Newtonian Limit

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The gravitational force on a test particle of constant proper mass m₀ in Newtonian gravity is

In component form this becomes

In the Newtonian limit where the field is weak and the particles are moving slowly, dt/dt  @ 1. More generally proper time is linearly proportional to coordinate. I.e.

where a and b are constants. We may take a = 1 and b = 0.  Substituting Eq. (3) into Eq. (2) results in

Compare Eq. (4) with the geodesic equation

If follows that the only non-zero Christoffel symbol is

Recall the definition of the Riemann tensor

Evaluate R^j_0k0

All other components of the Riemann tensor vanish. The only non-vanishing component of the Ricci tensor is therefore

Recall Poisson’s equation

We therefore have the equality

Consider once again the geodesic deviation equation

Evaluate Eq. (12) using spatial Cartesian coordinates. Then the term on the left becomes

Evaluate the acceleration of each particle at the same time, i.e. set x⁰ = 0. For slowly moving particles dt/dt  @ 1.  This reduces Eq. (13) to

Since the only nonvanishing components of the Riemann tensor are R^j_0k0Eq. (14) reduces to

Compare Eq. (15) to the Newtonian expression for tidal acceleration

Therefore the Riemann tensor is related to the Newtonian tidal force tensor as

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