EQUATION OF MOTION USING FRACTIONAL CALCULUS

by

KIHONG KWON, B.S., M.S.

A DISSERTATION

IN

PHYSICS

Submitted to the Graduate Faculty of Texas Tech University in

Partial Fulfillment of the Requirements for

the Degree of

DOCTOR OF PHILOSOPHY

Approved

Accepted

August, 1991

Pioy^nc/^f

^ '^ ' ' ACKNOWLEDGEMENTS

I would like to thank my advisor Dr. Young N. Kim for guidance. I

also would like to thank Dr. Raymond W. Mires, Dr. Charles W. Myles,

Dr. Randall D. Peters, and Dr. Thomas L. Gibson for serving as my

committee members. I appreciate the Department of Physics for

providing financial support through teaching assistantships during

my study.

I thank my parents. Finally, I thank my wife.

^h

11

TABLE OF CONTENTS

ACKNOWLEDGEMENTS ii

LIST OF TABLES i v

LIST OF FIGURES v

CHAPTERS

I. INTRODUCTION 1

I I . FRACmONAL CALCULUS 8

2.1 Basic Theory 8

2.2 Examples 1 6

2.3 Numerical Algorithm 1 9

2.4 Small Parameter Expansion 2 0

I I I . EQUATION OF MOTION 2 4

3.1 Geodesic Equation 2 4

3.2 The Schwartzschild Solution 3 0

3.3 Equation of Motion for Free Fall 3 3

3.4 Approximation 3 7

IV. RESULTS 4 1

V. DISCUSSION 4 9

REFERENCES 5 2

APPENDIX 5 6

i l l

LIST OF TABLES

Table 1. Proper time t versus fractional parameter 5 (x < 0.2) .... 4 8

Table 2. Proper time x versus fractional parameter e (x < 0.2) .... 4 8

IV

LIST OF HGURES

Fig. 1. Plot of radial coordinate r versus proper time x 4 4

Fig. 2. Plot of time t versus proper time x 4 4

Fig. 3. Plot of fractional parameter e versus proper time x 4 5

Fig. 4. Plot of fractional parameter 5 versus proper time x 4 6

Fig. 5. Plot of fractional parameter e versus radial coordinate r ... 4 7

Fig. 6. Plot of fractional parameter 5 versus radial coordinate r .. 47

CHAPTER I

INTRODUCmON

The equation of motion for a free particle is given by

^ = 0, (1.1) dx^

where ;c / (// = 0, 1, 2, 3) are the Cartesian spacetime coordinates,

x^ = cty x^ = X, x^ = y, x^ = z.

The proper time interval dr is defined by

c2 dT^ = c^dt^-dx^- dy^ - dz^, (1.2)

where c is the speed of light. Eq. (1.1) can be derived from the

following relativistic equation, which is the modified form of

Newton's second law.Ci]

^ =/- , (1.3) dr

where p^ is the particle's 4-momentum and / ^ is the 4-force acting

on the particle. The prefix 4- serves to distinguish vectors in

spacetime from those in space, which we call 3-vectors.

^ dr

= m {yc, yy)

= (Elc,jp),

where m is the particle's rest mass, 7= (1 - vVc^)'^''^^ y is the particle's

3-velocity, E is the energy of the particle, and p is the particle's 3-

momentum. In terms of the 3-force F , / / ^ is given by

^^=7(F-v/c,F).

When there is no force acting on the particle, f^ = 0. Since m is a

constant, from Eq. (1.3) we get Eq. (1.1).

When the gravitational force is acting on the particle, it follows

the geodesic path in spacetime according to general relativity. The

path of a free particle in the flat spacetime of special relativity is a

straight line and this generalizes to the geodesic in curved spacetime.

The equation of motion is given by

dV + r^dxrdif=o, (1.4) dr'^ dr dr

where x^ is general coordinate and the Einstein summation

convention is used. Any expression involving a twice-repeated index

(occurring twice as a superscript, twice as a subscript, or once as a

subscript and once as a superscript) stands for its sum over the

values 0, 1, 2, 3 of the repeated index. The proper time T is defined

by

c^dT^ = g^y dx^'dx^.

where g^y is the metric tensor and a function of coordinates.

can be written as

c'^dx^ = r\^y dx^'dx^,

where /I 0 0 0^

„ _ 0 - 1 0 0 ^ ^ ^ " 0 0 - 1 0

\0 0 0 - 1 /

Eq. (1.2)

The essential feature of the spacetime of special relativity is that we

may always introduce the Cartesian coordinate system about any

point, so that g^y = rj^y . We can construct a coordinate system about

any point P of general relativistic spacetime, in which

gfiV = rifiy + (do^p gfiy) X«X^. aa = ax«,

This approximation is valid for small x/ around P.

The Christoffel symbol of the second kind Ivo is defined byt ^

Ka = l^g^" ^ggg ^ ^gav _ ^ gv(7

ax' dx<^ ax«

where g^^^ is the inverse of g^y. The Ricci tensor of the first kind R^y

is defined by

R^V — + t ^ftl py — i ^yl fia dx^ ax«

After raising a subscript to define the Ricci tensor of the second kind,

and then contracting, we get the Ricci (or scalar) curvature.

R^y = g^^Rav.

R = R a •

The Einstein tensor Guy is

G^v — R^v ~W^ Sfiv

Now we approximate the metric tensor by

where the h^y is small. We neglect the second-order terms and

assume a slowly moving particle in a nearly inertial non-rotating

coordinate system. Substituting the above approximation into the

geodesic equation, we get

^ = - — , (/ = 1,2,3) dt^ ax'

where V = (1/2) c^hoo + const. This is the Newtonian equation of

motion for a particle moving in the gravitational field of potential V.

This gives

^00 = 1 + ^ . r-Z

When V = 0, goo reduces to its flat spacetime value. To get the

Christoffel symbol, we differentiate the metric tensor once with

respect to the coordinates. The r ^ is related to the curvature of

spacetime and the gravitational force of Newtonian physics. In the

case of flat spacetime, when we use the Cartesian coordinate, Ivo is 0

and Eq. (1.4) becomes Eq. (1.1).

We suppose that the second term in Eq. (1.4) is the small

correction term to the motion of the free particle in the presence of

gravitating body. By the free particle, we mean that the particle is

moving under the gravity alone. In general relativity, the gravity is

not considered as a force and its effect is included in the metric

tensor.

Fractional calculust^^^^ is the study of differentiation and

integration of arbitrary order. Mathematically it is applied to the

summation of series,t^l differential equations,t8.9] and special

functions.[10] It is also applied to the diffusion and Boltzmann

equations in physics.ti^-^^] Wyss solved the following one-

dimensional diffusion equation in terms of Fox functions.

, a T{x,t) a T(x,t) ax2 3 A

where 0 < ^ < 1. In Ref. 4, the diffusion on a curved surface is

discussed. The equation is

a z(x, r) ,2 ^ z( ' 0 dt ax4

This equation can be solved using the method of fractional calculus

as well as the conventional techniques.

The concept of a fractional physical quantity is not new. Wilczek

suggested fractional spin of the quantum particle.^4-16] in the

operational calculus, Namias introduced the fractional order Fourier

transform and applied it to quantum mechanical problems.H7]

Consider the meaning of ordinary differentiation. When we

differentiate the coordinate with respect to the time twice, we get

the acceleration which is related to the force in Newtonian

physics.[181 Differentiating three times, we get the physical quantity

called jerk.[191 Schot showed that, just as the normal component of

the acceleration vector contains the expression for the radius of

curvature of a path, the normal component of the jerk vector can be

expressed in terms of the aberrancy. The aberrancy measures the

asymmetry of a curve about its normal.

We propose the use of fractional calculus in the equation of

motion of the particle. Using the fractional differentiation operator,

Eq. (1.4) can be expressed as

The gravitational effect is included in e of the fractional operator.

When £ = 0, we get Eq. (1.1) in the case of the Cartesian coordinate.

The purpose of this study is to find e which makes the above

equation and Eq. (1.4) coincident in the first approximation and to

relate physical meanings to e.

We speculate that the fractional operator is useful for describing

the physical phenomena in the curved spacetime. In this respect we

use the known solution of the general relativistic equation for free

fall. To simplify the analysis, one-dimensional motion is used. The

radial coordinate r and the time coordinate t are functions of the

proper time r. The e and S which satisfy the following equations are

calculated numerically from the known r{r) and ^(T).

^ ^ = 0, dr^^^

(1.5)

^ ^=0. dT^^^

Also, by the perturbative expansion of the fractional operator, we get

the approximate forms for e and 8 . The physical meanings of e and

5 are guessed from the approximate forms. Possible application of

the fractional operator in quantum physics is suggested in Chapter V.

CHAPTER II

FRACTIONAL CALCULUS

2.1 Basic Theory

Fractional calculus is the study of differentiation and integration

of arbitrary order. In ordinary calculus, d^yldx'^ has a meaning

when n is an integer. In fractional calculus n can be any number

including irrational, fractional, or complex number.

The modern theory of fractional calculus is intimately connected

with the theory of operators. We use the following notation.[20.21]

C D / / ( ^ ) , V > 0

denotes integration of arbitrary order along the x- axis. The

subscripts c and x denote the limits of integration of a definite

integral which defines fractional integration.

We now consider the mathematical problem of defining

integration and differentiation of arbitrary order. For every

sufficiently general number v and every function / of a sufficiently

wide class, a new function cPx f is related to / by the following

criteria.

1. If/(xj is an analytic function, cPx f must be an analytic

function both of the variable x and of the order v.

2. The operation c^x must produce the same result as ordinary

differentiation when v is a positive integer n (in symbols

cPifOO=ji^)). When v is a negative integer, c^x^fix) must produce

8

9

the same result as ordinary n-fold integration. cPx^'fix) must vanish

together with all its n-1 derivatives at x = c.

3. The fractional operators must be linear.

(Px^'iafix) + b g^x)] = a cPx^'fix) + b ^D/^x) ,

where a, b are constants.

4. The operation of order zero must leave the function

unchanged.

5. The law of exponents must hold for integration of arbitrary

order.

A definition which fulfills these criteria named in honor of

Riemann and Liouville is

(v) L c^//(^) = ^ - {x-tY-^f{t)dt, (2.1)

r(v)

where r(v) is the gamma function.

When c = 0 and c = -««, we have the Riemann and the Liouville

definitions, respectively.

One method of obtaining Eq. (2.1) is by considering the n-fold

iterated integral.

F{x) = \ dxx\ dX2 \ dXn-l I dX2--- I dXnA I f{Xn) dXn •

This iterated integral can be written as a single integral by

integrating over an appropriate triangular region. The result is

10

(n) I Fix)=^-\ (X - Xnr^ fiXn) dXn

r(/

If we denote the operator of integration as

H'. • • dx.

We may v^htQ F(x) = QDX" f(x). Then letting Xn = t and generalizing n

with V, we get Eq. (2.1).

To obtain a definition of differentiation of arbitrary order, we

might try a formal replacement of v by -v in Eq. (2.1). But this yields

a divergent integral. The appropriate definition of differentiation of

arbitrary order, justified by analytic continuation, is

m nP cDifix) = cD? cD/f{x)

-A m

dx m Tip) f {x-t)P-'fit) dt

where m is the least integer greater than v, v = m - p, and 0< p < 1

11

Criterion 3 states that the operation of order zero leaves the

function unchanged. To show that we assume f(t) can be expanded

in a Taylor's series at t=x.

m=f(x) + (t-x)fix) + {t-x) if (X)

There is no loss of generality using the lower limit of integration 0

instead of c. When v approaches 0 in Eq. (2.1) we have

lim oD//(x) = lim v->0 v->0

(x-t) v-1

0 r(v)

-fix) dt

0 r(v)

V+/I-1 I (Y fV -fix) dt+--- + i-l)"\ ^^^ / " \ x ) dt

0 A2!r(v)

+ (-1) n.l /""'^ (0) ix-t) v+n

in+l) 0

r(v) dt

Because limr(v) = oo, all the terms of the right hand side vanish except

the first term. Thus,

lim (px^'fix) = lim ^^1£^ v^o v'-^or(v+l)

,0 (P^, fix) = fix).

12

Now we consider criterion 5.

p-v cD,^c^//(x)= c^r /W

By definition.

(x-sy -1

r(v) I is-t)""-'fit)dt ds.

Apply the Dirichlet formula for integration over a triangular region.

Above equation becomes

i fit)dt )r(v) X /

fit) dt I (x-5) -i (5-r)''-^ ds. Tip)Tiv)

Make the transformation

y = 5 - f

x-r

We have

'DTcPi''f= )r(v) A

ix-t)^^""-^ fit) dt H-y)^-^ y^-^ dy rou)r(v)

ix-t)^^"-'fit)dt np+v)

J -(/^+v') ,D:^^"•'^/:

13

Suppose we wish to establish the law of exponents for

differentiation of arbitrary order.

J)i i^D^f) = J)^^^f V >0, p>0.

Let V = n-p, p = m-G where n and m are the least integers larger than

V and p, respectively. Omitting subscripts for convenience we have

D''iD^=D''-PD'^-''f=D''D'PD'^D''^f

This suggests that we investigate the relation D'^D'P = D'^D"^ to

determine what restrictions have to be imposed on / to permit the

interchange of the orders of operation. From Eq. (2.1)

(P) 1 DPf=-^\ ix-t)P-'fit)dt.

Tip)

Integration by parts m times yields

T(p)\ P ^^' P(P+^)-^ *p(p+l)...(p+m-l)-' ^ 1

+ — ^ — ix-tf*"'-^ /"•' (r)dr, r(p+m) I

14

where the last term is /)-^P+'«)/'») = o-(P'^rn) Qmjr j ^ y g jf ^H terms in

the braces are zero, we have

D'^D-P f = D'^D-^f^""^D"" f = D'P 1^""^ = D'PD"" f.

So the law of exponents D''D^f=D''-'^f for differentiation of

arbitrary order holds if the function vanishes at the lower limit of

integration to a sufficiently high order,

/•(c)=/(c)=-..=/'"-^>(c) = 0.

A useful tool for computation of the fractional derivative of a

product is the generalized Leibniz rule.

cD^fix) ^(x) = S ( " ) cPxfix) cOr gix). n=0 ^

where Px is the ordinary differentiation operator, px is a

fractional operator and \ „ ) is the generalized binomial

coefficient.[22]

r(v+l)

" ' nlTiv-n+l)

(2.2)

(_l)n Tin-v) nl r (-V)

In the above Leibniz rule, the interchange of fix) and ^(x) on the

right-hand side is not obvious. A further generalization without such

a drawback is given by [ 1

15

oo

cPU{x)g{x)= X ( "^ Ic^r-^w c^r^^)' \n + p)

where p is an arbitrary number. When taking the arbitrary

derivative of a product, it is often convenient to choose the factors in

such a way that the series above terminates after n ordinary

differentiations. For example,

QDI'^ X ^X) = X oDi/2 ^x) +1 oD,-i/2 g^x)^

The following derivative is useful.[20.21] jf

V (x) = I ^ ^"' ^' • " ^ ds , J' g (x, s) u 0

where uis) is continuous in the interval c < s < x and ^(x, s) in the

triangle c < 5 < x < XQ and where 0 < v < 1, then we have

^, ^ g(x, c) u ic) ^ j Dxgix,s) + Dsgix.s) ^ (X-C)^ j^ (X-5)^

5

1 + , KO^iAu'^s)ds^ ix-sy

To show this, let us write

16

V,(;C)=| iS^^ljUi^ds (x-s)''

Since there is no longer a singularity in the integrand for the upper

limit of the integral, we can apply the ordinary rule for the

differentiation of a function defined by an integral. Then we have

tx-e

^ = fr'Y^'y^^^:c,.).(.)a/ax(x-.rl^^ dx \ (x-sY j

_ g(x, x-e) uix - e) £"

Since a/ax (x - 5 " = - a/a^ (x - 5)" , we can substitute this value in the

integrand of the preceding integral and integrate by parts. Then we

get.

dve (x) ^ I didx gix, s) uis) + dids [g(x, s) uis)] _ g(x, c) uic)

dx (x-sY ^ (x-cY

When £ approaches 0, we have the above formula.

2.2 Examples

Now we apply the fractional operation to x^ and a constant k. We

set the lower limit of integrarion c to 0 without the loss of generality.

The integral of order v is.

O^x :c« = nv) h

v-l *n ix-t)""-'t"" dt

17

r(Ai-n)

r(rt-hv+i)

We used the following formula.[23]

• n + v

r(,_o^^^,=E(^±i)i(^,....i^ Jo r(Z7-h^-h2)

Z 7 , 6 / > - l .

The derivative of order v is.

^U- = ^T(p'fx-

_ flf w

dx"^ J— ix-t)P-^ t"" dt

np) h

= d"" nn+\) ^.^^ ^^'^ r(«-hp+i)

i>±lLv«-v r(«-v+i)

The derivative of a constant /: is.

^ u-A 0Dxk = m

dx m yip) I ix-t)P-^ k dt

= A dx m

k xP

Tip) P

18

JC ^ -v X " . r(i-v)

Notice that the result is a function of x when v is not an integer.

As another example, consider the Abel's integral equation.

0(:c) = Jo i^-^y

v < l .

Here (p(x) is given and u(x) is unknown function, (pix) can be written

in the form

v-l 0(x) = r(l-v)oDr^w(x)

l-v Operating O^x' on both sides, we get

l-v oDi-"(K;c) = r(i-v)M(x).

From which we have

uix) - l _ ^ n i - v

r(i-v) (pr (i>(x)

- 1 d r(i-v) dx r(v) I .v-l ix-t)''-' (pit) dt

Since r(l-v) r(v) = ;r/sin ;rv,

i (x) = s in^_d_( TT ^ X »

^0

(x-r)''-^ 0(r) fifr.

19

2.3 Numerical Algorithm

For numerical calculation of fractional operation, Griinwald finite

difference formula is used.[24.25] Consider the following finite

difference quotients

fix) -fix-h)

fix) - 2fix-h) +fix-2h)

fix) - 3fix-h) + 3fix-2h) -fix-3h)

the limits of which, as h approaches zero, yield the first, second and

third derivatives with respect to x of f(x). Generalizing from these

formulas, Griinwald wrote

D^fix) = Um /z- X (-1)* ( r ) f(^-f^h). (2.3)

As n approaches infinity, we want h to approach zero. To see how

these two limits should take place simultaneously, it is instructive to

examine Eq. (2.3) for the special case in which v = -1.

cDx'f(x)= fiOdt. [ From Eq. (2.2),

i_lfl v U X l t l l ^ ^ ' k\r i-v)

Then,

20

(_ i ) . ( - iLn^=i . ^ ^ ^!r(i)

The R. H. S. of Eq. (2.3) becomes

fix-t) dt 0

i fin dt. x-limnA

But x-lim nh must equal c, which means that we should take

, X - c h =

n

Eq. (2.3) with the above h works for any number v.

2.4 Small Parameter Expansion

We want to expand OD|'*'^/(X) in the following form, e is a small

parameter (0 < e < 1) and a constant. The subscripts are omitted for

simplicity.

D^'^^f= D^f+£ [function ofx ] + £^ [function ofx ]+• • - .

Assume that f(x) can be expanded as a power series, f(^) = 2 ^" ^". n=0

D^^'f=D^D-^'-'^f

21

D<'-'^f=D-^'-'^^anXn n=Q

YanD-^'-'^x" n=0

„=o r in+2-£)

Operating D ^ on both sides, we get

„=o r («+2-e)

„=o r (rt-l-e)

2-e r - l - e . V T (/t-1-1) „ 9 ^

r ( - l - e ) r ( - £ ) n=2 r(AZ-l-£)

Notice that D^f= Y. an— x"-2. Neglecting e2 and higher order n=2 Tin-l)

terms, we have

x-e = g-£^r\ X

= 1 - e In X,

—^— =Zck (-£)^ r(-£) =1

= Ci i-£) = -£,

1 -1-e

r ( - i -£) r(-e)

= i-l-£) i-£) = £,

22

rin-l-£) = rin-l)-£^ dx x=n-l.

1 =—^ [1+£V^(«-1)], F in-l-£) r in-l)

where v^(xj is the psi function defined by V^ ) -r ( x )

ix) We also used

the relation —^=ZcicX^^[^6'\ Then, r(x) k=i

r (-i-£)

v-- l -e 1 fli ^ = -eai x-i,

r(-e)

oo °°

y an ^^^""^^ xn-^-^= I ^ „ 0 ^ ^ ; c - 2 [ l +£V^(;Z-l)][l-£ Inx] S r(rt-l-e) n=i r (n - l )

= 2- " n=2 Tin-\)

x" -2 [ l - e \n X + £\if in-\)]

Therefore, we get the following equation,

D^-^f^D^f+£^-£^-£r\nx + £Y^an Tin+\)

n=2 r in-\) \lAji-\)x n-l

23

(2.4)

The last term converges in the interval in which f{x) converges. We

can see it by the ratio test. Since \\fin) = \\iin-\) +-^,

lim n—>oo

an^\ ^ n+ V< ) n n-\ XjKn-l) is smaller than 1. Eq. (2.4) also works for the

negative values of e (I £ I < 1).

We can also derive Eq. (2.4) by the following method. Since

(Px Ax) is a function of x and v, we can write

(l>ix,v) = (p;^fix).

Then, (P^^'f=(pix,2+£)

= 0 (x, 2) -I- £ a^

av v=2

After some calculation, we get the same expression as Eq. (2.4).

CHAPTER III

EQUATION OF MOTION

3.1 Geodesic Equation

In Riemannian space which is coordinatized by x (/ = 1, 2, •••, «),

the distance ds between neighboring points is given by

d s'^ = gij dx^ dxJ,

where gij is the metric tensor. A geodesic may be defined by the

zero curvature conditions or by the condition that for any two of its

points sufficiently close together, its length between the two points is

least among all curves joining those points.[2] The minimum length

development employs a variational argument.

Let X' = x ' (0 represent the shortest curve (geodesic) passing

through A = x^ia) and B = x^ib), where ^ - a is as small as necessary.

Consider one parameter family of C 2 curves (the parametric

functions have continuous second order derivatives) passing through

A and B.

X' it, u) = X' it) + it -a) ib-t)u (l)\t),

where the multipliers (p ' (0 are arbitrary twice differentiable

functions. The length of a curve in this family is given by

24

25

ax' ax ^ ( w ) = | ^ 1 gij dt=\ iwit, u) dt. [

We assumed a positive definite metric. Since X' (^ 0) = x' (0, the

function L{u) must have a local minimum at w = 0. From the calculus

f of variation, >'(x) which makes I Fix, y, / ) dx an extremum.

satisfies the necessary condition.

dx dy «^ \ay'

In our case, F corresponds wi/2^ where

w^wit,0)=gij ^ ^ . (3.1) ^ dt dt

dF_ Then _ becomes

By

L^-m^ = L w-m^ dxL d± 2 ax* 2 ax* ^ dt

dF ~~,\ becomes

L^-m^ = \^.Mg•.,d±^ d^ 2 9^* 2 \ df dt >

dt

We used the symmetry property of gij. Then the necessary

condition is

26

^-m^Jil dxL dxL = Ail w-^i'^ gik ^ 1 . ^ x^ dt dt dt dt

The right-hand side is

dt dt \^xJ dt j

2v / ^ + 2w-l/2^.^^-2^ dt df

_ a gij Multiplying both sides by w 1/2 and using the notation gijk = » w e

ax* ge t

g..^dxL dxL ^ _^ - i g dw. dxL + 2giki^ ^ + 2gik^^, ' ^ dt dt ^'* dt dt ' ^ dt dt ^'* ^ 2

which rearranges to

dh' a-., dxL dxL + 0 n.^dxL dxL -;; silk , , ^ - gik] . , — '^gik^^- gij dt' dt dt dt dt

J_ n-.dw_ dxj_ ^ ^ dt dt

Using gij = gji, the third term on the left is split into two similar

terms, then

2 g i k ^ + (-gijk + gjki + gkij) dt-

dxL dxL = 1. Qv^^ ^^ dt dt ^ ^ dt dt

Multiplying by gP^I2, summing on k, and using the relation

gP^gik = 5?> we get

dt^ 2 ^ ^ dt dt 2w dt dt

27

d'^xP iP.dxL dxL = JL_ dw_ dxL ^ 2 '^ dt dt 2w dt dt

Changing index, we have

d^^f Md^^^d^dxL^ ii=U2,.-.,n) (3.2) dt^ dt dt 2w dt dt

Eq. (3.2), with w defined by Eq. (3.1), are the differential

equations for the geodesies of Riemannian space in terms of the

arbitrary curve parameter t. We may choose t = s = arc length, then

w = p = 1 , ^ ^ = 0. \dtl ds

Eq. (3.2) becomes

dfxL + f^dxL dxi = 0. (3.3) ds^ ^ ds ds

Since L '(0) = 0 is only a necessary condition for minimum length, the

geodesies are found among the solutions of Eq. (3.2) or Eq. (3.3).

In the case of null geodesic, we use the zero curvature condition.

The following system of n +1 ordinary differential equations in the n

unknown functions xKO will determine the null geodesic.

28

dbd.^fd^d^^O, ii=l,2,.-;n) dt^ ^ dt dt

gij^^=0. dt dt

In general relativity, p in XP has the range 0, 1, 2, 3. A free

particle follows a timelike geodesic (ds^ > 0) and the proper time is

the curve parameter. A photon follows a null geodesic. Since there

is no change in proper time along the path of a photon, T cannot be

used as a parameter. But we can still use an affine parameter.

In order to obtain x^is) of an affinely parametrized geodesic, we

must solve the system of differential equations Eq. (3.3). These

equations are second order and require 2n conditions to determine a

unique solution. It would seem to be a complicated procedure just to

set up the geodesic equations. There is a procedure which produces

the equations as well as the Fy^Ji]

Consider the Lagrangian L,

L(x',x'^)^^gij(x^)x^ xJ,

which we regard as a function of 2n independent variables

. . k (_ dx^ \ ^ ' ^ \~~d^j. The Lagrange equation is

d ^ ^ \ ^^

ds axN ax*

This reduces to the geodesic equation, as we now show.

Differentiating the Lagrangian, we have

29

= ^gikX' + irgkjX^ = gkjXJ dx k 2

dL

ax* 2 1 • i • i

= irgijkX xJ - ^giJ

Eq. (3.4) is

fyjxi)-^gijkx' x^ =^,

Using

We get

'rJ 4- o, •• v ' r> _ ^gijkX^ X-' = 0 . gkjX' + gkjiX X' - ^

gkjix' x^ = ^gkijx' xJ + ^gjkix' xJ,

gkj x^ + Ugkij + gjki - gijk)x' x^ = 0.

Multiplying by g^^ and summing on k, we get

x + li;x' x^ = 0,

which is the equation of an affinely parametrized geodesic.

If g.. does not depend on some particular coordinate x^ then Eq.

(3.4) is

it^ - "•

30

which implies that —7 = gijX^ is constant along the geodesic. dx

3.2 The Schwartzschild Solution

Einstein's field equation is

RPy .iRgpv = KT^'^ Am

where the left-hand side is the Einstein tensor GP^, TP^ is the energy

momentum stress tensor, and K = -%nGlc ^ iG is the gravitational

constant). An alternative form for the field equation is

where T = T^^ . The TP^ contains all forms of energy and

momentum. For example, if there is electromagnetic radiation

present, then this must be included in TP^. A region of spacetime in

which TP^ = 0 is called empty, and such a region is not only devoid of

matter, but of radiative energy and momentum also. From the above

equation, the empty spacetime field equation is

/ ?MV= 0 .

To solve the field equation and discover g^y, Schwartzschild made

the following assumptions. The field was static and spherically

symmetric. And the spacetime was empty and asymptotically flat.

This field represents the static spherically symmetric gravitational

3 1

field in the empty spacetime surrounding some massive spherical

object like a star. He also assumed that spacetime could be

coordinatized by coordinates (r, r, 6, 0), where t was a timelike

coordinate, 6 and 0 were polar angles, and r was some radial

coordinate. He postulated

c^dr^ = Air)dt^- Bir)dr^ - r'^ dO^ - r^sin^Odip^,

as a form for the line element, where A(r) and 5 ( r ) were some

unknown functions of r to be obtained by solving the field equation.

With the g^y obtained from the line element, we can calculate T^a

and R^y . Then solve R^y = 0 and use Newtonian potential in the

asymptotically flat region. The Schwartzschild solution for the empty

spacetime outside a spherical body of mass M is

c^dr^ = c2fl - 2GK\dt^ drl r^ d9^ - r^sin^Odip^. (3.5) \ c^rl 1 - 2GMIic^r)

This solution is asymptotically flat, and in no way incorporates

the gravitational effect of distant matter in the universe.

Nevertheless, it seems reasonable to adopt it as a model for the

gravitational field in the vicinity of a spherical massive object such

as a star, where the star's mass is the principal contributor to the

gravitational field. When M is 0, the Schwartzschild line element

reduces that of flat spacetime in spherical polar coordinates. The

coordinates t and r then have simple physical meanings: t is the

time as measured by clocks which are stationary in the reference

32

system employed, and r is the radial distance from the origin. With

increasing M, curvature is introduced. Spacetime is no longer flat,

and there is no reason to assume that the coordinates have the

simple physical meanings they had in flat spacetime.

When dt = dd = d(p = 0 in Eq. (3.5), the infinitesimal radial distance

is

^ = (i _ M)r '« dr,

where fn = GMIc^. So dR > dr and r no longer measures radial

distance. For a clock at a fixed point in space (r, 6, (p constant),

infinitesimal proper time interval is

dx = {\-2m.Y^dt.

As r -^ oo, dR ^ dr and dr -^ dt. Asymptotically the coordinate distance

dr coincides with the actual distance dR, and the coordinate time dt

with the proper time dr.

The Schwartzschild solution is the basis for the four tests of

general relativity, namely perihelion advance, the bending of light,

time delay in radar sounding, and the geodesic effect. Spectral shift

is more a test of the principle of equivalence than of general

relativity- But it can be discussed in the context of the

Schwartzschild solution.

3 3

3.3 Equation of Motion for Free Fall

The path of particle with mass moving in the vicinity of a

spherical massive object is given by the timelike geodesic of

spacetime. We assume that the particle is a test particle. This means

any curvature produced by the particle following the geodesic is

ignored. It does not have any effect on the body producing the

gravitational field.

For a timelike geodesic, we may use its proper time r as an affine

parameter. The geodesic equation is given by Eq. (3.4).

dr\dx^^ dL

dxP = 0,

where

L(x^,xo)^^g^yX^x^

1 2

c2(i _ 2m.)t^ _ (i _ 2m.y^ y^ _ ^2(0^ + sin^^^^

Here dots denote derivatives with respect to r, the coordinates are

x® = t,x^ = r, x2 = e,x'^ = (p, and m = GMIc 2.

Because of spherical symmetry, there is no loss of generality in

confining our attention to particles moving in the equatorial plane

given by e = nil. With this value for Q, geodesic equation for ^ = 1 is

(1 _ 2m)-^ r + ^ ^ r ^ - (l - ^ j ' ^ ^ r ^ - r0^ = 0. (3.6)

34

Since t and (p are cyclic coordinates, we have integrals of the two

remaining equations,

dL dL —7 = const, —7 = const. ^t d(p

With e = nil these are

(1 - ^ ) r = k, (3.7)

r^(p = h, (3.8)

where k and h are integration constants. We also have Eq. (3.5)

which defines r. With 6 = nl2 this becomes

c2(l - 2^)r^ - (1 - ^Y r2 - r2 0^ = c2, (3.9)

and may often be used in place of the rather complicated Eq. (3.6).

Eq. (3.7) gives the relation between the coordinate time t and the

proper time T. Eq. (3.8) is clearly analogous to the equation of

conservation of angular momentum. Eq. (3.9) yields an equation

analogous to the conservation of energy.

Eq. (3.5) gives

C2

(P w/ 0

and substituting for (p and r from Eqs. (3.7) and (3.8) gives

3 5

^ f + r2ll±clrA(i_2m] _ c2^2^4 ^ ^ dd \ h^ r '^ h^

If we putM=l/rand m =GMIc^, this reduces to

W + u^ = E +2GKu + 2GKu\ \d(p h 2 c2

where E = c'^ ik^-\)lh^. This corresponds to an energy equation. The

last term on the right is a relativistic correction, and this gives the

advance of the perihelion in planetary orbits.

For vertical free fall, (p is constant and h = 0 from Eq. (3.8). With

0 = 0 andf from Eq. (3.7) substituted, Eq. (3.9) becomes

r^ - c^k'^ + c2(l - 2^) = 0. (3.10)

This equation gives a meaning to the integration constant k. If the

particle is at rest ir = 0) when r = ro, then

^2 . 1 _ 2m '"0

Since T increases with t, Eq. (3.7) shows that

k = (i-^y\ (3.11)

In particular, i f r - > 0 as r-> oo, then ^ = 1. Differentiating Eq. (3.10)

gives

r + GM = 0. (3.12) r^

3 6

This equation has exactly the same form as its Newtonian

counterpart. But in Eq. (3.12) the coordinate r is not the vertical

distance and dots are derivatives with respect to proper time. In

Newtonian version r is vertical distance and dots are derivatives

with respect to the universal time.

Using Eq. (3.11), Eq. (3.10) becomes

i r ^ = O M ( i - i - ) . (3.13) ^0

Since the left-hand side is positive, Eq. (3.13) holds when r < ro- It

has the same form as the Newtonian equation expressing the fact

that a particle of unit mass falling from rest at r = ro gains a kinetic

energy equal to the loss in gravitational potential energy. Using Eq.

(3.13), we can calculate the proper time experienced by the particle

in falling from rest at r = TQ. If T = 0 when r = TQ, then

T = — L - I \-^^'^ dr. V2GM

Using '* = rosin^i/^, the integral becomes

.Tia

, 2 . , . W2 '•y^ I 7 ^ ^^0 sin v^cos Ydy/

1 - sin2 y/j

Jw

= 2r /2 j sin^\f/dy^

37

= '•o^^d - ¥+ ^sin2vA).

Since sin ^ = {^f\ cos v = (l - -^f^, r becomes

3/2

(2GA/)i^2 L 2 \rol \roi \ ro/ J

From Eqs. (3.7) and (3.13), we have

(3.15) dt _ dr 1 •

dr _ dr

k _ (l-2m/ro)i/2 - 2w/r 1 - 2mlr

-(2GM)•/^(^f'^ (3.16)

The above two equations are used to calculate r and t numerically.

3.4 Approximation

When r approaches 0, r can be approximated as

r = ro-AT2,

where A is a constant. We can find A by expanding terms in

Eq. (3.14).

ro ro

Assume ^ T 2 « 1, then

f n i / 2 ^ i _ ^ ^ 2 \W 2ro

3 8

We neglected the terms higher than T2 order. Substituting the above

approximations into Eq. (3.14), we get

A=-GM 2ro2

Now we calculate the approximate form for t.

dt ^ k dr 1 _ 2z7L '

where ^ = (^-7^f^^' We approximate ^ = 1 - ^ ,

( l _ 2 ^ ) - i s l + 2 m . .

Then

dr ^ ^oi^ r I

^ 1 _m. + 2m. 0 r

Since

39

- M - g ^ l

*" '•ol 2ro3 J

cfr ~ ^ + ZM. + mOM ^2 dr ^0 VQ^

Now r can be approximated as

\ rn) o». 4 ^ol 3ro^

The above equation satisfies the initial condition, ^ ( T = 0) = 0.

Now we calculate D^-^^ r, D'^-^^ t using Eq. (2.4). We use the

dimensionless forms for r and t.

r = ao + a2r'-.

t = bi r+b3 r^. (3.17)

where ao, ^2, ^i» and b^ are constants. Using Eq. (2.4), we get

D^^^r = D^r + £ ao 2a2^n r + a22 i-'f) LT^

D^^^t=D^t + 5 —-6/?3T In T-I-^3 6 (1-}^ r

w here y is the Euler constant (7 = 0.5772 •••)• Since D2r = 2^2,

40

D^t = 6Z?3T, equating

D2+^r = 0,

D2+^r = 0,

we get

£= -2^2

^ - 2 ^ 2 In T-2^2 7

1 ^^ -I- In T -t- 7

2^2 T

g_ -6^3 T (3.18)

1--6^3 Tin T +Z?3 6(1-'}^ T

1

^i— + l n T - ( l - } ^ 6^3 r

According to Eq. (3.18), £ and 8 depend on r. But we assumed £ is a

constant when deriving Eq. (2.4). Eq. (3.18) may be used for

comparison with the numerical results. When r approaches 0, we

h a v e

£ =

8 =

2a2

6^3 bi

T2,

T2.

(3.19)

The physical meanings of £ and 8 are discussed in Chapter V.

CHAPTER IV

RESULTS

We used the following dimensionless variables.

r : r X 10^0 meter

t : t X 106 sec.

T: T X 106 sec.

In the calculation, the parameters of the sun and the Mercury were

used. The gravitational constant G is 0.6673 X 10-10 Nm2/kg2. The

mass of the sun M is 1.989 X 10^0 kg. Then m = GMIc^ is

0.1477 X 10" meter, where c is the speed of light. Initial radial

coordinate ro is 6.98 X lOio m, which corresponds the aphelion

distance of the Mercury.

For calculation of r(T), Eq. (3.14) was used. For tir), Eqs. (3.15)

and (3.16) were used with the fourth order Runge-Kutta

algorithm.[27] Fig. 1 and Fig. 2 represent r(T) and t(r) with the

dimensionless variables. The dimensionless x ~ 1 corresponds 12

days of free falling. From Eq. (3.15), m/ro and mir are order of 10-8,

so we get the result that the time t and the proper time r are almost

same. This means the fractional parameter 8, which satisfies D 2+5/ =

0, should be a small number.

For calculation of the fractional parameters £, which satisfies

D 2+£r= 0, and 8, we used Griinwald algorithm, n in Eq. (2.3) was 30.

(In Ref. [4] and [28], they used n =32). The tolerance for calculation

of 8 was 10-13, for £, it was 5-10-7. The search and the false position

41

4 2

methods were used to find £ and 8. Fig. 3 and Fig. 4 are the results.

The £ and 8 are positive and increase as r increases. As we expected

5 is a small number of order 10-8. The increasing r means r is

decreasing, and we expect the curvature effect also increases as r

decreases. Fig. 5 and Fig. 6 represent £ and 8 as functions of r. The

dotted lines in Fig. 3 and Fig. 4 represent the analytical

approximation for £ and 8. It was calculated using Eq. (3.18). The

dimensionless form for Eq. (3.17) is

,(^) = ,,_1Q0GM^2^ 2ro2

(7) = (1 + ZZLW + i m J/L GMT^ , V rof 3 ro"*

where ro = 6.98, G = 0.6673, and M = 1.989. The above equation

becomes numerically

r(T) = 6.98-1.36T2,

r(T)=(l -H2.12- 1 0 - ^ ) T + 2 . 7 5 - IO'^T^.

The dimensionless form for Eq. (3.19) is

£ = 1 0 0 ^ ^ T 2 , ro

^=200 ^ ^ ^ T2. ro' tl +w/ro)

(4.1)

In terms of r, Eq. (4.1) is

43

, 0 (1 . ^ ) I ro) 8 = r n / l -I-

''0

= 2^£.

The above equations hold when r is near 0.

In £ case, the analytical and the numerical results are of the same

order. In 8 case, they are same order when r is larger than 0.16.

Table 1 and Table 2 represent 8 and £ for r smaller than 0.2.

in

^

t o u

Fig.l. Plot of radial coordinate r versus proper time r

44

t o u

Fig. 2. Plot of time t versus proper time r

45

Fig. 3. Plot of fractional parameter £ versus proper time r. The solid line represents data from numerical algorithm. The dotted line represents data from approximate equation.

46

Fig. 4. Plot of fractional parameter 8 versus proper time r. The solid line represents data from numerical algorithm. The dotted line represents data from approximate equation.

47

CD

Q)

o o

5.0 6.0

r

7.0

Fig. 5. Plot of fractional parameter £ versus radial coordinate, r

Fig. 6. Plot of fractional parameter 5 versus radial coordinate r

4 8

Table 1. Proper time r versus fractional parameter 8 {r < 0.2)

numerical 8

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

0.20

1.824

1.824

5.170

2.289

2.289

3.052

3.433

9.087

2.755

4.637

X

X

X

X

X

X

X

X

X

X

10-14

10-14

10-26

10-14

10-14

10-14

10-14

10-11

10-10

10-10

6.600

2.640

5.940

1.056

1.650

2.376

3.234

4.224

5.346

6.600

X

X

X

X

X

X

X

X

X

X

10-12

10-11

10-11

10-10

10-10

10-10

10-10

10-10

10-10

10-10

Table 2. Proper time r versus fractional parameter £ (r < 0.2)

r numerical £ approximate £

0.02 1.509 X 10-4 1.560 X 10-4

0.04 6.036 X 10-4 6.245 X 10-4

0.06 1.358 X 10-3 1.407 X 10-3

0.08 2.414 X 10-3 2.506 X 10-3

0.10 3.772 X 10-3 3.923 X 10-3

0.12 5.432 X 10-3 5.660 X 10-3

0.14 7.393 X 10-3 7.720 X 10-3

0.16 9.656 X 10-3 1.010 X 10-2

0.18 1.222 X 10-2 1.281 X 10-2

0.20 1.509 X 10-2 1.584 X 10-2

CHAPTER V

DISCUSSION

We studied the one-dimensional motion of a particle in

gravitational field using the fractional calculus. The r(r) and t(r) from

the general relativistic equations of motion are used to find £ and 8

which satisfy the following equations.

D2+^r(T) = 0,

D^^^tir) = 0.

The £ and 8 are functions of the proper time r and incorporate the

gravitational effect. Eq. (4.1), which is approximate form of £ and 8 ,

is

e=100GM^2 ro'

8=200 rn^^ T2. ro\l +mlro)

We can see that £ and 8 depend on the initial radial coordinate ro and

the mass M of a gravitating body. When ro approaches the infinity, £

and 8 become 0. This is the expected result. When the particle is at

infinite distance from the massive body, there is no gravitational

field. We expect that £ and 8 may be large numbers in the vicinity of

a black hole. When the repulsive force is acting on the particle, a2 in

49

5 0

Eq. (3.17) is positive and £ becomes a negative number according to

Eq. (3.19).

One possible application of the fractional operator formalism is as

a a follows. We may use fractional operator ~— instead of ~ as a

ox "'" ox

momentum operator in commutation relations of quantum physics.

We use

.. a Px = -i-h

axi+^

Then, by the first-order approximation, we get

[x,Px] = ifi + £[ ].

In the above equation, the empty bracket is a quantity, which should

be studied. The commutation relation means the uncertainty

principle. The £ in fractional operator D i+^ represents the spacetime

curvature effect, which is the gravitation according to general

relativity. In this way, we may combine the quantum physics and

the gravitation.

Another generalization of commutation relations is as follows.[29]

Snyder introduced the natural unit of length a to remove the

divergence trouble of quantum field theory.

[x,Px\ = i^b+(alhfpx^],

5 1

[t,p:\ = ifi[l-{a^fpA

[x, Py] = ly,Px] = ifi (alhfpxPy.

[x, pi\ = c2 \p^, t] = it (ainfpxPf

If all the components of the momentum are small compared to t/a

and the energy is small compared to ficla, then these relations

approach those which are given in ordinary quantum mechanics.

Further, when a approaches 0, we get the ordinary results. A

possible candidate for a is the Planck length

((G^C3)I/2 = 1.6 . 10-35m).[30] Then,

tia ~ 10 kg m/s,

ficia ~ 10 J.

One may speculate that non-local theory of Yukawa[3i.32] js

related with this kind of formalism. The use of fractional operator in

quantum physics needs more research.

REFERENCES

[1] J. Foster and J. D. Nightingale, A short course in General Relativity (Longman, New York, 1979).

[2] D. C. Kay, Theory and Problems of Tensor Calculus (McGraw-Hill, New York, 1988).

[3] B. Ross (Ed.), Fractional Calculus and Its Applications, Lecture Notes in Math. 457 (Springer-Verlag, New York, 1975).

[4] K. B. Oldham and J. Spanier, The Fractional Calculus (Academic Press, New York, 1974).

[5] S. G. Samko, A. A. Kilbas, and O. I. Marichev, Integrals and Derivatives of Fractional Order and Some of Their Application (Nauka i Tekhnika, Minsk, 1987).

[6] H. M. Srivastava and S. Owa (Eds.), Univalent Functions, Fractional Calculus, and Their Applications (Ellis Horwood, Chichester,1989).

[7] B. Ross and S. L. Kalla, "The Method of Fractional Operators Applied to Summation," Real Analysis Exchange 11 , 271-282 (1985) .

[8] L. Campos, "On the Solution of Some Simple Fractional Differential Equations," Internal. J. Math. & Math. Sci. 13, 481-496 (1990).

[9] M. A. Al-Bassam, "Some Applications of Generalized Calculus to Differential and Integro Differential Equations," in Mathematical Analysis and its Applications, edited by S. M. Mazhar (Pergamon, Oxford, 1988), pp. 61-76.

[10] J. L. Lavoie, T. J. Osier, and R. Tremblay, "Fractional derivatives and special functions," SIAM Rev. 18, 240-268 (1976).

52

5 3 [11] W. Wyss, "The Fractional Diffusion Equation," J. Math. Phys. 27,

2782-2785 (1986). W. R. Schneider and W. Wyss, "Fractional Diffusion and Wave Equations," J. Math. Phys. 30, 134-144 (1989) .

[12] R. R. Nigmatullin, "The Realization of the Generalized Transfer Equation in a Medium with Fractal Geometry," Phys. Stat. Sol. B 133 , 425-430 (1986).

[13] T. F. Nonnenmacher and D. J. F. Nonnenmacher, "Towards the Formulation of a Nonlinear Fractional Extended Irreversible Thermodynamics," Acta Physica Hungarica 66, 145-154 (1989). T. F. Nonnenmacher, "Fractional Integral and Differential Equations for a Class of Levy-type Probability Densities," J. Phys. A. 23 L697-L700 (1990).

[14] F. Wilczek, "Magnetic Flux, Angular Momentum, and Statistics," Phys. Rev. Lett. 48, 1144-1146 (1982). "Quantum Mechanics of Fractional-Spin Particles," Phys. Rev. Lett. 49, 957-959 (1982).

[15] J. Q. Liang and X. X. Ding, "New Model of Fractional Spin," Phys. Rev. Lett. 63, 831-833 (1989). Y. Aharonov, C. K. Au, and L. Vaidman, "Comment on 'New Model of Fractional Spin'," Phys. Rev. Lett. 66, 1638 (1991).

[16] Y. N. Kim, "Spin and Statistics of Elementary Particles," in Mathematical Foundations of Quantum Theory, edited by A. R. Marlow (Academic Press, New York, 1978), pp. 347-349.

[17] V. Namias, "The Fractional Order Fourier Transform and its Application to Quantum Mechanics," J. Inst. Maths. Applies. 25, 241-265 (1980). A. C. McBride and F. H. Kerr, "On Namias's Fractional Fourier Transforms," IMA J. Appl. Math. 39, 159-175 (1987) .

[18] L. Eisenbud, "On the Classical Laws of Motion," Am. J. Phys. 26, 144-159 (1958).

[19] S. H. Schot, "Jerk: The time rate of change of acceleration," Am. J. Phys. 46, 1090-1094 (1978). "Aberrancy: Geometry of the Third Derivative," Math. Mag. 5 1 , 259-275 (1978).

54

[20] H. T. Davis, "Fractional Operations as Applied to a Class of Volterra Integral Equations," Am. J. Math. 46, 95-109 (1924). "The Application of Fractional Operators to Functional Equations," Am. J. Math. 49, 123-142 (1927).

[21] H. T. Davis, The Theory of Linear Operators (Principia Press, Bloomington, Indiana, 1936).

[22] A. Erdelyi (Ed.), Higher Transcendental Functions Vol. I, (McGraw-Hill, New York, 1953), p. 52.

[23] H. B. Dwight, Tables of Integrals and Other Mathematical Data (Macmillan, New York, 1961), 4th ed., p. 213.

[24] C. Lubich, "Discretized fractional calculus," SIAM J. Math. Anal. 17, 704-719 (1986). "Fractional Linear Multistep Methods for Abel-Volterra Integral Equations of the First Kind," IMA J. Numer. Anal. 7, 97-106 (1987).

[25] R. F. Cameron and S. McKee, "The Analysis of Product Integration Methods for Abel's Equation using Discrete Fractional Differentiation," IMA J. Numer. Anal. 5, 339-353 (1985) .

[26] M. Abramowitz and I. A. Stegun (Eds.), Handbook of Mathematical Functions (Dover, New York, 1965), p. 256.

[27] S. E. Koonin, Computational Physics (Benjamin/Cummings, Menlo Park, 1986).

[28] F. G. Lether, D. M. Cline, and O. Evans, "An Error Analysis for the Calculation of Semiintegrals and Semiderivatives by the RL Algorithm," Appl. Math. Comp. 17, 45-67 (1985).

[29] H. S. Snyder, "Quantized Space-Time," Phys. Rev. 71 , 38-41 (1947). "The Electromagnetic Field in Quantized Space-Time," Phys. Rev. 72, 68-71 (1947).

55 [30] M. A. Markov, "Can the Gravitational Field Prove Essential for

the Theory of Elementary Particles?," Suppl. Prog. Theor. Phys., Extra number, 85-95 (1965).

[31] W. A. McKinley, "Search for a Fundamental Length in Microscopic Physics," Am. J. Phys. 28, 129-134 (1960).

[32] H. Yukawa, "Atomistics and the Divisibility of Space and Time," Suppl. Prog. Theor. Phys., Nos. 37 & 38, 512-523 (1966).

APPENDIX

FORTRAN PROGRAMS

C ***** THIS PROGRAM CALCULATES TIME T AS FUNCTION C OF TAU BY RUNGE-KUTTA METHOD C C dt/d tau = P, dr/d tau = G C STEP SIZE H = 0.01 C T,R,TAU,P,G ARE DIMENSIONLESS. C ALSO CHECKS FRACTIONAL PARAMETER FOR T(TAU) C

IMPLICIT REAL*8(A-H,0-Z) DIMENSION T(1000),R(1000) COMMON RO,N,T

R0=6.98D0 H=0.01D0 N=30

Q ***•* INITIAL CONDITIONS ***** T(1)=0.0D0 R(1)=R0 T(2)=H*P(R(1) )

C ***** R(2) FROM ANOTHER PROGRAM R(2)=6.97 98 63787188257D0

DO 1=2,150

RK1=H*P(R(I)) RL1=H*G(R(I))

RK2=H*P(R(I)+RL1/2.GDC) RL2=H*G(R(I)+RL1/2.GDC)

RK3=H*P(R(I)+RL2/2.GDG) RL3=H*G(R(I)+RL2/2.GDG)

RK4=H*P(R(I)+RL3) RL4=H*G(R(I)+RL3)

T ( I + l ) = T ( I ) + ( R K l + 2 . G D G * ( R K 2 + R K 3 ) + R K 4 ) / 6 . G D G R ( I + 1 ) = R ( I ) + ( R L l + 2 . G D G * ( R L 2 + R L 3 ) + R L 4 ) / 6 . G D G

C W R I T E ( 6 , * ) T ( I ) , R ( I )

END DO

56

57 C ***** CALCULATION OF FRACTIONAL PARAMETER C WITH INTERPOLATION OF T() VALUE USING C SUBROUTINE INPOL. SUBROUTINE ACA CALCULATES C BINOMIAL COEFFICIENT, D(2+E)T. C TO FIND E WHICH MAKES D(2+E)T = G WITH C TOLERANCE(IG(-13)), SEARCH METHOD IS USED. C INTERVAL IS REDUCED TO 1/2 WHEN THERE IS C A SIGN CHANGE. C T(J*H) = T[J+1] C

T0L=1.GD-13

DO J=2,5G,2

TAU=J*H E=-G.3D-8 DE=l.GD-9 Y1=1.GDG 1=1

* * • * * SEARCH METHOD

DO WHILE(DABS(Yl).GT.TOL) CALL ACA(J,E,Y) E=E+DE CALL ACA(J,E,Y1)

C WRITE(6,*) E,Y1

1=1 + 1 IF(Y*Y1.GT.G.GDG) GO TO IG E=E-DE DE=DE/2.GDG

IG END DO WRITE(6,5 6) TAU,E,Y1,I

5 6 FORMAT(2X,F6.2,2X,E12.4,2X,E12.4,2X, 14) END DO

STOP END

Q ***** ACA RECEIVES J,E, GIVES RES C J SPECIFIES TAU BY TAU = J*H C 2+E IS EXPONENT OF FRACTIONAL OPERATOR C RES = D(2+E) T WITH CHANGING TAU C A(K) IS BINOMIAL COEFFICIENT. C

SUBROUTINE ACA(J,E,RES) IMPLICIT REAL*8(A-H,0-Z) DIMENSION T(IGOG),S(5GG),A(5GG) COMMON RG,N,T AL=2.GDG+E

58 A(1)=AL

DO K=1,N A(K+1)=A(K)*(AL-K)/(K+l.GDO)

END DO

S(1)=T(J+1)

DO K=1,N CALL INPOL(J,K,VAL) S(K+l)=S(K) + ( (-1) **K) *A(K) *VAL END DO H=G.G1DG X=J*H RES=S(N+1)*N*N/(X*X)

RETURN END

C C C C C C

***** INPOL CALCULATES INTERPOLATION OF T [ J USING LAGRANGE TWO POINT FORMULA. J ; INDEX FOR CHANGE OF TAU IN MAIN K ; " K IN ACA VAL ; VALUES RETURNED

SUBROUTINE INPOL(J,K,VAL) IMPLICIT REAL*8(A-H,0-Z) DIMENSION T (IGGG) COMMON RG,N,T RN=N L=IDINT(J*(1.GDG-K/RN)) Q=J*(1.GDG-K/RN)-L

VAL=(1.GDG-Q)*T(L+1)+Q*T(L+2)

RETURN END

C ***** FUNCTIONS NEEDED FOR RUNGE-KUTTA METHOD

DOUBLE PRECISION FUNCTION P(X) IMPLICIT REAL*8(A-H,0-Z) COMMON RG ZM=G.1477D-6 ZK=DSQRT(1.GDG-2.GDG*ZM/RG) P=ZK/ (1.GDG-2.GDG*ZM/X) RETURN END

59

DOUBLE PRECISION FUNCTION G(X) IMPLICIT REAL*8(A-H,0-Z) COMMON RG RG=G.6673DG RM=1.98 9D0 G=-DSQRT(2.GD2*RG*RM*(RG-X)/(RG*X)) RETURN END

C ***** THIS PROGRAM CALCULATES RADIAL COORDINATE R C AS FUNCTION OF TAU AND FRACTIONAL PARAMETER C E WHICH MAKES D(2+E)R = G. C SUBROUTINE EQ FINDS R WHICH SATISFIES C F(TAU,R) = G WITH GIVEN TAU. C TOLERANCE OF EQ IS lG(-9). C E IS FOUND BY FALSE POSITION METHOD. C

IMPLICIT REAL*8(A-H,0-Z) COMMON RG,N RG=6.98DG N=3G H=G.G1DG

C DO 1=1,IG C TAU=H*(I-1)

C CALL EQ(TAU,R) C WRITE(6,*) TAU,R C 52 FORMAT(2X,F6.2,2X,F12.6) C END DO

Q ***** CALCULATION OF FRACTIONAL PARAMETER C USING SUBROUTINE ACA(J,E,RES)

T0L=l.GD-7 J=5G TAU=J*H EG=1.GDG CALL ACA(J,EG,YG) E1=G.57DG CALL ACA(J,E1,Y1)

C WRITE(6,*) TAU,E1,Y1 E2=E1-Y1*(El-EG)/(Yl-YG) 1 = 1 Y2=1.GDG

Q ***** FALSE POSITION METHOD. El IS FIXED.

DO WHILE(DABS(Y2).GT.TOL)

60

1=1+1 EG=E2 CALL ACA(J,EG,YG) E2=E1-Y1*(El-EG)/(Yl-YG) CALL ACA(J,E2,Y2) WRITE (6,*) TAU,E2,Y2,I

END DO

STOP END

SUBROUTINE ACA(J,E,RES) IMPLICIT REAL*8(A-H,0-Z) DIMENSION S(5GG),A(5G0) COMMON RG,N AL=2.GDG+E A(1)=AL

DO K=1,N A(K+1)=A(K)*(AL-K)/(K+l.GDG) END DO

RN=N X=J*G.G1DG HH=X/RN

CALL EQ(X,R) S(1)=R

DO K=1,N CALL EQ(X-K*HH,R) S(K+1)=S(K) + ( (-1) **K) *A(K) *R END DO

RES=S(N+1)*((N/X)**AL)

RETURN END

Q ***** EQ CALCULATES R FROM GIVEN TAU C USING FUNCTION F. C FALSE POSITION METHOD IS USED. C

SUBROUTINE EQ(T,R) IMPLICIT REAL*8(A-H,0-Z) COMMON RG

T0L=l.GD-9

A=4.GDG B=RG

X=A-F(T,A) *(B-A)/(F(T,B)-F(T,A) )

DO WHILE(DABS(F(T,X)).GT.TOL)

X=A-F (T, A) * (B-A) / (F (T,B) -F (T, A) ) A=X END DO R=X

RETURN END

C ***** THIS IS FOR SUBROUTINE EQ. C

DOUBLE PRECISION FUNCTION F(TAU,R) IMPLICIT REAL*8(A-H,0-Z) COMMON RG

G=G.6673DG RM=1.98 9DG C=(RG**1.5DG)/(1G.GDG*DSQRT(2.GDG*G*RM)) R1=R/RG R2=DSQRT(R1) F=C* (DASINd .GDG) -DASIN(R2) +DSQRT(R1* (1 .GDG-Rl) ) ) *TAU

RETURN END

61

C ***** THIS PROGRAM CALCULATES ANALYTICAL C APPROXIMATION FOR FRACTIONAL PARAMETER C DEL,E WITH CHANGING TAU. C D(2+DEL)T = G, D(2+E)R = G. C

IMPLICIT REAL*8(A-H,0-Z) Bl=l.GDG+2.12D-8 B3=2.75D-9 GAM=G.577215 6 64 9G153DG

AG=6.98DG A2=-1.36DG

DO 1=1,75

P ***** T MEANS TAU

62

T=G.G2DG*I

F 1 = B 1 / ( 6 * B 3 * T * T ) F2=DL0G(T) F3=-1.GD0+GAM

F = F 1 + F 2 + F 3 DEL=1.GDG/F D 1 = 1 . G D G / F 1

C W R I T E ( 6 , 5 2 ) T , F , F 1 , F 2 , F 3 C W R I T E ( 6 , 5 4 ) T ,DEL,D1

52 F O R M A T ( 2 X , F 6 . 2 , 4 ( 2 X , E 1 0 . 3 ) ) 54 F O R M A T ( 2 X , F 6 . 2 , 2 ( 2 X , E 1 2 . 4 ) )

U 1 = A G / ( - 2 * A 2 * T * T ) U2=DL0G(T) U3=GAM U=U1+U2+U3 E=1.GDG/U E1=1 .GDG/U1

W R I T E ( 6 , 5 4 ) T , E , E 1 C W R I T E ( 6 , 5 2 ) T , U , U 1 , U 2 , U 3

END DO

STOP END