Light’s bending angle in the equatorial plane of a Kerr black hole

We present here a detailed derivation of an explicit spin-dependent expression for the bending angle of light as it traverses in the equatorial plane of a spinning black hole. We show that the deflection produced in the presence of the black hole angular momentum explicitly depends on whether the motion of the light ray is in the direction, or opposite to the spin. Compared to the zero-spin Schwarzschild case, the bending angle is greater for direct orbits, and smaller for retrograde orbits, confirming our physical intuition about the loss of left-right symmetry from a lensing perspective. In addition, we show that for higher spins, the effect is more pronounced resulting in tighter winding of direct orbits with respect to the axis of rotation, and a higher degree of unwinding of retrograde orbits. A direct consequence of this effect is a shift in image positions in strong gravitational lensing.

Figure 1

Sign convention for orbits as viewed from above. The spin axis points into the page in this figure.

Figure 2

The critical impact parameter in units of ${\mathsf{m}}_{•}$ as a function of the spin parameter $\widehat{a}$. The Schwarzschild critical impact parameter is shown for reference.

Figure 3

The critical radius in units of ${\mathsf{m}}_{•}$ plotted as a function of the spin parameter $\widehat{a}$. The Schwarzschild radius, the radius of the Schwarzschild photon sphere, and the Kerr inner and outer horizon radii are shown for reference.

Figure 4

Impact parameter for $\widehat{a}=0.5$. The strong deflection regime is towards the left of this plot as $b^{\prime }$ goes to zero, and the weak deflection regime is as $b^{\prime }\to \infty$. The Schwarzschild bending angle is shown for reference.

Figure 5

Impact parameter for $\widehat{a}=0.99$. The strong deflection regime is towards the left of this plot as $b^{\prime }$ goes to zero, and the weak deflection regime is as $b^{\prime }\to \infty$. The Schwarzschild bending angle is shown for reference.

Figure 6

Bending angle plotted as a function of spin for $b^{\prime }=0.5$. The Schwarzschild case is shown for reference.