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Within General Relativity , 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 limit of GR.
A reader who is familiar with electromagnetism will benefit from the following analogy. In EM, one is familiar with the electrostatic potential ϕ E M {\displaystyle \phi ^{EM}} and the magnetic vector potential A → E M {\displaystyle {\vec {A}}^{EM}}. Together, they combine into the 4-vector potential A μ E M ↔ {\displaystyle A_{\mu }^{EM}\leftrightarrow \left} , 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 v 2 / c 2 {\displaystyle v^{2}/c^{2}} , where v {\displaystyle v} is a typical velocity and c {\displaystyle c} is the speed of light. This expansion is known as the post-Coulombic expansion. Within this expansion, ϕ E M {\displaystyle \phi ^{EM}} contributes to the two-body potential already at 0th order, while A → E M {\displaystyle {\vec {A}}^{EM}} contributes only from the 1st order and onward, since it couples to electric currents and hence the associated potential is proportional to v 2 / c 2 {\displaystyle v^{2}/c^{2}}.