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Tetrads in General Relativity

V. Parallel Transport

Newtonian Limit

   In the limit of a slowly moving particle in a weak stationary gravitational field, the geodesic equation (5.13) reduces to the Newtonian form. This limit involves three approximations:

  1. weak field: deviations from the Minkowskian metric ${h_{\mu \nu }}(x): = {g_{\mu \nu }}(x) - {\eta _{\mu \nu }}$ are small, so that only terms linear in ${h_{\mu \nu }}$ need be retained;
  2. stationary field: the deviations are independent of time, ${h_{\mu \nu }}(t,\vec x) = {h_{\mu \nu }}(\vec x)$;
  3. non-relativistic limit: velocities are much smaller than the speed of light: $v \ll c$ , ${v^2}/{c^2} \leqslant {h_{\mu \nu }}$.

   The dominant term of the metric in this approximation is ${g_{00}} = 1 + {h_{00}}$, which leads to Newtons equation of motion:

5.14

\[\frac{{{d^2}{\mathbf{x}}}}{{d{t^2}}} = - \frac{1}{2}{c^2}\nabla {h_{00}}{\qquad}{h_{00}} \simeq 2\frac{\Phi }{{{c^2}}} + {\text{const}}\]

Here $\Phi $ is the Newtonian gravitational potential, assumed to be small: $\Phi \ll {c^2}$.

   The additive constant to the potential is usually defined in such a way that $\Phi $ vanishes when $g_{00}$ assumes its Minkowski value $\eta_{00}$. It is worthwhile remarking that, to the level of approximation adopted, only $g_{00}$ enters the equation of motion, although the deviations of the other metric coefficients may be of the same order of magnitude. It is this circumstance that allows to describe non-relativistic gravity, to a first order, by means of a single scalar potential.

   This result might suggest that the connection term in the geodesic equation (5.13) is a representation of the gravitational field in GRT. Einstein strongly opposed this view because the whole left side of the geodesic equation (5.13) is tensorial (with respect to arbitrary coordinate transformations), whereas the two terms separately are not. He was of the opinion “that there is no need whatsoever to distinguish ‘inertial terms’ and ‘gravitational terms’ in the geodesic equation”. [The Princeton lectures (1921), published as the ‘The Meaning of Relativity’.]

   In GRT there is no concept of gravitational force. Curvature is used to geometrize the gravitational interaction. The gravitational interaction in this case is described by letting particles follow the curvature of spacetime. Geometry replaces the concept of force, and trajectories are determined, not by force equations, but by geodesics.