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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.

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 $$g_{\mu\nu}$$, is redefined and decomposed into the non-relativistic gravitational (NRG) fields $$g_{\mu\nu} \leftrightarrow \left( \phi, \vec{A}, \sigma_{ij}\right)$$ : $$\phi$$ is the Newtonian potential, $$\vec{A}$$ is known as the gravito-magnetic vector potential, and finally $$\sigma_{ij}$$ is a 3d symmetric tensor known as the spatial metric perturbation. The field redefinition is given by $$ds^2\equiv g_{\mu \nu}dx^\mu dx^\nu = e^{2 \phi}(dt-2\, \vec{A} \cdot d\vec{x})^2-e^{-2 \phi}(\delta_{ij} + \sigma_{ij})\, dx^i\, dx^j$$In components, this is equivalent to$$\begin{array}{ccl} g_{00} &=& e^{2 \phi} \\ g_{0i} &=& -2\, e^{2 \phi} \, A_i \\ g_{ij} &=& -e^{-2 \phi}\, (\delta_{ij}+\sigma_{ij}) + 4 \, e^{2 \phi} \,A_i \, A_j \end{array}$$where $$i,j=1,2,3$$.

Counting components, $$g_{\mu\nu}$$ has 10, while $$\phi$$ has 1, $$\vec{A}$$ has 3 and finally $$\sigma_{ij}$$ has 6. Hence, in terms of components, the decomposition reads $$10 = 1 + 3 + 6$$.

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. Taking the approximation of time independence, the idea of a Kaluza-Klein reduction was adapted to apply to the time direction rather than a spatial compact direction as in the original context. 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 $$\vec{A}$$ 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 $$g_{\mu \nu} = \eta_{\mu \nu} + h_{\mu \nu}$$, where $$\eta_{\mu \nu} $$ is the Minkowski metric, we find the standard weak field decomposition into a scalar, vector and tensor $$h_{\mu\nu} \to \left( h_{00}, h_{0i}, h_{ij} \right)$$, 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 approximation.

Physical interpretation
The scalar field $$\phi$$ is interpreted as the Newtonian gravitational potential.

The vector field $$\vec{A}$$ is interpreted as the gravito-magnetic vector potential. It is magnetic-like, or analogous to the magnetic vector potential in electromagnetism (EM). In particular, it is sourced by massive currents (the analogue of charge currents in EM), namely by momentum.

As a result, the gravito-magnetic vector potential is responsible for current-current interaction, which appears at the 1st post-Newtonian order. In particular, it generates a repulsive force between parallel massive currents. However, this repulsion is dominated by the standard Newtonian gravitational attraction, since in gravity a current "wire" must always be massive (charged).

Moreover, a spinning object is the analogue of an electromagnetic current loop, which forms as magnetic dipole, and as such it creates a magnetic-like dipole field in $$\vec{A}$$.

The symmetric tensor $$\sigma_{ij}$$ is known as the spatial metric perturbation. It must be accounted for starting from the 2nd post-Newtonian order. If one restricts to the 1st post-Newtonian order, $$\sigma_{ij}$$ can be ignored, and relativistic gravity is described by the $$\phi$$, $$\vec{A}$$ fields, and it becomes a strong analogue of electromagnetism, an analogy known as gravitoelectromagnetism.

Applications and generalizations
The two body problem in general relativity holds both intrinsic interest and observational, astrophysical interest. In particular, it is used to describe the motion of binary compact objects, which are the sources for gravitational waves. As such, the study of this problem is essential for both detection and interpretation of gravitational waves.

Within this two body problem, the effects of GR are captured by the two body effective potential, which is expanded post-Newtonian approximation. Non-relativistic gravitational fields were found to economize the determination of this two body effective potential.

Generalizations
In higher dimensions, with an arbitrary spacetime dimension $$d$$, the definition of non-relativistic gravitational fields generalizes into

$$ds^2 = e^{2 \phi}(dt-2\, \vec{A} \cdot d\vec{x})^2-e^{-2 \phi/(d-3)}(\delta_{ij} + \sigma_{ij}) dx^i dx^j$$Substituting $$d=4$$ reproduces the standard 4d definition above.