General relativistic analogs of Poisson’s equation and gravitational binding energy

Abstract

Employing the quasi-Maxwell form of the Einstein field equations in the context of gravitoelectromagnetism, we introduce a general relativistic analog of Poisson's equation as a natural outcome of the corresponding spacetime decomposition formalism. The active density introduced in this formalism, apart from the matter-energy density and pressure, includes a third component which is the gravitoelectromagnetic energy density. This general relativistic analog of Poisson's equation is compared with another analog introduced by Ehlers et al. in [1]. Introduction of the cosmological constant and its effect on the active mass, are also discussed for both exterior and interior static spacetimes. In the stationary case, we consider the kerr spacetime with a special choice for its interior metric. * Electronic address: r.gharechahi@ut.ac.ir † Electronic address: jkoohbor@ut.ac.ir ‡ Electronic address: nouri@ut.ac.ir (Corresponding author)

Static and stationary dark fluid universes: A gravitoelectromagnetic perspective

“…in which ln √ h and A g are the so-called gravitoelectric and gravitomagnetic potentials respectively [10]. We notice that gravitoelectric part of the GEM Lorentz force is the general relativistic version of the gravitational force in Newtonian gravity [11], while its gravitomagnetic part has no counterpart in Newtonian gravity. Obviously by their definition, they satisfy the following constraints…”

Frame-Dragging: Meaning, Myths, and Misconceptions

“…The analogous expression for F 1,2 follows by interchanging 1 ↔ 2 in (33). Observe that it is F 2,1 (not F GM2,1 ) which should be regarded (in the context of a PN approximation) as the "reaction" to the gravitomagnetic force F GM1,2 in Equation ( 32), as these are the ones that depend on J 1 ; and note that they do not cancel out:…”

“…We will derive here the exact equations describing the different types of frame-dragging (equations of "gravitoelectromagnetism", or GEM), considering stationary spacetimes, where their formulation (the so called 1+3 "quasi-Maxwell" formalism [26][27][28][31][32][33][34]38]) is particularly simple and intuitive. A generalization for arbitrary time-dependent fields is given in Appendix A.…”

“…Consider a (point-like) test particle of worldline x α (τ), 4-velocity dx α /dτ ≡ U α and mass m. The space components of the geodesic equation DU α /dτ = 0 yield 3 , for the line element (1) [26,27,[31][32][33][34],…”

“…In order to gain intuition into the "frame-dragging" effects, two analogies have been put forth: the electromagnetic analogy, in particular, between the magnetic field and the general relativistic Coriolis field (thus dubbed "gravitomagnetic field"), and the fluiddragging analogy. The former is based on solid equations, best known in weak field slow motion approximations [7,[20][21][22][23][24][25], but with exact versions [13,[26][27][28][29][30][31][32][33][34][35] holding in arbitrarily strong fields, and is known for providing a familiar and reliable formalism. The fluid analogy, initially proposed in [5,36], and then supported by other authors (e.g., [7,12,13]), consists of an analogy drawn between the effects created by the rotation of a body immersed in a fluid, and the frame-dragging effects.…”