Assume the test body moves slowly, $dx^i/dt=O(\epsilon)$, and that the field is quasistatic, so time derivatives are of higher order. The spatial <geodesic equation>, parametrized by coordinate time, then reduces at leading order to
$$
\frac{d^2x^i}{dt^2}=-\Gamma^i{}_{00}
=\frac12\partial_i h_{00}+O(\epsilon^3).
$$
Defining the Newtonian potential by
$$
\boxed{h_{00}=-2\Phi}
$$
gives
$$
\boxed{\frac{d^2\mathbf x}{dt^2}=-\boldsymbol\nabla\Phi},
$$
the Newtonian equation of motion. Terms involving $h_{0i}$, spatial velocity, or time derivatives enter beyond the stated order.

Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 309 / 2 / a / i / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/iii/paper-309/2/a/i/solution