[Paper Review] Quadrupolar gravitational fields described by the $q-$metric

Hernando Quevedo, Saken Toktarbay+1 more|arXiv (Cornell University)|Oct 20, 2013

Black Holes and Theoretical Physics1 references5 citations

TL;DR

This paper proposes the $q$-metric as a description of the exterior gravitational field of static, deformed compact objects with independent mass and quadrupole moment. Using Synge's method, it derives a physically reasonable interior solution matched to the $q$-metric, showing that energy conditions and matching constraints can be satisfied numerically, enabling a complete spacetime description despite naked singularities in the exterior solution.

ABSTRACT

We investigate the Zipoy-Voorhees metric ($q-$metric) as the simplest static, axially symmetric solution of Einstein's vacuum field equations that possesses as independent parameters the mass and the quadrupole moment. In accordance with the black holes uniqueness theorems, the presence of the quadrupole completely changes the geometric properties of the corresponding spacetime that turns out to contain naked singularities for all possible values of the quadrupole parameter. The naked singularities, however, can be covered by interior solutions that correspond to perfect fluid sources with no specific equations of state. We conclude that the $q-$metric can be used to describe the entire spacetime generated by static deformed compact objects.

Motivation & Objective

• To reinterpret the $q$-metric as describing the gravitational field of a static, deformed mass distribution with independent mass and quadrupole moment.
• To investigate whether physically reasonable interior solutions can be matched to the $q$-metric, which otherwise features naked singularities for non-zero quadrupole moments.
• To apply Synge's method—postulating an interior metric and deriving the corresponding energy-momentum tensor via Einstein's equations—without assuming a specific equation of state.
• To verify that the resulting matter distribution satisfies energy conditions and matching conditions at the boundary with the exterior $q$-metric.

Proposed method

• The $q$-metric is derived as a limit of the Zipoy-Voorhees transformation applied to the Schwarzschild solution, with the quadrupole parameter $q$ as an independent parameter.
• The exterior $q$-metric is expressed in isotropic coordinates and expanded to first order in $q$, yielding a perturbative form suitable for matching.
• A spherically symmetric interior metric ansatz is postulated in isotropic coordinates, with metric functions $\mu(r)$, $\alpha(r)$, and $\beta(r,\theta)$ to describe a perfect fluid source.
• The Einstein tensor is computed from the interior metric, and the resulting energy-momentum tensor is derived via Einstein's equations, enabling analysis of physical consistency.
• Matching conditions are imposed at a radial boundary $r = r_m$, requiring continuity of the first fundamental form and alignment of metric components.
• Energy conditions ($T^t_t \geq 0$, $T^t_t - T^r_r \geq 0$) and boundary conditions are applied to constrain the solution space, with numerical analysis indicating feasible physical solutions.

Experimental results

Research questions

• RQ1 Can the $q$-metric, which describes a static, axially symmetric spacetime with a non-zero quadrupole moment, be matched to a physically viable interior solution for a deformed compact object?
• RQ2 Does Synge's method—postulating an interior metric and deriving the corresponding energy-momentum tensor—yield a matter distribution that satisfies energy conditions and boundary constraints?
• RQ3 What are the mathematical and physical conditions under which the $q$-metric's naked singularity can be hidden by a realistic fluid source without requiring a specific equation of state?
• RQ4 Can the system of differential equations for the interior metric functions be solved numerically or analytically under physical constraints, particularly for small $q$?
• RQ5 Is the resulting interior solution consistent with the expected behavior of energy density and pressure in a compact, deformed object?

Key findings

• The $q$-metric describes a spacetime with a naked singularity for all non-zero values of the quadrupole parameter $q$, except in the Schwarzschild limit ($q=0$), where an event horizon forms.
• The singularity is localized near the origin and lies within a region of radius comparable to the Schwarzschild radius for astrophysical compact objects.
• By applying Synge's method, a physically reasonable interior metric was constructed that matches the exterior $q$-metric at a boundary $r = r_m$, satisfying continuity of the first fundamental form.
• The energy conditions $T^t_t \geq 0$ and $T^t_t - T^r_r \geq 0$ were found to be compatible with the derived matter distribution, indicating a physically plausible energy-momentum tensor.
• Numerical analysis confirms that solutions exist which satisfy both the matching conditions and energy conditions simultaneously, with pressure and energy density profiles consistent with physical expectations.
• The system of differential equations for the interior metric functions is compatible and admits potential analytical solutions in the limit of small $q$, suggesting a path toward exact solutions for slightly deformed sources.