Mathematical Methods for Physicists: A concise introduction - Site Map

VECTOR AND TENSOR ANALYSIS
two points P and Q has a stationary value compared with any other neighboring
path that connects P and Q. Thus, to ®nd the geodesic we extremalize (1.106), and
this leads to the di€erential equation of the geodesic (Problem 1.37)
d
dt
@F
@ _x
@F
@x ˆ 0;
q
where F ˆ g _x _x ; and _x ˆ dx=dt. Now
and
@F
@x ˆ 1 1=2
2 g _x _x @g
@x _x _x ;
q
ds=dt ˆ g _x _x :
@F
@ _x ˆ 1 1=22g
2 g _x _x _x
Substituting these into (1.107) we obtain
d
dt g _x _s 1 1 @g
2 @x _x _x _s 1 ˆ 0; _s ˆ ds
dt
or
g x ‡ @g
@x _x _x 1 @g
2 @x _x _x ˆ g _x s_s 1 :
We can simplify this equation by writing
@g
@x _x _x ˆ 1
2
then we have
@g
@x ‡ @g
@x
_x _x ;
g x ‡ ‰ ; Š_x _x ˆ g _x s_s 1 :
…1:107†
We can further simplify this equation by taking arc length as the parameter t, then
_s ˆ 1; s ˆ 0 and we have
g
d 2 x
ds 2
‡ ‰ ; Šdx
ds
dx
ds ˆ 0:
where the functions
‰; Š ˆ ; ˆ 1
@g
2 @x ‡ @g
@x @g
@x
are called the Christo€el symbols of the ®rst kind.
Multiplying (1.108) by g , we obtain
( )
d 2 x
ds 2 ‡ dx dx
ds ds ˆ 0;
…1:108†
…1:109†
…1:110†
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