Visualizing Spacetime Curvature via Gradient Flows II: An Example of the Construction of a Newtonian analogue

Abstract

This is the first in a series of papers in which the gradient flows of fundamental curvature invariants are used to formulate a visualization of curvature. We start with the construction of strict Newtonian analogues (not limits) of solutions to Einstein's equations based on the topology of the associated gradient flows. We do not start with any easy case. Rather, we start with the Curzon - Chazy solution, which, as history shows, is one of the most difficult exact solutions to Einstein's equations to interpret physically. We show that the entire field of the Curzon - Chazy solution, up to a region very "close" to the the intrinsic singularity, strictly represents that of a Newtonian ring, as has long been suspected. In this regard, we consider our approach very successful. As regrades the local structure of the singularity of the Curzon - Chazy solution within a fully general relativistic analysis, however, whereas we make some advances, the full structure of this singularity remains incompletely resolved.

12 pages twocolumn revtex 4-1 9 figures. Expanded and corrected

References in corpus (1)

• Visualizing Spacetime Curvature via Gradient Flows I: Introduction

Cited by in corpus (6)

• Exact Teleparallel Gravity of Binary Black Holes
• Invariant characterization of the Kerr spacetime: Locating the horizon and measuring the mass and spin of rotating black holes using curvature invariants
• On geometry of deformed black holes: I. Majumdar-Papapetrou binary
• On geometry of deformed black holes: II. Schwarzschild hole surrounded by a Bach-Weyl ring
• Visualizing Spacetime Curvature via Gradient Flows III: The Kerr Metric and the Transitional Values of the Spin Parameter
• Slowly rotating Curzon-Chazy Metric