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GR Sessions 7: Black Holes

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GR Sessions 7: Black Holes

Wednesdays November 21, 2012

1. Conformal diagrams. The solution to Einstein’s equations with a positive cosmological constant can be written as

ds2= `2 −dτ2+ cosh2τ dψ2+ sin2ψ dθ2+ sin2θdφ2

= `2 −dτ2+ cosh2τ dΩ23 ,

where ψ and θ range from [0, π] and φ ranges from [0, 2π]. The quantity in parentheses is the metric on the 3-sphere. Using the coordinate transformation cosh τ = sec T , draw the Penrose (conformal) diagram of de Sitter space. You can find information about Penrose diagrams in Appendix H of Carroll.

2. Kerr black holes An observer orbits a Kerr black hole of Mass M and angular momentum (per unit mass) a in the equatorial plane.

(a) Consider a constant r orbit and define Ω = dt to be her angular velocity as measured by a very distant and stationary observer. Show that the observer’s four velocity is given by

vµ= v0(1, 0, 0, Ω) , where

v0=



1 − 2GM

r +4GM a r Ω −



r2+ a2+2GM a2 r

 Ω2

−1/2 . (b) Consider the polynomial

Y ≡ −1 +2GM

r −4GM a r Ω +



r2+ a2+2GM a2 r

 Ω2.

Using part 2a show that Y is always negative.

(c) Using this result show that Ω is nonzero in the ergosphere. Also show that the observer can not stay fixed at constant radius once she crosses the outer horizon r+.

(d) Show that Kepler’s law Ω2=GMr3 holds for circular orbits around a Schwarzschild black hole.

(e) Derive an analogous result for equatorial orbits around a Kerr black hole. Hint: You can save a lot of time by first showing the geodesic equation reduces to

Γµνρ

dxν

dxρ dτ = 0 , where Γµνρ= (∂νgµρ+ ∂ρgµν− ∂µgνρ).

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