[Paper Review] Optical Geometry of the Kerr Space-time

Cameron Bloomer|arXiv (Cornell University)|Nov 21, 2011

Relativity and Gravitational Theory4 references3 citations

TL;DR

This paper applies the Gauss-Bonnet theorem to the optical geometry of the Kerr spacetime, deriving the gravitational deflection angle of light in the equatorial plane using differential geometry. It shows that the curvature contribution from the pseudo-Riemannian part of the Kerr metric yields a second-order term in the angular momentum parameter $ a $, specifically $ \frac{8}{3}\frac{M a^2}{b^3} $, while first-order effects arise from Finsler geometry, not the Riemannian curvature.

ABSTRACT

We pursue a geometrical approach to gravitational lensing theory. We present a survey of the background theory of General Relativity, including particular properties of the Schwarzschild and Kerr solutions. Next we outline a proof of the Gauss Bonnet theorem and its applications to surfaces in optical geometry, as developed by G. W. Gibbons and M. C. Werner. Finally, we attempt to extend this geometrical approach to the axially symmetric Kerr spacetime, and arrive at an expression for the gravitational deflection angle in the equatorial plane.

Motivation & Objective

• To extend the geometric lensing framework, previously applied to Schwarzschild spacetime, to the axially symmetric Kerr spacetime.
• To compute the gravitational deflection angle of light in the equatorial plane of a Kerr black hole using optical geometry and differential topology.
• To determine the contribution of the Riemannian curvature part of the Kerr metric to the deflection angle, particularly its dependence on the spin parameter $ a $.
• To clarify the origin of first-order and second-order terms in $ a $ in the deflection angle, distinguishing between Riemannian and Finsler geometric contributions.

Proposed method

• Utilizes the Gauss-Bonnet theorem to relate the global topology of the optical surface to local curvature, enabling deflection angle computation.
• Constructs the optical metric from the Kerr spacetime by projecting null geodesics onto a spatial surface, using $ g^{\text{opt}}_{mn} = g_{mn} / (-g_{00}) $.
• Calculates the Gauss curvature $ K $ of the optical surface using the Riemann curvature tensor and Christoffel symbols derived from the optical metric.
• Evaluates the double integral $ \int\int_D K \, dA $ over the domain $ D $, approximating the deflection angle via asymptotic limits and weak-field expansion.
• Applies the weak-deflection approximation, setting $ b \gg M $, to simplify the integral and extract leading-order terms in $ M $ and $ a $.
• Compares the derived $ a^2 $-dependent term with existing results from the literature to isolate the Riemannian contribution.

Experimental results

Research questions

• RQ1 How does the Gauss-Bonnet theorem apply to the optical geometry of the Kerr spacetime?
• RQ2 What is the contribution of the Riemannian curvature of the optical surface to the gravitational deflection angle in the Kerr metric?
• RQ3 Why does the first-order term in the spin parameter $ a $ not appear in the Riemannian curvature contribution?
• RQ4 Can the optical geometry approach reproduce known deflection angles in the weak-field limit of Kerr spacetime?

Key findings

• The Riemannian part of the Kerr metric contributes to the deflection angle at second order in $ a $, yielding $ \frac{8}{3}\frac{M a^2}{b^3} $.
• The first-order term in $ a $, which depends on the direction of photon motion (prograde vs. retrograde), does not originate from the Riemannian curvature but from Finsler geometry.
• The total deflection angle includes a dominant $ \frac{4M}{b} $ term and a $ \frac{4M}{b^3} $ term from the $ a^2 $ contribution, consistent with known weak-field expansions.
• The optical geometry framework successfully isolates the geometric origin of the $ a^2 $-dependent deflection, validating the use of Gauss-Bonnet in non-spherically symmetric spacetimes.
• The negative curvature of the optical surface implies local geodesic divergence, but global topology enables convergence, enabling gravitational lensing.