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.
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.
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.
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
contributes only from the 1st order and onward, since it couples to electric currents and hence the associated potential is proportional to
Going beyond the strict limit, corrections can be organized into a perturbation theory known as the post-Newtonian expansion.
, is redefined and decomposed into the non-relativistic gravitational (NRG) fields
is a 3d symmetric tensor known as the spatial metric perturbation.
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,[1] interpreted in the context of the post-Newtonian expansion in,[2] and finally the normalization of
was changed in [3] to improve the analogy between a spinning object and a magnetic dipole.
By definition, the post-Newtonian expansion assumes a weak field approximation.
is the Minkowski metric, we find the standard weak field decomposition into a scalar, vector and tensor
, 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.
It is magnetic-like, or analogous to the magnetic vector potential in electromagnetism (EM).
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 contribution to the force between parallel massive currents.
However, this repulsion is overturned by the standard Newtonian gravitational attraction, since in gravity a current "wire" must always be massive (charged) -- unlike EM.
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
Hence it becomes a strong analogue of electromagnetism, an analogy known as gravitoelectromagnetism.
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.
Non-relativistic gravitational fields were found to economize the determination of this two body effective potential.
, the definition of non-relativistic gravitational fields generalizes into [1]