Conserved Quantities
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To find
conserved quantities for a particle moving on a geodesic we take an approach
almost identical to that used for the gravitational
force, i.e. start with the geodesic equation, which can be written as
Multiply
each side by (m0)2
and note that Pm
= m0Um.
Substitute
into Eq.
(2) to obtain
This can
now be simplified by substituting in the expression for the Christoffel symbols,
i.e.
We
therefore have the following theorem
If all of the gravitational potentials, gbl, are independent of xa then Pa is a constant of motion along any geodesic.
In
an inertial frame of reference P0
= P0
º
mc = m0c
dt/dt
= gm0c
= E/c2
where E is the particle’s inertial energy. In a non-inertial
frame of reference P0
¹
P0.
If the gravitational potentials, gbl,
are time independent then it is P0
which is constant and not P0.
Therefore it is P0
that is referred to as the inertial
energy of the particle i.e. E = P0.