invariant under arbitrary changes in the spatial coordinates, so we can evaluate it just as well in co-moving Robertson–Walker coordinates. This can
be done directly, using Eq. (1.1.13), but to save work, suppose we adopt a
spatial coordinate system in which the particle position is near the origin
x
i
= 0, where g˜ij = δij + O(x
2
), and we can therefore ignore the purely
spatial components
i
jk
of the affine connection. General relativity gives
the equation of motion
d
2
x
i
dτ
2
= −
i
µν
dx
µ
dτ
dx
ν
dτ
= −
2
a
da
dt
dx
i
dτ
dt
dτ
.
Multiplying with
...more
