The Penrose inequality for the perturbed Schwarzschild initial data J. Tafel University of Warsaw Jurekfest 2019 joint work with J. Kopi´ nski Jurekfest 2019 joint work with J. Kopi´ nski 1 J. Tafel (University of Warsaw) The Penrose inequality for the perturbed Schwarzschild initial data / 14
The Penrose inequality The surface area of the Kerr horizon � m 2 − a 2 ) , | S h | = 8 π m ( m + hence (the Penrose inequality) � | S h | m ≥ 16 π . Jurekfest 2019 joint work with J. Kopi´ nski 2 J. Tafel (University of Warsaw) The Penrose inequality for the perturbed Schwarzschild initial data / 14
The Penrose inequality The surface area of the Kerr horizon � m 2 − a 2 ) , | S h | = 8 π m ( m + hence (the Penrose inequality) � | S h | m ≥ 16 π . Stronger version 16 π + 4 π J 2 E 2 − p 2 ≥ | S h | | S h | , where E , p , J are, respectively, the total energy, momentum and angular momentum. Jurekfest 2019 joint work with J. Kopi´ nski 2 J. Tafel (University of Warsaw) The Penrose inequality for the perturbed Schwarzschild initial data / 14
Consider the Cauchy data with a horizon. If a future singularity is surrounded by the event horizon (physically realistic data, the cosmic censorship conjecture) configuration tends to a stationary state then the no-hair theorem etc. (almost) imply that the end state is the Kerr metric with E ∞ and S ∞ h E ∞ ≤ E (because of radiation) | S ∞ h | ≥ | S h | (BH thermodynamics) Jurekfest 2019 joint work with J. Kopi´ nski 3 J. Tafel (University of Warsaw) The Penrose inequality for the perturbed Schwarzschild initial data / 14
Consider the Cauchy data with a horizon. If a future singularity is surrounded by the event horizon (physically realistic data, the cosmic censorship conjecture) configuration tends to a stationary state then the no-hair theorem etc. (almost) imply that the end state is the Kerr metric with E ∞ and S ∞ h E ∞ ≤ E (because of radiation) | S ∞ h | ≥ | S h | (BH thermodynamics) Conclusion: the Penrose inequality should be satisfied on the initial surface Jurekfest 2019 joint work with J. Kopi´ nski 3 J. Tafel (University of Warsaw) The Penrose inequality for the perturbed Schwarzschild initial data / 14
Vacuum initial data with a horizon Constraints on g ′ ij , K ′ ij K ′ i j − H ′ δ i � � ∇ i = 0 j R ′ + H ′ 2 − K ′ 2 = 0 where H ′ = K ′ i i and K ′ 2 = K ′ ij K ′ ij and R ′ is the Ricci scalar of g ′ ij . Jurekfest 2019 joint work with J. Kopi´ nski 4 J. Tafel (University of Warsaw) The Penrose inequality for the perturbed Schwarzschild initial data / 14
Vacuum initial data with a horizon Constraints on g ′ ij , K ′ ij K ′ i j − H ′ δ i � � ∇ i = 0 j R ′ + H ′ 2 − K ′ 2 = 0 where H ′ = K ′ i i and K ′ 2 = K ′ ij K ′ ij and R ′ is the Ricci scalar of g ′ ij . Horizon: compact surface with vanishing expansion of outer null rays (marginal outer trapped surface - MOTS) H ′ − K ′ nn + h = 0 , where n i is the unit normal vector and h = ∇ i n i is the mean curvature of the surface. Jurekfest 2019 joint work with J. Kopi´ nski 4 J. Tafel (University of Warsaw) The Penrose inequality for the perturbed Schwarzschild initial data / 14
Vacuum initial data with a horizon Constraints on g ′ ij , K ′ ij K ′ i j − H ′ δ i � � ∇ i = 0 j R ′ + H ′ 2 − K ′ 2 = 0 where H ′ = K ′ i i and K ′ 2 = K ′ ij K ′ ij and R ′ is the Ricci scalar of g ′ ij . Horizon: compact surface with vanishing expansion of outer null rays (marginal outer trapped surface - MOTS) H ′ − K ′ nn + h = 0 , where n i is the unit normal vector and h = ∇ i n i is the mean curvature of the surface. If H ′ = h = 0 then the Penrose inequality follows from the Hamiltonian constraint (Geroch, ...., Huisken and Ilmanen). Jurekfest 2019 joint work with J. Kopi´ nski 4 J. Tafel (University of Warsaw) The Penrose inequality for the perturbed Schwarzschild initial data / 14
The conformal approach j + 1 g ′ ij = ψ 4 g ij , K ′ i j = ψ − 6 A i 3 H ′ δ i j , where A i i = 0. Jurekfest 2019 joint work with J. Kopi´ nski 5 J. Tafel (University of Warsaw) The Penrose inequality for the perturbed Schwarzschild initial data / 14
The conformal approach j + 1 g ′ ij = ψ 4 g ij , K ′ i j = ψ − 6 A i 3 H ′ δ i j , where A i i = 0. The momentum constraint j = 2 3 ψ 6 ∇ j H ′ . ∇ i A i Jurekfest 2019 joint work with J. Kopi´ nski 5 J. Tafel (University of Warsaw) The Penrose inequality for the perturbed Schwarzschild initial data / 14
The conformal approach j + 1 g ′ ij = ψ 4 g ij , K ′ i j = ψ − 6 A i 3 H ′ δ i j , where A i i = 0. The momentum constraint j = 2 3 ψ 6 ∇ j H ′ . ∇ i A i The Hamiltonian constraint (the Lichnerowicz equation) △ ψ = 1 8 R ψ − 1 8 A ij A ij ψ − 7 + 1 12 H ′ 2 ψ 5 . Jurekfest 2019 joint work with J. Kopi´ nski 5 J. Tafel (University of Warsaw) The Penrose inequality for the perturbed Schwarzschild initial data / 14
The conformal approach j + 1 g ′ ij = ψ 4 g ij , K ′ i j = ψ − 6 A i 3 H ′ δ i j , where A i i = 0. The momentum constraint j = 2 3 ψ 6 ∇ j H ′ . ∇ i A i The Hamiltonian constraint (the Lichnerowicz equation) △ ψ = 1 8 R ψ − 1 8 A ij A ij ψ − 7 + 1 12 H ′ 2 ψ 5 . The MOTS condition n i ∂ i ψ + 1 2 h ψ − 1 4 A nn ψ − 3 + 1 6 H ′ ψ 3 = 0 on S h . Jurekfest 2019 joint work with J. Kopi´ nski 5 J. Tafel (University of Warsaw) The Penrose inequality for the perturbed Schwarzschild initial data / 14
The conformal approach j + 1 g ′ ij = ψ 4 g ij , K ′ i j = ψ − 6 A i 3 H ′ δ i j , where A i i = 0. The momentum constraint j = 2 3 ψ 6 ∇ j H ′ . ∇ i A i The Hamiltonian constraint (the Lichnerowicz equation) △ ψ = 1 8 R ψ − 1 8 A ij A ij ψ − 7 + 1 12 H ′ 2 ψ 5 . The MOTS condition n i ∂ i ψ + 1 2 h ψ − 1 4 A nn ψ − 3 + 1 6 H ′ ψ 3 = 0 on S h . Existence theorems under extra conditions (D. Maxwell, Anglada). Jurekfest 2019 joint work with J. Kopi´ nski 5 J. Tafel (University of Warsaw) The Penrose inequality for the perturbed Schwarzschild initial data / 14
Approximate constraint equations Assumption: the preliminary metric g ij is flat and S h is the sphere with radius m / 2. Jurekfest 2019 joint work with J. Kopi´ nski 6 J. Tafel (University of Warsaw) The Penrose inequality for the perturbed Schwarzschild initial data / 14
Approximate constraint equations Assumption: the preliminary metric g ij is flat and S h is the sphere with radius m / 2. If K ′ ij = 0 then ψ = ψ 0 = 1 + m 2 r and the conformal transformation leads to the Schwarzschild initial metric on t = const dr ′ 2 g ′ = + r ′ 2 ( d θ 2 + sin 2 θ d ϕ 2 ) 1 − 2 m r ′ The Penrose inequality is saturated. Jurekfest 2019 joint work with J. Kopi´ nski 6 J. Tafel (University of Warsaw) The Penrose inequality for the perturbed Schwarzschild initial data / 14
Approximate constraint equations Assumption: the preliminary metric g ij is flat and S h is the sphere with radius m / 2. If K ′ ij = 0 then ψ = ψ 0 = 1 + m 2 r and the conformal transformation leads to the Schwarzschild initial metric on t = const dr ′ 2 g ′ = + r ′ 2 ( d θ 2 + sin 2 θ d ϕ 2 ) 1 − 2 m r ′ The Penrose inequality is saturated. Let ǫ be a small parameter and ψ = ψ 0 + ψ 1 + ψ 2 + ... A ij = B ij + ... H ′ = B + ... where ψ n ∼ ǫ n and B ij , B ∼ ǫ . Jurekfest 2019 joint work with J. Kopi´ nski 6 J. Tafel (University of Warsaw) The Penrose inequality for the perturbed Schwarzschild initial data / 14
The approximate constraints j = 2 ∇ i B i 3 ψ 6 0 ∇ j B △ ψ = − 1 + 1 8 B ij B ij ψ − 7 12 B 2 ψ 5 0 . 0 Jurekfest 2019 joint work with J. Kopi´ nski 7 J. Tafel (University of Warsaw) The Penrose inequality for the perturbed Schwarzschild initial data / 14
The approximate constraints j = 2 ∇ i B i 3 ψ 6 0 ∇ j B △ ψ = − 1 + 1 8 B ij B ij ψ − 7 12 B 2 ψ 5 0 . 0 Remarks: The momentum constraint is a linear underdetermined system for B ij and B . Jurekfest 2019 joint work with J. Kopi´ nski 7 J. Tafel (University of Warsaw) The Penrose inequality for the perturbed Schwarzschild initial data / 14
The approximate constraints j = 2 ∇ i B i 3 ψ 6 0 ∇ j B △ ψ = − 1 + 1 8 B ij B ij ψ − 7 12 B 2 ψ 5 0 . 0 Remarks: The momentum constraint is a linear underdetermined system for B ij and B . A relation between total energy and area of horizon should follow from the Lichnerowicz equation. Jurekfest 2019 joint work with J. Kopi´ nski 7 J. Tafel (University of Warsaw) The Penrose inequality for the perturbed Schwarzschild initial data / 14
The approximate constraints j = 2 ∇ i B i 3 ψ 6 0 ∇ j B △ ψ = − 1 + 1 8 B ij B ij ψ − 7 12 B 2 ψ 5 0 . 0 Remarks: The momentum constraint is a linear underdetermined system for B ij and B . A relation between total energy and area of horizon should follow from the Lichnerowicz equation. ψ 1 is harmonic and ψ 2 satisfies the Poisson equation with the Neumann type boundary condition. Jurekfest 2019 joint work with J. Kopi´ nski 7 J. Tafel (University of Warsaw) The Penrose inequality for the perturbed Schwarzschild initial data / 14
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