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PHZ 6607 November 11
Integrals for conserved quantities Q, M, J. These are essential for the Hamiltonian formulation and also crops up in the initial value formulation
Z
q
2
d x γ (2) nµ σν F µν
Q=−
∂Σ
B.H. generally play a very important role (like H atom for Q.M.). ∃
several B.H. solutions - no more, because asymptotically B.H.’s can be completely characterized by M,J,Q
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Kerr Black Hole
Kerr Black Hole (M,J) - stationary (not static)
2GM
ds = − 1 −
ρ2
2
+
dt2 −
4GM ar sin2 θ
dtdθ
ρ2
sin2 θ 2
ρ2 2
2 2
2
2
2
[(r
+
a
)
−
a
4
sin
θ]dφ
+
dr + ρ2 dθ2
ρ2
4
where
4 = r2 − 2GM r + a2
ρ2 = r2 + a2 cos2 θ
J = aM
Where does gtt go to zero? Where does grr go to ∞? Where does g tt go to
∞?
gtt goes to zero when
r2 + a2 cos2 θ − 2GM r = 0
r2 + a2 (1 − sin2 θ) − 2GM r = 0
42 − a2 sin2 θ = 0
grr goes to zero at 4 = 0 then
4 = (r − r+ )(r − r− )
r+ > r−
r+ and r− are both “event” horizons
x = (r2 + a2 )1/2 sin θ cos φ
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y = (r2 + a2 )1/2 sin θ sin φ
z = r cos θ
In the neighborhood of r+ , it is linear, need to do a coordinate transformation
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