Gravitational Force in a Uniform Field

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The kth component of the gravitational Force (an inertial force) is defined as

Define g º dt/dt. Then the value of the components of the 4-velocity U are given by

where v^a and v_a are used in Eq. (1) above and will be used again below. The metric for a uniform gravitational field is

The values of U_k for this metric can easily be shown to be equal to

The only non-vanishing Christoffel Symbols for a Uniform Field are

Therefore the only non-vanishing G_k is G₃

Substitute v⁰ = c into Eq. (6) to obtain

The next step is to calculate g º dt/dt. To determine this value let ds = cdt in Eq. (1)

Now divide Eq. (3) by c²dt² which gives

Substituting Eq. (6) into Eq. (4) gives

The quantity

is sometimes referred to as the local velocity [1]. Our final result is therefore

which has a familiar form to it. In fact this form is identical to the result obtained in Newtonian mechanics for a uniform gravitational field with the mass replaced with the relativistic mass!

References:

[1] Basic Relativity, by Richard A. Mould, Springer-Verlag, (1994), p. 253.

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