[Paper Review] Shapiro Delays at the Quadrupole Order for Tests of the No-Hair Theorem Using Pulsars around Spinning Black Holes

Pierre Christian, Dimitrios Psaltis+1 more|arXiv (Cornell University)|Nov 5, 2015

Pulsars and Gravitational Waves Research4 citations

TL;DR

This paper derives a closed-form analytic expression for Shapiro time delays at quadrupole order in the spacetime of a spinning black hole with an arbitrary quadrupole moment, using the Butterworth-Ipser metric. The method enables fast, precise computation of pulse arrival time delays for testing the no-hair theorem via pulsar timing around Sgr A*, with key results showing that deviations in the quadrupole moment from Kerr predictions can be detected through high-precision timing analysis.

ABSTRACT

One avenue for testing the no-hair theorem is obtained through timing a pulsar orbiting close to a black hole and fitting for quadrupolar effects on the time-of-arrival of pulses. If deviations from the Kerr quadrupole are measured, then the no-hair theorem is invalidated. To this end, we derive an expression for the light travel time delay for a pulsar orbiting in a black-hole spacetime described by the Butterworth-Ipser metric, which has an arbitrary spin and quadrupole moment. We consider terms up to the quadrupole order in the black-hole metric and derive the time-delay expression in a closed analytic form. This allows for fast computations that are useful in fitting time-of-arrival observations of pulsars orbiting close to astrophysical black holes.

Motivation & Objective

• To develop a fast, analytic method for computing Shapiro time delays in spacetimes with arbitrary quadrupole moments, essential for testing the no-hair theorem.
• To model light propagation delays in the vicinity of a spinning black hole beyond the Kerr metric, incorporating deviations in the quadrupole moment.
• To enable high-precision timing analysis of pulsars orbiting Sgr A* to detect potential violations of the no-hair theorem.
• To provide a computationally efficient framework for fitting time-of-arrival data from pulsars near supermassive black holes.

Proposed method

• Uses the Butterworth-Ipser metric, which describes a stationary, axisymmetric black hole spacetime with arbitrary spin and quadrupole moment, up to second order in the gravitational potential.
• Applies an iterative solution method to the Hamilton-Jacobi equation for null geodesics, allowing derivation of the light travel time delay in closed analytic form.
• Converts the metric to Cartesian coordinates and uses geometric units (G = c = 1) to simplify calculations and ensure consistency with astrophysical applications.
• Derives the time delay as a sum of geometric and relativistic (Shapiro) contributions, with explicit second-order terms in mass and quadrupole moment.
• Performs coordinate transformations between Schwarzschild, isotropic, and de Donder gauges to validate consistency with prior results in the first-order limit.
• Simplifies the second-order quadrupole delay term under the assumption that the z-component of the separation is negligible, yielding a compact expression in terms of the dimensionless parameter β.

Experimental results

Research questions

• RQ1 Can an analytic, closed-form expression for Shapiro time delays be derived at quadrupole order in a black hole spacetime with arbitrary quadrupole moment?
• RQ2 How do deviations from the Kerr metric’s quadrupole moment affect pulse arrival times in pulsar systems near supermassive black holes?
• RQ3 To what extent can the quadrupole moment of a black hole be measured using high-precision pulsar timing observations?
• RQ4 Is the iterative method for solving the Hamilton-Jacobi equation for null geodesics viable for higher-order relativistic corrections in non-Kerr spacetimes?
• RQ5 Can the derived time delay formula be simplified for astrophysically relevant configurations, such as when the pulsar is much closer to the black hole than the observer?

Key findings

• The paper derives a fully analytic expression for the Shapiro time delay that includes second-order contributions from mass and quadrupole moment, valid for arbitrary spin and quadrupole moment.
• The derived delay formula is consistent with first-order results in the literature when transformed to the same coordinate system, validating the approach.
• The second-order mass contribution to the time delay matches known results in the de Donder gauge, confirming the correctness of the coordinate transformation and iterative method.
• For configurations where the z-component of the separation is negligible, the second-order quadrupole delay simplifies to Δ²_quad = -β/r_A (n_A · n_B), a compact and computationally efficient expression.
• In the limit r_B ≫ r_A, the quadrupole delay reduces to Δ²_quad ≈ -β/r_A (n_A · n_B), enabling fast fitting of time-of-arrival data in pulsar timing arrays.
• The method provides a robust, fast, and accurate tool for testing the no-hair theorem by detecting deviations in the black hole’s quadrupole moment from the Kerr prediction.