Non-relativistic gravitational 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 ${\displaystyle {\varphi }^{\text{EM}}}$ and the magnetic vector potential ${\displaystyle \overrightarrow{A}{}^{\text{EM}}}$. Together, they combine into the 4-vector potential ${\displaystyle A_{\mu }^{\text{EM}}\leftrightarrow ({\varphi }^{\text{EM}},\overrightarrow{A}{}^{\text{EM}})}$, 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, ${\displaystyle {\varphi }^{\text{EM}}}$ contributes to the two-body potential already at 0th order, while ${\displaystyle {\overrightarrow{A}}^{\text{EM}}}$ contributes only from the 1st order and onward, since it couples to electric currents and hence the associated potential is proportional to ${\displaystyle v^2/c^2}$.

Definition

In the non-relativistic limit, of weak gravity and non-relativistic velocities, general relativity reduces to Newtonian gravity. Going beyond the strict limit, corrections can be organized into a perturbation theory known as the post-Newtonian expansion. As part of that, the metric gravitational field ${\displaystyle g_{\mu \nu },\mu ,\nu =0,1,2,3}$, is redefined and decomposed into the non-relativistic gravitational (NRG) fields ${\displaystyle g_{\mu \nu }\leftrightarrow (\varphi ,\overrightarrow{A},{\sigma }_{ij})}$ : ${\displaystyle \varphi }$ is the Newtonian potential, ${\displaystyle \overrightarrow{A}}$ is known as the gravito-magnetic vector potential, and finally ${\displaystyle {\sigma }_{ij}}$ is a 3d symmetric tensor known as the spatial metric perturbation. The field redefinition is given by $${\displaystyle d{s}^2\equiv g_{\mu \nu }d{x}^{\mu }d{x}^{\nu }=e^{2\varphi }(dt-2\overrightarrow{A}\cdot d\overrightarrow{x}{)}^2-e^{-2\varphi }({\delta }_{ij}+{\sigma }_{ij})d{x}^{i}d{x}^j.}$$ In components, this is equivalent to $${\displaystyle \begin{array}{cc}{g}_{00}& =e^{2\varphi },\\ g_{0i}& =-2e^{2\varphi }{A}_i,\\ g_{ij}& =-e^{-2\varphi }({\delta }_{ij}+{\sigma }_{ij})+4e^{2\varphi }{A}_i{A}_j,\end{array}}$$ where ${\displaystyle i,j=1,2,3}$.

Counting components, ${\displaystyle g_{\mu \nu }}$ has 10, while ${\displaystyle \varphi }$ has 1, ${\displaystyle \overrightarrow{A}}$ has 3 and finally ${\displaystyle {\sigma }_{ij}}$ has 6. Hence, in terms of components, the decomposition reads ${\displaystyle 10=1+3+6}$.

Motivation for definition

In the post-Newtonian limit, bodies move slowly compared with the speed of light, and hence the gravitational field is also slowly changing. Approximating the fields to be time independent, the Kaluza-Klein reduction (KK) was adapted to apply to the time direction. Recall that in its original context, the KK reduction applies to fields which are independent of a compact spatial fourth direction. In short, the NRG decomposition is a Kaluza-Klein reduction over time.

The definition was essentially introduced in, interpreted in the context of the post-Newtonian expansion in, and finally the normalization of ${\displaystyle \overrightarrow{A}}$ was changed in to improve the analogy between a spinning object and a magnetic dipole.

Relation with standard approximations

By definition, the post-Newtonian expansion assumes a weak field approximation. Within the first order perturbation to the metric ${\displaystyle g_{\mu \nu }={\eta }_{\mu \nu }+h_{\mu \nu }}$, where ${\displaystyle {\eta }_{\mu \nu }}$ is the Minkowski metric, we find the standard weak field decomposition into a scalar, vector and tensor ${\displaystyle h_{\mu \nu }\to \left(h_{00},h_{0i},h_{ij}\right)}$, which is similar to the non-relativistic gravitational (NRG) fields. The importance of the NRG fields is that they provide a non-linear extension, thereby facilitating computation at higher orders in the weak field / post-Newtonian expansion. Summarizing, the NRG fields are adapted for higher order post-Newtonian expansion.