In the Zipoy-Voorhees spacetime, the Shapiro time delay and the Shirokov oscillation frequencies acquire corrections from the quadrupole deformation parameter q, with the delay correction appearing at first order in q.
A stationary q-metric
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abstract
We present a stationary generalization of the static $q-$metric, the simplest generalization of the Schwarzschild solution that contains a quadrupole parameter. It possesses three independent parameters that are related to the mass, quadrupole moment and angular momentum. We investigate the geometric and physical properties of this exact solution of Einstein's vacuum equations, and show that it can be used to describe the exterior gravitational field of rotating, axially symmetric, compact objects.
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Constraining quadrupole deformations with relativistic effects
In the Zipoy-Voorhees spacetime, the Shapiro time delay and the Shirokov oscillation frequencies acquire corrections from the quadrupole deformation parameter q, with the delay correction appearing at first order in q.