Gravitational
Force on a Falling Particle
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The gravitational
force on a particle in free fall is given by
The velocity terms (which are not 4-vectors) are defined as
Consider
the spacetime around a spherically symmetric body. The spacetime is then
described by the Schwarzschild metric, which is defined as
where
Consider
the case of pure radial motion, i.e.
. The metric then becomes
Eq. (1)
then the radial component of the force becomes
The
first term in Eq. (6) becomes
The
second term in Eq. (6) becomes
Substituting
Eq. (7) and (8) into Eq. (6) gives
The spatial
distance, ds,
between two points is defined as the negative of value of the metric when dt = 0. The
metric in Eq. (5) then becomes
The
spatial velocity is defined as ds/dt
and therefore
Substituting
Eq. (11) into Eq. (9) gives
The
mass, m, can be calculated in terms of dt/dt by using the
metric once again. Divide Eq. (5) through by c2dt2
to obtain
where F
º
-GM/r. The mass is then given by
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