From the static isotropic metric for general relativity,
where
For radial infall, J = 0 and
For a particle starting at rest at R = r,
Switch to non-dimensionalized coordinates with
, then
where the constant has been chosen such that
. Now introduce the "cycloid parameter" defined by
giving
so combining (9) and (8) gives
]
Now,
Here
and
so
Plugging these in
Note that
so
(1)
where
(2)
(3)
(4)
For radial infall, J = 0 and
(5)
For a particle starting at rest at R = r,
(6)
(7)
(8)
(9)
Switch to non-dimensionalized coordinates with
, then(10)
where the constant has been chosen such that
. Now introduce the "cycloid parameter" defined by (11)
giving
(12)
(13)
(14)
so combining (9) and (8) gives
]
(15)
(16)
Now,
(17)
Here
andso
Plugging these in
(18)
Note that
(19)
so
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