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 ...
Local approaches to black holes What is a trapped surface ? 1/ Expansion of a surface along a normal vector field 1 Consider a spacelike 2-surface S (induced metric: q) 2 Take a vector field v defined on S and normal to S at each point 3 ε being a small parameter, displace the point p by the vector εv to the point p ′ 4 Do the same for each point in S, keeping the value of ε fixed 5 This defines a new surface S ′ (Lie dragging) Eric Gourgoulhon (LUTH) Black holes: trapping horizons CERN, 17 March 2010 10 / 38
Local approaches to black holes What is a trapped surface ? 1/ Expansion of a surface along a normal vector field 1 Consider a spacelike 2-surface S (induced metric: q) 2 Take a vector field v defined on S and normal to S at each point 3 ε being a small parameter, displace the point p by the vector εv to the point p ′ 4 Do the same for each point in S, keeping the value of ε fixed 5 This defines a new surface S ′ (Lie dragging) At each point, the expansion of S along v is defined from the relative change in the area element δA: θ (v) := lim ε→0 1 ε δA ′ − δA δA = Lv ln √ q = q µν ∇µvν Eric Gourgoulhon (LUTH) Black holes: trapping horizons CERN, 17 March 2010 10 / 38
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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> CERN, 17 March 2010 10 / 38
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