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 ...
Viscous fluid analogy Original Damour-Navier-Stokes equation Hyp: H = null hypersurface (particular case: black hole event horizon) Then h = ℓ (C = 0) Damour (1979) has derived from Einstein equation the relation S Lℓ Ω (ℓ) + θ (ℓ) Ω (ℓ) = Dκ − D · σ (ℓ) + 1 2 Dθ(ℓ) + 8πq ∗ T · ℓ or equivalently S Lℓ π + θ (ℓ) π = −DP + 2µD · σ (ℓ) + ζDθ (ℓ) + f (∗) with π := − 1 8π Ω(ℓ) momentum surface density P := κ 8π pressure µ := 1 shear viscosity 16π ζ := − 1 bulk viscosity 16π f := −q ∗ T · ℓ external force surface density (T = stress-energy tensor) (∗) is identical to a 2-dimensional Navier-Stokes equation Eric Gourgoulhon (LUTH) Black holes: trapping horizons CERN, 17 March 2010 23 / 38
Viscous fluid analogy Original Damour-Navier-Stokes equation (con’t) Introducing a coordinate system (t, x 1 , x 2 , x 3 ) such that then t is compatible with ℓ: Lℓ t = 1 H is defined by x 1 = const, so that x a = (x 2 , x 3 ) are coordinates spanning St ℓ = ∂ + V ∂t with V tangent to St: velocity of H’s null generators with respect to the coordinates x a [Damour, PRD 18, 3598 (1978)]. Then θ (ℓ) = DaV a + ∂ ∂t ln √ q q := det qab σ (ℓ) ab 1 = 2 (DaVb + DbVa) − 1 2 θ(ℓ) qab + 1 ∂qab 2 ∂t compare Eric Gourgoulhon (LUTH) Black holes: trapping horizons CERN, 17 March 2010 24 / 38
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Viscous fluid analogy<br />
Original Damour-Navier-S<strong>to</strong>kes equation<br />
Hyp: H = null hypersurface (particular case: black hole <strong>event</strong> horizon)<br />
Then h = ℓ (C = 0)<br />
Damour (1979) has derived <strong>from</strong> Einstein equation the relation<br />
S Lℓ Ω (ℓ) + θ (ℓ) Ω (ℓ) = Dκ − D · σ (ℓ) + 1<br />
2 Dθ(ℓ) + 8πq ∗ T · ℓ<br />
or equivalently<br />
S<br />
Lℓ π + θ (ℓ) π = −DP + 2µD · σ (ℓ) + ζDθ (ℓ) + f (∗)<br />
with π := − 1<br />
8π Ω(ℓ) momentum surface density<br />
P := κ<br />
8π pressure<br />
µ := 1<br />
shear viscosity<br />
16π<br />
ζ := − 1<br />
bulk viscosity<br />
16π<br />
f := −q ∗ T · ℓ external force surface density (T = stress-energy tensor)<br />
(∗) is identical <strong>to</strong> a 2-dimensional Navier-S<strong>to</strong>kes equation<br />
Eric Gourgoulhon (<strong>LUTH</strong>) <strong>Black</strong> <strong>holes</strong>: <strong>trapping</strong> <strong>horizons</strong> CERN, 17 March 2010 23 / 38
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