APPENDIX C

Determination of the Equation of Free Planar Motion as a Function of the Proper Time of the Gravitating Mass.


It is first noted for future reference that substitution for 
×
m

m
, derived from (3.19), into (3.8) gives



×
w
 
= - w æ
ç
è
2
×
s

s
-
×
s

u
du

ds
ö
÷
ø
(C.1)

First the second order variation of radial position with respect to the proper time of the gravitating mass is computed thus

d2s

dtp2
 =  dt

dtp
d

dt
æ
è
dt

dtp
ds

dt
ö
ø
(C.2)

which from (2.15) becomes

d2s

dtp2
 =  æ
ç
è
1 -
×
s
2
 

c2u2
 -  w2s2

c2
ö
÷
ø
-1/2

 
d

dt
ì
í
î
×
s
 
æ
ç
è
1
×
s
2
 

c2u2
- w2s2

c2
ö
÷
ø
 - 1/2

 
ü
ý
þ
(C.3)

working this out yields

d2s

dtp2
 = 

×
×
s

æ
è
1
×
s
2

c2u2
 -  w2s2

c2
ö
ø
 + 
×
s
  æ
ç
è
×
s
××
s

c2u2

  - 
×
s
3
 

c2u3
  
du

ds

 + 

w
×
w
s2

c2

 + 

w
2s
×
s

c2
ö
÷
ø

æ
ç
è
1
×
s
2
 

c2u2
 -  w2s2

c2
ö
÷
ø
2

 
(C.4)


Substitution for 
×
w
 
  from (C.1) then gives after reduction


d2s

dtp2
 = 

×
×
s
æ
è

1
w2s2

c2
ö
ø

 - 
w2s
×
s

2

 

c2

 - 
×
s

4

 

c2u3
  
du

ds

 + 
w2s2
×
s
2
 

c2u
  
du

ds

æ
ç
è
1
×
s
2
 

c2u2
 -  w2s2

c2
ö
÷
ø
2

 
(C.5)


Now substitution for  
××
s
 
 from (3.18) gives


d2s

dtp2
 = -c2u du

ds
 + 
u2w2s - uw2s2  du

ds

æ
ç
è
1
×
s
2
 

c2u2
 -  w2s2

c2
ö
÷
ø
(C.6)

But

w df

dt
df

dtp
dtp

dt
 = w¢ æ
ç
è
1 -
×
s
2
 

c2u2
 -  w2s2

c2
ö
÷
ø
1/2

 
(C.7)

and insertion of this into (C.6) finally gives

d2s

dtp2
 = - c2u du

ds
 - uw¢2s2 du

ds
 + u2w¢2s
(C.8)

which is the required relationship, expressed in the axes of D1 and the parameter u. Substitution for u, (from (4.7)) and its spatial gradient, reduces (C.8) to

d2s

dtp2
= - ac2

s2
 + ( s - 3a )w¢2
(C.9)

being the equation of motion as a function of the proper time expressed fully in the axes of D1.

Note that because the time dilatation effect is embodied in the proper time tp, the reactive acceleration term due to this in (3.18), is not present in (C.9).



G1 Version 2.2.4
Ó P.G.Bass, November 2009
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