Shadow of A Black Hole Part 1: Kerr BH (Planetary Science)

In 1973, James Bardeen initiated his research on gravitational lensing by spinning black holes. Bardeen gave a thorough analysis of null geodesics (light-ray propagation) around a Kerr black hole.

The Kerr solution had been discovered in 1962 by the New Zealand physicist Roy Kerr and since then focused the attention of many researchers in General Relativity, because it represents the most general state of equilibrium of an astrophysical black hole.

The Kerr spacetime’s metric depends on two parameters : the black hole mass “M” and its normalized angular momentum “a”. An important difference with usual stars, which are in differential rotation, is that Kerr black holes are rotating with perfect rigidity : all the points on their event horizon move with the same angular velocity. There is however a critical angular momentum, given by a = M (in units where G=c=1) above which the event horizon would “break up”: this limit corresponds to the horizon having a spin velocity equal to the speed of light. For such a black hole, called “extreme”, the gravitational field at the event horizon would cancel, because the inward pull of gravity would be compensated by huge repulsive centrifugal forces.

In the last twenty years increasing evidence has been found for the existence of a supermassive black hole at the center of our galaxy. It is expected that a distant observer should “see” this black hole as a dark disk in the sky which is known as the “shadow”. It is sometimes said that the shadow is an image of the event horizon.

James Bardeen was the first to correctly calculate the shape of the shadow of a Kerr black hole. He computed how the black hole’s rotation would affect the shape of the shadow that the event horizon casts on light from a background star field. For a black hole spinning close to the maximum angular momentum, the result is a D-shaped shadow.

Apparent shape of an extreme Kerr black hole as seen by a distant observer in the equatorial plane, if the black hole is in front of a source of illumination with an angular size larger than that of the black hole. The shadow bulges out on the side of the hole moving away from the observer (at right) and squeezes inward and flattens on the side moving toward the observer (at left). © Bardeen

Reference: Bardeen, J. M. 1973, Timelike and null geodesics in the Kerr metric, in Black Holes”, Gordon and Breach, Science Publishers, Inc; New York; 23, 6(5), pp. 215–239, 1973.

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