# Order of magnitude of time derivatives of gravitational potential and gravitational vector potential on Earth

[This question is connected with this one. Since the estimation and measurement of spatial gradients and time derivatives have very different levels of difficulties, I thought it best to ask two separate questions]

In post-Newtonian approximation of gravitation, the metric on and around the Earth is taken to have the expression $$\begin{bmatrix} -1+\frac{2}{c^2}U+ \frac{2}{c^4}(\psi-U^2) + \mathrm{O}(c^{-5}) & -\frac{4}{c^3} V_x + \mathrm{O}(c^{-5}) & -\frac{4}{c^3} V_y + \mathrm{O}(c^{-5}) & -\frac{4}{c^3} V_z + \mathrm{O}(c^{-5}) \\\\ -\frac{4}{c^3} V_x + \mathrm{O}(c^{-5}) & 1+ \frac{2}{c^2} U + \mathrm{O}(c^{-4}) & 0 & 0 \\\\ -\frac{4}{c^3} V_y + \mathrm{O}(c^{-5}) & 0 & 1+ \frac{2}{c^2} U + \mathrm{O}(c^{-4}) & 0 \\\\ -\frac{4}{c^3} V_z + \mathrm{O}(c^{-5}) & 0 & 0 & 1+ \frac{2}{c^2} U + \mathrm{O}(c^{-4}) \end{bmatrix}$$ in a coordinate system $$(ct, x, y,z)$$, where $$U$$, $$\psi$$, $$V_i$$ depend on all coordinates, including time. $$U$$ is related to the Newtonian gravitational potential, and $$(V_j)$$ is the so-called gravitational vector potential. See eg Poisson & Will 2014, eqns (8.2). This metric is used for example for GPS purposes, see eg Petit & Luzum 2010.

These potentials on Earth have a time dependence both because Earth is non-static and because of the movement of massive bodies in the Solar System.

Can anyone provide an order of magnitude for the time derivative of the gravitational potential and the vector potential: $$\frac{\partial U}{\partial t} \approx \mathord{?}\ \mathrm{m^2/s^3} \qquad \frac{\partial V_j}{\partial t} \approx \mathord{?}\ \mathrm{m^3/s^4}$$ I've been looking in the references above, in the references given in this question and its answer, but I don't manage to find it. This article has potentially useful information, but I didn't manage to extract estimates from it. Cheers!

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