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.


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