Extended gravitoelectromagnetism. II. Metric perturbation

Abstract

This article makes the connection between the new gravitoelectromagnetic theory presented in the part I paper and general gravity in its weak formulation. The perturbation in the metric tensor is obtained in terms of the flat-space total energy–momentum tensor given by the sum of the consistent fully relativistic fluid and gravitoelectromagnetic field contributions. Expressions for the perturbed metric in the internal (fluid) and external (vacuum) regions are given in terms of the fluid and field variables. This formulation of gravitoelectromagnetism is compatible with the formation of gravitational waves. The geodesic equation is obtained in both the internal and external regions, including new terms neglected in the standard gravitomagnetic formulation. It is shown that the new terms reproduce the anomalous shift of planetary precession, which results from a balance between the gravitoelectric and gravitomagnetic forces.

Appendix A: Nomenclature

Constants and standard notation

$c=299,792,458$ m s$^{-1}$ velocity of light

$G=6.67408\times 10^{-11}$ m$^{3}$ kg$^{-1}$ s$^{-2}$ gravitational constant

$k_{B}=1.38064852\times 10^{-23}$ m$^{2}$ kg s$^{-2}$ K$^{-1}$ Boltzmann constant

$\left( \mu ,\nu ,\rho \ldots \right)$ Greek letters for spacetime indices assume the values 0, 1, 2, 3

$\left( i,j,k\ldots \right)$ Latin letters for space indices assume the values 1, 2, 3

$\left( -+++\right)$ metric signature; $\eta ^{\mu \nu }$ Minkowski tensor

$\overline{\overline{{\varvec{I}}}}$ unit dyadic; $\overline{\overline{\overline{\epsilon }}}$ Levi–Civita tensor in three dimensions; $\delta _{ij}$ Kronecker delta

GE, GM, GEM meaning gravitoelectric, gravitomagnetic and gravitoelectromagnetic, respectively

Section 2

t time variable; ${\varvec{r}}$ position variable in three dimensions; m particle mass

${\varvec{u}}$ fluid velocity; $\beta =u/c$ normalized fluid velocity; $\gamma =1/\sqrt{1-\beta ^{2}}$ Lorentz factor

p pressure; $\rho$ mass density; n number density; T absolute temperature; $\gamma _{A}$ “adiabatic” coefficient

${\varvec{E}}_{g}$ GE field; ${\varvec{B}}_{g}$ GM field; ${\varvec{j}}$ mass current density

$\phi _{g}$ GE (Newtonian) scalar potential; ${\varvec{A}}_{g}$ GM vector potential

$j^{\mu }$ contravariant mass four-current density; $u^{\mu }$ contravariant fluid four-velocity

$F_{g}^{\mu \nu }$ GEM field tensor; $A_{g}^{\mu }$ four-vector GEM potential

$\overset{\circ }{U}$ proper energy density; $\overset{\circ }{n}$ rest frame number density; $\overset{\circ }{T}$ rest frame temperature; s specific entropy

$U_{f}$ fluid energy density; ${\varvec{G}}_{f}$ fluid momentum density; $\overline{\overline{{\varvec{T}}}}_{f}$ fluid stress density

$U_{g}$ GEM energy density; ${\varvec{G}}_{g}$ GEM momentum density; $\overline{\overline{{\varvec{T}}}}_{g}$ GEM stress density; ${\varvec{S}} _{g}$ GEM Poynting vector

$T_{f}^{\mu \nu }$ fluid energy–momentum; $T_{g}^{\mu \nu }$ GEM field energy–momentum; $T^{\mu \nu }$ total energy–momentum

Section 3

$x^{\mu }$, $\xi ^{\mu }$ contravariant four-vector coordinates; $g_{\mu \nu }$ metric tensor;

$h_{\mu \nu }$ metric tensor perturbations; ${\widetilde{h}}_{\mu \nu }$ trace-reversed metric perturbations; h trace of $h_{\mu \nu }$; $\widetilde{h}$ trace of ${\widetilde{h}}_{\mu \nu }$

$\phi _{\text {eff}}$, ${\varvec{A}}_{\text {eff}}$, $\overline{\overline{\psi }}_{\text {eff}}$ effective GEM scalar potential, vector potential and stress dyadic in the fluid region

${\varvec{E}}_{\text {eff}}$, ${\varvec{B}}_{\text {eff}}$ effective GE and GM fields

$\overline{\overline{\psi }}_{g}$ GEM stress dyadic formed by juxtaposition of vectors ${\varvec{a}}$ and ${\varvec{e}}$ in the vacuum region

Section 4

ds space–time element; $\tau$ proper time element; S action

$\Gamma _{\mu \nu }^{\rho }$ Christoffel symbol; ${\varvec{v}}$ particle velocity

$\left( r,\theta ,\varphi \right)$ spherical coordinates

Section 5

M, m central and orbiting masses, respectively

${\varvec{r}}_{m}$, ${\varvec{v}}_{m}$, $\rho _{m}$, ${\varvec{j}}_{m}$ position vector, velocity, mass density and current density associated with mass m

${\varvec{r}}_{M}$, ${\varvec{v}}_{M}$, $\rho _{M}$, ${\varvec{j}}_{M}$ position vector, velocity, mass density and current density associated with mass M

$\phi _{g,m}$, ${\varvec{A}}_{g,m}$, ${\varvec{E}}_{g,m}$, ${\varvec{B}} _{g,m}$ GEM potentials and fields produced by mass m

$\phi _{g,M}$, ${\varvec{A}}_{g,M}$, ${\varvec{E}}_{g,M}$, ${\varvec{B}} _{g,M}$ GEM potentials and fields produced by mass M

${\varvec{L}}_{m}$, ${\varvec{L}}_{M}$ angular momentum of masses m and M

$\gamma _{m}$, $\omega _{T}$, $f_{T}$ Lorentz factor, Thomas precession and kinematic Thomas factor associated with mass m

$\phi _{g}$, ${\varvec{A}}_{g}$, ${\varvec{E}}_{g}$, ${\varvec{B}}_{g}$ simplified notation for the potentials and fields produced by mass M in the position of m

${\varvec{r}}$, ${\varvec{v}}$, ${\varvec{L}}$ simplified notation for the position, velocity and angular momentum of m

$\beta$, $\gamma$ simplified notation for the normalized velocity and Lorentz factor of m

$\ell$, $\omega _{0}$, a specific angular momentum, angular frequency and radial distance variables

$\tau$, $\xi$, normalized time and radial distance variables

$\epsilon$, $\alpha$ dimensionless relativistic correction factor and auxiliary variable

$v_{0}$, $T_{0}$ velocity and period of the Kepler circular orbit

F, f, $\eta$, $\omega$, $\psi$ auxiliary variables used in the integration of the equation of motion

$\Delta \varphi$, $\sigma$ precession shift of the perihelion for a half revolution and total precession per orbit

${\varvec{P}}$, ${\varvec{J}}$ canonical momentum and canonical angular momentum of mass m