APPENDIX B

Reduction of Selected Relativistic Gravitational Expressions to their Classical Equivalents.

This exercise is only affected for those expressions not previously so treated in [1]. Note that to reduce these equations to their Special Relativistic equivalents, it is only necessary to put u = 1. Subsequent reduction to the classical equivalents is achieved by putting c = ¥. Note also from [1] Eq(G1) when u = 1, a = 0 and therefore s = r.

Section 2.

The four spatial/temporal reaction forces,

   (i) The spatial term

m
××
s
 
ê
ê

u
= 1 
= m
××
r
(B.1)

    (ii) The temporal term

- m æ
è
×
s
 
××
s
 
 - c2u
×
s
 
du

ds
ö
ø

æ
è
c2u2
×
s
 
2
 
ö
ø
1/2
 
ê
ê
ê
ê



u
= 1 

- m
×
r
 
××
r
 

c æ
è
1
×
r
2

c2
ö
ø
1/2

 
(B.2)

    (iii) The temporal term

×
m
 
æ
è
c2u2
×
s
 
2
 
ö
ø
1/2
 
ê
ê
ê
ê



u
= 1
= c æ
ç
è
1
×
r
 
2
 

c2
ö
÷
ø
1/2

 
dm

dt
(B.3)

    (iv) The spatial term

×
m
 
×
s
 
|
|


u
= 1
 
=
×
r
 
dm

dt
(B.4)

All of the above reduced expressions are as the Special Relativity equivalents as derived in [2] Section 3.

Section 3.

    (v) Eq(3.18), Inertial mass in D1

      (a) To the special relativistic equivalent, putting u = u0 = 1

ma
m
0

æ
è
1
×
r
 
2

c2
ö
ø
3/2

 
(B.5)

       (b) To the classical equivalent, in (B.5) putting c = ¥.

ma = m = m0
(B.6)

   (vi) Eq(3.17) artificially induced acceleration in D1

      (a) To the special relativistic equivalent, putting u = 1.


××
r
 

F æ
è
1
×
r
2

c2
ö
ø
3/2

 

m0
(B.7)

where [2], Eq(3.6) has been subsequently inserted for m.

      (b) To the classical equivalent, in (B.7) putting c = ¥.

××
r
 
= F

m0
(B.8)

    (vii) Eq(4.16), loss of energy by a gravitating mass when brought to rest. From (B.6) m1 = m0 so that (4.16) becomes

Ek = m0 c2
æ
è
1
×
r1
2

c2
ö
ø
1/2

 

æ
è
1
×
r
2

c2
ö
ø
1/2

 
 - m0 c2
(B.9)

and after the mass has been brought to rest

Ek = m0 c2 ì
í
î
æ
ç
è
1 -
×
r
 
2
1 

c2
ö
÷
ø
1/2

 
 - 1 ü
ý
þ
(B.10)


This is a negative quantity and exactly the kinetic energy that would have been lost by the mass had it been decelerated to stop from the velocity 
×
r
 

1
 in D0, Pseudo-Euclidean Space-Time.



G2 Version 1.1.1
Ó P.G.Bass, March 2009
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