Gravitoelectromagnetism: Why Equation Differs in Sources

[olgerm]

In article "A note on the gravitoelectromagnetic analogy" by Matteo Luca Ruggiero (https://arxiv.org/pdf/2111.09008v1.pdf) equation number 18 is $\nabla \dot\ E=4 \pi G \rho$
, but corresponding equation in the wikipediapage(https://en.wikipedia.org/wiki/Gravitoelectromagnetism#Equations) is $\nabla \dot\ E_g=-4 \pi G \rho_g$
. $E$ notes same thing in the article as $E_g$ on the wikipedia page. $\rho$ note same thing in the article as $\rho_g$ on the Wikipedia page. Why is this equation different in these sources? To me seems that Wikipedia equation is correct, because from it follows that direction of gravitational field is directed to (not away from) bodies(with positive mass).

 

[robphy]

Possibly useful: https://en.wikipedia.org/wiki/Gravitoelectromagnetism#Scaling_of_fields

The literature does not adopt a consistent scaling for the gravitoelectric and gravitomagnetic fields, making comparison tricky.
For example, to obtain agreement with Mashhoon's writings, all instances of Bg in the GEM equations must be multiplied by −1/2c and Eg by −1. These factors variously modify the analogues of the equations for the Lorentz force. There is no scaling choice that allows all the GEM and EM equations to be perfectly analogous. The discrepancy in the factors arises because the source of the gravitational field is the second order stress–energy tensor, as opposed to the source of the electromagnetic field being the first order four-current tensor.
This difference becomes clearer when one compares non-invariance of relativistic mass to electric charge invariance. This can be traced back to the spin-2 character of the gravitational field, in contrast to the electromagnetism being a spin-1 field. (See Relativistic wave equations for more on "spin-1" and "spin-2" fields).