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

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Viscous fluid analogy Generalized Damour-Navier-Stokes equation From the contracted Ricci identity applied to the vector m and projected onto St: (∇µ∇νm µ − ∇ν∇µm µ ) q ν α = Rµνm µ q ν α and using Einstein equation to express Rµν, one gets an evolution equation for Ω (ℓ) along H: S Lh Ω (ℓ) + θ (h) Ω (ℓ) = Dκ − D · σ (m) + 1 2 Dθ(m) − θ (k) DC + 8πq ∗ T · m Ω (ℓ) : normal fundamental form of St associated with null normal ℓ θ (h) , θ (m) and θ (k) : expansion scalars of St along the vectors h, m and k respectively D : covariant derivative within (St, q) κ : component of ∇hh along ℓ σ (m) : shear tensor of St along the vector m C : half the scalar square of h Eric Gourgoulhon (LUTH) Black holes: trapping horizons CERN, 17 March 2010 27 / 38
Viscous fluid analogy Null limit (event horizon) If H is a null hypersurface, h = m = ℓ and C = 0 and we recover the original Damour-Navier-Stokes equation: S Lℓ Ω (ℓ) + θ (ℓ) Ω (ℓ) = Dκ − D · σ (ℓ) + 1 2 Dθ(ℓ) + 8πq ∗ T · ℓ Eric Gourgoulhon (LUTH) Black holes: trapping horizons CERN, 17 March 2010 28 / 38
Page 1 and 2: Black holes: from event horizons to
Page 3 and 4: Outline Concept of black hole and e
Page 5 and 6: Concept of black hole and event hor
Page 15 and 16: Local approaches to black holes Loc
Page 17 and 18: Local approaches to black holes Wha
Page 27 and 28: Local approaches to black holes Lin
Page 29 and 30: Local approaches to black holes Con
Page 35 and 36: Local approaches to black holes Exa
Page 37 and 38: Local approaches to black holes Fol
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Page 41 and 42: Local approaches to black holes Evo
Page 43 and 44: Outline Viscous fluid analogy 1 Con
Page 45 and 46: Viscous fluid analogy Concept of bl
Page 47 and 48: Viscous fluid analogy Original Damo
Page 49 and 50: Viscous fluid analogy Original Damo
Page 51: Viscous fluid analogy Generalizatio
Page 55 and 56: Viscous fluid analogy Case of futur
Page 57 and 58: Viscous fluid form Viscous fluid an
Page 59 and 60: Angular momentum and area evolution
Page 69 and 70: Outline Applications to numerical r
Page 71: Review articles References A. Ashte

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