Black holes: from event horizons to trapping horizons - LUTH ...

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Angular momentum and area evolution laws Area evolution law for an event horizon A(t) : area of the 2-surface St; S ɛ : volume element of St; ¯κ(t) := 1 κ A(t) St S ɛ Integrating the null Raychaudhuri equation on St, one gets d2 A − ¯κdA = − dt2 dt 8πT (ℓ, ℓ) + σ (ℓ) :σ (ℓ) − (θ(ℓ) ) 2 2 + (¯κ − κ)θ(ℓ) S ɛ (1) St Simplified analysis : assume ¯κ = const > 0 : Cauchy problem =⇒ diverging solution of the homogeneous equation: dA = α exp(¯κt) ! dt correct treatment: impose dA = 0 at t = +∞ (teleological !) dt dA dt = +∞ D(u)e ¯κ(t−u) du D(t) : r.h.s. of Eq. (1) t Non causal evolution [Damour, 1979] Eric Gourgoulhon (LUTH) Black holes: trapping horizons CERN, 17 March 2010 34 / 38
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)] Eric Gourgoulhon (LUTH) Black holes: trapping horizons CERN, 17 March 2010 35 / 38
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Black holes: from event horizons to
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Page 43 and 44: Outline Viscous fluid analogy 1 Con
Page 45 and 46: Viscous fluid analogy Concept of bl
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Page 55 and 56: Viscous fluid analogy Case of futur
Page 57 and 58: Viscous fluid form Viscous fluid an
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Page 71: Review articles References A. Ashte
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Black holes: from event horizons to trapping horizons - LUTH ...

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