The Origin of Inertia (PDF)

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The Origin of Inertia

M. Molloy

Department of Electrical Engineering, University of South Florida,4202 E. Fowler Ave, Tampa, FL 33620

H. Hickman

5010 Dante Ave., Tampa, FL 33629

Abstract

It may be that electric charges in the past and future can exert forces on electric charges

in the present. A naive mathematical analysis, based on the premise that an electriccharge experiences self forces from all of its past and future "editions", leads to someinteresting predictions: 1) In terms of the self-force, only a few microseconds into thepast or future can have an appreciable effect on the present. 2) As an electric chargeapproaches the instant of its own annihilation, it will attempt to "save itself" byaccelerating towards the past. 3) When a charge moves with constant velocity, the self-force from the future exactly balances the self-force from the past. But when the charge issuddenly accelerated, the self forces are no longer in balance. This out of balancecondition suggests an answer to the question of why a charge must be pushed in order tomake it accelerate. i.e. - It suggests an origin for the phenomena of inertia.

Introduction

In 1915 Albert Einstein presented his field equations for the General Theory ofRelativity [1]. Beginning with Kurt Gdel in 1949 [2], several researchers have foundsolutions to those equations that apparently admit the possibility of travel into the past orinto the future [3-6]. Implicit in each solution is the idea that the past and future actuallyexist. In other words, if you can go there, then it must be there. If the past and future doexist, it may be that an electric charge can exert a force on itself, both from the past andfrom the future.

Coulomb's law for the force on one electric charge due to another electric chargecan be expressed as

||||4

1

21

21

221

2121

r

r

r

qqF

!

!

!

!

=

, (1)


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2

where 12F!

is the force from charge 1q onto charge 2q , represents the permittivity of

whatever medium is between the charges, and 21r!

is the distance vector that stretches

from point 1 to point 2.

( ) ( ) ( )2122

122

1212 zzyyxxr ++=!

= the Pythagorean distance from point 1to point 2. (2)

The purpose of this paper is to explore the consequences of replacingPythagorean distance by spacetime interval so that our expanded Coulomb's law becomes

||||41

21

21

221

2121

ss

sqqF

!

!

!

!

=

, (3)

where 21s!

is a Minkowski vector that stretches from event 1 to event 2, and

( ) ( ) ( ) ( )[ ]212212212212212 zzyyxxttcs ++=!

= the timelike flatspace interval from event 1 to event 2 (4)

Specifically, we will examine the idea that a charge in the future can exert a force on its

past self, and vice versa ( qqq = 21 ).

(Note that the minus sign in front of equation (3) is necessary so that the

expanded Coulomb's law matches the original Coulomb's law when 012 = tt .)

Two Immediate Results

For an isolated stationary charge, equation (3) reduces to,

( )( )t

ttc

qFself

4

12

122

2

=

!

, (5)


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where t is a unit vector pointing in the time direction, assumed to be orthonormal to the

,, ji and k unit vectors that point in the space directions.

Since162 109=c , and that number appears in the denominator of equation

(5), it looks like the magnitude of selfF!

will be extremely small for any lengthy 12 tt .

In other words: In terms of the self force, it looks like only a few microseconds into thepast or future can appreciably affect the present.

Also from equation (5),2q in the numerator is always greater than zero,

regardless of whether q is positive or negative. So the initial impression is that the self

force is always repulsive. The minus sign in front of equation (3) however, reverses that

situation, so that the self force is always attractive. When 12 tt > , the force is coming

from the past, and the direction of 12F

!

is ( ) ( )tt =+

. So the force from the past triesto pull the charge into the past. When 12 tt < , the force is coming from the future, and

the direction of 12F!

is ( ) ( )tt += . So the force from the future tries to pull thecharge into the future.

This 'always attractive' aspect of the self force leads to a fascinating prediction.The next section will show that the self forces from the past and the future exactlybalance for a charge moving with uniform velocity. But that analysis is predicated on theidea that the charge is going to exist in the future. As a charge approaches the instant ofits own annihilation, the self force from the future should begin to disappear. The self

force from the past begins to dominate, and it looks like the charge will attempt to"escape destruction" by accelerating towards the past. This prediction is experimentallyverifiable. In fact, it may have already been verified by the formation of positronium.Positronium is formed when a positron and an electron, on a course towards annihilation,"hesitate" long enough to orbit each other briefly. It may be that they hesitate becausethey are trying to accelerate into the past. That same "hesitation" could also play asignificant role in the ultimate explanation as to why the electrons inside an atom do notfall into the nucleus.

A Moving Charge

In the study of electrostatics, the electric field from a point charge is defined as

r

r

r

q

q

FE

!

!

!

!

!

==

24

1

. (6)

For a continuous line of charge, a differential element ofE!

becomes


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4

r

r

r

dqEd

!

!

!

!

=

24

1

, (7)

so that the electric field from the entire line would be

=

linethealong r

r

r

dlE

!

!

!

!

24

1

, (8)

where dldq = , and is the line charge density in Coulombs per meter.

A single charge, moving with a slow (non-relativistic) constant velocity (whichcould be zero), traces out a sort of "line", as it moves through time and one dimension ofspace, as shown in Figure 1. By way of analogy with the electrostatic case, we intend to

define a sort of "temporal" line charge density, t , in Coulombs per second. We will

then express in terms of t, and integrate to sum up the contributions to selfE!

from all

of the past editions, and all of the future editions, of q . The total self force on q should

then be

)futurefromselfpastfromselftotalself EEqF!!!

+= . (9)

A question arises as to how many q 's should be spread out over one second. In

seems intuitive to assume that the present has some amount of duration. (Which isanother way of saying that "now" lasts longer than zero seconds.) We will arbitrarily take

the duration of now to be the Planck time,43101 Pt seconds. If it became

necessary to obtain an actual value for t , we could then sayP

t tq

= . Allowing

now to have duration, also means that our integrations will go up to2

Pt , instead of

going right up to zero.

Building on equation (3), and by analogy with equation (8), we surmise that

=

linethealong

t

selfs

s

s

dlE

!

!

!

!

24

1

. (10)


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For pastfromselfE!

,s

s!

!

needs to be a unit vector pointing from the past towards

now. The s!

vector should stretch from ( )ixct, almost to ( )0,0 , where ( )ixct, is a

point on the line locatedbelow the origin in Figure 1. Expressing x as

vtx = , (11)

where v is a constant, and we are allowing negative times, it seems reasonable tosuppose that

(12)

The t variable in equation (12) is always negative below the origin, so any value of twill change both cap vector signs to positive, and it looks like the direction ofs

!

is OK.The magnitude ofs

!

has already been defined as the timelike spacetime interval, whichreduces to

222 xtcs =!

(13)

for one spatial dimension. Substituting vtx = into equation (13), and factoring out a t

results in

( ) ( )ivct

ivtt

vct

ct

s

s 2222

+

=!

!

. (14)

The t's cancel in equation (14), leaving

( ) ( )ivc

ivt

vc

c

s

s 2222

+

=!

!

. (15)

And all of a sudden there is a problem. There are no variables in equation (15), so it is aconstant unit vector. Further, both terms in front of the cap vectors are positive, and thereare no terms left to reverse the minus signs. By canceling the t's, we have created asituation in which equation (15) now points in the wrong direction! The correct unitvector should point from the past towards now, and that unit vector is expressed as

( )( ) ( )( )

( ) ( )iivttctiivttcts

00

+=

+++=!


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( ) ( )ivc

ivt

vc

c

s

s 2222+

++

=!

!

. (16)

( By now the reader will have noticed that we are using a geometry in which s!

( ) ( )22 ixct += , instead of the usual = sss!!!

. We are doing this so that our

results will match up with the definition of the spacetime interval.)

From Figure 1 and equation (11),

( ) ( ) 2222 vcdtidxcdtdl =+= , (17)

and we are finally ready to assemble pastfromselfE!

based on equation (10).

( )( ) ( )

+

++

=

i

vc

ivt

vc

c

vct

vcdtE

Pt

tsfp

4

12222

22

222

22

!

, (18)

where is some arbitrary number of seconds. Factoring out the constants, equation (18)reduces to

= 2

2

1Ptsfp dt

tKE

!!

, (19)

where ( ) ( )

+

++

= i

vc

ivt

vc

cK t

4 2222

!. Evaluating the integral in equation

(19) leads to

+=

12

P

pastfromselft

KE!!

. (20)

For futurefromselfE

!

, we need a unit vector that points from the future towards now.The tail of that vector is above the origin in Figure 1, so that

( ) ( )ivc

ivt

vc

c

s

s 2222

+

=!

!

. (21)


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Assembling the appropriate elements into the form of equation (10), we have

( )

( ) ( )

( )

.21

1

4

1

22

22222

2222

22

+=

=

+

=

+

+

+

+

P

t

tt

sff

tK

dtt

K

ivc

ivt

vc

c

vct

vcdtE

P

P

!

!

!

(22)

We are finally ready to substitute into equation (9).

)

( ) .00

2112

==

++

+=

+=

q

tK

tKq

EEqF

PP

futurefromselfpastfromselftotalself

!!

!!!

(23)

Evidently the self forces that are trying to pull the charge towards the past, and the selfforces that are trying to pull the charge towards the future, exactly cancel when thecharge is moving with a slow uniform velocity. That result was certainly to be expected,since it is in accordance with Newton's 1st law.

Things change however, when the charge is suddenly accelerated. Consider a

situation in which the charge has been moving with uniform velocity from to

2Pt , and then experiences constant acceleration from

2Pt+ to + . Expressing x

in terms of t,

+


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We have already calculated the self force coming in from the past, and that result isgiven by equation (20). For the self force coming in from the future, ours

!

vector

stretches from

+vtatict

2

2

1, almost to ( )0,0 , and all of the t values are

positive this time. So out new unit vector should be

( )( )

( )( )i

vatct

vatitt

vatct

ct

s

s

21

21

21

2222

+

+

+

+

=!

!

. (25)

Again the t's cancel, but this time the direction remains correct, and we have

( )( )

( )( )i

vatc

vatit

vatc

c

s

s

21

21

21 2222

+

+

+

+

=!

!

. (26)

Along the line, our differential element of length will be

( ) ( )[ ] ( )222222

1 vatcdtvtatidcdtdl +=++= , (27)

so that

( )( )

+

+

+

+=

2222

22

214 P

tt

sffss

vatct

vatcdtE!

!

!

, (28)

wheres

s!

!

is given by equation (26).

Equation (28) cannot be evaluated closed form, but clearly it is not going to givethe same result as the evaluation of equation (22) (unless of course 0=a ). So the

magnitude of the self force from the charge when it is being accelerated in the future, isno longer equal to the magnitude of the self force from the charge when it was movingwith constant velocity in the past. For a suddenly accelerated charge, the two self forcesare no longer in balance, and the total self force no longer equals zero.

When the charge moves with constant velocity, = 0selfF!

. When the charge

starts to be accelerated, 0selfF!

. It certainly looks like an externally applied force


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would be necessary to bring about that change. i.e. - This looks like the reason why acharge has to be pushed in order to make it accelerate.

Conclusions

Certain calculations based on the theory of General Relativity imply that the pastand the future actually exist. If they do exist, then it may be that electric charges in thepast and future, can exert forces on electric charges in the present. Pushing that idea alittle further, we suppose that a charge in the present can experience "self forces" from allof its past and future "editions". A first-cut mathematical analysis leads to some eyebrowraising predictions: In terms of self force, not very far into the past or future can affect thepresent. A charge that approaches the instant of its own annihilation will experience a"rescue force" that tries to pull it towards the past. The total self forces from the past andfuture exactly balance when a charge moves with a slow constant velocity. But when thecharge is suddenly accelerated, the self force from the past is no longer exactly offset bythe self force from the future. That last prediction looks like the reason why a charge

must be pushed in order to make it accelerate. In other words, it suggests an explanationfor the age old mystery of inertia.


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Figure 1. A single charge moving with a slow constant velocity

dlcdt

idx

ct

ix


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Figure 2. The charge moves with constant velocity before t= 0, then withconstant acceleration aftert= 0.

ct

ix


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References

1. A. Einstein, Sitzungsberichte, Preussische Akademie der Wissenschaften, 844 (1915).

2. K. Gdel, Rev. Mod. Phys., 21(3), 447 (1949).

3. F.J. Tipler, Phys. Rev. D, 9(8), 2203 (1974).

4. M.S. Morris, K.S. Thorne, U. Yurtsever, Phys. Rev. Lett., 61(13), 1446 (1988).

5. J.R. Gott, Phys. Rev. Lett., 66(9), 1126 (1991).

6. A. Ori, Phys. Rev. Lett., 71(16), 2517 (1993).

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The Origin of Inertia (PDF)