[Paper Review] Light deflection in Kerr field for off-equatorial source

Sarani Chakraborty, A. Sen|arXiv (Cornell University)|Apr 13, 2015

Stellar, planetary, and galactic studies3 citations

TL;DR

This paper derives the analytical expression for light deflection in the Kerr metric for an off-equatorial source, assuming the light ray remains at a constant height $ l $ above the equatorial plane. Using a weak-field approximation and perturbative integration up to second order in rotation parameter $ h $, the authors show that deflection increases with source height $ l $ and decreases with impact parameter $ u $, with distinct prograde and retrograde dependencies due to frame-dragging effects.

ABSTRACT

Deflection angle for a light ray travelling in the equatorial plane of a rotating Kerr mass has been already calculated by various investigators. Considering the light ray to be travelling only slightly above the equatorial plane, calculations have been made for such a ray for its deflection angle. In this paper, we calculate deflection angles for the light ray at various heights, which are small compared to the impact parameter and derive corresponding analytical expressions for deflection angle.

Motivation & Objective

• To extend existing light deflection calculations in Kerr geometry beyond the equatorial plane to include off-equatorial sources.
• To investigate the influence of source height $ l $ above the equatorial plane on the deflection angle $ \Delta\phi $ in a rotating gravitational field.
• To derive a closed-form analytical expression for $ \Delta\phi $ valid in the weak-field limit, incorporating rotation and source height effects.
• To verify the consistency of the derived expression with known equatorial results under limiting conditions.
• To quantify the dependence of deflection on $ \psi \simeq l/u $, the ratio of source height to projected impact parameter.

Proposed method

• Adopting a weak-field approximation in the Kerr metric, the authors model the light ray trajectory as maintaining a constant latitude $ \theta $, corresponding to a fixed height $ l $ above the equatorial plane.
• The deflection angle $ \Delta\phi $ is computed via integration of the null geodesic equation, retaining terms up to second order in the rotation parameter $ h $, with $ h \sim a/M $.
• The derivation uses a perturbative expansion in $ h $, integrating the differential equation for $ d\phi/dr $, and applying boundary conditions at infinity and closest approach.
• The resulting expression for $ \Delta\phi $ is expressed in terms of mass $ M $, rotation parameter $ a $, and $ \psi \simeq l/u $, where $ u $ is the projected impact parameter.
• The method incorporates frame-dragging effects through the Kerr metric’s off-diagonal component $ g_{t\phi} $, which influences the photon’s angular momentum and trajectory.
• The solution is validated by taking the limit $ \psi \to 0 $, recovering the known equatorial deflection formula from Aazami et al. [16], confirming consistency.

Experimental results

Research questions

• RQ1 How does the deflection angle of light in the Kerr field change when the source is located off the equatorial plane?
• RQ2 What is the analytical dependence of the deflection angle on the source height $ l $ and the projected impact parameter $ u $?
• RQ3 How does frame-dragging from rotation modify the deflection angle for off-equatorial light rays compared to equatorial ones?
• RQ4 What is the quantitative impact of prograde versus retrograde motion on the deflection angle in the off-equatorial configuration?
• RQ5 Does the derived expression reduce to known equatorial results in the limit $ l \to 0 $? If so, how is this consistency demonstrated?

Key findings

• The deflection angle $ \Delta\phi $ increases monotonically with $ \psi \simeq l/u $, indicating that higher source heights lead to greater bending for both prograde and retrograde light rays.
• For both the Sun and the pulsar PSR J 1748-2446, the deflection angle decreases with increasing $ u/2m $, confirming the expected weak-field behavior.
• The derived expression (Eq. 38) reduces exactly to the equatorial deflection formula of Aazami et al. [16] when $ \psi \to 0 $, validating the consistency of the model.
• The prograde deflection is consistently larger than the retrograde deflection at the same $ \psi $, due to frame-dragging enhancing bending in the direction of rotation.
• The second-order terms in $ h $ (rotation parameter) contribute significantly to the deflection, particularly through terms involving $ \hat{a}^2 $, the square of the Kerr parameter.
• The analysis confirms that for $ l \ll u $, the assumption of constant $ \psi $ is valid for analytical tractability, though the full $ r $-dependence of $ \psi $ is left for future work.