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 ...
Black holes: from event horizons to trapping horizons Eric Gourgoulhon Laboratoire Univers et Théories (LUTH) CNRS / Observatoire de Paris / Université Paris Diderot 92190 Meudon, France eric.gourgoulhon@obspm.fr http://luth.obspm.fr/~luthier/gourgoulhon/ Laboratoire d’Astrophysique de Toulouse-Tarbes Toulouse, France 10 December 2009 Eric Gourgoulhon (LUTH) Black holes: trapping horizons LATT, Toulouse, 10 December 2009 1 / 38
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<strong>Black</strong> <strong>holes</strong>:<br />
<strong>from</strong> <strong>event</strong> <strong>horizons</strong> <strong>to</strong> <strong>trapping</strong> <strong>horizons</strong><br />
Eric Gourgoulhon<br />
Labora<strong>to</strong>ire Univers et Théories (<strong>LUTH</strong>)<br />
CNRS / Observa<strong>to</strong>ire de Paris / Université Paris Diderot<br />
92190 Meudon, France<br />
eric.gourgoulhon@obspm.fr<br />
http://luth.obspm.fr/~luthier/gourgoulhon/<br />
Labora<strong>to</strong>ire d’Astrophysique de Toulouse-Tarbes<br />
Toulouse, France<br />
10 December 2009<br />
Eric Gourgoulhon (<strong>LUTH</strong>) <strong>Black</strong> <strong>holes</strong>: <strong>trapping</strong> <strong>horizons</strong> LATT, Toulouse, 10 December 2009 1 / 38
Plan<br />
1 Concept of black hole and <strong>event</strong> horizon<br />
2 Local approaches <strong>to</strong> black <strong>holes</strong><br />
3 Viscous fluid analogy<br />
4 Angular momentum and area evolution laws<br />
5 Applications <strong>to</strong> numerical relativity<br />
Eric Gourgoulhon (<strong>LUTH</strong>) <strong>Black</strong> <strong>holes</strong>: <strong>trapping</strong> <strong>horizons</strong> LATT, Toulouse, 10 December 2009 2 / 38
Outline<br />
Concept of black hole and <strong>event</strong> horizon<br />
1 Concept of black hole and <strong>event</strong> horizon<br />
2 Local approaches <strong>to</strong> black <strong>holes</strong><br />
3 Viscous fluid analogy<br />
4 Angular momentum and area evolution laws<br />
5 Applications <strong>to</strong> numerical relativity<br />
Eric Gourgoulhon (<strong>LUTH</strong>) <strong>Black</strong> <strong>holes</strong>: <strong>trapping</strong> <strong>horizons</strong> LATT, Toulouse, 10 December 2009 3 / 38
Concept of black hole and <strong>event</strong> horizon<br />
What is a black hole ?<br />
... for the astrophysicist: a very deep gravitational potential well<br />
[J.A. Marck, CQG 13, 393 (1996)]<br />
Eric Gourgoulhon (<strong>LUTH</strong>) <strong>Black</strong> <strong>holes</strong>: <strong>trapping</strong> <strong>horizons</strong> LATT, Toulouse, 10 December 2009 4 / 38
Concept of black hole and <strong>event</strong> horizon<br />
What is a black hole ?<br />
... for the astrophysicist: a very deep gravitational potential well<br />
Binary BH in galaxy NGC 6240<br />
d = 1.4 kpc<br />
[Komossa et al., ApJ 582, L15 (2003)]<br />
Binary BH in radio galaxy 0402+379<br />
d = 7.3 pc<br />
[Rodriguez et al., ApJ 646, 49 (2006)]<br />
Eric Gourgoulhon (<strong>LUTH</strong>) <strong>Black</strong> <strong>holes</strong>: <strong>trapping</strong> <strong>horizons</strong> LATT, Toulouse, 10 December 2009 4 / 38
Concept of black hole and <strong>event</strong> horizon<br />
What is a black hole ?<br />
... for the mathematical physicist:<br />
B := M − J − (I + )<br />
i.e. the region of spacetime where light<br />
rays cannot escape <strong>to</strong> infinity<br />
(M , g) = asymp<strong>to</strong>tically flat<br />
manifold<br />
I + = future null infinity<br />
J − (I + ) = causal past of I +<br />
<strong>event</strong> horizon: H := ∂J − (I + )<br />
(boundary of J − (I + ))<br />
H smooth =⇒ H null hypersurface<br />
Eric Gourgoulhon (<strong>LUTH</strong>) <strong>Black</strong> <strong>holes</strong>: <strong>trapping</strong> <strong>horizons</strong> LATT, Toulouse, 10 December 2009 5 / 38
Concept of black hole and <strong>event</strong> horizon<br />
What is a black hole ?<br />
... for the mathematical physicist:<br />
B := M − J − (I + )<br />
i.e. the region of spacetime where light<br />
rays cannot escape <strong>to</strong> infinity<br />
(M , g) = asymp<strong>to</strong>tically flat<br />
manifold<br />
I + = future null infinity<br />
J − (I + ) = causal past of I +<br />
<strong>event</strong> horizon: H := ∂J − (I + )<br />
(boundary of J − (I + ))<br />
H smooth =⇒ H null hypersurface<br />
Eric Gourgoulhon (<strong>LUTH</strong>) <strong>Black</strong> <strong>holes</strong>: <strong>trapping</strong> <strong>horizons</strong> LATT, Toulouse, 10 December 2009 5 / 38
Concept of black hole and <strong>event</strong> horizon<br />
Drawbacks of the classical definition<br />
not applicable in cosmology, for in general (M , g) is not asymp<strong>to</strong>tically flat<br />
Eric Gourgoulhon (<strong>LUTH</strong>) <strong>Black</strong> <strong>holes</strong>: <strong>trapping</strong> <strong>horizons</strong> LATT, Toulouse, 10 December 2009 6 / 38
Concept of black hole and <strong>event</strong> horizon<br />
Drawbacks of the classical definition<br />
not applicable in cosmology, for in general (M , g) is not asymp<strong>to</strong>tically flat<br />
even when applicable, this definition is highly non-local:<br />
Eric Gourgoulhon (<strong>LUTH</strong>) <strong>Black</strong> <strong>holes</strong>: <strong>trapping</strong> <strong>horizons</strong> LATT, Toulouse, 10 December 2009 6 / 38
Concept of black hole and <strong>event</strong> horizon<br />
Drawbacks of the classical definition<br />
not applicable in cosmology, for in general (M , g) is not asymp<strong>to</strong>tically flat<br />
even when applicable, this definition is highly non-local:<br />
Determination of ˙<br />
Moreover this is not locally linked with the notion of strong gravitational field:<br />
[Ashtekar & Krishnan, LRR 7, 10 (2004)]<br />
J − (I + ) requires the knowledge of the entire future null infinity.<br />
Example of <strong>event</strong> horizon in a flat region of<br />
spacetime:<br />
Vaidya metric, describing incoming radiation<br />
<strong>from</strong> infinity:<br />
ds 2 = −<br />
<br />
1 − 2m(v)<br />
<br />
dv 2 + 2dv dr + r 2 (dθ 2 +<br />
r<br />
sin 2 θdϕ 2 )<br />
with m(v) = 0 for v < 0<br />
dm/dv > 0 for 0 ≤ v ≤ v0<br />
m(v) = M0 for v > v0<br />
Eric Gourgoulhon (<strong>LUTH</strong>) <strong>Black</strong> <strong>holes</strong>: <strong>trapping</strong> <strong>horizons</strong> LATT, Toulouse, 10 December 2009 6 / 38
Concept of black hole and <strong>event</strong> horizon<br />
Drawbacks of the classical definition<br />
not applicable in cosmology, for in general (M , g) is not asymp<strong>to</strong>tically flat<br />
even when applicable, this definition is highly non-local:<br />
Determination of ˙<br />
Moreover this is not locally linked with the notion of strong gravitational field:<br />
[Ashtekar & Krishnan, LRR 7, 10 (2004)]<br />
J − (I + ) requires the knowledge of the entire future null infinity.<br />
Example of <strong>event</strong> horizon in a flat region of<br />
spacetime:<br />
Vaidya metric, describing incoming radiation<br />
<strong>from</strong> infinity:<br />
ds 2 = −<br />
<br />
1 − 2m(v)<br />
<br />
dv 2 + 2dv dr + r 2 (dθ 2 +<br />
r<br />
sin 2 θdϕ 2 )<br />
with m(v) = 0 for v < 0<br />
dm/dv > 0 for 0 ≤ v ≤ v0<br />
m(v) = M0 for v > v0<br />
⇒ no local physical experiment whatsoever can<br />
locate the <strong>event</strong> horizon<br />
Eric Gourgoulhon (<strong>LUTH</strong>) <strong>Black</strong> <strong>holes</strong>: <strong>trapping</strong> <strong>horizons</strong> LATT, Toulouse, 10 December 2009 6 / 38
Concept of black hole and <strong>event</strong> horizon<br />
Another non-local feature: teleological nature of <strong>event</strong><br />
<strong>horizons</strong><br />
[Booth, Can. J. Phys. 83, 1073 (2005)]<br />
The classical black hole boundary, i.e.<br />
the <strong>event</strong> horizon, responds in advance<br />
<strong>to</strong> what will happen in the future.<br />
Eric Gourgoulhon (<strong>LUTH</strong>) <strong>Black</strong> <strong>holes</strong>: <strong>trapping</strong> <strong>horizons</strong> LATT, Toulouse, 10 December 2009 7 / 38
Concept of black hole and <strong>event</strong> horizon<br />
Another non-local feature: teleological nature of <strong>event</strong><br />
<strong>horizons</strong><br />
[Booth, Can. J. Phys. 83, 1073 (2005)]<br />
The classical black hole boundary, i.e.<br />
the <strong>event</strong> horizon, responds in advance<br />
<strong>to</strong> what will happen in the future.<br />
To deal with black <strong>holes</strong> as ordinary<br />
physical objects, a local definition would<br />
be desirable<br />
→ quantum gravity, numerical relativity<br />
Eric Gourgoulhon (<strong>LUTH</strong>) <strong>Black</strong> <strong>holes</strong>: <strong>trapping</strong> <strong>horizons</strong> LATT, Toulouse, 10 December 2009 7 / 38
Outline<br />
Local approaches <strong>to</strong> black <strong>holes</strong><br />
1 Concept of black hole and <strong>event</strong> horizon<br />
2 Local approaches <strong>to</strong> black <strong>holes</strong><br />
3 Viscous fluid analogy<br />
4 Angular momentum and area evolution laws<br />
5 Applications <strong>to</strong> numerical relativity<br />
Eric Gourgoulhon (<strong>LUTH</strong>) <strong>Black</strong> <strong>holes</strong>: <strong>trapping</strong> <strong>horizons</strong> LATT, Toulouse, 10 December 2009 8 / 38
Local approaches <strong>to</strong> black <strong>holes</strong><br />
Local characterizations of black <strong>holes</strong><br />
Recently a new paradigm has appeared in the theoretical approach of black<br />
<strong>holes</strong>: instead of <strong>event</strong> <strong>horizons</strong>, black <strong>holes</strong> are described by<br />
<strong>trapping</strong> <strong>horizons</strong> (Hayward 1994)<br />
isolated <strong>horizons</strong> (Ashtekar et al. 1999)<br />
dynamical <strong>horizons</strong> (Ashtekar and Krishnan 2002)<br />
slowly evolving <strong>horizons</strong> (Booth and Fairhurst 2004)<br />
All these concepts are local and are based on the notion of trapped surfaces<br />
Eric Gourgoulhon (<strong>LUTH</strong>) <strong>Black</strong> <strong>holes</strong>: <strong>trapping</strong> <strong>horizons</strong> LATT, Toulouse, 10 December 2009 9 / 38
Local approaches <strong>to</strong> black <strong>holes</strong><br />
What is a trapped surface ?<br />
1/ Expansion of a surface along a normal vec<strong>to</strong>r field<br />
1 Consider a spacelike 2-surface S<br />
(induced metric: q)<br />
Eric Gourgoulhon (<strong>LUTH</strong>) <strong>Black</strong> <strong>holes</strong>: <strong>trapping</strong> <strong>horizons</strong> LATT, Toulouse, 10 December 2009 10 / 38
Local approaches <strong>to</strong> black <strong>holes</strong><br />
What is a trapped surface ?<br />
1/ Expansion of a surface along a normal vec<strong>to</strong>r field<br />
1 Consider a spacelike 2-surface S<br />
(induced metric: q)<br />
2 Take a vec<strong>to</strong>r field v defined on<br />
S and normal <strong>to</strong> S at each<br />
point<br />
Eric Gourgoulhon (<strong>LUTH</strong>) <strong>Black</strong> <strong>holes</strong>: <strong>trapping</strong> <strong>horizons</strong> LATT, Toulouse, 10 December 2009 10 / 38
Local approaches <strong>to</strong> black <strong>holes</strong><br />
What is a trapped surface ?<br />
1/ Expansion of a surface along a normal vec<strong>to</strong>r field<br />
1 Consider a spacelike 2-surface S<br />
(induced metric: q)<br />
2 Take a vec<strong>to</strong>r field v defined on<br />
S and normal <strong>to</strong> S at each<br />
point<br />
3 ε being a small parameter,<br />
displace the point p by the<br />
vec<strong>to</strong>r εv <strong>to</strong> the point p ′<br />
Eric Gourgoulhon (<strong>LUTH</strong>) <strong>Black</strong> <strong>holes</strong>: <strong>trapping</strong> <strong>horizons</strong> LATT, Toulouse, 10 December 2009 10 / 38
Local approaches <strong>to</strong> black <strong>holes</strong><br />
What is a trapped surface ?<br />
1/ Expansion of a surface along a normal vec<strong>to</strong>r field<br />
1 Consider a spacelike 2-surface S<br />
(induced metric: q)<br />
2 Take a vec<strong>to</strong>r field v defined on<br />
S and normal <strong>to</strong> S at each<br />
point<br />
3 ε being a small parameter,<br />
displace the point p by the<br />
vec<strong>to</strong>r εv <strong>to</strong> the point p ′<br />
4 Do the same for each point in<br />
S, keeping the value of ε fixed<br />
Eric Gourgoulhon (<strong>LUTH</strong>) <strong>Black</strong> <strong>holes</strong>: <strong>trapping</strong> <strong>horizons</strong> LATT, Toulouse, 10 December 2009 10 / 38
Local approaches <strong>to</strong> black <strong>holes</strong><br />
What is a trapped surface ?<br />
1/ Expansion of a surface along a normal vec<strong>to</strong>r field<br />
1 Consider a spacelike 2-surface S<br />
(induced metric: q)<br />
2 Take a vec<strong>to</strong>r field v defined on<br />
S and normal <strong>to</strong> S at each<br />
point<br />
3 ε being a small parameter,<br />
displace the point p by the<br />
vec<strong>to</strong>r εv <strong>to</strong> the point p ′<br />
4 Do the same for each point in<br />
S, keeping the value of ε fixed<br />
5 This defines a new surface S ′<br />
(Lie dragging)<br />
Eric Gourgoulhon (<strong>LUTH</strong>) <strong>Black</strong> <strong>holes</strong>: <strong>trapping</strong> <strong>horizons</strong> LATT, Toulouse, 10 December 2009 10 / 38
Local approaches <strong>to</strong> black <strong>holes</strong><br />
What is a trapped surface ?<br />
1/ Expansion of a surface along a normal vec<strong>to</strong>r field<br />
1 Consider a spacelike 2-surface S<br />
(induced metric: q)<br />
2 Take a vec<strong>to</strong>r field v defined on<br />
S and normal <strong>to</strong> S at each<br />
point<br />
3 ε being a small parameter,<br />
displace the point p by the<br />
vec<strong>to</strong>r εv <strong>to</strong> the point p ′<br />
4 Do the same for each point in<br />
S, keeping the value of ε fixed<br />
5 This defines a new surface S ′<br />
(Lie dragging)<br />
At each point, the expansion of S along v is defined <strong>from</strong> the relative change in<br />
the area element δA: θ (v) := lim<br />
ε→0<br />
1<br />
ε<br />
δA ′ − δA<br />
δA<br />
= Lv ln √ q = q µν ∇µvν<br />
Eric Gourgoulhon (<strong>LUTH</strong>) <strong>Black</strong> <strong>holes</strong>: <strong>trapping</strong> <strong>horizons</strong> LATT, Toulouse, 10 December 2009 10 / 38
Local approaches <strong>to</strong> black <strong>holes</strong><br />
What is a trapped surface ?<br />
2/ The definition<br />
S : closed (i.e. compact without boundary) spacelike 2-dimensional surface<br />
embedded in spacetime (M , g)<br />
Eric Gourgoulhon (<strong>LUTH</strong>) <strong>Black</strong> <strong>holes</strong>: <strong>trapping</strong> <strong>horizons</strong> LATT, Toulouse, 10 December 2009 11 / 38
Local approaches <strong>to</strong> black <strong>holes</strong><br />
What is a trapped surface ?<br />
2/ The definition<br />
S : closed (i.e. compact without boundary) spacelike 2-dimensional surface<br />
embedded in spacetime (M , g)<br />
Being spacelike, S lies outside the light<br />
cone<br />
Eric Gourgoulhon (<strong>LUTH</strong>) <strong>Black</strong> <strong>holes</strong>: <strong>trapping</strong> <strong>horizons</strong> LATT, Toulouse, 10 December 2009 11 / 38
Local approaches <strong>to</strong> black <strong>holes</strong><br />
What is a trapped surface ?<br />
2/ The definition<br />
S : closed (i.e. compact without boundary) spacelike 2-dimensional surface<br />
embedded in spacetime (M , g)<br />
Being spacelike, S lies outside the light<br />
cone<br />
∃ two future-directed null directions<br />
orthogonal <strong>to</strong> S:<br />
ℓ = outgoing, expansion θ (ℓ)<br />
k = ingoing, expansion θ (k)<br />
In flat space, θ (k) < 0 and θ (ℓ) > 0<br />
Eric Gourgoulhon (<strong>LUTH</strong>) <strong>Black</strong> <strong>holes</strong>: <strong>trapping</strong> <strong>horizons</strong> LATT, Toulouse, 10 December 2009 11 / 38
Local approaches <strong>to</strong> black <strong>holes</strong><br />
What is a trapped surface ?<br />
2/ The definition<br />
S : closed (i.e. compact without boundary) spacelike 2-dimensional surface<br />
embedded in spacetime (M , g)<br />
Being spacelike, S lies outside the light<br />
cone<br />
∃ two future-directed null directions<br />
orthogonal <strong>to</strong> S:<br />
ℓ = outgoing, expansion θ (ℓ)<br />
k = ingoing, expansion θ (k)<br />
In flat space, θ (k) < 0 and θ (ℓ) > 0<br />
S is trapped ⇐⇒ θ (k) < 0 and θ (ℓ) < 0 [Penrose 1965]<br />
S is marginally trapped ⇐⇒ θ (k) < 0 and θ (ℓ) = 0<br />
Eric Gourgoulhon (<strong>LUTH</strong>) <strong>Black</strong> <strong>holes</strong>: <strong>trapping</strong> <strong>horizons</strong> LATT, Toulouse, 10 December 2009 11 / 38
Local approaches <strong>to</strong> black <strong>holes</strong><br />
What is a trapped surface ?<br />
2/ The definition<br />
S : closed (i.e. compact without boundary) spacelike 2-dimensional surface<br />
embedded in spacetime (M , g)<br />
Being spacelike, S lies outside the light<br />
cone<br />
∃ two future-directed null directions<br />
orthogonal <strong>to</strong> S:<br />
ℓ = outgoing, expansion θ (ℓ)<br />
k = ingoing, expansion θ (k)<br />
In flat space, θ (k) < 0 and θ (ℓ) > 0<br />
S is trapped ⇐⇒ θ (k) < 0 and θ (ℓ) < 0 [Penrose 1965]<br />
S is marginally trapped ⇐⇒ θ (k) < 0 and θ (ℓ) = 0<br />
trapped surface = local concept characterizing very strong gravitational fields<br />
Eric Gourgoulhon (<strong>LUTH</strong>) <strong>Black</strong> <strong>holes</strong>: <strong>trapping</strong> <strong>horizons</strong> LATT, Toulouse, 10 December 2009 11 / 38
Local approaches <strong>to</strong> black <strong>holes</strong><br />
Link with apparent <strong>horizons</strong><br />
A closed spacelike 2-surface S is said <strong>to</strong> be outer trapped (resp. marginally outer<br />
trapped (MOTS)) iff [Hawking & Ellis 1973]<br />
the notions of interior and exterior of S can be defined (for instance<br />
spacetime asymp<strong>to</strong>tically flat) ⇒ ℓ is chosen <strong>to</strong> be the outgoing null normal<br />
and k <strong>to</strong> be the ingoing one<br />
θ (ℓ) < 0 (resp. θ (ℓ) = 0)<br />
Eric Gourgoulhon (<strong>LUTH</strong>) <strong>Black</strong> <strong>holes</strong>: <strong>trapping</strong> <strong>horizons</strong> LATT, Toulouse, 10 December 2009 12 / 38
Local approaches <strong>to</strong> black <strong>holes</strong><br />
Link with apparent <strong>horizons</strong><br />
A closed spacelike 2-surface S is said <strong>to</strong> be outer trapped (resp. marginally outer<br />
trapped (MOTS)) iff [Hawking & Ellis 1973]<br />
the notions of interior and exterior of S can be defined (for instance<br />
spacetime asymp<strong>to</strong>tically flat) ⇒ ℓ is chosen <strong>to</strong> be the outgoing null normal<br />
and k <strong>to</strong> be the ingoing one<br />
θ (ℓ) < 0 (resp. θ (ℓ) = 0)<br />
Σ: spacelike<br />
hypersurface extending<br />
<strong>to</strong> spatial infinity<br />
(Cauchy surface)<br />
outer trapped region of Σ : Ω = set of points p ∈ Σ through which there is a<br />
outer trapped surface S lying in Σ<br />
apparent horizon in Σ: A = connected component of the boundary of Ω<br />
Proposition [Hawking & Ellis 1973]: A smooth =⇒ A is a MOTS<br />
Eric Gourgoulhon (<strong>LUTH</strong>) <strong>Black</strong> <strong>holes</strong>: <strong>trapping</strong> <strong>horizons</strong> LATT, Toulouse, 10 December 2009 12 / 38
Local approaches <strong>to</strong> black <strong>holes</strong><br />
Connection with singularities and black <strong>holes</strong><br />
Proposition [Penrose (1965)]:<br />
provided that the weak energy condition holds,<br />
∃ a trapped surface S =⇒ ∃ a singularity in (M , g) (in the form of a future<br />
inextendible null geodesic)<br />
Proposition [Hawking & Ellis (1973)]:<br />
provided that the cosmic censorship conjecture holds,<br />
∃ a trapped surface S =⇒ ∃ a black hole B and S ⊂ B<br />
Eric Gourgoulhon (<strong>LUTH</strong>) <strong>Black</strong> <strong>holes</strong>: <strong>trapping</strong> <strong>horizons</strong> LATT, Toulouse, 10 December 2009 13 / 38
Local approaches <strong>to</strong> black <strong>holes</strong><br />
Local definitions of “black <strong>holes</strong>”<br />
A hypersurface H of (M , g) is said <strong>to</strong> be<br />
a future outer <strong>trapping</strong> horizon (FOTH) iff<br />
(i) H foliated by marginally trapped 2-surfaces<br />
(θ (k) < 0 and θ (ℓ) = 0)<br />
(ii) Lk θ (ℓ) < 0 (locally outermost trapped surf.)<br />
[Hayward, PRD 49, 6467 (1994)]<br />
Eric Gourgoulhon (<strong>LUTH</strong>) <strong>Black</strong> <strong>holes</strong>: <strong>trapping</strong> <strong>horizons</strong> LATT, Toulouse, 10 December 2009 14 / 38
Local approaches <strong>to</strong> black <strong>holes</strong><br />
Local definitions of “black <strong>holes</strong>”<br />
A hypersurface H of (M , g) is said <strong>to</strong> be<br />
a future outer <strong>trapping</strong> horizon (FOTH) iff<br />
(i) H foliated by marginally trapped 2-surfaces<br />
(θ (k) < 0 and θ (ℓ) = 0)<br />
(ii) Lk θ (ℓ) < 0 (locally outermost trapped surf.)<br />
[Hayward, PRD 49, 6467 (1994)]<br />
a dynamical horizon (DH) iff<br />
(i) H foliated by marginally trapped 2-surfaces<br />
(ii) H spacelike<br />
[Ashtekar & Krishnan, PRL 89 261101 (2002)]<br />
Eric Gourgoulhon (<strong>LUTH</strong>) <strong>Black</strong> <strong>holes</strong>: <strong>trapping</strong> <strong>horizons</strong> LATT, Toulouse, 10 December 2009 14 / 38
Local approaches <strong>to</strong> black <strong>holes</strong><br />
Local definitions of “black <strong>holes</strong>”<br />
A hypersurface H of (M , g) is said <strong>to</strong> be<br />
a future outer <strong>trapping</strong> horizon (FOTH) iff<br />
(i) H foliated by marginally trapped 2-surfaces<br />
(θ (k) < 0 and θ (ℓ) = 0)<br />
(ii) Lk θ (ℓ) < 0 (locally outermost trapped surf.)<br />
[Hayward, PRD 49, 6467 (1994)]<br />
a dynamical horizon (DH) iff<br />
(i) H foliated by marginally trapped 2-surfaces<br />
(ii) H spacelike<br />
[Ashtekar & Krishnan, PRL 89 261101 (2002)]<br />
a non-expanding horizon (NEH) iff<br />
(i) H is null (null normal ℓ)<br />
(ii) θ (ℓ) = 0 [Há´jiček (1973)]<br />
Eric Gourgoulhon (<strong>LUTH</strong>) <strong>Black</strong> <strong>holes</strong>: <strong>trapping</strong> <strong>horizons</strong> LATT, Toulouse, 10 December 2009 14 / 38
Local approaches <strong>to</strong> black <strong>holes</strong><br />
Local definitions of “black <strong>holes</strong>”<br />
A hypersurface H of (M , g) is said <strong>to</strong> be<br />
a future outer <strong>trapping</strong> horizon (FOTH) iff<br />
(i) H foliated by marginally trapped 2-surfaces<br />
(θ (k) < 0 and θ (ℓ) = 0)<br />
(ii) Lk θ (ℓ) < 0 (locally outermost trapped surf.)<br />
[Hayward, PRD 49, 6467 (1994)]<br />
a dynamical horizon (DH) iff<br />
(i) H foliated by marginally trapped 2-surfaces<br />
(ii) H spacelike<br />
[Ashtekar & Krishnan, PRL 89 261101 (2002)]<br />
a non-expanding horizon (NEH) iff<br />
(i) H is null (null normal ℓ)<br />
(ii) θ (ℓ) = 0 [Há´jiček (1973)]<br />
an isolated horizon (IH) iff<br />
(i) H is a non-expanding horizon<br />
(ii) H’s full geometry is not evolving along the<br />
null genera<strong>to</strong>rs: [Lℓ , ˆ ∇] = 0<br />
[Ashtekar, Beetle & Fairhurst, CQG 16, L1 (1999)]<br />
Eric Gourgoulhon (<strong>LUTH</strong>) <strong>Black</strong> <strong>holes</strong>: <strong>trapping</strong> <strong>horizons</strong> LATT, Toulouse, 10 December 2009 14 / 38
Local approaches <strong>to</strong> black <strong>holes</strong><br />
Local definitions of “black <strong>holes</strong>”<br />
A hypersurface H of (M , g) is said <strong>to</strong> be<br />
BH in equilibrium = IH<br />
(e.g. Kerr)<br />
BH out of equilibrium = DH<br />
generic BH = FOTH<br />
a future outer <strong>trapping</strong> horizon (FOTH) iff<br />
(i) H foliated by marginally trapped 2-surfaces<br />
(θ (k) < 0 and θ (ℓ) = 0)<br />
(ii) Lk θ (ℓ) < 0 (locally outermost trapped surf.)<br />
[Hayward, PRD 49, 6467 (1994)]<br />
a dynamical horizon (DH) iff<br />
(i) H foliated by marginally trapped 2-surfaces<br />
(ii) H spacelike<br />
[Ashtekar & Krishnan, PRL 89 261101 (2002)]<br />
a non-expanding horizon (NEH) iff<br />
(i) H is null (null normal ℓ)<br />
(ii) θ (ℓ) = 0 [Há´jiček (1973)]<br />
an isolated horizon (IH) iff<br />
(i) H is a non-expanding horizon<br />
(ii) H’s full geometry is not evolving along the<br />
null genera<strong>to</strong>rs: [Lℓ , ˆ ∇] = 0<br />
[Ashtekar, Beetle & Fairhurst, CQG 16, L1 (1999)]<br />
Eric Gourgoulhon (<strong>LUTH</strong>) <strong>Black</strong> <strong>holes</strong>: <strong>trapping</strong> <strong>horizons</strong> LATT, Toulouse, 10 December 2009 14 / 38
Local approaches <strong>to</strong> black <strong>holes</strong><br />
Example: Vaidya spacetime<br />
[Ashtekar & Krishnan, LRR 7, 10 (2004)]<br />
The <strong>event</strong> horizon crosses the flat region<br />
The dynamical horizon lies entirely outside<br />
the flat region<br />
Eric Gourgoulhon (<strong>LUTH</strong>) <strong>Black</strong> <strong>holes</strong>: <strong>trapping</strong> <strong>horizons</strong> LATT, Toulouse, 10 December 2009 15 / 38
Local approaches <strong>to</strong> black <strong>holes</strong><br />
Dynamics of these new <strong>horizons</strong><br />
The <strong>trapping</strong> <strong>horizons</strong> and dynamical <strong>horizons</strong> have their own dynamics, ruled by<br />
Einstein equations.<br />
In particular, one can establish for them<br />
existence and (partial) uniqueness theorems<br />
[Andersson, Mars & Simon, PRL 95, 111102 (2005)],<br />
[Ashtekar & Galloway, Adv. Theor. Math. Phys. 9, 1 (2005)]<br />
first and second laws of black hole mechanics<br />
[Ashtekar & Krishnan, PRD 68, 104030 (2003)], [Hayward, PRD 70, 104027 (2004)]<br />
a viscous fluid bubble analogy (“membrane paradigm”, as for the <strong>event</strong><br />
horizon)<br />
[EG, PRD 72, 104007 (2005)], [EG & Jaramillo, PRD 74, 087502 (2006)]<br />
Eric Gourgoulhon (<strong>LUTH</strong>) <strong>Black</strong> <strong>holes</strong>: <strong>trapping</strong> <strong>horizons</strong> LATT, Toulouse, 10 December 2009 16 / 38
Local approaches <strong>to</strong> black <strong>holes</strong><br />
Foliation of a hypersurface by spacelike 2-surfaces<br />
hypersurface H = submanifold of<br />
spacetime ⎧(M<br />
, g) of codimension 1<br />
⎨ spacelike<br />
H can be null<br />
⎩<br />
timelike<br />
H = <br />
t∈R<br />
St<br />
St = spacelike 2-surface<br />
Eric Gourgoulhon (<strong>LUTH</strong>) <strong>Black</strong> <strong>holes</strong>: <strong>trapping</strong> <strong>horizons</strong> LATT, Toulouse, 10 December 2009 17 / 38
Local approaches <strong>to</strong> black <strong>holes</strong><br />
Foliation of a hypersurface by spacelike 2-surfaces<br />
hypersurface H = submanifold of<br />
spacetime ⎧(M<br />
, g) of codimension 1<br />
⎨ spacelike<br />
H can be null<br />
⎩<br />
timelike<br />
H = <br />
t∈R<br />
St<br />
St = spacelike 2-surface<br />
⇐= 3+1 perspective<br />
Eric Gourgoulhon (<strong>LUTH</strong>) <strong>Black</strong> <strong>holes</strong>: <strong>trapping</strong> <strong>horizons</strong> LATT, Toulouse, 10 December 2009 17 / 38
Local approaches <strong>to</strong> black <strong>holes</strong><br />
Foliation of a hypersurface by spacelike 2-surfaces<br />
hypersurface H = submanifold of<br />
spacetime ⎧(M<br />
, g) of codimension 1<br />
⎨ spacelike<br />
H can be null<br />
⎩<br />
timelike<br />
H = <br />
t∈R<br />
St<br />
St = spacelike 2-surface<br />
intrinsic viewpoint adopted here (i.e. not<br />
relying on extra-structure such as a 3+1<br />
foliation)<br />
q : induced metric on St (positive<br />
definite)<br />
D : connection associated with q<br />
Eric Gourgoulhon (<strong>LUTH</strong>) <strong>Black</strong> <strong>holes</strong>: <strong>trapping</strong> <strong>horizons</strong> LATT, Toulouse, 10 December 2009 17 / 38
Local approaches <strong>to</strong> black <strong>holes</strong><br />
Evolution vec<strong>to</strong>r on the horizon<br />
Vec<strong>to</strong>r field h on H defined by<br />
(i) h is tangent <strong>to</strong> H<br />
(ii) h is orthogonal <strong>to</strong> St<br />
(iii) Lh t = h µ ∂µt = 〈dt, h〉 = 1<br />
NB: (iii) =⇒ the 2-surfaces St are Lie-dragged<br />
by h<br />
Eric Gourgoulhon (<strong>LUTH</strong>) <strong>Black</strong> <strong>holes</strong>: <strong>trapping</strong> <strong>horizons</strong> LATT, Toulouse, 10 December 2009 18 / 38
Local approaches <strong>to</strong> black <strong>holes</strong><br />
Evolution vec<strong>to</strong>r on the horizon<br />
Vec<strong>to</strong>r field h on H defined by<br />
(i) h is tangent <strong>to</strong> H<br />
(ii) h is orthogonal <strong>to</strong> St<br />
(iii) Lh t = h µ ∂µt = 〈dt, h〉 = 1<br />
NB: (iii) =⇒ the 2-surfaces St are Lie-dragged<br />
by h<br />
Define C := 1<br />
h · h<br />
2<br />
H is spacelike ⇐⇒ C > 0 ⇐⇒ h is spacelike<br />
H is null ⇐⇒ C = 0 ⇐⇒ h is null<br />
H is timelike ⇐⇒ C < 0 ⇐⇒ h is timelike.<br />
Eric Gourgoulhon (<strong>LUTH</strong>) <strong>Black</strong> <strong>holes</strong>: <strong>trapping</strong> <strong>horizons</strong> LATT, Toulouse, 10 December 2009 18 / 38
Local approaches <strong>to</strong> black <strong>holes</strong><br />
Normal null frame associated with the evolution vec<strong>to</strong>r<br />
The foliation (St)t∈R entirely fixes the<br />
ambiguities in the choice of the null normal<br />
frame (ℓ, k), via the evolution vec<strong>to</strong>r h:<br />
there exists a unique normal null frame (ℓ, k)<br />
such that<br />
h = ℓ − Ck and ℓ · k = −1<br />
Normal fundamental form: Ω (ℓ) := −k · ∇q ℓ or Ω (ℓ)<br />
α := −kµ∇νℓ µ q ν α<br />
Evolution of h along itself: ∇hh = κ ℓ + (Cκ − Lh C)k − DC<br />
NB: null limit : C = 0, h = ℓ =⇒ ∇ℓℓ = κ ℓ =⇒ κ = surface gravity<br />
Eric Gourgoulhon (<strong>LUTH</strong>) <strong>Black</strong> <strong>holes</strong>: <strong>trapping</strong> <strong>horizons</strong> LATT, Toulouse, 10 December 2009 19 / 38
Outline<br />
Viscous fluid analogy<br />
1 Concept of black hole and <strong>event</strong> horizon<br />
2 Local approaches <strong>to</strong> black <strong>holes</strong><br />
3 Viscous fluid analogy<br />
4 Angular momentum and area evolution laws<br />
5 Applications <strong>to</strong> numerical relativity<br />
Eric Gourgoulhon (<strong>LUTH</strong>) <strong>Black</strong> <strong>holes</strong>: <strong>trapping</strong> <strong>horizons</strong> LATT, Toulouse, 10 December 2009 20 / 38
Viscous fluid analogy<br />
Concept of black hole viscosity<br />
Hartle and Hawking (1972, 1973): introduced the concept of black hole<br />
viscosity when studying the response of the <strong>event</strong> horizon <strong>to</strong> external<br />
perturbations<br />
Damour (1979): 2-dimensional Navier-S<strong>to</strong>kes like equation for the <strong>event</strong><br />
horizon =⇒ shear viscosity and bulk viscosity<br />
Thorne and Price (1986): membrane paradigm for black <strong>holes</strong><br />
Eric Gourgoulhon (<strong>LUTH</strong>) <strong>Black</strong> <strong>holes</strong>: <strong>trapping</strong> <strong>horizons</strong> LATT, Toulouse, 10 December 2009 21 / 38
Viscous fluid analogy<br />
Concept of black hole viscosity<br />
Hartle and Hawking (1972, 1973): introduced the concept of black hole<br />
viscosity when studying the response of the <strong>event</strong> horizon <strong>to</strong> external<br />
perturbations<br />
Damour (1979): 2-dimensional Navier-S<strong>to</strong>kes like equation for the <strong>event</strong><br />
horizon =⇒ shear viscosity and bulk viscosity<br />
Thorne and Price (1986): membrane paradigm for black <strong>holes</strong><br />
Shall we restrict the analysis <strong>to</strong> the <strong>event</strong> horizon ?<br />
Can we extend the concept of viscosity <strong>to</strong> the local characterizations of black hole<br />
recently introduced, i.e. future outer <strong>trapping</strong> <strong>horizons</strong> and dynamical <strong>horizons</strong> ?<br />
NB: <strong>event</strong> horizon = null hypersurface<br />
future outer <strong>trapping</strong> horizon = null or spacelike hypersurface<br />
dynamical horizon = spacelike hypersurface<br />
Eric Gourgoulhon (<strong>LUTH</strong>) <strong>Black</strong> <strong>holes</strong>: <strong>trapping</strong> <strong>horizons</strong> LATT, Toulouse, 10 December 2009 21 / 38
Viscous fluid analogy<br />
Navier-S<strong>to</strong>kes equation in New<strong>to</strong>nian fluid dynamics<br />
i ∂v<br />
ρ<br />
∂t + vj∇jv i<br />
<br />
= −∇ i P + µ∆v i <br />
+<br />
or, in terms of fluid momentum density πi := ρvi,<br />
ζ + µ<br />
3<br />
<br />
∇ i (∇jv j ) + f i<br />
∂πi<br />
∂t + vj ∇jπi + θπi = −∇iP + 2µ∇ j σij + ζ∇iθ + fi<br />
where θ is the fluid expansion:<br />
θ := ∇jv j<br />
and σij the velocity shear tensor:<br />
σij := 1<br />
2 (∇ivj + ∇jvi) − 1<br />
3<br />
θ δij<br />
P is the pressure, µ the shear viscosity, ζ the bulk viscosity and fi the density of<br />
external forces<br />
Eric Gourgoulhon (<strong>LUTH</strong>) <strong>Black</strong> <strong>holes</strong>: <strong>trapping</strong> <strong>horizons</strong> LATT, Toulouse, 10 December 2009 22 / 38
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 />
Eric Gourgoulhon (<strong>LUTH</strong>) <strong>Black</strong> <strong>holes</strong>: <strong>trapping</strong> <strong>horizons</strong> LATT, Toulouse, 10 December 2009 23 / 38
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> LATT, Toulouse, 10 December 2009 23 / 38
Viscous fluid analogy<br />
Original Damour-Navier-S<strong>to</strong>kes equation (con’t)<br />
Introducing a coordinate system (t, x 1 , x 2 , x 3 ) such that<br />
then<br />
t is compatible with ℓ: Lℓ t = 1<br />
H is defined by x 1 = const, so that x a = (x 2 , x 3 ) are coordinates spanning St<br />
ℓ = ∂<br />
+ V<br />
∂t<br />
with V tangent <strong>to</strong> St: velocity of H’s null genera<strong>to</strong>rs with respect <strong>to</strong> the<br />
coordinates x a [Damour, PRD 18, 3598 (1978)].<br />
Then<br />
θ (ℓ) = DaV a + ∂<br />
∂t ln √ q q := det qab<br />
σ (ℓ)<br />
ab<br />
1<br />
=<br />
2 (DaVb + DbVa) − 1<br />
2 θ(ℓ) qab + 1 ∂qab<br />
2 ∂t<br />
compare<br />
Eric Gourgoulhon (<strong>LUTH</strong>) <strong>Black</strong> <strong>holes</strong>: <strong>trapping</strong> <strong>horizons</strong> LATT, Toulouse, 10 December 2009 24 / 38
Viscous fluid analogy<br />
Negative bulk viscosity of <strong>event</strong> <strong>horizons</strong><br />
From the Damour-Navier-S<strong>to</strong>kes equation, ζ = − 1<br />
< 0<br />
16π<br />
This negative value would yield <strong>to</strong> a dilation or contraction instability in an<br />
ordinary fluid<br />
It is in agreement with the tendency of a null hypersurface <strong>to</strong> continually contract<br />
or expand<br />
The <strong>event</strong> horizon is stabilized by the teleological condition imposing its expansion<br />
<strong>to</strong> vanish in the far future (equilibrium state reached)<br />
Eric Gourgoulhon (<strong>LUTH</strong>) <strong>Black</strong> <strong>holes</strong>: <strong>trapping</strong> <strong>horizons</strong> LATT, Toulouse, 10 December 2009 25 / 38
Viscous fluid analogy<br />
Generalization <strong>to</strong> the non-null case<br />
Starting remark: in the null case (<strong>event</strong> horizon), ℓ plays two different roles:<br />
evolution vec<strong>to</strong>r along H (e.g. term S Lℓ )<br />
normal <strong>to</strong> H (e.g. term q ∗ T · ℓ)<br />
When H is no longer null, these two roles have <strong>to</strong> be taken by two different<br />
vec<strong>to</strong>rs:<br />
evolution vec<strong>to</strong>r: obviously h<br />
vec<strong>to</strong>r normal <strong>to</strong> H: a natural choice is m := ℓ + Ck<br />
Eric Gourgoulhon (<strong>LUTH</strong>) <strong>Black</strong> <strong>holes</strong>: <strong>trapping</strong> <strong>horizons</strong> LATT, Toulouse, 10 December 2009 26 / 38
Viscous fluid analogy<br />
Generalized Damour-Navier-S<strong>to</strong>kes equation<br />
From the contracted Ricci identity applied <strong>to</strong> the vec<strong>to</strong>r m and projected on<strong>to</strong> St:<br />
(∇µ∇νm µ − ∇ν∇µm µ ) q ν α = Rµνm µ q ν α and using Einstein equation <strong>to</strong> express<br />
Rµν, one gets an evolution equation for Ω (ℓ) along H:<br />
S Lh Ω (ℓ) + θ (h) Ω (ℓ) = Dκ − D · σ (m) + 1<br />
2 Dθ(m) − θ (k) DC + 8πq ∗ T · m<br />
Ω (ℓ) : normal fundamental form of St associated with null normal ℓ<br />
θ (h) , θ (m) and θ (k) : expansion scalars of St along the vec<strong>to</strong>rs h, m and k<br />
respectively<br />
D : covariant derivative within (St, q)<br />
κ : component of ∇hh along ℓ<br />
σ (m) : shear tensor of St along the vec<strong>to</strong>r m<br />
C : half the scalar square of h<br />
Eric Gourgoulhon (<strong>LUTH</strong>) <strong>Black</strong> <strong>holes</strong>: <strong>trapping</strong> <strong>horizons</strong> LATT, Toulouse, 10 December 2009 27 / 38
Viscous fluid analogy<br />
Null limit (<strong>event</strong> horizon)<br />
If H is a null hypersurface,<br />
h = m = ℓ and C = 0<br />
and we recover the original Damour-Navier-S<strong>to</strong>kes equation:<br />
S Lℓ Ω (ℓ) + θ (ℓ) Ω (ℓ) = Dκ − D · σ (ℓ) + 1<br />
2 Dθ(ℓ) + 8πq ∗ T · ℓ<br />
Eric Gourgoulhon (<strong>LUTH</strong>) <strong>Black</strong> <strong>holes</strong>: <strong>trapping</strong> <strong>horizons</strong> LATT, Toulouse, 10 December 2009 28 / 38
Viscous fluid analogy<br />
Case of future <strong>trapping</strong> <strong>horizons</strong><br />
Definition [Hayward, PRD 49, 6467 (1994)] :<br />
H is a future <strong>trapping</strong> horizon iff θ (ℓ) = 0 and θ (k) < 0.<br />
The generalized Damour-Navier-S<strong>to</strong>kes equation reduces then <strong>to</strong><br />
S Lh Ω (ℓ) + θ (h) Ω (ℓ) = Dκ − D · σ (m) − 1<br />
2 Dθ(h) − θ (k) DC + 8πq ∗ T · m<br />
[EG, PRD 72, 104007 (2005)]<br />
NB: Notice the change of sign in the − 1<br />
2 Dθ(h) term with respect <strong>to</strong> the original<br />
Damour-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> LATT, Toulouse, 10 December 2009 29 / 38
Viscous fluid analogy<br />
Case of future <strong>trapping</strong> <strong>horizons</strong><br />
Definition [Hayward, PRD 49, 6467 (1994)] :<br />
H is a future <strong>trapping</strong> horizon iff θ (ℓ) = 0 and θ (k) < 0.<br />
The generalized Damour-Navier-S<strong>to</strong>kes equation reduces then <strong>to</strong><br />
S Lh Ω (ℓ) + θ (h) Ω (ℓ) = Dκ − D · σ (m) − 1<br />
2 Dθ(h) − θ (k) DC + 8πq ∗ T · m<br />
[EG, PRD 72, 104007 (2005)]<br />
NB: Notice the change of sign in the − 1<br />
2 Dθ(h) term with respect <strong>to</strong> the original<br />
Damour-Navier-S<strong>to</strong>kes equation<br />
The explanation: it is θ (m) which appears in the general equation and<br />
<br />
<strong>event</strong> horizon (m = h) : (m) (ℓ) θ = θ<br />
θ (m) + θ (h) = 2θ (ℓ) =⇒<br />
<strong>trapping</strong> horizon (θ (ℓ) = 0) : θ (m) = −θ (h)<br />
Eric Gourgoulhon (<strong>LUTH</strong>) <strong>Black</strong> <strong>holes</strong>: <strong>trapping</strong> <strong>horizons</strong> LATT, Toulouse, 10 December 2009 29 / 38
Viscous fluid form<br />
Viscous fluid analogy<br />
S<br />
Lh π + θ (h) π = −DP + 1<br />
8π D · σ(m) + ζDθ (h) + f<br />
with π := − 1<br />
8π Ω(ℓ) momentum surface density<br />
P := κ<br />
8π pressure<br />
1<br />
8π σ(m) shear stress tensor<br />
ζ := 1<br />
bulk viscosity<br />
16π<br />
f := −q ∗ T · m + θ(k)<br />
DC external force surface density<br />
8π<br />
Similar <strong>to</strong> the Damour-Navier-S<strong>to</strong>kes equation for an <strong>event</strong> horizon, except<br />
the New<strong>to</strong>nian-fluid relation between stress and strain does not hold:<br />
σ (m) /8π = 2µσ (h) , rather σ (m) /8π = [σ (h) + 2Cσ (k) ]/8π<br />
Eric Gourgoulhon (<strong>LUTH</strong>) <strong>Black</strong> <strong>holes</strong>: <strong>trapping</strong> <strong>horizons</strong> LATT, Toulouse, 10 December 2009 30 / 38
Viscous fluid form<br />
Viscous fluid analogy<br />
S<br />
Lh π + θ (h) π = −DP + 1<br />
8π D · σ(m) + ζDθ (h) + f<br />
with π := − 1<br />
8π Ω(ℓ) momentum surface density<br />
P := κ<br />
8π pressure<br />
1<br />
8π σ(m) shear stress tensor<br />
ζ := 1<br />
bulk viscosity<br />
16π<br />
f := −q ∗ T · m + θ(k)<br />
DC external force surface density<br />
8π<br />
Similar <strong>to</strong> the Damour-Navier-S<strong>to</strong>kes equation for an <strong>event</strong> horizon, except<br />
the New<strong>to</strong>nian-fluid relation between stress and strain does not hold:<br />
σ (m) /8π = 2µσ (h) , rather σ (m) /8π = [σ (h) + 2Cσ (k) ]/8π<br />
positive bulk viscosity<br />
This positive value of bulk viscosity shows that FOTHs and DHs behave as<br />
“ordinary” physical objects, in perfect agreement with their local nature<br />
Eric Gourgoulhon (<strong>LUTH</strong>) <strong>Black</strong> <strong>holes</strong>: <strong>trapping</strong> <strong>horizons</strong> LATT, Toulouse, 10 December 2009 30 / 38
Outline<br />
Angular momentum and area evolution laws<br />
1 Concept of black hole and <strong>event</strong> horizon<br />
2 Local approaches <strong>to</strong> black <strong>holes</strong><br />
3 Viscous fluid analogy<br />
4 Angular momentum and area evolution laws<br />
5 Applications <strong>to</strong> numerical relativity<br />
Eric Gourgoulhon (<strong>LUTH</strong>) <strong>Black</strong> <strong>holes</strong>: <strong>trapping</strong> <strong>horizons</strong> LATT, Toulouse, 10 December 2009 31 / 38
Angular momentum and area evolution laws<br />
Angular momentum of <strong>trapping</strong> <strong>horizons</strong><br />
Definition [Booth & Fairhurst, CQG 22, 4545 (2005)]: Let ϕ be a vec<strong>to</strong>r field on H which<br />
is tangent <strong>to</strong> St<br />
has closed orbits<br />
has vanishing divergence with respect <strong>to</strong> the induced metric: D · ϕ = 0<br />
(weaker than being a Killing vec<strong>to</strong>r of (St, q) !)<br />
For dynamical <strong>horizons</strong>, θ (h) = 0 and there is a unique choice of ϕ as the<br />
genera<strong>to</strong>r (conveniently normalized) of the curves of constant θ (h)<br />
[Hayward, PRD 74, 104013 (2006)]<br />
The generalized angular momentum associated with ϕ is then defined by<br />
J(ϕ) := − 1<br />
<br />
〈Ω<br />
8π<br />
(ℓ) , ϕ〉 S ɛ,<br />
Remark 1: does not depend upon the choice of null vec<strong>to</strong>r ℓ, thanks <strong>to</strong> the<br />
divergence-free property of ϕ<br />
Remark 2:<br />
coincides with Ashtekar & Krishnan’s definition for a dynamical horizon<br />
coincides with Brown-York angular momentum if H is timelike and ϕ a<br />
Killing vec<strong>to</strong>r<br />
St<br />
Eric Gourgoulhon (<strong>LUTH</strong>) <strong>Black</strong> <strong>holes</strong>: <strong>trapping</strong> <strong>horizons</strong> LATT, Toulouse, 10 December 2009 32 / 38
Angular momentum and area evolution laws<br />
Angular momentum flux law<br />
Under the supplementary hypothesis that ϕ is transported along the evolution<br />
vec<strong>to</strong>r h : Lh ϕ = 0, the generalized Damour-Navier-S<strong>to</strong>kes equation leads <strong>to</strong><br />
<br />
d<br />
J(ϕ) = − T (m, ϕ)<br />
dt S ɛ − 1<br />
<br />
σ<br />
16π<br />
(m) : Lϕ q − 2θ (k) <br />
ϕ · DC<br />
Sɛ<br />
St<br />
St<br />
[EG, PRD 72, 104007 (2005)]<br />
Eric Gourgoulhon (<strong>LUTH</strong>) <strong>Black</strong> <strong>holes</strong>: <strong>trapping</strong> <strong>horizons</strong> LATT, Toulouse, 10 December 2009 33 / 38
Angular momentum and area evolution laws<br />
Angular momentum flux law<br />
Under the supplementary hypothesis that ϕ is transported along the evolution<br />
vec<strong>to</strong>r h : Lh ϕ = 0, the generalized Damour-Navier-S<strong>to</strong>kes equation leads <strong>to</strong><br />
<br />
d<br />
J(ϕ) = − T (m, ϕ)<br />
dt S ɛ − 1<br />
<br />
σ<br />
16π<br />
(m) : Lϕ q − 2θ (k) <br />
ϕ · DC<br />
Sɛ<br />
St<br />
Two interesting limiting cases:<br />
St<br />
[EG, PRD 72, 104007 (2005)]<br />
Eric Gourgoulhon (<strong>LUTH</strong>) <strong>Black</strong> <strong>holes</strong>: <strong>trapping</strong> <strong>horizons</strong> LATT, Toulouse, 10 December 2009 33 / 38
Angular momentum and area evolution laws<br />
Angular momentum flux law<br />
Under the supplementary hypothesis that ϕ is transported along the evolution<br />
vec<strong>to</strong>r h : Lh ϕ = 0, the generalized Damour-Navier-S<strong>to</strong>kes equation leads <strong>to</strong><br />
<br />
d<br />
J(ϕ) = − T (m, ϕ)<br />
dt S ɛ − 1<br />
<br />
σ<br />
16π<br />
(m) : Lϕ q − 2θ (k) <br />
ϕ · DC<br />
Sɛ<br />
St<br />
Two interesting limiting cases:<br />
H = null hypersurface : C = 0 and m = ℓ :<br />
<br />
d<br />
J(ϕ) = − T (ℓ, ϕ)<br />
dt S ɛ − 1<br />
<br />
16π<br />
St<br />
St<br />
St<br />
[EG, PRD 72, 104007 (2005)]<br />
σ (ℓ) : Lϕ q S ɛ<br />
i.e. Eq. (6.134) of the Membrane Paradigm book (Thorne, Price &<br />
MacDonald 1986)<br />
Eric Gourgoulhon (<strong>LUTH</strong>) <strong>Black</strong> <strong>holes</strong>: <strong>trapping</strong> <strong>horizons</strong> LATT, Toulouse, 10 December 2009 33 / 38
Angular momentum and area evolution laws<br />
Angular momentum flux law<br />
Under the supplementary hypothesis that ϕ is transported along the evolution<br />
vec<strong>to</strong>r h : Lh ϕ = 0, the generalized Damour-Navier-S<strong>to</strong>kes equation leads <strong>to</strong><br />
<br />
d<br />
J(ϕ) = − T (m, ϕ)<br />
dt S ɛ − 1<br />
<br />
σ<br />
16π<br />
(m) : Lϕ q − 2θ (k) <br />
ϕ · DC<br />
Sɛ<br />
St<br />
Two interesting limiting cases:<br />
H = null hypersurface : C = 0 and m = ℓ :<br />
<br />
d<br />
J(ϕ) = − T (ℓ, ϕ)<br />
dt S ɛ − 1<br />
<br />
16π<br />
St<br />
St<br />
St<br />
[EG, PRD 72, 104007 (2005)]<br />
σ (ℓ) : Lϕ q S ɛ<br />
i.e. Eq. (6.134) of the Membrane Paradigm book (Thorne, Price &<br />
MacDonald 1986)<br />
H = future <strong>trapping</strong> horizon :<br />
d<br />
J(ϕ) = −<br />
dt<br />
<br />
St<br />
T (m, ϕ) S ɛ − 1<br />
16π<br />
<br />
St<br />
σ (m) : Lϕ q S ɛ<br />
Eric Gourgoulhon (<strong>LUTH</strong>) <strong>Black</strong> <strong>holes</strong>: <strong>trapping</strong> <strong>horizons</strong> LATT, Toulouse, 10 December 2009 33 / 38
Angular momentum and area evolution laws<br />
Area evolution law for an <strong>event</strong> horizon<br />
A(t) : area of the 2-surface St; S ɛ : volume element of St; ¯κ(t) := 1<br />
<br />
κ<br />
A(t) St<br />
S ɛ<br />
Integrating the null Raychaudhuri equation on St, one gets<br />
d2 <br />
A<br />
− ¯κdA = −<br />
dt2 dt<br />
<br />
8πT (ℓ, ℓ) + σ (ℓ) :σ (ℓ) − (θ(ℓ) ) 2<br />
2 + (¯κ − κ)θ(ℓ) S<br />
ɛ (1)<br />
St<br />
[Damour, 1979]<br />
Eric Gourgoulhon (<strong>LUTH</strong>) <strong>Black</strong> <strong>holes</strong>: <strong>trapping</strong> <strong>horizons</strong> LATT, Toulouse, 10 December 2009 34 / 38
Angular momentum and area evolution laws<br />
Area evolution law for an <strong>event</strong> horizon<br />
A(t) : area of the 2-surface St; S ɛ : volume element of St; ¯κ(t) := 1<br />
<br />
κ<br />
A(t) St<br />
S ɛ<br />
Integrating the null Raychaudhuri equation on St, one gets<br />
d2 <br />
A<br />
− ¯κdA = −<br />
dt2 dt<br />
<br />
8πT (ℓ, ℓ) + σ (ℓ) :σ (ℓ) − (θ(ℓ) ) 2<br />
2 + (¯κ − κ)θ(ℓ) S<br />
ɛ (1)<br />
St<br />
Simplified analysis : assume ¯κ = const > 0 :<br />
Cauchy problem =⇒ diverging solution of the homogeneous equation:<br />
dA<br />
= α exp(¯κt) !<br />
dt<br />
[Damour, 1979]<br />
Eric Gourgoulhon (<strong>LUTH</strong>) <strong>Black</strong> <strong>holes</strong>: <strong>trapping</strong> <strong>horizons</strong> LATT, Toulouse, 10 December 2009 34 / 38
Angular momentum and area evolution laws<br />
Area evolution law for an <strong>event</strong> horizon<br />
A(t) : area of the 2-surface St; S ɛ : volume element of St; ¯κ(t) := 1<br />
<br />
κ<br />
A(t) St<br />
S ɛ<br />
Integrating the null Raychaudhuri equation on St, one gets<br />
d2 <br />
A<br />
− ¯κdA = −<br />
dt2 dt<br />
<br />
8πT (ℓ, ℓ) + σ (ℓ) :σ (ℓ) − (θ(ℓ) ) 2<br />
2 + (¯κ − κ)θ(ℓ) S<br />
ɛ (1)<br />
St<br />
Simplified analysis : assume ¯κ = const > 0 :<br />
Cauchy problem =⇒ diverging solution of the homogeneous equation:<br />
dA<br />
= α exp(¯κt) !<br />
dt<br />
correct treatment: impose dA<br />
= 0 at t = +∞ (teleological !)<br />
dt<br />
dA<br />
dt =<br />
+∞<br />
D(u)e ¯κ(t−u) du D(t) : r.h.s. of Eq. (1)<br />
t<br />
Non causal evolution<br />
[Damour, 1979]<br />
Eric Gourgoulhon (<strong>LUTH</strong>) <strong>Black</strong> <strong>holes</strong>: <strong>trapping</strong> <strong>horizons</strong> LATT, Toulouse, 10 December 2009 34 / 38
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 />
Eric Gourgoulhon (<strong>LUTH</strong>) <strong>Black</strong> <strong>holes</strong>: <strong>trapping</strong> <strong>horizons</strong> LATT, Toulouse, 10 December 2009 35 / 38
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> LATT, Toulouse, 10 December 2009 35 / 38
Outline<br />
Applications <strong>to</strong> numerical relativity<br />
1 Concept of black hole and <strong>event</strong> horizon<br />
2 Local approaches <strong>to</strong> black <strong>holes</strong><br />
3 Viscous fluid analogy<br />
4 Angular momentum and area evolution laws<br />
5 Applications <strong>to</strong> numerical relativity<br />
Eric Gourgoulhon (<strong>LUTH</strong>) <strong>Black</strong> <strong>holes</strong>: <strong>trapping</strong> <strong>horizons</strong> LATT, Toulouse, 10 December 2009 36 / 38
Applications <strong>to</strong> numerical relativity<br />
Applications <strong>to</strong> numerical relativity<br />
Initial data: isolated <strong>horizons</strong> (helical symmetry)<br />
[EG, Grandclément & Bonazzola, PRD 65, 044020 (2002)]<br />
[Grandclément, EG & Bonazzola, PRD 65, 044021 (2002)]<br />
[Cook & Pfeiffer, PRD 70, 104016 (2004)]<br />
A posteriori analysis: estimating mass, linear and angular momentum of<br />
formed black <strong>holes</strong><br />
[Schnetter, Krishnan & Beyer, PRD 74, 024028 (2006)]<br />
[Cook & Whiting, PRD 76, 041501 (2007)]<br />
[Krishnan, Lous<strong>to</strong> & Zlochower, PRD 76, 081501(R) (2007)]<br />
Numerical construction of spacetime: inner boundary conditions for a<br />
constrained scheme with “black hole excision”<br />
[Jaramillo, EG, Cordero-Carrión, & J.M. Ibáñez, PRD 77, 047501 (2008)]<br />
[Vasset, Novak & Jaramillo, PRD 79, 124010 (2009)]<br />
Eric Gourgoulhon (<strong>LUTH</strong>) <strong>Black</strong> <strong>holes</strong>: <strong>trapping</strong> <strong>horizons</strong> LATT, Toulouse, 10 December 2009 37 / 38
Review articles<br />
References<br />
A. Ashtekar and B. Krishnan : Isolated and dynamical <strong>horizons</strong> and their<br />
applications, Living Rev. Relativity 7, 10 (2004) ;<br />
http://www.livingreviews.org/lrr-2004-10<br />
I. Booth : <strong>Black</strong> hole boundaries, Canadian J. Phys. 83, 1073 (2005) ;<br />
http://arxiv.org/abs/gr-qc/0508107<br />
E. Gourgoulhon and J.L. Jaramillo : A 3+1 perspective on null hypersurfaces<br />
and isolated <strong>horizons</strong>, Phys. Rep. 423, 159 (2006) ;<br />
http://arxiv.org/abs/gr-qc/0503113<br />
E. Gourgoulhon and J. L. Jaramillo : New theoretical approaches <strong>to</strong> black<br />
<strong>holes</strong>, New Astron. Rev. 51, 791 (2008) ;<br />
http://arxiv.org/abs/0803.2944<br />
S.A. Hayward : Dynamics of black <strong>holes</strong>, Adv. Sci. Lett. 2, 205 (2009);<br />
http://arxiv.org/abs/0810.0923<br />
B. Krishnan : Fundamental properties and applications of quasi-local black<br />
hole <strong>horizons</strong>, Class. Quantum Grav. 25, 114005 (2008);<br />
http://arxiv.org/abs/0712.1575<br />
Eric Gourgoulhon (<strong>LUTH</strong>) <strong>Black</strong> <strong>holes</strong>: <strong>trapping</strong> <strong>horizons</strong> LATT, Toulouse, 10 December 2009 38 / 38