Exact Relativistic Newtonian Representation of Gravitational Static Spacetime Geometries

Abstract

We construct a self-consistent relativistic Newtonian analogue corresponding to gravitational static spherical symmetric spacetime geometries, staring directly from a generalized scalar relativistic gravitational action in Newtonian framework, which gives geodesic equations of motion identical to those of the parent metric. Consequently, the derived velocity-dependent relativistic scalar potential, which is a relativistic generalization of Newtonian gravitational potential, exactly reproduces the relativistic gravitational features corresponding to any static spherical symmetric spacetime geometry in its entirety, including all the experimentally tested gravitational effects in the weak field up to the present. This relativistic analogous potential is expected to be quite useful in studying wide range of astrophysical phenomena, especially in strong field gravity.

Spherical accretion in giant elliptical galaxies: multitransonicity, shocks, and implications on AGN feedback

“…Although the SdS geometry generally describes BH in a spatially inflated Universe, however, it can be applied to describe the spacetime outside any spherical or nearly spherical matter distribution, or even outside any mass distribution at length-scales where the deviation from spherical symmetry of the mass distribution can be ignored (e.g., SS11). Owing to the difficulty of studying complex astrophysical phenomena in the framework of full general relativity, many complex astrophysical phenomena has been studied through the use of pseudo-Newtonian potentials (PNPs), which are prescribed to approximately mimic corresponding GR effects, and has been extensively used in the astrophysical literature (e.g., Ghosh & Mukhopadhyay 2007;Ghosh et al 2014;Sarkar et al 2014;Ghosh et al , 2016. Although few PNPs exist in literature that can well mimic SDS spacetime (Stuchlík & Kovář 2008;Stuchlík et al 2009;Sarkar et al 2014), here, we focus on the PNP prescribed in Stuchlík et al (2009), which is given by ΨPN…”

A geometric relativistic dynamics under any conservative force

“…More recently, the above shortcomings were addressed in [15]. Using a metric approach and hypothesizing a generic relativistic gravitational action and a corresponding Lagrangian, the authors derive a velocity-dependent relativistic potential which generalizes the classical Newtonian potential.…”

“…This means that in order to reproduce relativistic effects, one can no longer distinguish between potential and kinetic energy, as in Newtonian dynamics. This also explains the need to include the velocity in the modified Newtonian potentials proposed in [22,14,23,24,15]. The mixed term in (25) is approximately β 2 U(x) and is therefore only seen for high velocities or in high-precision experiments.…”

Relativity from the geometrization of Newtonian dynamics

“…For radial motion, the geometrization of Newton's second law involves only replacing the time parameter with proper time parameter. From (20), we obtain c 2 f (r) 2 = dU dr m . Thus, f (r) = 2U mc 2 + const, and from (11),…”

“…This means that in order to reproduce relativistic effects, one can no longer distinguish between potential and kinetic energy, as in Newtonian dynamics. This also explains the need to include the velocity in the modified Newtonian potentials proposed in [16][17][18][19][20].…”