Black holes: from event horizons to trapping horizons - LUTH ...
Black holes: from event horizons to trapping horizons - LUTH ... Black holes: from event horizons to trapping horizons - LUTH ...
Angular momentum and area evolution laws Area evolution law for a dynamical horizon Dynamical horizon : C > 0; κ ′ := κ − Lh ln C; ¯κ ′ (t) := 1 κ A(t) St ′ S ɛ From the (m, h) component of Einstein equation, one gets d2A dA + ¯κ′ dt2 dt = 8πT (m, h) + σ (h) :σ (m) + (θ(h) ) 2 + (¯κ 2 ′ − κ ′ )θ (h) S ɛ (2) St [EG & Jaramillo, PRD 74, 087502 (2006)] Simplified analysis : assume ¯κ ′ = const > 0 (OK for small departure from equilibrium [Booth & Fairhurst, PRL 92, 011102 (2004)]): Standard Cauchy problem : dA dt = dA dt t + t=0 0 D(u)e ¯κ′ (u−t) du D(t) : r.h.s. of Eq. (2) Causal evolution, in agreement with local nature of dynamical horizons Eric Gourgoulhon (LUTH) Black holes: trapping horizons CERN, 17 March 2010 35 / 38
Outline Applications to numerical relativity 1 Concept of black hole and event horizon 2 Local approaches to black holes 3 Viscous fluid analogy 4 Angular momentum and area evolution laws 5 Applications to numerical relativity Eric Gourgoulhon (LUTH) Black holes: trapping horizons CERN, 17 March 2010 36 / 38
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Angular momentum and area evolution laws<br />
Area evolution law for a dynamical horizon<br />
Dynamical horizon : C > 0; κ ′ := κ − Lh ln C; ¯κ ′ (t) := 1<br />
<br />
κ<br />
A(t) St<br />
′ S ɛ<br />
From the (m, h) component of Einstein equation, one gets<br />
d2A dA<br />
+ ¯κ′<br />
dt2 dt =<br />
<br />
8πT (m, h) + σ (h) :σ (m) + (θ(h) ) 2<br />
+ (¯κ<br />
2<br />
′ − κ ′ )θ (h) S<br />
ɛ (2)<br />
St<br />
[EG & Jaramillo, PRD 74, 087502 (2006)]<br />
Simplified analysis : assume ¯κ ′ = const > 0<br />
(OK for small departure <strong>from</strong> equilibrium [Booth & Fairhurst, PRL 92, 011102 (2004)]):<br />
Standard Cauchy problem :<br />
dA<br />
dt<br />
= dA<br />
dt<br />
<br />
<br />
t<br />
<br />
+<br />
t=0 0<br />
D(u)e ¯κ′ (u−t) du D(t) : r.h.s. of Eq. (2)<br />
Causal evolution, in agreement with local nature of dynamical <strong>horizons</strong><br />
Eric Gourgoulhon (<strong>LUTH</strong>) <strong>Black</strong> <strong>holes</strong>: <strong>trapping</strong> <strong>horizons</strong> CERN, 17 March 2010 35 / 38
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