pp-09-11

7/29/2019 PP-09-11

1/5

Volume 2 PROGRESS IN PHYSICS April, 2007

Numerical Solution o Time-Dependent Gravitational Schrodinger Equation

Vic Christianto, Diego L. Rapoport and Florentin Smarandache

Sciprint.org a Free Scientifc Electronic Preprint Server, http://www.sciprint.orgE-mail: [email protected]

Dept. o Sciences and Technology, Universidad Nacional de Quilmes, Bernal, ArgentinaE-mail: [email protected]

Department o Mathematics, University o New Mexico, Gallup, NM 87301, USAE-mail: [email protected]

In recent years, there are attempts to describe quantization o planetary distance

based on time-independent gravitational Schrodinger equation, including Rubcic &

Rubcics method and also Nottales Scale Relativity method. Nonetheless, there is

no solution yet or time-dependent gravitational Schrodinger equation (TDGSE). In

the present paper, a numerical solution o time-dependent gravitational Schrodinger

equation is presented, apparently or the rst time. These numerical solutions

lead to gravitational Bohr-radius, as expected. In the subsequent section, we also

discuss plausible extension o this gravitational Schrodinger equation to include

the efect o phion condensate via Gross-Pitaevskii equation, as described recently

by Mofat. Alternatively one can consider this condensate rom the viewpoint

o Bogoliubov-deGennes theory, which can be approximated with coupled time-

independent gravitational Schrodinger equation. Further observation is o course

recommended in order to reute or veriy this proposition.

1 Introduction

In the past ew years, there have been some hypotheses sug-

gesting that quantization o planetary distance can be derived

rom a gravitational Schrodinger equation, such as Rubcic

& Rubcic and also Nottales scale relativity method [1, 3].Interestingly, the gravitational Bohr radius derived rom this

gravitational Schrodinger equation yields prediction o new

type o astronomical observation in recent years, i.e. extra-

solar planets, with unprecedented precision [2].

Furthermore, as we discuss in preceding paper [4], using

similar assumption based on gravitational Bohr radius, one

could predict new planetoids in the outer orbits o Pluto

which are apparently in good agreement with recent observa-

tional nding.. Thereore one could induce rom this observ-

ation that the gravitational Schrodinger equation (and gravi-

tational Bohr radius) deserves urther consideration.

In the meantime, it is known that all present theories

discussing gravitational Schrodinger equation only take its

time-independent limit. Thereore it seems worth to nd out

the solution and implication o time-dependent gravitational

Schrodinger equation (TDGSE). This is what we will discuss

in the present paper.

First we will nd out numerical solution o time-inde-

pendent gravitational Schrodinger equation which shall yield

gravitational Bohr radius as expected [1, 2, 3]. Then we ex-

tend our discussion to the problem o time-dependent grav-

itational Schrodinger equation.

In the subsequent section, we also discuss plausible ex-

tension o this gravitational Schrodinger equation to include the

efect o phion condensate via Gross-Pitaevskii equation,

as described recently by Mofat [5]. Alternatively one can

consider this phion condensate model rom the viewpoint o

Bogoliubov-deGennes theory, which can be approximated

with coupled time-independent gravitational Schrodinger

equation. To our knowledge this proposition o coupled time-independent gravitational Schrodinger equation has never

been considered beore elsewhere.

Further observation is o course recommended in order

to veriy or reute the propositions outlined herein.

All numerical computation was perormed using Maple.

Please note that in all conditions considered here, we use

only gravitational Schrodinger equation as described in Rub-

cic & Rubcic [3], thereore we neglect the scale relativistic

efect or clarity.

2 Numerical solution o time-independent gravitational

Schrodinger equation and time-dependent gravita-tional Schrodinger equation

First we write down the time-independent gravitational

Schrodinger radial wave equation in accordance with Rubcic

& Rubcic [3]:

d2R

dr2+

2

r

dR

dr+

8m2E

H2R +

+2

r

42GMm2

H2R ( + 1)

r2R = 0 .

(1)

When H, V, E represents gravitational Planck constant,Newtonian potential, and the energy per unit mass o the

56 V. Christianto, D. L. Rapoport and F. Smarandache. Numerical Solution o Time-Dependent Gravitational Schrodinger Equation

7/29/2019 PP-09-11

2/5

April, 2007 PROGRESS IN PHYSICS Volume 2

orbiting body, respectively, and [3]:

H = h

2 f

Mmnm20

, (2)

V(r) = GMmr

, (3)

E =E

m. (4)

By assuming that R takes the orm:

R = er (5)

and substituting it into equation (1), and using simplied

terms only o equation (1), one gets:

= 2 er 2 er

r+

8GMm2 er

r H2. (6)

Ater actoring this equation (7) and solving it by equat-

ing the actor with zero, yields:

RR = 2

4GMm2 H2

2H2= 0 , (7)

or

RR = 4GMm2 H2 = 0 , (8)and solving or , one gets:

a =42GMm2

H2. (9)

Gravitational Bohr radius is dened as inverse o this

solution o , then one nds (in accordance with Rubcic &Rubcic [3]):

r1 =H2

42GMm2, (10)

and by substituting back equation (2) into (10), one gets [3]:

r1 =

2 f

c

2GM . (11)

Equation (11) can be rewritten as ollows:

r1 =GM

20, (11a)

where the specic velocity or the system in question canbe dened as:

0 =

2 f

c

1= g c . (11b)

The equations (11a)-(11b) are equivalent with Nottales

result [1, 2], especially when we introduce the quantization

number: rn = r1 n2 [3]. For complete Maple session o these

all steps, see Appendix 1. Furthermore, equation (11a) may

be generalised urther to include multiple nuclei, by rewrit-

ing it to become: r1 =(GM)/v2 r1 =(G M)/v2, where

M represents the sum o central masses.Solution o time-dependent gravitational Schrodinger

equation is more or less similar with the above steps, except

that we shall take into consideration the right hand side

o Schrodinger equation and also assuming time dependent

orm o r:

R = er(t)

. (12)Thereore the gravitational Schrodinger equation now

reads:

d2R

dr2+

2

r

dR

dr+

8m2E

H2R +

+2

r

42GMm2

H2R ( + 1)

r2R = H

dR

dt,

(13)

or by using Leibniz chain rule, we can rewrite equation

(15) as:

H dRdr (t)

dr (t)

dt+

d2R

dr2+

2

r

dR

dr+

8m2E

H2R +

+2

r

42GMm2

H2R ( + 1)

r2R = 0 .

(14)

The remaining steps are similar with the aorementioned

procedures or time-independent case, except that now one

gets an additional term or RR:

RR = H3

d

dtr(t)

r(t) 2r(t)H2 +

+ 8GMm2 2H2 = 0 .(15)

At this point one shall assign a value or ddt

r(t) term,because otherwise the equation cannot be solved. We choosed

dtr(t) = 1 or simplicity, then equation (15) can be rewritten

as ollows:

RR : =rH3

2+

rH22

2+ 42GMm2H2 = 0 . (16)

The roots o this equation (16) can be ound as ollows:

a1 : = r2H+2H+

r4H44H3r+4H232rGMm22

2rH,

a2 : = r2H+2H

r4H44H3r+4H232rGMm22

2rH.

(17)

Thereore one can conclude that there is time-dependent

modication actor to conventional gravitational Bohr radius

(10). For complete Maple session o these steps, see Ap-

pendix 2.

3 Gross-Pitaevskii efect. Bogoliubov-deGennes appro-

ximation and coupled time-independent gravitational

Schrodinger equation

At this point it seems worthwhile to take into consideration a

proposition by Mofat, regarding modication o Newtonian

acceleration law due to phion condensate medium, to include

Yukawa type potential [5, 6]:

a(r) = GMr2

+ Kexp ( r)

r2(1 + r) . (18)

V. Christianto, D. L. Rapoport and F. Smarandache. Numerical Solution o Time-Dependent Gravitational Schrodinger Equation 57

7/29/2019 PP-09-11

3/5


Thereore equation (1) can be rewritten to become:

d2R

dr2+

2

r

dR

dr+

8m2E

H2R +

+ 2r

42

GMKexp( r)(1 + r)m2H2

R

( + 1)r2

R = 0 ,

(19)

or by assuming = 20 = 0 r or the exponential term,equation (19) can be rewritten as:

d2R

dr2+

2

r

dR

dr+

8m2E

H2R +

+2

r

42

GMKe20(1+0r)

m2

H2R(+1)

r2R = 0 .

(20)

Then instead o equation (8), one gets:

RR = 8GMm22H282m2Ke0(1+) = 0 . (21)Solving this equation will yield a modied gravitational

Bohr radius which includes Yukawa efect:

r1 =H2

42(GMKe20)m2 (22)

and the modication actor can be expressed as ratio between

equation (22) and (10):

=GM

(GM

Ke20)

. (23)

(For complete Maple session o these steps, see Appendix 3.)

A careul reader may note that this Yukawa potential

efect as shown in equation (20) could be used to explain

the small discrepancy (around 8%) between the observeddistance and the computed distance based on gravitational

Bohr radius [4, 6a]. Nonetheless, in our opinion such an

interpretation remains an open question, thereore it may be

worth to explore urther.

There is, however, an alternative way to consider phion

condensate medium i.e. by introducing coupled Schrodinger

equation, which is known as Bogoliubov-deGennes theory

[7]. This method can be interpreted also as generalisation o

assumption by Rubcic-Rubcic [3] o subquantum structurecomposed o positive-negative Planck mass. Thereore,

taking this proposition seriously, then one comes to hypo-

thesis that there shall be coupled Newtonian potential, in-

stead o only equation (3).

To simpliy Bogoliubov-deGennes equation, we neglect

the time-dependent case, thereore the wave equation can be

written in matrix orm [7, p.4]:

[A] [] = 0 , (24)

where [A] is 22 matrix and [] is 21 matrix, respectively,which can be represented as ollows (using similar notation

with equation 1):

A =

8GMm2er

rH22er

2er

r

2

e

r

2er

r

8GMm2er

rH2

(25)

and

=

f(r)

g (r)

. (26)

Numerical solution o this matrix diferential equation

can be ound in the same way with the previous methods,

however we leave this problem as an exercise or the readers.

It is clear here, however, that Bogoliubov-deGennes ap-

proximation o gravitational Schrodinger equation, taking

into consideration phion condensate medium will yield non-

linear efect, because it requires solution o matrix diferen-

tial equation

(21) rather than standard ODE in conventionalSchrodinger equation (or time-dependent PDE in 3D-

condition). This perhaps may explain complicated structures

beyond Jovian Planets, such as Kuiper Belt, inner and outer

Oort Cloud etc. which o course these structures cannot be

predicted by simple gravitational Schrodinger equation. In

turn, rom the solution o (21) one could expect that there are

numerous undiscovered celestial objects in the Oort Cloud.

Further observation is also recommended in order to

veriy and explore urther this proposition.

4 Concluding remarks

In the present paper, a numerical solution o time-dependentgravitational Schrodinger equation is presented, apparently

or the rst time. This numerical solution leads to gravita-

tional Bohr-radius, as expected.

In the subsequent section, we also discuss plausible ex-

tension o this gravitational Schrodinger equation to include

the efect o phion condensate via Gross-Pitaevskii equation,

as described recently by Mofat. Alternatively one can con-

sider this condensate rom the viewpoint o Bogoliubov-

deGennes theory, which can be approximated with coupled

time-independent gravitational Schrodinger equation.

It is recommended to conduct urther observation in order

to veriy and also to explore various implications o our pro-

positions as described herein.

Acknowledgment

The writers would like to thank to Pros. C. Castro or valu-

able discussions.

Submitted on January 08, 2007

Accepted on January 30, 2007

Corrected ater revision on March 06, 2007

For recent articles discussing analytical solution o matrix diferential

equations, the reader is reerred to Electronic Journal o Diferential Equa-

tions (ree access online on many mirrors as http://ejde.math.txstate.edu,

http://ejde.math.unt.edu, http://www.emis.de/journals/EJDE etc.).


7/29/2019 PP-09-11

4/5

April, 2007 PROGRESS IN PHYSICS Volume 2

Reerences

1. Nottale L. et al. Astron. & Astrophys., 1997, v.322, 1018.

2. Nottale L., Schumacher G. and Levere E. T. Astron. & Astro-

phys., 2000, v. 361, 379389; accessed online on http://daec.obspm.r/users/nottale.

3. Rubcic A. and Rubcic J. The quantization o solar like

gravitational systems. Fizika, B-7, 1998, v. 1, No. 113.

4. Smarandache F. and Christianto V. Progress in Physics, 2006,

v.2, 6367.

5. Mofat J. arXiv: astro-ph/0602607.

6. Smarandache F. and Christianto V. Progress in Physics, 2006,

v.4, 3740; [6a] Christianto V. EJTP, 2006, v. 3, No. 12, 117

144; accessed onle on http://www.ejtp.com.

7. Lundin N. I. Microwave induced enhancement o the Joseph-

son DC. Chalmers University o Technology & Gotterborg

University Report, p. 47.

8. Grin M. arXiv: cond-mat/9911419.

9. Tuszynski J. et al. Physica A, 2003, v. 325, 455476; accessed

onle on http://sciencedirect.com.

10. Toussaint M. arXiv: cs.SC/0105033.

11. Fischer U. arXiv: cond-mat/9907457; [8a] arXiv: cond-mat/

0004339.

12. Zurek W. (ed.) In: Proc. Euroconerence in Formation and

Interaction o Topological Deects, Plenum Press, 1995;

accessed online: arXiv: cond-mat/9502119.

13. Volovik G. arXiv: cond-mat/0507454.

Appendix 1 Time-independent gravitational Schrodinger equation

> restart;> with (linalg);> R: = exp((alpha*r));

R := e r

> D1R:=dif (R,r); D2R:=dif(D1R,r);

D1R := e r

D2R := 2er

> SCHEQ1:=D2R+D1R*2/r+8*pi2*m*E*R/h2 +8*pi2*G*M*m2*R/(r*h2)l*(l+1)*R/r2=0;> XX1:=actor(SCHEQ1);

> #Using simplied terms only rom equation (A*8, o Rubcic & Rubcic, 1998)> ODESCHEQ:=D2R+D1R*2/r+8*pi2*G*M*m2*R/(r*h2) =0;

ODESCHEQ := 2e r

2e r

r+

82GMm2e r

rH2= 0

> XX2:=actor(SCHEQ2);

XX2 :=e r

2rH2 2H2+ 82GMm2

rH2

= 0

> RR:= solve (XX2, r);

RR := 2(42GMm2 H2)

2H2

> #Then solving or RR=0, yields:

> SCHEQ3:= 4*pi2*G*M*m2h2*alpha=0;

SCHEQ3 := 42GMm

2H

2 = 0

> a:= solve(SCHEQ3, alpha);

a :=42GMm2

H2

> #Gravitational Bohr radius is dened as inverse o alpha:> gravBohrradius:=1/a;

rgravBohr :=H2

42GMm2

Appendix 2 Time-dependent gravitational Schrodinger equation

> #Solution o gravitational Schrodinger equation (Rubcic, Fizika 1998);> restart;> #with time evolution (Hagendorns paper);> S:=r(t); R:=exp((alpha*S)); R1:=exp((alpha*r));

S := r(t)

R := er

> D4R:=dif(S,t); D1R:=alpha*exp((alpha*S)); D2R:=alpha2*exp((alpha*S)); D5R:=D1R*D4R;

D4R :=d

dtr(t)

D1R := er(t)

D2R := 2er(t)

D1R := er(t) d

dtr(t)

> #Using simplied terms only rom equation (A*8)> SCHEQ3:=h*D5R+D2R+D1R*2/S+8*pi2*G*M*m2*R/(S*h2);> XX2:=actor(SCHEQ3);

XX2 := e

r(t)H

3

dr(t)

dt r(t)

2

r(t)H

2

2H

2

+ 8

2

GMm

2r(t)H2

= 0

> #From standard solution o gravitational Schrodinger equation, we know (Rubcic,Fizika 1998):> SCHEQ4:=4*pi2*G*M*m2h2*alpha;

SCHEQ4 := 42GMm

2H

2

> #Thereore time-dependent solution o Schrodinger equation may introduce newterm to this gravitational Bohr radius.> SCHEQ5:=(XX2*(S*h2)/(exp((alpha*S))))2*SCHEQ4;

ODESCHEQ5 := H3dr(t)

dtr(t)

2r(t)H

2

> #Then we shall assume or simplicity by assigning value to d[r(t)]/dt:> D4R:=1;

> Thereore SCHEQ5 can be rewritten as:> SCHEQ5:= H3*alpha*r/2 +alpha2*r*H2/24*pi2*G*M*m2H2*alpha=0;

SCHEQ5 :=rH3

2+rH22

2+ 4

2GMm

2H

2 = 0

> Then we can solve again SCHEQ5 similar to solution o SCHEQ4:> a1:=solve(SCHEQ5,alpha);

a1 : =r2H+ 2H+

r4H4 4H3r + 4H2 32rGMm22

2rH

a2 : =r2H+ 2H

r4H4 4H3r + 4H2 32rGMm22

2rH

> #Thereore one could expect that there is time-dependent change o gravitationalBohr radius.

V. Christianto, D. L. Rapoport and F. Smarandache. Numerical Solution o Time-Dependent Gravitational Schrodinger Equation 59

7/29/2019 PP-09-11

5/5


Appendix 3 Time-independent gravitational Schrodinger equation

with Yukawa potential [5]

> #Extension o gravitational Schrodinger equation (Rubcic, Fizika 1998);> restart;> #departure rom Newton potential;

> R:=exp((alpha*r));R := e

r

> D1R:= dif(R,r); D2R:= dif (D1R,r);

D1R := e r

D2R := 2er

> SCHEQ2:=D2R+D1R*2/r+8*pi2*(G*MK*exp(2*mu)*(1+mu*r))*m2*R/(r*h2)=0;

ODESCHEQ := 2er

2er

r+

+82(GMKe2(1 + r))m2e r

rH2= 0

> XX2:=actor(SCHEQ2);> RR1:=solve(XX2,r);

RR1 := 2(H2 + 42GMm2 42m2Ke2)

2H2 + 82m2Ke2

> #rom standard gravitational Schrodinger equation we know:> SCHEQ3:=4*pi2*G*M*m2h2*alpha=0;> a:=solve(SCHEQ3, alpha);> #Gravitational Bohr radius is dened as inverse o alpha:> gravBohrradius: =1/a;

rgravBohr :=H2

42GMm2

> #Thereore we conclude that the new terms o RR shall yield new terms (YY) intothis gravitational Bohr radius:> PI:= (RR*(alpha2*h2)(8*pi2*G*M*m2 +2*h2*alpha));> #This new term induced by pion condensation via Gross-Pitaevskii equation may

be observed in the orm o long-range potential efect. (see Mofat J., arXiv: astro-ph/0602607, 2006; also Smarandache F. and Christianto V. Progress in Physics, v.2,

2006, & v. 1, 2007, www.ptep-online.com)> #We can also solve directly:> SCHEQ5:=RR*(alpha2*h2)/2;

SCHEQ5 :=2H2(H2 + 42GMm2 42m2Ke2)

2H2 + 82m2Ke2

> a1:=solve(SCHEQ5, alpha);

a1 := 0, 0,42m2(GMKe2)

H2

> #Then one nds modied gravitational Bohr radius in the orm:> modigravBohrradius: =1/(4*pi2*(G*MK*exp (2*mu))*m2/h2);

rmodified.gravBohr :=H2

42m2(GMKe2)

> #This modication can be expressed in chi-actor:

> chi:=modigravBohrradius/gravBohrradius;

:=GM

GMKe2


pp-09-11

Documents