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The Location of Trapped Surfaces

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The Location of Trapped Surfaces The location of trapped surfaces José M M Senovilla Department of Theoretical Physics and History of Science University of the Basque Country, Bilbao, Spain Mathematical Relativity in Lisbon, Lisboa, 18th June 2009 Outline 1 Trapped surfaces. Formal definitions 2 Some simple questions on black holes 3 The future-trapped region T and its boundary B 4 B = AH 6 5 Closed trapped surfaces are clairvoyant ! 6 Fundamental general results 7 Application to Robertson-Walker spacetimes 8 Application to Vaidya: the hypersurface Σ 9 Where is B in Vaidya? 10 Conclusions and outlook References I Bengtsson and JMMS, Note on trapped surfaces in the Vaidya solution, Phys. Rev. D 79 024027 (2009) [arXiv:0809.2213] I Bengtsson and JMMS, The boundary of the region with trapped surfaces in spherical symmetry, Preprint JMMS, On the boundary of the region containing trapped surfaces, AIP Conf. Proc. 1122 72 (2009) [arXiv:0812.2767] Φ: S xµ = Φµ(λA): −! V Thus, the tangent vectors (seen on ) are: V µ µ @Φ ~eA Φ0(@ A ) e = ≡ λ () A @λA First fundamental form: µ ν γAB(λ) = g S(~eA;~eB) = gµν(Φ)e e j A B is positive-definite. So, S is assumed to be SPACELIKE. Then, x S 8 2 Tx = TxS TxS? V ⊕ called tangent and normal parts. The trapped surface fauna: Definitions Let ( ; g) be a 4-dimensional Lorentzian manifold with metric V tensor g of signature ( ; +; +; +). − First fundamental form: µ ν γAB(λ) = g S(~eA;~eB) = gµν(Φ)e e j A B is positive-definite. So, S is assumed to be SPACELIKE. Then, x S 8 2 Tx = TxS TxS? V ⊕ called tangent and normal parts. The trapped surface fauna: Definitions Let ( ; g) be a 4-dimensional Lorentzian manifold with metric V tensor g of signature ( ; +; +; +). − Φ: S xµ = Φµ(λA): −! V Thus, the tangent vectors (seen on ) are: V µ µ @Φ ~eA Φ0(@ A ) e = ≡ λ () A @λA Then, x S 8 2 Tx = TxS TxS? V ⊕ called tangent and normal parts. The trapped surface fauna: Definitions Let ( ; g) be a 4-dimensional Lorentzian manifold with metric V tensor g of signature ( ; +; +; +). − Φ: S xµ = Φµ(λA): −! V Thus, the tangent vectors (seen on ) are: V µ µ @Φ ~eA Φ0(@ A ) e = ≡ λ () A @λA First fundamental form: µ ν γAB(λ) = g S(~eA;~eB) = gµν(Φ)e e j A B is positive-definite. So, S is assumed to be SPACELIKE. The trapped surface fauna: Definitions Let ( ; g) be a 4-dimensional Lorentzian manifold with metric V tensor g of signature ( ; +; +; +). − Φ: S xµ = Φµ(λA): −! V Thus, the tangent vectors (seen on ) are: V µ µ @Φ ~eA Φ0(@ A ) e = ≡ λ () A @λA First fundamental form: µ ν γAB(λ) = g S(~eA;~eB) = gµν(Φ)e e j A B is positive-definite. So, S is assumed to be SPACELIKE. Then, x S 8 2 Tx = TxS TxS? V ⊕ called tangent and normal parts. K~ is called the shape tensor or second fundamental form vector of S in . V OBS: K~ is orthogonal to S. K~ : X(S) X(S) X(S) × −! ? where X(S) (X(S)?) is the set of vector fields tangent (perpendicular) to S. The second fundamental form of S in ( ; g) V relative to any ~n X(S) is: 2 ? µ KAB(~n) nµK : ≡ AB These are 2-covariant symmetric tensor fields on S. Notation: extrinsic curvature C In particular, we have ~e ~eB = Γ ~eC K~ AB r A AB − OBS: K~ is orthogonal to S. K~ : X(S) X(S) X(S) × −! ? where X(S) (X(S)?) is the set of vector fields tangent (perpendicular) to S. The second fundamental form of S in ( ; g) V relative to any ~n X(S) is: 2 ? µ KAB(~n) nµK : ≡ AB These are 2-covariant symmetric tensor fields on S. Notation: extrinsic curvature C In particular, we have ~e ~eB = Γ ~eC K~ AB r A AB − K~ is called the shape tensor or second fundamental form vector of S in . V The second fundamental form of S in ( ; g) V relative to any ~n X(S) is: 2 ? µ KAB(~n) nµK : ≡ AB These are 2-covariant symmetric tensor fields on S. Notation: extrinsic curvature C In particular, we have ~e ~eB = Γ ~eC K~ AB r A AB − K~ is called the shape tensor or second fundamental form vector of S in . V OBS: K~ is orthogonal to S. K~ : X(S) X(S) X(S) × −! ? where X(S) (X(S)?) is the set of vector fields tangent (perpendicular) to S. Notation: extrinsic curvature C In particular, we have ~e ~eB = Γ ~eC K~ AB r A AB − K~ is called the shape tensor or second fundamental form vector of S in . V OBS: K~ is orthogonal to S. K~ : X(S) X(S) X(S) × −! ? where X(S) (X(S)?) is the set of vector fields tangent (perpendicular) to S. The second fundamental form of S in ( ; g) V relative to any ~n X(S) is: 2 ? µ KAB(~n) nµK : ≡ AB These are 2-covariant symmetric tensor fields on S. + + K~ AB = KAB(~k−) ~k KAB(~k ) ~k− − − The mean curvature vector: AB + + X(S)? H~ γ K~ AB = θ−~k θ ~k− 3 ≡ − − AB θ γ KAB(~k ) are called the future null expansions ± ≡ ± Mean curvature vector. Null expansions For a spacelike surface there are two independent normal vector fields, we can choose them to be future-pointing and null, µ ~k± X(S)?; adding k+µk = 1, there remains the freedom 2 − − + + 2 2 ~k ~k0 = σ ~k; ~k− ~k0− = σ− ~k− −! −! The mean curvature vector: AB + + X(S)? H~ γ K~ AB = θ−~k θ ~k− 3 ≡ − − AB θ γ KAB(~k ) are called the future null expansions ± ≡ ± Mean curvature vector. Null expansions For a spacelike surface there are two independent normal vector fields, we can choose them to be future-pointing and null, µ ~k± X(S)?; adding k+µk = 1, there remains the freedom 2 − − + + 2 2 ~k ~k0 = σ ~k; ~k− ~k0− = σ− ~k− −! −! + + K~ AB = KAB(~k−) ~k KAB(~k ) ~k− − − AB θ γ KAB(~k ) are called the future null expansions ± ≡ ± Mean curvature vector. Null expansions For a spacelike surface there are two independent normal vector fields, we can choose them to be future-pointing and null, µ ~k± X(S)?; adding k+µk = 1, there remains the freedom 2 − − + + 2 2 ~k ~k0 = σ ~k; ~k− ~k0− = σ− ~k− −! −! + + K~ AB = KAB(~k−) ~k KAB(~k ) ~k− − − The mean curvature vector: AB + + X(S)? H~ γ K~ AB = θ−~k θ ~k− 3 ≡ − − Mean curvature vector. Null expansions For a spacelike surface there are two independent normal vector fields, we can choose them to be future-pointing and null, µ ~k± X(S)?; adding k+µk = 1, there remains the freedom 2 − − + + 2 2 ~k ~k0 = σ ~k; ~k− ~k0− = σ− ~k− −! −! + + K~ AB = KAB(~k−) ~k KAB(~k ) ~k− − − The mean curvature vector: AB + + X(S)? H~ γ K~ AB = θ−~k θ ~k− 3 ≡ − − AB θ γ KAB(~k ) are called the future null expansions ± ≡ ± Future-trapped surfaces: H~ is future on S The main cases are: H~ Expansions Type of surface + zero θ = θ− = 0 stationary or minimal + null and future θ = 0; θ− < 0 marginally f-trapped + null and future θ < 0; θ− = 0 marginally f-trapped + timelike future θ < 0; θ− < 0 f-trapped Schwarzchild (in units with G = c = 1) 2M ds2 = 1 dv2 + 2dvdr + r2dΩ2 − − r M is a constant: the total mass. Here v is advanced (null) time. It is easily checked that the round spheres defined by constant values of v and r are trapped if and only if r < 2M. And they are "traditional" if r > 2M. The Event Horizon EH: r 2M = 0 − Example: the Schwarzschild solution in Eddington-Finkelstein advanced coordinates It is easily checked that the round spheres defined by constant values of v and r are trapped if and only if r < 2M. And they are "traditional" if r > 2M. The Event Horizon EH: r 2M = 0 − Example: the Schwarzschild solution in Eddington-Finkelstein advanced coordinates Schwarzchild (in units with G = c = 1) 2M ds2 = 1 dv2 + 2dvdr + r2dΩ2 − − r M is a constant: the total mass. Here v is advanced (null) time. The Event Horizon EH: r 2M = 0 − Example: the Schwarzschild solution in Eddington-Finkelstein advanced coordinates Schwarzchild (in units with G = c = 1) 2M ds2 = 1 dv2 + 2dvdr + r2dΩ2 − − r M is a constant: the total mass. Here v is advanced (null) time. It is easily checked that the round spheres defined by constant values of v and r are trapped if and only if r < 2M. And they are "traditional" if r > 2M. Example: the Schwarzschild solution in Eddington-Finkelstein advanced coordinates Schwarzchild (in units with G = c = 1) 2M ds2 = 1 dv2 + 2dvdr + r2dΩ2 − − r M is a constant: the total mass. Here v is advanced (null) time. It is easily checked that the round spheres defined by constant values of v and r are trapped if and only if r < 2M. And they are "traditional" if r > 2M. The Event Horizon EH: r 2M = 0 − The Penrose, or conformal, diagram r = 0 singularity i+ M J + = 2 r v = i0 −∞ J − i− The Event Horizon EH The boundary of the past of J +. By definition, this is always a null hypersurface. The event horizon (EH) in general 0 In asymptotically flat situations —with J ± and i —, one can define the region from where J + cannot be reached by any causal means.
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