DEVELOPMENT OF ECE2: GRAVITATIONAL FIELD EQUATIONS, . ANTISYMMETRY, EQUIVALENCE PRINCIP_LES, COUNTER GRAVITATION

AND AHARONOV BOHM EFFECTS.

by

M. W. Evans, H. Eckardt and D. W. Lindstrom,

Civil List, AlAS and UPITEC

(www.webarchive.org.uk, www.aias.us, W\:VW.upitec.org, www.et3m.net,

www.atomicprecision.com)

ABSTRACT

The gravitational field I potential equations are derived of ECE2 theory and

applied to counter gravitation. It is shown that the gravitational force between m and M can

vanish under well defined conditions, and can become positive, so m repels M, and a craft

may be lifted off the ground by an on board electric device. The antisymmetry laws ofECE2

are derived and used to derive the Newtonian equivalence principle form geometry. The

theory of spin connection resonance is developed and shown to result in zero gravitation. The

Aharonov Bohm effects of vacuum ECE2 theory are developed.

Keywords: Development ofECE2 theory, gravitational field I potential equations, counter

gravitation, antisymmetry laws, derivation of the equivalence principle, vacuum ECE2 theory.

1. INTRODUCTION

In recent papers of this series { 1 - 12} the ECE2 theory has been developed and

applied to derive the field I potential equations of electrodynamics. In Section 2, ECE2 is

applied to find the field I potential equations of gravitation. It is shown that the latter can

produce zero or repulsive gravitation under well defined conditions, and that counter

gravitation can be designed with an on board electrical device. The antisymmetry laws of

ECE2 are derived and shown to result immediately in the Newtonian equivalence principle,

thus deriving it from geometry. The EC2 theory allows potentials to exist in the absence of

fields, Aharonov Bohm effects which are used to define the vacuum ECE2 theory needed to

understand the principles of energy from spacetime.

This paper should be read with its background notes posted with UFT318 on

www.aias.us. Note 318(1) derives the gravitational field I potential equations from geometry

and the methods of UFT313 to UFT317. Note 318(2) derives the anti symmetry principles

from which the Newtonian equivalence principle follows directly. Note 318(3) gives the

ECE2 theory of spin connection resonance, which is applied to counter gravitation in Note

318( 4 ). The ECE2 theory is applied to the Aharonov Bohm effects in Note 318( 5), giving the

basics of the vacuum ECE2 theory. Notes 318( 6) and 318(7) develop the theory of zero

gravitational force between m and M, and repulsion between m and M.

Section 3 is a numerical and graphical analysis of the main results of Section 2.

2. DEVELOPMENT OF ECE2

Following the principles and methods developed in the immediately preceding

papers UFT313 to UFT31 7 it is possible to infer the gravitational field equations of ECE2

theory, which have the same structure as the electromagnetic field equations. The four

gravitational field equations of ECE2 theory are as follows:

'J • ') 7- '" • ') - 4-~ &-~ - ( i)

n -j_ J_Q "- =-(cvc • ...Q +-~ )<.."\.\-:. ~& ~ ')(.~ T -;t - 1.) C

- c~)

- - - (}) - -

Here g is the gravitational field, G is Newton's constant, ;? is the mass density, J - I,..... -h-.

is the current of mass density, g is the gravitomagnetic field, (n is the

-.) gravitomagnetic mass density, and .......-..Sl is the current of gravitomagnetic mass density. In

- these equations:

\t'(o -::.. J ( ~. - (c;) ( 0 ')

(

and

(~.) s) - (~) \{ d.. -(

where the tetrad and spin connection four vectors are respectively

and

~r .. ( ~ 0 ' - ~) - h)

The field potential relations are

1- :l ( cw.~ -(~ dQ cp - '1 1?

_c.,

~ --- - -- dt --1- d_ ~ ?<.~ - (,~ and

SL - '\J X Q -·

and where the gravitational v~ :ur po~entir~ ~ _ J c Q) . _ ( \ ~) In electrodynamics it is almost always assumed that the magnetic charge current

density is zero. The parallel assumption in ECE2 gravitational theory leads to a simpler set of

equations:

- (~>) \j .52_ - 0 -- - -(\~) \j ")<-. ~ 1- Jil - 0 - -- di -']_· d \tC -~ ":..- ~rfh -(ts)

- - - "- 4-y &- 2 ~ _..

dC). \(( ~Q \ ~~SL -- ) ---- -- ,. -~ - c ---..... - df c -{t&)

whose structure is superficially similar to the Maxwell Heaviside (MH) equations of

electrodynamics. However, Eqs. ( \ ~ ) to ( \ b ) are equations of a generally covariant

unified field theory (ECE2 theory).

From Eqs. ( l S) and ( \ b ): ( ) ~ / d (r-, \ 1..-&q.") ~-.Ll-1·"6- -:--~~- ') ':!"-;, ~ ~""' c :l dt - v"l +fl:

L

0 - (\'i) so it follows that

I.e. D

which is the equation of conservation of mass current density in ECE2 theory.

In the Newtonian universal gravitation, it is known experimentally that:

--m~ ~~ - c~~ <

so it follows that:

d.j (J_ '\S _CJ( _ -dMb- .Lev -"{ d)( ~M&- - - (.) ( - r (o) f -. < 'l ( - (

l

~ ( -(~h) and

~ - ( d.:l) \{( -- -· (

Eq. ( ~).)shows how ECE2 reduces to the N~wtonian theory. In the latter theory, the only

field equations used are:

- _\J (f - . ---and:

- -~

together with the Newtonian equivalence principle:

1/\....m& ,. .

Eq. ()...))in the Newtonian theory is a theoretically unproven assumption. It happens to

produce the experimental data. In ECE2 however Eq. ( lS) can be derived from geometry

and antisymmetry as follows.

Consider the ECE2 generalizations of the Coulomb and Newton laws:

5!. . £ -;: \'( . ~ "' r:- -( :l0 0

~ ~ _(JI) and

--where ~ is the electric charge density, E the electric field strength in volts per metre,

and f 0 the vacuum S. I. permittivity. The electric field strength is defined by:

where the electromagnetic four potential is:

Pt~ cr~

The anti symmetry laws { 1 - 12} of ECE2 are therefore

- 'lfe + )c<:.J. 8_ -= d~ and

In the absence of a vector potential A and gravitomagnetic potential (Q :

and

---The Newtonian equivalence principle ( :l,S ) follows immediately from Eq. ( 3>3 ):

- f ~ lhj -;. - Vk.'l 'f' ::. - d..n.- (f S- - (14-) -where the gravitational potential ofNewtonian universal gravitation is:

-w - - [JLCs- - (:,9 ' HereM is the mass of an object that attracts the mass m. So:

-

It follows that the spin connection vector is:

Similarly in electrostatics:

I o:_ -e_ ~ ~ - e ']__ cfe - - )_~ 1 e 0-. - {~i)

1e ~ -~ -C~~) \.vrr f" (

where:

- - :)_.el (J ~c~) ~ -\.r'IT fo ('

so:

and the spin connection vector is again:

- (4-~ -

In the absence of a vector potential:

~ -=- - :(} ,[ --;)_ j G - (~;}\ _ ~ }e 'fe - ')

and:

~ · s ~ ~ 1 t 0 c ~~ so: ('J )_ T ~:) f~ - -~/to _(4Lr)

Eq. ( 4~) becomes an undamped Euler Bernoulli equation with the following choice of

electric charge density:

whose solution is:

Eq. ( ~0\.) reduces to the Poisson equation ofNewtonian dynamics: .( l(? _ ( Scl~

'J ~ 't~ ~ - ~ I... ) when

The acceleration due to the gravity of a laboratory mass m in ECE2 theory is:

-

In the Z axis: - (s~)

so:

it follows that:

and

0

In this condition there is no gravitational attraction between m and M.

The gravitational potential energy in joules of the mass m is:

U::~~-- -:. ~ "t: ~ - l (, t) and the electrostatic potential energy of a charge e is:

\A: k'.. "" e_ fe Therefore:

and

This equation shows that gravitation can be engineered by an on board device that

-\ ~

In the Z axis:

and so:

giving the ac;_eleration due to gravity: W,_

1\. -::. .L _f.\ ~'l. s:·hl ~1.-w)_ -lCJ (o;(\ 1\ +- e J/,~ + ~ r~ 6 1. .,.._ ~ "l - ~ ' '!, 1. J J L "t ~ ~o ~L ~

-·(l~ There is no gravitational force between d M - ) m an under the conditions

T~ ( ~L L) ~ ;;t<j_ ~ b~) ~1.

and

Numerical solutions ofthese equations are given in Section 3.

From Eqs. ( \ \ ) and ( b 8 ) the gravitational potential is:

There is no gravitational force between m and M if In.._ is zero, so in this case:

f - J2 - ("11 \ {_ f.,(_ ~"l.:l_~;) )

and the electric field strength needed for condition ( tl) is:

s__ - 1_ f {_ -)_~ r e - h9 When the electric field strength of an on board device contained within a vehicle of mass m is

tuned to condition ( ll ), the g forces between m and M vanish. The vehicle is no longer

> attracted to the earth's mas·M. Finally the condition for a positive g is a negative spin

•'

connection , so usmg:

--

it becomes positive, or repulsive, when

d-.0_ Lo > In this condition the mass m lifts off the ground.

Finally in this section, the ECE2 vacuum theory is developed briefly with the

equations:

_... 11 fr

~ -~ 1- Ja· T ~ ( cuo 8_- f 0.) -=-.0 - .------- .jt -(g!) and

-(~~ \S 'J )' ~ -\-) ~ -r- B_ -=-0 - -- --which show immediately that 1 and (\ are non-zero when E and B are zero. These

I -- --are the conditions for the well known Aharonov Bohrn effects, potentials are observed

experimentally to exist in the absence of fields. These are the ECE2 vacuum potentials

describing the electromagnetic energy present in spacetime (the ECE2 vacuum). By

antisymmet~ '\J f -\- J_ C (,.) ~ A ""- _ ~ .e__ -

-- -- at _)f:::_

-{V>)

-(~~ and it follows that the vacuum potentials are defined by:

-t 1coe~8 0 -~

0

_ {lSJ

-(~6) - d

-- d.__ f ~ ""- 0

-If it is assumed for the sake of simplicity that:

there are three equations in three unknowns: _ _ (~g \ ~ '9_ 1 -t- d-(_uo B_ 0 )

-dfl- J;;t -)r~ ~ (J - tr:<1.J 'J ',(_ 1\ -\- 'l~ ,X. A - 2- - ( qo) - -

and these can be solved numerically, giving. r , .::2 and c.>

l ;~nerrm~:en~i~n:d~A:u:is~ t~~ l 6_) .- ( ~) The ECE2 vacuum can be thought of as being made up of photons with energy momentum:

~ ,M - .e_ A"'-=--r \'\: ,~. "0- t (sz ) \r( ).- ("'~ The Einstein I de Broglie equations of the ECE2 vacuum are:

~ - -e__ + "-~ CJ -=- 't V\-. v ")

and

---where m is the mass of the photon and where the Lorentz factor is:

~ (~-~:j-1/~-(~~ The ECE fermion equation { 1 - 12} can be used to describe the interaction of the ECE2

photons with a circuit modelled by one electron. In UFT311 the circuit design needed to take

energy from the ECE vacuum was described in all detail and excellent agreement found

between ECE and data.

3. NUMERICAL AND GRAPHICAL ANSL YSIS

(Section by co authors Horst Eckardt and Douglas Lindstrom)

ACKNOWLEDGMENTS

The British Government is thanked for a Civil List Pension and the staff of

AlAS and others for many interesting discussions. Dave Burleigh is thanked for posting,

feedback activity software and website maintenance and design, Robert Cheshire for

broadcasting, and Alex Hill for translation and broadcasting.

REFERENCES

{1} M .W. Evans, H. Eckardt, D. W. Lindstrom and S. J. Crothers, "The Principles ofECE

Theory" (UFT281 to UFT288, and in prep., New Generation 2015).

{2} M. W. Evans, Ed. J. Found. Phys. Chern., (open source on www.aias.us. and Cambridge

International Science Publishing, CISP, 2011 ).

{3} M .W. Evans, Ed. "Definitive Refutations of the Einsteinian General Relativity" (special

issue of ref. (3), open source on wvvw.aias.us).

{4} M. W. Evans, S. J. Crothers, H. Eckardt and K. Pendergast, "Criticisms ofthe Einstein

Field Equation" (UFT30 1 and CISP 201 0).

{5} M .W. Evans, H. Eckardt and D. W. Lindstrom, "Generally Covariant Unified Field

Theory" (Abramis Academic, 2005 to 2011, and open source on www.aias.us) in seven

volumes softback.

{ 6} L. Felker, "The Evans Equations of Unified Field Theory" (UFT302 and Abramis

Academic 2007).

{7} H. Eckardt, "The ECE and ECE2 Engineering Models" (UFT303).

{8} M .W. Evans, "Collected Scientometrics" (UFTJ07 and New Generation 2015, softback).

{9} M .W. Evans and L. B. Crowell, "Classical and Quantum Electrodynamics and the B(3)

Field" (World Scientific, 2001 and Omnia Opera section ofwww.aias.us).

Development of ECE2: gravitational �eld

equations, antisymmetry, equivalence principles,

counter gravitation and Aharonov Bohm e�ects

M. W. Evans∗, H. Eckardt†, G. J. Evans, D. W. LindstromCivil List, A.I.A.S. and UPITEC

(www.webarchive.org.uk, www.aias.us,www.atomicprecision.com, www.upitec.org)

3 Numerical and graphical analysis

The conditions that no gravitational force between masses m and M should existare given by Eqs.(74) and (75):

tan(kZZ) = 2ωZ

kZ, (96)

∂φe∂Z

= −2ωZφe. (97)

From Eq.(97) a solution for φe can be determined which has to be inserted inEq.(77) to give a charge density

ρe = φe(k2Z − k20). (98)

A consistency problem seems to remain because ρe was assumed to be strictlyperiodic by Eq.(46) which cannot be guaranteed by this approach, but we willsee that the above conditions need only to be ful�lled at a certain point Z whichis de�ned by kZ to let the gravitational force vanish.

In Eqs.(96-97) the spin connection ωZ appears for which we have to introducea model. For obtaining an Euler-Bernoulli equation, k0 has to be a constant.This means, that the divergence of ωz has to be constant as follows from Eq.(45).Therefore we make the �rst approach

ωZ =Z

Z20

(99)

with a new constant Z0. The squared constant in the denominator is requiredto obtain the right physical dimension 1/m. Eq.(97) then has the solution

φe (Z) = φ0 e− Z2

Z02 (100)

∗email: emyrone@aol.com†email: mail@horst-eckardt.de

1

with a constant φ0. From (96) follows

tan (kZ Z) =2ωZ

kZ=

2Z

kZ Z02 . (101)

This is a transcendental equation. Approximating the tangens function by the�rst two terms of its taylor expansion around zero:

tan (kZ Z) ≈ k0 Z +k30 Z

3

3(102)

leads to an equation of third order which has the only non-trivial, positivesolution

Z =

√3√2− kZ2 Z0

2

kZ2 Z0

. (103)

This means that the gravitational force can only be suppressed at this value ofthe Z coordinate. The potential φe has to be tuned to have the suitable valuefor this Z. Both depend on the wave number kZ which can be choosen freelyin principle. The dependence of this Z value on kZ is graphed in Fig. 1 forparameters Z0 = 1 m, φ0 = 100 V. It is important to note that Z is de�nedonly below a maximum value of kZ . The dependence of φe on kZ is shown inFig. 2. There is an onset at a minimal wave number. For the method to work,both Z(kZ) and φe(kZ) need an overlapping kZ range which is quite small inthis case at about 1.2/m. The curves additionally depend on Z0.

Another choice for ωZ could be

ωZ =1

2 Z(104)

which is valid for the Coulomb potential. Although this does not ful�ll condition(45) for constancy, we used this to show the e�ect. Maybe one can restrict toa small portion of Z where it is nearly constant. We furthermore use cartesiancoordinates. Then the solution for the electrical potential is

φe (Z) =A0

Z(105)

with a constant A0, i.e. a Coulomb-like potential. Eq.(96) gives a quarticequation in Z with the only real-valued, non-negative solution

Z =

√√21− 3√2 kZ

=0.88954361752413

kZ. (106)

The resulting functions Z(kZ) and φe(kZ) are graphed in Figs. 3 and 4. Thereis a broad overlapping range of kZ now.

As the third and last approach we derive ωZ directly from Eq.(96):

ωZ (Z) =1

2kz tan (kz Z) . (107)

Inserting this into (97) gives the astonishingly simple solution

φe (Z) = φ0 cos (kz Z) . (108)

By Eq.(98) this gives the original oscillating charge density ρe(Z). The tangensfunction is linear near to Z = 0, so condition (45) is ful�lled too. This seemsmore to be a consistency check because ωZ was chosen to ful�ll one of the zeroforce conditions automatically.

2

Figure 1: kZ dependency of Z coordinate for linear ωZ .

Figure 2: kZ dependency of φe for linear ωZ .

3

Figure 3: kZ dependency of Z coordinate for hyperbolic ωZ .

Figure 4: kZ dependency of φe for hyperbolic ωZ .

4

{ 10} M . W. Evans and S. Kielich, Eds., "Modern Nonlinear Optics" (Wiley Interscience, . New York, 1992, 1993, 1997, 2001) in six volumes ~nd two editions.

{ 11} M .W. Evans and J.-P. Vigier, "The Enigmatic Photon" (Kluwer, Dordrecht, 1994 to

2002, and Omnia Opera Section ofwww.aias.us) in five volumes hardback, five volumes

softback.

{ 12} M . W. Evans and A. A. Hasanein, "The Photomagneton in Qauntum field Theory"

(World Scientific 1994).