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arXiv · 1312.7501

Variational Results and Solutions in Gauge Gravity and a Bifurcation Analysis of Black Hole Orbital Dynamics

Bruce Herold Dean
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Abstract

An analysis of all known spherically symmetric solutions to the field equations originating from the Riemann tensor quadratic curvature Lagrangian is presented. A new exact solution is found for the field equation originating from the "energy-momentum" equation of the gauge gravity theory. Imposing equivalence between the Palatini and standard variational field equations yields an algebraic condition that restricts the number spacetime solutions to gauge gravity. An analysis of a new spherically symmetric solution to the conformal gravity field equations is also presented. Point particle orbital dynamics in both the Schwarzschild and Reissner-Nordstrom black hole spacetimes are analyzed as 2-d conservative bifurcation phenomena. The Schwarzschild dynamics exhibit both saddle-center and transcritical bifurcation points and a calculation of periastron precession is presented that incorporates a phase-plane analysis of the relativistic equations of motion. Level curves of constant energy are illustrated for both timelike and null geodesics and a phase-plane analysis of dynamical invariance between the proper and coordinate time reference frames is discussed. The Reissner-Nordstrom dynamics exhibit saddle-center, transcritical, pseudo-transcritical, and additional bifurcations that combine all three previous bifurcations in various combinations. Periastron precession in the Reissner-Nordstrom spacetime is analyzed using the phase-plane and bifurcation techniques and extended to include a bifurcation point of the dynamics. A numerical solution at these parameter values illustrates that such orbits typically yield a much larger precession value compared to the standard value for timelike, precession. The acausal geodesics considered by Brigman are also discussed and their precession value is calculated.

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Bruce Herold Dean. 2013-12-29. Variational Results and Solutions in Gauge Gravity and a Bifurcation Analysis of Black Hole Orbital Dynamics. https://arxiv.org/abs/1312.7501

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