APPENDIX B.

Derivation of the 4 Vector Curl of the Acceleration Potentials.


The derivations below are for the Acceleration Potentials external to the source. Derivations for the Potentials internal to the source follow the same process with the same results.

B.1 Curl of the Spatial Acceleration Potential Vector.

The spatial Acceleration Potential vector, As, has only one component, in the s direction, and therefore by the definition of curl, the 4-vector curl of As, is zero, i.e.


Ñ4 ×As = 0
(B.1)

B.2 Curl of the Temporal Acceleration Potential Vector.

The temporal Acceleration Potential vector At, has only one component, in the x0 direction and therefore by the definition of curl, its 4-vector curl is zero. However, because At, is a function of s, it is necessary to prove this.


Ñ4 × At = Ñ4 × ì
í
î
-   ( 1 - u2 ) 1/2

u
 As ü
ý
þ

            =  ì
í
î
-  ( 1 - u2 ) 1/2

u
ü
ý
þ
Ñ4 × As + Ñ4 ì
í
î
( 1 - u2 ) 1/2

u
ü
ý
þ
× As
(B.2)

By (B.1) this reduces to


Ñ4 × At = - Ñ4 ì
í
î
( 1 - u2 )1/2

u
ü
ý
þ
× As
(B.3)

Because u is only a function of s, the gradient in (B.3) can be written


Ñ4 ì
í
î
( 1 - u2 )1/2

u
ü
ý
þ
=

s
ì
í
î
( 1 - u2 )1/2

u
ü
ý
þ
s
(B.4)

Substituting for u and carrying out the partial differentiation gives


Ñ4 ì
í
î
( 1 - u2 )1/2

u
ü
ý
þ
= - ì
ï
í
ï
î


g
M

s2  æ
è
2gM

c2s
ö
ø
1/2

 
æ
è
1 -  2gM

c2s
ö
ø
3/2

 
ü
ï
ý
ï
þ
s
(B.5)

and therefore


Ñ4 × As = ì
ï
í
ï
î


g
M

s2  æ
è
2gM

c2s
ö
ø
1/2

 
æ
è
1 - 2gM

c2s
ö
ø
3/2

 
ü
ï
ý
ï
þ
s   × gM

s2
s = 0
(B.6)


G4 Version 1.0.0
Ó P.G.Bass, Agust 2009

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