Non-relativistic gravitational fields

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This is currently being merged. After a discussion, consensus to merge this into Parameterized post-Newtonian formalism was found. You can help implement the merge by following the instructions at Help:Merging and the resolution on the discussion. Process started in April 2023.

Within general relativity (GR), Einstein's relativistic gravity, the gravitational field is described by the 10-component metric tensor. However, in Newtonian gravity, which is a limit of GR, the gravitational field is described by a single component Newtonian gravitational potential. This raises the question to identify the Newtonian potential within the metric, and to identify the physical interpretation of the remaining 9 fields.

The definition of the non-relativistic gravitational fields provides the answer to this question, and thereby describes the image of the metric tensor in Newtonian physics. These fields are not strictly non-relativistic. Rather, they apply to the non-relativistic (or post-Newtonian) limit of GR.

A reader who is familiar with electromagnetism (EM) will benefit from the following analogy. In EM, one is familiar with the electrostatic potential ${\displaystyle \phi ^{\text{EM}}}$ and the magnetic vector potential ${\displaystyle {\vec {A}}{}^{\text{EM}}}$. Together, they combine into the 4-vector potential ${\displaystyle A_{\mu }^{\text{EM}}\leftrightarrow (\phi ^{\text{EM}},{\vec {A}}{}^{\text{EM}})}$, which is compatible with relativity. This relation can be thought to represent the non-relativistic decomposition of the electromagnetic 4-vector potential. Indeed, a system of point-particle charges moving slowly with respect to the speed of light may be studied in an expansion in ${\displaystyle v^{2}/c^{2}}$, where ${\displaystyle v}$ is a typical velocity and ${\displaystyle c}$ is the speed of light. This expansion is known as the post-Coulombic expansion. Within this expansion, ${\displaystyle \phi ^{\text{EM}}}$ contributes to the two-body potential already at 0th order, while ${\displaystyle {\vec {A}}^{\text{EM}}}$ contributes only from the 1st order and onward, since it couples to electric currents and hence the associated potential is proportional to ${\displaystyle v^{2}/c^{2}}$.