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Comparison of vacuum static quadrupolar metrics.

Francisco Frutos-Alfaro1, Hernando Quevedo2,3,4, Pedro A Sanchez2

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We explored Einstein's equations to find new solutions generalizing the Schwarzschild metric with a quadrupole parameter. The q-metric is identified as the simplest such generalization, offering insights into gravitational physics.

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Area of Science:

  • General Relativity
  • Theoretical Physics
  • Gravitational Physics

Background:

  • The Schwarzschild solution is a fundamental solution in general relativity describing vacuum spacetime around a spherical mass.
  • Generalizations of the Schwarzschild solution are crucial for understanding more complex gravitational scenarios.

Purpose of the Study:

  • To investigate static and axisymmetric vacuum solutions of Einstein's equations.
  • To generalize the Schwarzschild solution by introducing a quadrupole parameter.
  • To analyze the multipole structure of these generalized solutions.

Main Methods:

  • Solving Einstein's field equations for static and axisymmetric vacuum spacetimes.
  • Testing solutions for elementary and asymptotic flatness and curvature regularity.
  • Analyzing the multipole structure using the Geroch definition.

Main Results:

  • Identified a class of solutions generalizing the Schwarzschild metric with a quadrupole parameter.
  • Confirmed that these solutions are equivalent up to the quadrupole level.
  • The q-metric, a variant of the Zipoy-Voorhees metric, emerged as the simplest generalization.

Conclusions:

  • The q-metric provides the simplest generalization of the Schwarzschild metric incorporating a quadrupole parameter.
  • This work contributes to the understanding of non-spherical vacuum solutions in general relativity.