ps6 - Physics 523, Introduction to Relativity Homework 6...

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Unformatted text preview: Physics 523, Introduction to Relativity Homework 6 Due Tuesday, 6 th December 2011 Hans Bantilan Massless Particle near a Kerr Black Hole Consider the Kerr spacetime with mass M and angular momentum per unit mass a such that 0 a M , so that the metric can be expressed in Boyer-Lindquist coordinates ( t,r,, ) as g ij dx i dx j =- 1- 2 Mr dt 2- 4 Mar sin 2 dtd + dr 2 + d 2 + r 2 + a 2 + 2 Ma 2 rsin 2 d 2 with its inverse given by g ij x i x j =- A t 2- 4 Mar t + r 2 + 1 2 + - a 2 sin 2 sin 2 2 where ( r ) = r 2- 2 Mr + a 2 ( r, ) = r 2 + a 2 cos 2 A ( r, ) = ( r 2 + a 2 ) 2- a 2 ( r ) sin 2 . a. We are to find the trajectories of massless particles in Kerr that lie on the equator = / 2. These particles follow geodesics ( ) whose tangent vector field we denote by p = ( ). Since /t and / are Killing fields of the Kerr spacetime, we can immediately write down two conserved quantities of any geodesic in Kerr- e = g ( t ,p ) = p t l = g ( ,p ) = p . For a massless particle, the tangent vectors p are null, so we then have at each point on the geodesic = g ( p,p ) = g tt p t 2 + 2 g t p t p + g rr p r 2 + g p 2 + g p 2 = g tt p t 2 + 2 g t p t p + g rr ( g rr p r ) 2 + g ( g p ) 2 + g p 2 =- A e 2 + 4 Mar el + dr d 2 + - a 2 l 2 . In the last equality we have used the fact that the geodesic lies on the equator = / 2 with p = 0. So 2 dr d 2 = Ae 2- 4 Marel- (- a 2 ) l 2 = A e 2- 4 Mar A el + A (2 Mr- r 2 ) A 2 l 2 and we can simplify by completing the square, which yields 1 2 dr...
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This note was uploaded on 02/02/2012 for the course PHY 523 at Princeton.

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ps6 - Physics 523, Introduction to Relativity Homework 6...

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