the non-uniformity of the field. In the present theory, in

principle, only the potential difference between electrodes induces a frequency shift. The non-uniformity is

merely "double insurance".

Experiment 1 - Constant potential

The 354 MHz, 8.4 Tesla rig was chosen for the experiments. Tiffs unit has a field homogeneity of 0.1

ppm or about 35 Hz, more than sufficient to resolve the predicted 5ppm effect. The proton sample was

Benzene. The initial experiment, (Fig.2), was a simple free induction decay (FID) with the E-field on vs.

off. The sample was enclosed in a 2mm thick aluminum can placed at high potential. Thus the E-field will

vanish in the interior of the can at the sample but there will remain a constant potential. The voltage

terminal (sample chamber) could be set + or - with respect to ground and the NMR FID was monitored.

T

HO

H1

Figure 2. NMR "can" arrangement. External magnetic field, H0, is perpendicular to the radio

frequency field, H1, applied through a coil wrapped around the proton sample, S. The 5kV

electric field is applied outside the can leaving a constant 5kV potential inside.

An initial proton NMR line obtained from a water sample showed a field homogeneity of 0.3 ppm at

a field intensity of 8.4 T. The goal was a homogeneity of 0.1 ppm or better. A field homogeneity of 10

ppb, far exceeding our goal and sufficient to resolve the smallest effects was achieved by shimming the

magnetic field, reducing the sample size and confining the sample geometry to a long thin tube parallel to

the external magnetic field thereby reducing end effects. Figure 3 shows the full double can probe with

NASA/CR—2005-213999 3
innercancoverandoutergroundshieldremoved.Figure4showstheprobeheaddetailsbeforeandafter

thesamplegeometrychange.Figure5showstheprobeheadwiththeinnercaninplaceandouter ground

can removed.

Figure 3. Full probe assembly with double cans removed showing probe head.

Figure 4. NMR probe head with low homogeneity sample geometry (left) and high

homogeneity "needle" sample (right).

Figure 5. NMR probe head with constant potential inner can cover in place.

NASA/CR—2005-213999 4
Following the high homogeneity tuning, the first trial of three major experiments was completed. A

5kV electric field was applied between the inner shielding can in which the proton sample (benzene)

resided and the outer ground can. FID NMR was performed with field on and off in steady state. The

electrostatic potential in the shield can was thus a constant 0 or 5kV. A second test was performed using a

+3kV to -3kV potential reversal between the cans. In both tests the observed line shift, Figures 6a and 6b,

was A V / V = 0 --- 1.0x 10 -9 , consistent with a null prediction of both classical E&M and the present

theory.

3O

e--

e--

2O

10

0

-12 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0

Frequency (Hz)

Figure 6a. NMR proton line in Benzene sample showing zero shift upon application of 5kV potential

between shield can and gronnd can.

3O

.m

ee2O

10

0

-12 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0

Frequency (Hz)

Figure 6b. NMR proton line in Benzene sample showing zero shift upon reversal of 3kV potential

between shield can and gronnd can.

NASA/CR—2005-213999 5
Experiment 2 - Time-varying potential

Originally, considerations in the theory suggested that a time-varying potential in the metric might be

equivalent to a time-varying coordinate system and thus changes in clock rates. However subsequent

calculation of this effect clearly showed that even the new Einstein equations were invariant under a timevarying potential in g00 •

The NMR "spin-echo" teclmique was employed in tlfis experiment to monitor the precession. In the

spin-echo teclmique, the proton's nuclear spins, precessing at 354 MHz and aligned along the external 8.4T

magnetic field "z" direction, are first rotated 90 ° with respect to "z". This is accomplished by application of

an rf pulse (90 pulse) at the Larmor frequency to a coil around the sample oriented in that direction.

Following the rotation, the spin vectors dephase with characteristic time T2* and spread out in the "x-y"

plane. At a time "c later a second rf pulse (180 pulse) is applied that rotates all the in-plane spin vectors an

additional 180 ° wlfich causes them to reverse directions in the x-y plane and thus rephase. The net

rephased signal is the so-called "echo". Any miniscule line shifts occuring during the rephasing process

will affect the echo amplitude and phase because an exact time reversal will not occur. Figure 7 shows the

NMR spin-echo pulse sequence.

Figure 7.

m 90 pulse

time, t

180 pulse

NMR spin-echo timing sequence showing spin rephasing time "c.

In order to ensure that the experiment had sufficient sensitivity, a small milliamp-level calibration

current was injected into the NMR "shim coils" - those coils used to correct for small field gradients in the

region of the sample in order to improve the magnetic field homogeneity. The current injection simulated

the on-off and off-on step voltage procedure in order to quantify the expected NMR phase shift behavior.

This calibrated response could then be compared to the observed step-voltage response as a function of x.

Figure 8 shows the NMR system response to the current injection. The "on-on" and "off-off" states refer to

the current being constant - either on or off during the NMR acquisition period, x, between the "90" pulse

and the "180" pulse. These data overlap. There is no expected difference since these are steady states and

the phase increases linearly with x. Tlfis signal is the "control" data and represents the system response with

no effects present. The rising "on-off" and falling "off-on" signals result from the current injection and are

phase-reversed as expected. The phase difference between the control signal and the injected signal is 250 °

at x =1 second, corresponding to 0.7Hz or 2ppb.

The proton sample (H20) was shielded inside an aluminum can as before. This ensures that the

sample will see only a voltage potential, spatially constant inside the can, with zero electric field. The

voltage on the can, with respect to grotmd, was changed during the NMR signal acquisition. The phase

change, resulting from any magnetic line slfifts arising from clock rate changes, was measured to an

accuracy of approximately 2 degrees (Aqb= 0.035 tad.) corresponding to a frequency shift of 0.005 Hz

( Afmin -- A_/2nT max ; T max_ TI=I sec). Tlfis, in turn, corresponds to a frequency shift sensitivity of

0.005/354,000,000 or approximately 0.01ppb.. Upon application of a 5KV step-function (20 ms risetime)

between the aluminum can and ground in both turn-on and turn-off modes, zero shift was observed in the

NMR line to an accuracy of 0.01ppb.

NASA/CR—2005-213999 6
6OO

5OO

4OO

3OO

2OO

1O0

0

-1O0

-200

-300

i

500

i

1000

Tau (ms)

O0

Figure 8. NMR calibration spin-echo phase signal as a function of 90-180 separation

time Tau. Cycling refers to the switching of a small test magnetic field shift.

Figure 9 show the results of the second spin-echo experiment. For the control response, the "on-on"

and "off-off" electric field states are reproduced. In the "on-of£' state, the 5000V electric field is on and

turned off at time x. In the "off-on" state, the electric field is off and turned on at time x. The risetime is

approximately 30ms. x was varied from zero to 1000 ms. No change from the control signal was observed

to an accuracy of 0.01 ppb as evidenced by the same phase for the cases at each Tau.

250

.e 200

o')

15o

_. 100

>

N 5o

I1:

0 I I

0 500 1000 1500

Tau (ms)

° on.on

_,on.off

off.on

off.off

Figure 9. NMR data spin-echo phase signal as a function of 90-180 pulse separation

time Tau. Cycling refers to the switching of the electric field.

10 NASA/CR—2005-213999 7
Experiment 3 - Physical displacement of Hydrogen atom through high electric field

For the third and final experiment, hydrogen in benzene at room temperature was gravity-flowed

between two electrodes, an upper one at ground potential and a lower one at +5000 V, both situated in the

NMR coil in the external 8.4T magnetic field while NMR was performed with the electric field on and off.

The electrodes were copper discs placed in the 3mm I.D. of glass tubing and connected to a high voltage

source through glass/epoxy seals. The electrode spacing was 1.0 cm giving an average electric field of

5000V/cm. The 2-turn NMR coil diameter was 1.5 cm ensuring that the HV region was inside the coil. The

coil was untuned to avoid radiation damping since the signal was already very large. The electrode spacing

was chosen for two reasons: (1) to avoid the situation of "nniform electric field" which would have been

approximately the case for spacings of 1-2mm or less. Cassenti has shown [Appendix B] that the case of

uniform electric field for tiffs theory is reducible to a Riemann-flat space, thus making interpretation of

results more difficult or questionable. A non-uniform field, as in the present electrode geometry, ensures

that this does not occur and that only the potential difference between electrodes - which is the same

regardless of electric field - is relevant. (2) the larger spacing avoids electric breakdown effects, creating

unwanted currents, in the benzene. Figure 10(a) shows the NMR coil system schematic. Figure 10(b)

shows the actual assembly lying on its side and opened for inspection.

OW

. _---0 ,,,_

B

(a)

:::::::::::::::::::::::::::::::::::::::::::::::::::::::<+,:c:::: c:::::< : c :::+:: ::<: : : :: ::::::::::: :: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

::::::::::::::::::::::::::::::::::::::::::::::::::::::::::>_ :::::::<+:+:c _ .... +> :+:+:+: _c::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

::::::::::::::::::::::::::::::::::::::::::::::::::::::c::-_ q:::_q::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

::::::::::::::::::::::::::::::::::::::::::_q:::::_ _ _:_ ..... ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

Figure 10. NMR coil and flow arrangement for E-field experiment: (a) schematic, (b) opened

assembly on its side (flow left to right).

An NMR FID experiment was performed with and without flow. A 20-30 ms :/12was obtained by

careful adjustment of the B-field shim coils. The NMR line and effects of flow without the presence of an

E-field were modeled. The :/12value gives a line width of approximately 17 Hz ( Af = 1/TgT 2). Figure

1 l(a) shows the measured NMR line as a function of flow velocity through the coil varying from 3 cm/s to

50 cm/s. Since the FID has a time constant of 20-30 ms, the proton must stay in the H1 field at least this

long in order to contribute a significant time-slffft signal arising from the maximum 5kV potential change.

This corresponds to a flow speed of 15-30 cm/s, the mid-range of the chosen flows. Note that the

measured shift with zero volts is approximately 10 Hz, from 0 to 50 cm/s flow rate, in approximate

agreement with the theoretical calculation, Figure 1 l(b). Figure 12 shows the predicted lines for a

maximum potential of 5000 Volts. The shift is at least ten times that for zero volts.

11 NASA/CR—2005-213999 8
Fortheflowexperiment, theNMRlinewasobtainedforvoltageoff, voltageon,andadummy

voltageon(voltageonbutHVcabledisconnected). Theprobevoltagewasdischargedwhenthedummy

experiment wasperformed. ThisdataisshowninFigure14a-14f. A typical NMR FID signal is shown in

Figure 13 and compared to the calculated FID.

Resonance - 0 Volts

(1)

"0

"-t

D.

E

9O

7O

5O

10

Rz

•-.- 0 flow

3 cm/s

6 cm/s

..........12 cm/s

............. 25 cm/s

............ 50 cm/s

Figure 1l(a). Measured NMR lines for zero E-field and flows from 0 - 50 cm/s.

0..5

I I I

0

? / _ .

/ ......... s "" :.'-.,,

o I I I

80 95 110 125 140

frequency, Hz

Figure ll(b). Calculated NMR lines for zero E-field for flows of 0, 12, 25 and 50 cm/s.

12 NASA/CR—2005-213999 9
1.5

Iw l 1

i

Iw jl

Iw jl

I 05

0

0

I I I

100 200 300 400

_equency, Hz

Figure 12. Calculated NMR lines for 5000V/cm E-field for flows of 0, 12, 25 and 50 cm/s.

Free Induction Decay - 0 volts

lOOO

5oo

0

m

-500

-1000

time, ms

t n

Figure 13. Typical measured FID signal (left) compared to calculated signal for prediction(right).

We note in Figure 14 that there is no change in the line position for each flow greater than the FFT

resolution of _ 2 Hz corresponding approximately to a variation of 6 ppb, at least a hundred times smaller

than the predicted shift of Figure 12. Figure 11 is the overlay of Figs. 14 for zero flow.

When the experiment was completed, the probe was disassembled and carefully inspected to ensure

that all voltage connections were secure.

13 NASA/CR—2005-213999 10
Resonance - 0 flow

90

70

50

10

Hz

•-,,- 0 V

.....,,----.5 kV

....._.._.---- 5 kV dummy

Figure 14a. Variation in NMR line for zero flow and applied voltage.

Resonance - 3 cm/s flow

¢

.t.t.0..-

9O

70

3o

1 , 10

-130 -110 -90 -70

Hz

•-.- 0 V

...... ;,,,-----5kV

....._._----5kV dummy

Figure 14b. Variation in NMR line at 3cm/s flow and applied voltage.

14 NASA/CR—2005-213999 11
Resonance - 6 cm/s flow

7O

5O

3O

10

•-.-0 V

..... ._,....v5 kV

..... _,-_-----5kV dummy

Hz

Figure 14c. Variation in NMR line at 6cm/s flow and applied voltage.

Resonance - 12 cm/s flow

7O

5O

3O

10

•-.-0 V

---_,,-----5kV

....._,_-----5kV dummy

Hz

Figure 14d. Variation in NMR line at 12cm/s flow and applied voltage.

15 NASA/CR—2005-213999 12
Resonance - 25 cm/s flow

9O

7O

J

-130 -110 -90 -70

Hz

-,-OV

...... ,,,---o5 kV

....._,.----5kV dummy

Figure 14e. Variation in NMR line at 25cm/s flow and applied voltage.

Resonance - 50 cm/s flow

9O

7O

'-_ 50

i 10

-130 -110 -90 -70

Hz

•-,-0 V

_--_,----, 5 kV

.....,-,,,-----5kV dummy

Figure 14f. Variation in NMR line at 50cm/s flow and applied voltage.

16 NASA/CR—2005-213999 13
IV. Conclusions

Experimental Conclusions

All three experiments showed null results. The first experiment, constant potential applied to the

proton in the hydrogen atom, produced a null result as expected and explained earlier. The second

experiment, a time-varying potential applied to the proton in the hydrogen atom, also produced a null result

as expected and explained. From the second experiment we have learned that the present theory is

relativistically Gauge Invariant in the context of Maxwell's equations - as it should be.

The third experiment, physical displacement of a proton in a hydrogen atom through an electric field,

was expected to produce a frequency shift in the NMR line. No change was observed to high precision.

This is explained as follows. When a hydrogen atom is placed in an electric field, classically, only one

effect is produced - an induced dipole moment of the atom resulting from the stretching of the electron

orbital. Otherwise, no work is done on the hydrogen atom (except for the exceedingly small amount

involved in stretching) since the work done on the proton is cancelled by the opposite work done on the

electron(due to its negative charge) in the electric field. Furthermore, it is straightforward to show

(Appendix C) that no work is actually done on the proton itself since the electric force on it is precisely

balanced an opposing force, in the field direction, from the electron.

Implied Conclusions

The fact that no work is done on the proton provides a clue to explaining the present null result. The

present theory is a single particle theory and does not take into account effects of other charged particles on

the proton. It simply places the proton in a potential field and assumes there will be a metric effect similar

to that in GR, which is also a single particle theory. It ignores the nearby electron as simply an additional

non-interacting "test particle" in an external electric field. But that is wrong. The electron clearly interacts

dynamically with the proton to cancel forces. Note the term forces.

The same can be stated for GR. If there is a nearby interacting massive particle, then GR will be

inadequate in its present form to deal with this. This is a particularly glaring issue in GR since GR is

independent of the mass of the test particle. GR will simply ask; what is the potential at the test particle

location? If there is a nearby interacting particle in the same external field, GR will simply add the two

potentials (one arising from the source and one from the second particle) at the location of the initial test

particle to predict the effect on clocks. This, however, clearly ignores the dynamical effect of the second

particle on the first. The metric, after all, can only accommodate one particle. That is the difficulty with

the so-called "two body problem" in GR. The source of this difficulty seems to be the lack of a solid

Hamiltonian formalism for GR that can accommodate forces.

A possible way out of this difficulty for both GR and the present theory is to use the concept of work

done on the test particle by the external field and any other interacting particle. This approach still allows a

single-particle metric theory, but accommodates the dynamics of a multiple-particle system as a cumulative

effect on the single particle metric. Thus for the present theory, since zero work is done on the proton,

there should be no dynamical effects and hence no clock change.

In the next section, we shall quantify this approach and show that it is consistent with GR in its present

form, does not alter any known predictions of GR and can be generalized to include the present theory as

well as predict new results within GR.

Relation of the Metric to the Lagrangian and Work Done

Let/_ be defined as the specific Lagrangian or Lagrangian per unit rest energy,

£_ (T-V) (5)

mc 2

where T, V and m are the kinetic energy, potential energy and rest mass of the test particle respectively.

17 NASA/CR—2005-213999 14
Supposeweareatrestatsomeheightz in a gravitational field, so that T=O and V=mgz, then

= gz / ¢2. We have shown (Appendix E) for weak fields and nonrelativistic speeds that the proper time

element can be written in terms of the Specific Lagrangian and coordinate time element as

dr = _ 2[, dr. (6)

Consider two positions in the gravitational field z1 and z 2 = z 1 + h. Assume a proper time interval d'/"1

at z 1 and d'/" 2 at z 2 . Then, approximating the square root for weak fields,

dr2_ 1 g(z2-zl) _l gh

dr 1 c 2 c 2

We can rewrite this in terms of the change in potential energy and Lagrangian:

(7)

dr2 - 1 mgh av)

dr 1 mc m pC2

In general, a variation in the Specific Lagrangian results in a change in the clock rate.

(8)

Referring more concisely to the metric for a single particle, p, we may express the Lagrangian

change more clearly in terms of the net conservative work, We, done on p.

aL -aL +we

P m 2

pC

The change in proper time intervals for the particle between two locations is thus

(9)

dr'

- 1- ALp. dr

The weak-field, non-relativistic, metric for a given particle, i, acted upon by forces and hence net

conservative work, Wci ,done upon it by all other particles in the field is given by

(10)

2Wci )

dgi 2= 1 c2dti2-dxi2-dYi2-dzi 2 (11)

me2

W_i =f_i.aFi . (12) where

We have rewritten the metric in this way because potential is not well-defined except through

potential energy and work, where it is defined as work per trait mass in gravitation and work per unit

charge in electromagnetism.

It can be shown that this formulation of GR does not alter the major known GR experirnents, the

perihelion shift of Mercury, the bending of light and the gravitational red shift to within the current

experimental error. Indeed, the change to the perihelion shift very nearly removes the remaining 0.25%

discrepancy. The measured precession of Mercury's perihelion is 43.11" _+0.21"/century [5]. GR predicts

42.98". The present work corrects this to 43.18"/century, in closer agreement with the measurement.

Application to the hydrogen atom

The advantage of the work formulation is that it is true for any field. In particular, it is true for gravity

as well as the present theory. There are now two possible interpretations for changes in clock rates. The

first is that the potential enters directly through the metric and the second uses the work done on the

18 NASA/CR—2005-213999 15
particle.Assumethefirstiscorrect,andconsiderthecaseofneutralmonatomichydrogenatomsmovingat

aconstant speedthroughanelectricfield.Theratiooftheclockrates,usingequation(8),aregivenby(for

weakfieldsandsmallvelocities)

2

dr

-- = 1 _ (13)

dt mc 2 2c 2 '

where e is the electric charge of the proton, C is a constant, the potential energy is from an electric field that

varies only with the z coordinate, and the velocity is constant and in the z-direction. If the clock rates are

set to be the same at z -- zo (i.e., dT/dt=l) then equation (13) becomes

Note that the constant is given by

dr e[_(z)-_(Zo)]

i = 1 (14)

dt mc 2

C = (p(Zo) mv2 (15)

2e

For the nuclear magnetic resonance experiment we can take dt = 1/ f and dr = 1/ fo where f0 is now

the rest frame frequency. Then

f-1 e[q_(z)-q_(z°)] (16)

fo mc2

Thus, from a potential description only, we are left with an ill-defined consideration of the potential

difference between two points affecting time and would conclude that the proton's clock should change.

This approach clearly does not take into account the forces of the electron upon the proton. Hence we go to

the next approach, consideration of forces and work done to define potential energy change for the proton.

An interpretation based on the work done requires a description of the forces acting on the proton.

Consider the hydrogen atom oriented as in Figure 15. Here the electron is assumed to be a classical point

particle. The electron and the proton orbit about a line fllrough the common center of mass, with the

centers displaced along the direction of the electric field.

P

Figure 15. Hydrogen atom in an electric field

Note that there are two forces acting on either the proton or the electron. There is the internal attractive

force acting on a line between the proton and the electron and the force due to the external electric field.

The component of the internal force along the electric field exactly balances the force due to the external

field. The remaining component of the internal force acts perpendicular to the motion for either the proton

or the electron.

Hence, if the work interpretation is used no work is done on the proton and the frequency remains

the same, resulting in the null experiment (see full discussion in Appendix E).

NASA/CR—2005-213999 1916
The Clock Principle

Potentials, potential energy and work

In the previous section we found a simple Lagrangian formulation that places gravitation on an equal

footing with our theory in regards to changes in the temporal portion of the metric. This serves to simplify

the interpretation of our results by a direct comparison to what is expected of standard gravitation theory

under similar conditions. The Lagrangian formulation deals with kinetic and potential energy changes.

Clocks raised in a gravitational field are at rest in the two positions and can be slowly moved between

them. Thus the Lagrangian becomes simply the negative change of potential energy of the clock, moved

from the lower position to the upper. But, for conservative fields, this is precisely the conservative work

done on the clock. However, we must be careful and can no longer use the word "clock" loosely. When

we refer to "clock" henceforth, we mean the mechanism of the clock. Thus we mean that work is

performed on the mechanism. Clearly all clock mechanisms are driven by energy changes. What is not as

obvious is that the mechanism of any clock must reflect the proper time variations in a gravitational field.

For example, the mass-spring mechanism of a simple clock must somehow change with different heights in

a gravitational field. Similarly, a pendulum clock must exhibit changes in its mechanism. Even an atomic

clock is subject to this consideration. This brings us to the first clock postulate:

1) Every clock has a mechanism which must be held accountable for observed changes in its

measurement of time.

We shall now examine the relationship between work and changes in clock time in a gravitational

field. We use the notation "nc" to mean nonconservative and "c" for conservative. Conservative forces,

by definition can be represented by a gradient of potential energy, V:

/_c = -_TV (17)

Einstein Rocket -- Equivalence Principle for clock changes in a Gravitational Field

We shall employ the famous Einstein rocket "gedanken-experiment" to demonstrate our concepts.

Consider a rocket lifting a mass, m, by a stiff wire in a gravitational field. Let us suppose that we adjust its

thrust to precisely oppose the graviational pull on the mass. The rocket-mass system is now balanced and

hovers, for example, at the surface of the earth. An arbitrarily small external force, {, may now raise the

system to a height, h. This is shown in Figure 16. The force { does no work on the system since it can be

made arbitrarily small. However, upon rising a height h, due to the infinitesimal assistance of the guiding

force, the rocket does work on the mass m. The potential energy of the mass is V1 at the surface and V2at

height h.

Figure 16.

2 m

h

V1-- S Fnc Fc=-mgmg

Balanced force configuration of a rocket-mass system

2O NASA/CR—2005-213999 17
Intlfissituation,thenonconservative forceisthatoftherocket.Tlfisisequalandoppositeto

gravity.Wemakethefollowingdefinitions:

Zz-Z l=h ; d_=dz/_ ; _'=-g/_

; ;,,c---;c

(18)

Then the non-conservative work performed on the mass m by the rocket is

Z 2

W_c = f ff.c "d_ = mgh.

Z 1

and the conservative work performed on the mass m by gravity is

Z 2

Wc = f tic" d_ = -mgh.

Z 1

Now, from the definition of conservative work we have,

(19)

(20)

AV = V 2 -VI = -W c = mgh.

Summarizing, we find for the conservative work done on m:

(21)

W c = -mgh = -AV (22)

Einstein rocket replaced by negative mass

Let us now reconsider the rocket of Figure 16. We can equally well replace the rocket by negative

mass equal in magnitude to the lower positive mass - Fig. 17. The earth's gravitational field repels negative

matter and thus an amount -m will precisely balance the force of attraction on m, a situation equivalent to

the rocket. Note, however, that the inertial response of -m to this is to fall downward along with m.

Nevertheless, since only forces enter the definition of work, no work is done to maintain the position of the

mass dipole and, indeed, just as in the case of the rocket, no work is done by an arbitrary small external

force, 8, that raises the rocket a height h in the gravitational field.

h

F_ m

F

g

Figure 17. Balanced force configuration for rocket replaced by negative mass

We evaluate the work done in this situation. There are two forces acting on rn, /ff

g

time both forces are conservative. Therefore,

Wg =-AVg and W_ m =-AV_ m .

and F_,,. This

(23)

21 NASA/CR—2005-213999 18
Thetotalworkdoneonthemassmiszerobecausethereisnochangeinthekineticenergy,since

=Ipg Ipm ar =0 (24)

However, now the total work done on m is conservative since, unlike with the rocket, there are no nonconservative forces acting. The rigid wire transmits the conservative force from m. Thus we find:

wc +w_., =o (25)

This model of a massive neutral dipole is exactly equivalent to an electric neutral dipole such as the proton

and electron of a hydrogen atom in an electric field. In the latter case, the two forces acting on the proton,

the external electric field and the opposing force of the electron are both conservative. The total

conservative work on the proton is zero when the atom is moved a distance h through a known potential

difference [Appendix E]. That is precisely our experiment. There was no clock change observed.

This leads us to the second and third Clock Postulates:

2) External conservative work done on a clock mechanism is responsible for changes in its rate

when the motion of the clock can be neglected.

3) When the clock is not at rest, the change in its rate arises more generally from the change in

the Specific Lagrangian for the mechanism.

The third postulate takes into account the change when there is kinetic energy present [Appendix E].

These postulates reconcile the observations and considerations of the present experiment with those

observed for gravity. Therefore one should not expect to see a clock change in the imagined massive

dipole experiment since there was no net conservative work done on m.

Inconsistency between General Relativity and existence of a neutral mass dipole

Now consider a clock on the mass m attached to the rocket. The mass m might itself be part of a

clock mechanism. Upon reaching height h, the clock will change.

Assume a proper time interval dz" 1at z I and dz" 2 at z 2 = zI + h. Then, we have shown that the

relation between the two intervals is given by

d_:2 _ 1 gh _ 1 mgh (26)

d,c 1 c 2 mc 2

and is related to the work, -mgh, done by gravity on m. But the mass m cannot distinguish between the

pull of the rocket on the wire from that of the negative mass. In both instances the clock is displaced a

height h and must change according to the single particle metric of GR. In the case of the negative mass,

the Clock Principle states that since no net conservative work is done on m, no clock change will be

observed.

Since we assume GR to be true, either the Clock Principle is false or a mass dipole cannot exist as the

analogue of a charge dipole. Thus, if the clock principle is true, negative mass cannot exist.

If the clock principle is false, then work and energy change are not related in general to time change.

It follows that a metric theory of forces other than gravitation cannot be constructed.

NASA/CR—2005-213999 2219
Clock Conclusions

We have arrived at the following three postulates regarding behavior of clocks and time:

1) Every clock has a mechanism that must be held accountable for observed changes in its

measurement of time.

2) External conservative work done on the clock mechanism is responsible for changes in its rate

when the motion of the clock can be neglected.

3) When the clock is not at rest, the change in its rate arises more generally from the change in

the Specific Lagrangian, defined in terms of work, for the mechanism.

It is generally agreed that in order for a clock to indicate the passage of time, energy change in the

mechanism is required. The mechanism itself is always reducible to a mass or energy. Thus, it appears that

space is coupled to time through energy. When a clock is raised in a gravitational field, work is performed

on the mechanism. Conservative work, Wc, performed on the mechanism changes its rate. Wc includes the

net work done by all conservative forces acting on the mechanism. If W_ =0 so that applied forces

opposing conservative forces are also conservative, then there will be no observed clock change. In general,

a change in the specific Lagrangian L will result in a change in the clock rate.

If conservative work changes clocks then negative mass cannot exist.. This does not preclude the

existence of negative energy since the general energy-momentum relation has two roots,

1

E = +_[p2c2 +

m2c 4

of which the positive root, in the rest frame of a particle of mass m (p=0), is the famous Einstein energymass relation.

V. Future Directions

The third experiment performed in this work leaves open the possibility that this theory and all metric

theories of electromagnetism are invalid. One may inquire as to whether there is a definitive experiment,

having learned from the present work, that would settle this issue. In fact we have identified two possible

experiments - both involving free charges as opposed to the bound systems of the present work. These

would provide unambiguous tests of the theory. Both test the Lagrangian formalism which is the

generalization of the Clock Principle. These are; (1) an Electron Spin Resonance test of the Lagrangian

formulation, and (2) a classical measurement of the quantum Aharonov-Bolma Effect. We will focus on an

experimental description of (1) since (2) is considerably more exotic and exceedingly difficult.

(1) Microwave Cavity Test of the Single Particle Lagrangian Formulation

A conceptually simple, but technically difficult, test of the Lagrangian interpretation for a classical

theory unifying gravity and electromagnetism would utilize a diode vacuum tube- consisting of two

charged plates, at a constant potential, with electrons between the plates. The electrons accelerating from

rest between the cathode to the anode would have a constant total energy (T+U=0) but the Lagrangian (TU) would change by a factor of two in the metric. If Electron Spin resonance (ESR) measurements were

performed on the electrons in the diode, then there would be a relative change in the resonant frequency of

each electron as it moved from the cathode to the anode given by Eq. 10, dr" / dr = 1+ AL = 1+ 2T / mc 2.

p

The result would be a broadening of the line by a factor of two over the shift due to the Lorentz time

dilation - a readily distinguishable effect in the proper experiment.

Fairly large voltages (5kV or greater) would have to be used to produce an observable broadening. But

large voltages will also result in large accelerations and short resident times for the electrons in the diode.

NASA/CR—2005-213999 2320
Henceit willbenecessary totraptheelectronsbetweentheplates.Tiffscouldbeaccomplished by

alternatingvoltage between the plates at a sufficiently high frequency to prevent most electrons from

reaching either plate. If the electrons are to be confined to within a centimeter, then the field strengths

required for an observable shift in the ESR frequency will result in microwave frequencies. A microwave

magnetron would be a more sensitive device in which to test these ideas. Advantage could be taken of the

resonant cavity to excite the electrons but the measurement must still utilize ESR within the cavity so as to

be performed in the electron rest frame as required by the theory.

(2) A classical measurement of the quantum Aharonov-Bohm Effect

In the Aharonov-Bolun Effect [9], an electron beam is split and impinges on a long, fine, solenoid,

whereupon the two electron-wave components are permitted to interfere on a distant screen. Tiffs is all

arranged in an electron microscope. A long solenoid is equivalent to a so-called "infinite solenoid". The

internal field, B, is constant with flux _. Although the external field is zero, the vector potential is

nonzero. Only magnetic fields generate observable forces - vector potentials are unobservable and do notso there are no forces on the electron beam outside the solenoid. With the magnetic field off in the solenoid,

the two interfering beams create an interference pattern consisting of bright and dark bands on the screen.

With the field on, the pattern is observed to shift even though there are no forces acting on the beam. Tiffs

effect is attributed to the quantum phase of the correlated beams which is directly affected by the vector

potential. Tiffs is seen as a strictly quantum effect. However, we have shown (Appendix G) that the

present theory exactly accounts for it flarough the Lagrangian approach. The expected effect is a small time

delay of one beam relative to the other as they pass the solenoid. In order to convert tiffs real time delay to

a phase shift, to agree with QM, one must associate a frequency with the electron beam. That frequency is

the so-called "zitterbewegung" frequency and is the natural frequency to choose.

NASA/CR—2005-213999 2421

Appendix A. On the Compatibility of Gravitation and Electrodynamics

The present work is a highly condensed summary of a theory that is shown to correctly represent both

gravity and electromagnetism classically within a Pseudo-Riemalmian geometry[ 1]. It is grotmded upon a

new affine connection which derives from an electrodynamic torsion acting upon charged particles in an

electromagnetic field. The geometry is that seen by a charged particle and depends upon the

electromagnetic potential at the particle location within a space-time that can include gravity. The present

theory applies to weak fields and therefore does not couple the two fields. Present gravity experiments

remain unaffected and the correct electromagnetic potentials derive from metrical solutions of the field

equations. Thus both the Einstein equations and the Maxwell equations are satisfied by the metric solutions.

It is shown, however, that pure electromagnetic geometry should produce effects similar to those of

gravity. These effects are resolved within the particle reference frame since that represents the simplest

solution. The result is a consistent field theory with solutions and an experimental prediction.

In ref.[3] the new connection is derived from first principles but can be found effectively from

introducing an electrodynamic torsion. The same connection was fonnd by Schr0dinger [2] but not

identified or recognized as such - indeed he discarded the antisymmetic part of his connection(pure torsion)

since it did not contribute to the motion.

This torsion, given by

_'_ = K___u'_F/iv, (1)

[,,v] 2

does not alone contribute to the Lorentz motion of a charged particle. Its properly symmetrized

contribution to the connection (eqn. 2 below) does, however. The pure torsion, eqn. (1) alone, was shown

to be the source of the well-known Thomas Precession term in the "classical spin" equation of motion. It

cannot be physically ignored. It can be said that the connection is fully determined by the metric tensor and

the torsion as seen in eqn.(2). J. Vargas [6] has independently found the same torsion tensor from a

Differential Forms approach.

The electrodynamic connection is:

Fx = -lg_(Ovgv_ +Ovgw -O_gvv)

vv 2

F_. = Av;_ - A_;_1

(2)

u_ is the test particle 4-velocity and v; = -e/mc 2 .

Torsion is traditionally identified with angular momentum, which, it seems, has not borne physical

fruit in the present context. The present work departs from tradition and clearly identifies it with

electromagnetic objects.

In a second paper[3], the new Einstein tensor is developed from that connection. This follows the

work of J. Schouten [7] exactly. The new Einstein tensor has both charged and "displacement" currents as

sources. The electromagnetic energy density is ignored as contributing only higher order massive effects

on the geometry in comparison to the new current terms. Solutions of the new field equations are presented

that yield the classical electromagnetic potentials together with gravitational potentials in the appropriate

cases. These solutions include (1) spherical electric plus gravitational field; (2) cylindrical line charge

25 NASA/CR—2005-213999 23
fieldand(3),cylindrical uniformmagneticfield.

Riemann and Einstein tensors with torsion:

The Riemann tensor is given by:

g e_

-E : + l¢---2-(uvF/_;o +u/fv_;o +u_F/_v.o-uoF/_.v-u/_Fo_;v-u_F/_o.v) R, vo R_vo , , ,

This Riemann tensor satisfies the following Cyclic Identity upon alternating indices :

(3)

(4)

In deriving (3), covariant differentiation was passed through 4-velocities. Thus the identity (4) implicitly

assumes this. Eqn. (4) can be derived from a Differential forms approach, as shown in the Addendum,

without this assumption. The Einstein tensor is given by:

K" r

G,v = G_v +T(u, Fv;, + uvF/2, + u'F/_v;, - 2u'F/_,;v - 2g,vu'F_:.% ) (5)

The Einstein tensor separates into symmetric and antisymmetric parts(eq.6). The symmetric part provides

the metrical field equation while its conservation yields the Maxwell source equations as our second field

equation. The antisymmetric part yields the homogeneous Maxwell equations as a solution for arbitrary u

since it must vanish. Eq. (4) also yields the homogeneous Maxwell equations.

~ K" °F

- -gvo ,;r g,o v;r G(,.) =G,v Tu ( vo;; + F_v;o F_ - F_ + 2g,vF,_;_)

- 1¢ °F , ,

where real currents in (6) are defined from,

.3

F_;r =-J/l

(6)

Following Schr6dinger, the Einstein tensor on the right of eq.(6) is taken to vanish since this is purely

gravitational and the electromagnetic field is assumed to contribute a negligible energy density compared to

the massive source for this solution. This leaves G-tilda on the left with no apparent structure except as

determined by its electromagnetic torsion effects on the right. Since we have shown that this tensor is

symmetric and pseudo-Riemannian we assert that its structure will be that of the usual Einstein tensor but

that it now is a more general function of a metric tensor that includes the effects of electromagnetic

torsion. We can drop the tilda notation since g-tildas on the right will only contribute in second order. This

leads us to the new Einstein equation.

Field equations and solutions:

The governing equations are chosen to be (for vanishing charge current sources):

=-Tu +

F_£. =0

(7)

These equations are functions not only of the metric and electromagnetic fields but also of the test particle

4-velocity. The dependence on the 4-velocity is not unexpected since in classical correspondence the

velocity-dependent Lorentz force must be accounted for and the connection describes precisely those

26 NASA/CR—2005-213999 24
forces.Thishasbeenaboneofcontention, butthisisaclassical theoryandthatislife. Thetestchargerest

frameischosenforsimplicityinthesolutions.Conservation oftheEinsteintensoryieldsexactlythewave

equationfortheEMfieldandthusMaxwell'sourcequations:

Gw ;_ = _ __K2u ° (Fvo;_ ( + Jro, - Jo ;v ) (8)

This vanishes identically since it can be shown that the wave eq. is

F/,o;r; r = Jo;,, - J,,;o

and thus the new Einstein tensor is conserved. Solutions of Eq.7 for the three cases described above are:

(1) Spherical Gravity plus Electric Field

The spherical interval is:

dr 2 = e" dt 2 _ e_dr 2 _ r2dO 2 _ r 2 sin 20dO 2 .

We ignore charge-mass (QM) product terms for our "zero coupling approximation" to find solutions:

= ; = 1- -* (9)

r r

Q E =-

r 2

(2) Line Charge Electric Field

The cylindrical interval is given by:

The solutions are:

dr 2 = eV dt 2 _ e_ dr 2 _ r2 dO 2 _ dz 2

e _ =1

e v =e4_rAln(r/R)_ l+41cAln(R )

2A

Er

(10)

We have taken KA << 1. R is a constant of integration. A is the line charge density. The temporal

metric coefficient fits the standard electric potential form, 1-2Kqbel, analogous to the gravitational form,

l+2qbg r which was also the case for the spherical electric field solution, qbel is precisely the Maxwell line

charge potential.

(3) Cylindrical Uniform Magnetic Field

For this case, the classic Rotating Frame metric was found to be an exact solution of the new Einstein

equation. It can be stated that the magnetic field is equivalent to a rotating spatial frame. The interval is

given by:

dz "2 = (1 - c02r2)dt2 - dr 2 - r2 d O 2 - 2cor2 d Odt - dz 2

NASA/CR—2005-213999 2725
Thesolutionsare:

Fol_o_, = coBr

go2 = 1¢Br2

Fo =(_x/3) =coB •

l ntui r r , F1 =-B 2ntui

(11)

m = -_:cB is the cyclotron frequency for the orbit. EM field solutions are shown here with respect to

standard unit vectors. B is a constant, the magnetic field. Note that the solutions are given with respect to

standard "unit vectors". The solutions as found from the theory are with respect to "base vectors". The

relation between unit and base vectors is given by:

Thus, for example,

Fl_vbos_ = _]lgm_g w F/_v.._' (not summed)

in the magnetic solution:

F12_o_ = -Br ; Fol_o_, = o)Br

Mathematical issues

In order to obtain the above results, covariant derivatives were passed through the 4- velocities. The

motivation for this, other than it greatly simplifies the work and gives physically correct results, is that the

4-velocity is a parallel vector field on the curve (geodesics) hence invariant to order _: in the field

equations on the right. This was not a rigorous assumption at the time, just a compelling one. Below is a

rigorous proof that 4-velocities effectively pass through covariant differentiation with respect to the

coordinates in the Appendix using a Differential Forms approach [8]. The proof derives the fundamental

Identity (eqn. 3) that Riemann satisfies and clearly shows that the four velocities fall out "as though they

are constant". It therefore asserted in this work that differentiation can pass through the 4-velocities. A full

rigorous approach might use a Finsler Space context (J. Vargas uses such an approach in his works) in

which the 4-velocities are independent variables. But that is beyond the scope of the present work.

ADDENDUM: Validation of field equations using Differential Forms

The curvature 2-form for a differentiable manifold X n referred to local coordinates xv endowed with a

general nonsymmetric connection is given by:

f2.,, = R._,odff /x dx ° (A1)

The torsion 2-form is given by:

f2 e = 1 S._c_dxV A dxC_ (A2)

2

where,

°

FI/_o I = S./_o 2

(A3)

is the torsion tensor of this work. This torsion, although a direct product of a vector field on a curve in X n

with a tensor field on X n is nevertheless valid with regard to defining a differentiable manifold since each

field is itself differentiable on its domain. It can be shown that the exterior covariant derivative of the

torsion 2-form is related to the curvature 2-form from the well known result [8]:

NASA/CR—2005-213999 2826
E

DF_6" = - F_._ /x dx _ (A4)

The crucial point of this result is that with differential forms, the covariant derivative is the absolute

derivative, so that partial covariant derivatives, relevant in coordinate representations, need not be

considered at all at this point.

Now the vector ue in general is the sum of a geodesic gravitational component and an electromagnetic

contribution of order _:. We can ignore the electromagnetic contribution to the geometry since it will enter

the field equations as order _:2. Thus, since ue is then a vector on a geodesic, we may conclude without

hesitation or question that

Du6" = 0 (A5)

We now expand (A4) using the above results to find:

DS._o/x dxV/x dx ° = 1R._vdx_/x dxV/x dx °

(A6)

Now utilizing the covariant partial on the spatial field F yields:

lcu F_v;odx A dx A dx ° = t_.o_vax A ax A ax

(A7)

In component form this becomes precisely the equation, (4) above,

= •

(A8)

Thus Eq.(3) is a valid result and the fact that the partial covariant derivative passes through ue is an artifact

of the component representation of a possible Differential Forms approach to this theory. I choose to

adhere to the component approach and adopt the rule that covariant partial differentiation with respect to

local coordinates, xv, in X n will pass through vector fields defined on a geodesic in X n.

Theory summary and conclusions:

We have shown that classical electrodynamics, neglecting radiative effects, can be embedded in a

geometric framework in a self consistent way through the solutions of the field equations for the

appropriate metrical and electromagnetic field variables. In the process, Maxwell's equations fall out

naturally from conservation and symmetry requirements. From these solutions, not only are the correct

electromagnetic fields fotmd for a spherical electric field plus gravity, a line charge electric field and a

uniform magnetic field, but also the expected electromagnetic potentials appear in the metric tensor

alongside gravitational potentials. The procedure by which this is accomplished is partly grounded in

Schr6dinger's affine theory through a new "electrodynamic connection".

All that we have shown here is consistent with what is currently observed. Coexisting electric and

gravitational fields act independently, within the scope of present measurements, on charged test particles,

yet appear to share similar geometries. A classical neutral particle can pass with impunity through an

electromagnetic field suggesting that electromagnetic fields do not influence the global geometry. In a

sense we have a "relativity of geometry" since test charges with different K experience correspondingly

scaled geometries in their rest frames. Indeed, electromagnetic forces are velocity-dependent which stems

from the nature of the Lorentz transformation.

Finally, it should be stressed that we have made several simplifications. We have only included order

K terms from the start. We have ignored the energy-momentum tensor. For the electric field solution we

chose the rest frame and assumed spherical symmetry which is not strictly correct. We also ignored weak

coupling terms.

NASA/CR—2005-213999 2927

Appendix B. Metric for Constant Fields

For constant electromagnetic fields (i.e., constant in space and time) a metric accurate to first order

in the fields can be written as

e----_(ll,/,Fvr + u,vF/_r )x _ . (1)

g/_v = riley + mc 2

Where ///1v is the Lorentz metric in rectangular coordinates, Fpv is a constant anti-symmetric tensor that

will be identified with the electromagnetic fields, u/_=dx_/ds is the four velocity of the particle, x _ are the

coordinates (ct,x,y,z) and

= p v

ds 2 g uvdx dx . (2)

Then to highest order in the fields

g/_" = 17/_ ___ e (u/_FV _ + uVF/_ )xr. (3)

mc 2

To first order in the fields the connections are

Fa _ e T]a2(bl/_Fv2 +blvF/_2). (4)

_v 2mc=

FCruv

The geodesics become

Note, to first order in the fields the Lorentz metric, r/p_, raises and lowers indices. Eq. (4) can be written as

2mc=

du a e

2

ds mc

which are the correct equations of motion.

V O"

U F v (6)

The rank four Riemann curvature tensor can now be calculated. The Riemann tensor contains

both products of the connections and partial derivatives of the connections with respect to the coordinates.

The products can be neglected since they are second order in the fields, and the derivatives vanish. Hence,

the Riemann curvature vanishes and the space is Riemann flat (i.e., it is equivalent to a flat Lorentz space.)

Consider now the case of an electric field aligned along z. Then

I°°°!1 0 0 0

Fptv =

0 0 0

-E 0 0

For arbitrary four velocities the metric becomes

(7)

g/.tv =

1 _ 2eu°Ez eulEz eu2Ez e(u°Ect - u3Ez)-

mc 2 mc 2 mc 2 mc 2

euiEz - 1 0 eu_Ez

mc 2 mc 2

euzEz 0 - 1 euzEz

2

mc mc 2

_ e(uoEct-u3Ez ) eu_Ez euzEz -1 2eu3Ect

mc 2 mc 2 mc 2 mc 2

(8)

30 NASA/CR—2005-213999 29
Notethathecomponent

isasexpected.

2euoEz

g00 = 1 q (9)

mc 2

The metric in equation (8) can be shown to be Riemann flat. Hence, the metric can be transformed

to that of a freely falling (inertial) reference frame.

Consider the case of a particle starting at an arbitrary velocity. The components in the x and y directions

can be made to vanish by rotating the coordinates so that 141=142=0. The metric then becomes

guy =

1 + 2eu°Ez 0 0

2

mc

0 -1 0

0 0 -1

- e(u°Ect - u3Ez) 0 0

2

mc

e(uoEct - u3Ez )

mc 2

0

0

2eu3Ect -1

mc 2

(9)

Performing a Lorentz transformation so that the velocity in the z direction vanishes(then uo=l), the metric

becomes

g[_tv =

1 + 2eEz

2

mc

0

0

eEct

mc 2

eEct0 0

mc 2

-1 0 0

0 -1 0

0 0 -1

(10)

Equation (10) can be made diagonal by a change of coordinates. If we set

z = z'+ eE(ct')2

2mc 2

(11)

while t' = t, X' = X, and y' = y the new metric becomes

p

g/a v =

1 + 2eEz" 0 0 0

mc 2

0 -1 0 0

0 0 -1 0

0 0 0 -1

(12)

31 NASA/CR—2005-213999 30
Appendix C. Potential Due to Two Oppositely Charged Circular Loops and Plates

Consider two charged loops with charge per trait length ___O- as shown in Figure 1. The loops have a

radius, b, and are centered on the z-axis with a separation of L along z. The wires of each loop consist of a

circular cross-section of radius a.

Y

Fig. 1 Charged Loop Geometry

The centers of the loops are described by

L

x=bcosqz, y = b sin qz, z =+--

2

The potential, (,0, at all points is given by

l 2Jr

(x-bcos_ff) 2 + (y-bsin_ff) 2 + (z-L2) 2

21r

1 ! obdN 4_reo _/(x-bcosN) _+(y-bsinN) _+(z+L/2) _

(1)

(2)

Using polar coordinates, ( X = r cos 0 and y = r sin 0 ) and letting ¢' = (7/" - _ + 0) / 2, equation (2)

becomes

(7

(p=--

7(_ 0

(7

;rt'8o

_!: dO

_(l+b)_ +(

_ __ ,_!2 de

(1 + b) _ + (z+L/2,2 b ) - 4rsin2 b ¢

z-L/2) 2 _4rsin 2 ¢

b b

(3)

The integrals in equation (3) are complete elliptic integrals of the first kind, K(m), where

K(m) =

_/2 dO

fo _/1-msin2¢

(4)

NASA/CR—2005-213999 3231
Equation(3)cannowbewrittenas

(5)

b 2.)2 4(r / b) where /9_+ = (l 4- ) 2 4- ( z 4- L / , and m+ -

b

Figures 2 and 3 illustrate the potential between the loops. The potentials are related to the charge density

through

V0 1

- [(p(r =b+a,z =L/2)+fp(r =b-a,z =L/2)] (6)

2 2

Figure 2 demonstrates that the potential is fairly linear over the center forty percent of the axial distance

between the loops and does not vary significantly over the radial coordinates. Figure 3 shows a variation of

less than ten percent in the potential at the surface of the loops.

The solution between two charged wire loops can be readily extended to two finite disks by

integrating over the radius of the loops. If the voltage is constant on the surface of the disks then a charge

distribution that varies with the radius will be required. For the case of two disks of outer radius b and

tlfickness h the potential is given by

For a charge distribution given by

p(r') = Po , (8)

the constant P0 can be set to obtain the correct voltage on the plates. Figure 4 shows the variation in the

potential between the plates for the apparatus at Washington University.

O

cO.

5.00E-01

4.00E-01

3.00E-01

2.00E-01

1.00E-01

O.OOE+O0

-1.00E-01

-2.00E-01

-3.00E-01

-4.00E-01

-5.00E-01

-1.00E+O0 -5.00E-01 O.OOE+O0 5.00E-01

--db=O.O

" r/b=0.1

r/b=O. 2

....... r/b=0.3

-- r/b=0.4

--r/b=O.5

1.00E+O0

z/L

Fig. 2 Potential Between Two Wire Loops for a=O.25mm, b=1.5mm, L=lOmm

NASA/CR—2005-213999 3332
0.60

0

0.50

0.40

0.30

0.20

0.10

0.00

/

/

60 120 180 240 300

Angle - deg

Fig. 3 Potential at Loop Surface for a=O.25mm, b=1.5mm, L=lOmm

360

6000

|

"6

n

4000

2OOO

-2000

-4000

-6000

-- r=O.5mm

-- r=l .Omm

.........r=1.5mm

.........r=2.0mm

-- r=2.5mm

-1.0

Fig. 4

-0.5 0.0 0.5

Z/L

Potential for Disks, b=1.5ram, L=lO.mm

1.0

NASA/CR—2005-213999 3433

Appendix D. Potential for Hydrogen in Electric Fields

For a single hydrogen atom in an electric field, P, the energy, E> is

lmev_ 1 2 + -_ mp Vp -Jr

q pqe

4/rc 0 _p -F_

t-qp_.P+qe_.P=E _. (1)

qp is the charge on the proton (+e), qe is charge on the electron (-e), mp and me are the masses of the proton

and electron respectively, Vp and v_ are the speeds of the proton and electron respectively, and _, and

are the position vectors of the proton and the electron respectively. In rectangular coordinates the position

vectors can be written as

rp = Xpl + y pJ + Zp£: , (2)

and

_e= x/+ yeJ +ze£

Changing the velocities to momentum and substituting for the charges, equation (1) becomes

(3)

Where

p2 p_ e 2

+ t-eF'.(Fp - Fe) : E_. (4)

2m e 2rap 4roe0 r

r = _/(Xp --Xe) 2 + (yp - ye) 2 + (Zp - Ze) 2 .

For a field in the z direction (/_ = F£ ), the wave equation, from equation (4), becomes

(5)

.... h2 V2qb h2 Vp_2 e 2qb _-eF(zp - z e )_ = ih O____

2m e 2rap 4rCgor Ot

(6)

Where qb is the wave function,

3 2 3 2 3 2

V 2 -

e ax_ +7-7+ aye az_ '

and

3 2 3 2 3 2

V 2 - +-U-T + .

Converting to center-of-mass position, R, and relative position, ?" coordinates, where

(7)

(8)

and

Then

(m_+ me)_ =m/_ + m/e,

r =G -_p.

m e

mp+ m e

(9)

(10)

(11)

35 NASA/CR—2005-213999 35
and

__/_-_ mp

mp+ m e

The wave equation (6) now becomes

r. (12)

h2 2

Vzqb+ h_ VZqb__ eZqb eFz_=_ihO_____

2M 2/,t 4n-c0r Ot

(13)

Where

M =mp + m e , (14)

rupee

It - , (15)

M

02 02 02

_2_

OX 2 + Oy 2 +mOZ 2

(16)

and

02 02 02

V 2 -

0X 2 _ 0Z2 '

(17)

For the steady response we can use separation of variables, where

• (X, Y, Z, x, y, z, t) = q_(X, Y, Z)qz(x, y, z)e i(g+u)'/r' . (18)

h2 h2 e 2

--V2_F +--V2qz-_ eF z = -E + E .

2MW 2/,t q/ 4n-c0r

(19)

We can take

Then

h 2

IV2_ =E_.

2M

--h2 V 2_ff_[ e 2_ eFz qz = E qz

2/,t q/ 4n-c0r

(20)

(21)

Classically the potential energy at the electron's position, Ve is

e 2 m

Ve - P eFz - eFZ,

4n-80 r M

(22)

and at the proton's, Vp, is

e 2 m

Vp - e eFz + eFZ. (23)

4n-e 0r M

Note that the total potential energy is the sum of the last two terms in each equation (22) and (23)

plus one-half the sum of the first terms in equations (22) and (23). That is the total potential energy is the

36 NASA/CR—2005-213999 36
sumofthepotential energiesduetotheexternalforcesplusone-halfthesumoftheinternalforces,since

theinternalforcescanonlybecountedonce.Weshouldalsonotethatthetotalenergyfortheelectronand

theprotonmustincludethekineticenergyofeach.

Thepotential,(p, appearinginthemetricingoo is

v

(/9--

mc 2

The potential for the electron or the proton can be found by substituting Ve or Vp for V in equation (24)

respectively. The wave function can now be used to find the expected values for the potential of the

electron and the proton using respectively,

(24)

(25)

and

q/_op#d d .

RF

Where U-is the complex conjugate of u, and the wave functions have been normalized using

(26)

and

_q'dk 3 =1,

R

_ _dF _ = 1.

F

From equations (24) through (27), the expected potentials for the electron and the proton become

(27)

(28)

and

e2____ f _ l _tdF3

(_)= 4¢v£0rnec2y- r

e 2 1

Mme mpeF ! eF f_Z_d_3 (29) c 2 _z _dF 3 - -------T

= meC

Mmpc 2 = mpC -

The first two terms in equation (29) are in the energy for the Stark effect. The first two terms in equation

(30) are at least me/mp times smaller. The change in energy for the Stark effect in Hydrogen is (given in

many texts) as

9 3 2

Z_E Star k =- aoF , (31)

4

where ao is the Bohr radius. For a field of 10keV/cm, the change in energy for the electron is 0.82x10 6

mS and for the proton is 0.45x10 9 mpC2. The last term in equations (29) and (30) represents the change in

the potential due to the motion of the center-of-mass of the hydrogen atom. Tlfis can occur due the forced

or free convection (i.e., flow) of the hydrogen or just due to the random motion of the atoms.

Consider the case of random motion of the atoms. The mean free path of the atoms at 300K,

10SPa, and using a cross-section of 2x10 10m is about 3.4x10 7 m. Then the collision frequency is about

0.14 ns and in 1 ps there will be about 7100 collisions. The total distance traveled during the 1 ps is 3x10 5

m. Hence, the potential change in the metric for the proton ((pp) is 3x10 s. For the electron, the potential

change in the metric ((p_) is 6x10 5.

NASA/CR—2005-213999 3737

Appendix E. Relation Between Metric, Lagrangian and Work on a Clock

We have seen that a possible explanation for our null result is that clock changes arise from work

done on the system. In this case the system is a proton in a hydrogen atom. In order to permit an

interpretation of the data it is necessary to relate the concept of mechanical work to the single particle

metric of GR. We shall see that this approach will enable us to generalize this concept to any physical

forces, in particular, the electromagnetic forces in the present theory. Below, we derive the relationship

between the work done on a particle and it's space-time metric.

The Schwarzchild metric in rectangular coordinates, for weak fields and small velocities (i.e.,

[d(ct)] 2 >> dx 2 +dy 2 +dz2),is

We may relate the gravitational potential, ¢ ,to the potential energy of the test particle, g, by simply

rewriting this equation, recalling that inertial and gravitational mass are equivalent:

(la)

where

1+ 2V

ds 2=_ _ c2dt 2_dx 2_@2_dz 2,

GMm

V = me -

r

and m is the particle mass, c is the speed of light, G is the gravitational constant, M is the mass of the

central body and r is the distance from the central mass.

(lb)

(2)

The equations of motion for the metric above can be found by minimizing the action, I,

r8

I = I mcds

rA

where ds = cdI2. Substituting equation (1) into equation (3)

(3)

I:Im . j(1+2v 7Fjt J-7Lt ) ta -;l

rA _\ rA

(4)

L is the Lagrangian function for the particle.

The conjugate energy, E, and momentum, /3, are given by

dt OL dF 3L mc2(l+ 2V "d-r - dr

E- (--_t _: t ---m-_c 2 and _b- (dF_-m (5a,b)

The metric can now be used to write a relationship between the energy and the momentum with the aid of

the expressions in equations (5):

1

m2c 4 -- __ E 2 _ p2c2

2V

14---

mc 2

(6)

NASA/CR—2005-213999 3839
Solvingfortheenergyandtakingthepositivesquarerootyields

I __ ( + 2V )o2c 2

E = mc 2 1+ 2V + 1

mc 2 _ f

(7)

For weak fields and small velocities equation (7) becomes

2

E = mc 2 +V + p---

2m

(8)

which is the correct nonrelativistic expression for the total energy.

Minimizing the action will result in equations of motion which are simply the geodesics.

Assuming small velocities (i.e., dt / dr = 1 and weak fields the equations of motion for the rectangular

components are

d_ _V

- (9)

dt m

Taking the dot product of the rectangular components of the velocity, "¢, on both sides of equation (9) and

integrating

2

! my2 + V = p + V = constant. (10)

2 2m

Hence, the energy, E, in equation (8) is also constant.

For small velocities and weak fields the last equation of motion is

d2(ct)_ 2 _'V. d?_ 2 dW (11)

ds 2 mc 2 cdr mc 2 cdr

where use has been made of ds = cdv and W is the work done on the particle. Integrating once

dt 2W

-- = C +-- (12)

dr mc 2 "

dt

The clock rates can be synchronized by setting -- = 1 at W = 0. Then C = 1 and equation (12) can be

dr

written,

dr 2W

- 1 (123

dt mc 2

A similar result can be obtained from the metric in equation (1) where

tY; t

(13)

(

NASA/CR—2005-213999 3940
Thisreducesto

dr __]_--q_-_T L

or --= _tl_L_ _ £-- 2 (15)

dt mc

where L is the "specific Lagrangian, and T is the kinetic energy. Note that there is an arbitrary constant

in the potential energy that can be set so that dT/dt=l when the clocks are synchronized. Equation (14) is

exactly the same as equation (12'). This can be shown by noting that dL=dT-dV, while from equation (10)

dT =- dV=dW. This makes dL=2dW or L=2W+C and equations (12") and (15) are identical.

Equation (15) is the desired relationship. For weak fields this can be approximated as

d_"

-- = 1-Z. (15a)

dt

4O NASA/CR—2005-213999 41

AppendixF. TimeDilationand Work Done for a Hydrogen Atom

The Clock Principle can be used to find the changes in the clock rates if the work done is substituted into

the space-time metric. Recall that the Schwarzschild metric, for small velocities and weak fields can be

written as

c2d_-2= 1-.}- 2V__lc2d[2_dx2_dy2_dE2 '

mc j

and that equation (1) can be written as

where L = T - V is the Lagrangian. The work done in the particle by all the forces is

w = P.df,

I

where dr = dx[ + dyj + dz/_. If the forces are conservative, then F = -VV. Hence,

W : -fvv. d?" : -V.

3

(1)

(2)

(3)

(4)

The constant resulting from the integration can be neglected by calibrating the clocks so the rates are equal

when V is zero. Using the work done the Lagrangian can now be written as

The metric now becomes

L=T+W.

dr 2=I1 2(T+W)-

L mc 2

(5)

dt 2 . (6)

Consider the case of a hydrogen atom in an electric field. A stable configuration exists with the hydrogen

atom oriented as in Figure 1. Let the proton be at

Fp = Xpi + yp} + zplc ,

and let the electron be at

_ = xe[+ yej+ ze/_.

The distance between them is

R 0 = _/(Xp --Xe) 2 + (yp -- ye) 2 + (Zp --Ze) 2 .

(7)

(8)

(9)

/

/

/

/

P+ _-R_

Y

Figure 1. Hydrogen Atom in an External Electric Field

41 NASA/CR—2005-213999 43
Thepotential energyatthelocationoftheprotonisgivenby

e2

V - eEzp,

4ZCoR 0

where e the magnitude of the charge on the electron, /_ is the electric field strength, and e0 is the

permittivity of free space. The force on the proton is given by

(10)

F=-V' v,

P

where V p is the gradient with respect to the proton coordinates xp, yp, and zp.

Then

/_= e2[(xp -xe)t+(YP-Ye)J+(ZP- ze)k] ',-eEk.

4rCSoR3/2

(11)

(12)

The equilibrium angle for the configuration in Figure 1 is given by

eE _ Zp - z e

sin cr =

(e2/ 4ze0R ) R0

The force now becomes

(13)

_= e2[(Xp-Xe)l+(Yp-Ye )_]

4zeoR3/2 (14)

In order to calculate the work done assume the projected orbits of the electron and the proton are circles in

the x-y plane, then

Fv = Rp [7"cos(c0t) + jsin(cot)]+ VotfC, (15)

p

The work done on the proton is

where t is the time, v0 is the speed of the proton in the z-direction, cois the angular velocity, and Rp is the

radius of the proton's orbit given by

me R 0 coso_ (16) Rp -

mp + m e

The quantities me and mp are the mass of the electron and proton respectively. The force can now be

written as

e2[[ cos(rot)+ _ sin(mt)]

(17)

4_CoR _

w:f;.drp :0 (18)

Since the kinetic energy is also constant, there will be no change in the clock rates associated with the

proton.

NASA/CR—2005-213999 4244
Appendix G. Quantum Aharonov-Bohm Effect Consistent with New Theory

The spacetime metric in the Theory of General Relativity is basic to an understanding of the four

dimensional motion of particles in a gravitational field• For small fields and small velocities the interval

can be written as

dr 2 = (1 + 2--_-_)dt 2 -l(dx2 + dy 2 + dz2). (1).

c _ c _

Where q9 is tiae potential energy per unit mass, 2" is the proper time, t is the coordinate time, x, y, z are tiae

rectangular coordinates, and c is the speed of light. An extension of General Relativity to electromagnetic

fields can be developed by considering the potential for a gravitational field due to a point mass M at the

origin• For tlfis case tiae potential is given by

-GM -GMm

(10 --

r mr

Where r2 = X2 + y2 + z2 is tile distance from the source mass to the particle of mass m, and G is the

gravitational constant. For small electromagnetic fields and small velocities,

(2)

e(-_+_)

_0- (3)

m

Where v is tiae particle velocity, _ is the electric potential A is the magnetic potential, and e is the charge

of the particle. For the case of no electric field the interval in equation (1) becomes

dz "2 : (1 + 2 ev--_3 )dt2mc--_ (dx2 + dy2 + dz2) "

(4)

The relative rates of proper time and coordinate time can now be readily found for small fields and small

particle velocities

dt _ l eYES4 l ( v ) 2 F , (5)

dr. mc 3 2t,7)

• The elapsed coordinate time between two spacetime points 1 and 2 is given in terms of the proper time by

t>2=f:[1 eFG4 l(v_21

_- de'.

mc 3 7(.7)

(6)

In the Aharonov-Bolma experiment over the two distinct patias (see Figure 1) tiae difference in arrival times

is

At=t3_4_t>2=f:[1 eFG4mc 3 t_l(v_21d l r-2_c) -f2 I1 1 eFG44-1(v_ 3 -2(7) 2]mc Jdr

or, 3 eF_ "]. 2(eFG4)/

(7)

NASA/CR—2005-213999 4345
Overthepathsfrom2-3and4-1theparticleisataninfinitedistancefromthesolenoidandthereisno

contributiontothedifferenceintimes.Hence,wecanwrite

( efi_dr ]l,r efiO:lg

At=_( mc3 f =_ mc----- S

(8)

Where C is the closed contour 1-2-4-3-1 in Figure 1. The quantity _ AE]J_ is just the magnetic flux

qb contained witlfin the contour C. Hence,

eqb

At - (9)

mc 3

In the Mmronov-Bolma experiment, the quantum phase angle difference, Ace is measured making equation

a_eqb

Ao_ = (oAt -

mc 3 •

Where cois the frequency of particle in the rest frame of the particle, that is

(10)

mc 2

(oh

(11)

Equations (10) and (11) give for the phase difference

eqb

AO_=--.

hc

(12)

This is the correct expression for the Mmronov-Bolma experiment. Note that General Relativity is

incapable of predicting the change in phase unless electromagnetic potentials are included in the metric.

Hence, the Maaronov-Bolma effect is an indication that there is a geometric interpretation for

electromagnetic effects.

Finally, the quantity eqb is quantized according to

e_ = 2xnhc (13)

Hence, from equation (9), for a particle of mass m, time and space may be quantized according to

2rmh 2rmh

At - As = cat -

mc 2 ' mc

(14)

4

i

]-- oo Solenoid + cx_

i

i

i i

1 2

Figure 1. Two particle paths past a perfect solenoid from points 1 and 3 at - oo to 2 and 4 at + oo

NASA/CR—2005-213999 4446
NASA/CR—2005-213999 47
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AIAA–2001–3906

E–15323

WBS–22–713–74–70

NAS3–00094

50

Search for Effects of an Electrostatic Potential on Clocks in the Frame

of Reference of a Charged Particle

Harry I. Ringermacher, Mark S. Conradi, and Brice N. Cassenti

Space propulsion; Gravitation; Electric field; NMR

Unclassified -Unlimited

Subject Categories: 20 and 70

KRONOTRAN Enterprises, LLC

1054 Main Street

Delanson, New York 12053

Prepared for the 37th Joint Propulsion Conference and Exhibit cosponsored by the AIAA, SAE, AIChE, and ASME, Salt

Lake City, Utah, July 8–11, 2001. Harry I. Ringermacher, KRONOTRAN Enterprises, LLC, 1054 Main Street, Delanson,

New York 12053; Mark S. Conradi, Washington University, Department of Physics, 1 Brookings Drive, St. Louis,

Missouri 63130; and Brice N. Cassenti, United Technologies Research Center, 411 Silver Lane, East Hartford, Connecticut 06108. Project Manager, Marc G. Millis, Propulsion Systems Division, NASA Glenn Research Center, organization

code RTP, 216–977–9535.

Results of experiments to confirm a theory that links classical electromagnetism with the geometry of spacetime are

described. The theory, based on the introduction of a Torsion tensor into Einstein’s equations and following the approach

of Schrödinger, predicts effects on clocks attached to charged particles, subject to intense electric fields, analogous to the

effects on clocks in a gravitational field. We show that in order to interpret this theory, one must re-interpret all clock

changes—both gravitational and electromagnetic—as arising from changes in potential energy and not merely potential.

The clock is provided naturally by proton spins in hydrogen atoms subject to Nuclear Magnetic Resonance trials. No

frequency change of clocks was observed to a resolution of 6310–9. A new “Clock Principle” was postulated to explain

the null result. There are two possible implications of the experiments: (a) The Clock Principle is invalid and, in fact, no

metric theory incorporating electromagnetism is possible; (b) The Clock Principle is valid and it follows that a negative

rest mass cannot exist.