A Note on the Gravitoelectromagnetic Analogy

Abstract

We discuss the linear gravitoelectromagnetic approach used to solve Einstein’s equations in the weak-field and slow-motion approximation, which is a powerful tool to explain, by analogy with electromagnetism, several gravitational effects in the solar system, where the approximation holds true. In particular, we discuss the analogy, according to which Einstein’s equations can be written as Maxwell-like equations, and focus on the definition of the gravitoelectromagnetic fields in non-stationary conditions. Furthermore, we examine to what extent, starting from a given solution of Einstein’s equations, gravitoelectromagnetic fields can be used to describe the motion of test particles using a Lorentz-like force equation.

On the Rotation Curve of Disk Galaxies in General Relativity

“…(see, e.g., Landau &Lifshitz 1971 andPoisson &Will 2014, for a detailed description of higher-order post-Newtonian expansions; see also Rindler 1997;Lynden-Bell & Nouri-Zonoz 1998;Mashhoon et al 1999;Clark & Tucker 2000;Ruggiero & Tartaglia 2002;Mashhoon 2008;Costa &Natário 2021, andRuggiero 2021). We indicate with ∧ the vector (cross) product, ∇ is the usual nabla operator, and j = ρv is the gravitational current density, ρ and v being the mass density and velocity field of the sources.…”

Galactic dark matter effects from purely geometrical aspects of General Relativity

“…Iorio et al [11]). However, this formalism has limitations (for instance the geodesic equation does not take a Lorentz-like form in non stationary conditions as discussed by Ruggiero [12]) and we should not forget that it is just an approximation of the full theory.…”

Friedmann-Lemaître-Robertson-Walker cosmology through the lens of gravitoelectromagnetism

“…It is well known (e.g., [1]) that linearized gravity can be recast formally as a Maxwell-like theory by introducing a gravitoelectric and a gravitomagnetic potential. Gravitoelectromagnetism has a long history and several applications (e.g., [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16] and references therein) and it is universally recognized as a characteristic of GR. Certain geometries that are solutions of the Einstein equations are usually not contemplated from the point of view of gravitoelectromagnetism in their weak-field limit.…”

“…describing a dust with energy density ρ and four-velocity field u µ . Gravitoelectromagnetism is introduced by noting that the linearized Einstein equations in the Lorentz gauge (2.4) assume the form of Maxwell equations and that the geodesic equation resembles the equation for the Lorentz force acting on a particle of unit charge [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15] (there are, however, subtleties in the Lorentz force equation when φ (g) and A (g) are time-dependent [17]). The line element is written as…”

“…from which one reads off the gravitoelectromagnetic potentials φ (g) and A (g) [2][3][4][5][6][7][8][9][10][11][12][13][14][15]. The 3-dimensional projection of the timelike geodesic equation for a massive particle of 3-velocity v assumes the form analogous to the Lorentz force equation [1]…”