Comparison of vacuum static quadrupolar metrics.

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.