Null Cones to Infinity, Curvature Flux, and Bondi Mass Arick Shao (joint work with Spyros Alexakis) University of Toronto May 22, 2013 Arick Shao (University of Toronto) Null Cones to Infinity May 22, 2013 1 / 1
Introduction Elements of Mathematical GR Mathematical General Relativity In general relativity, spacetime is modeled as 4-dimensional Lorentzian manifold ( M , g ) satisfying the Einstein equations : Ric g − 1 2 Scal g · g = T . Ric g , Scal g : Ricci and scalar curvature of ( M , g ) . T : stress-energy tensor for matter field. Vacuum spacetimes : no matter field ( T ≡ 0) Einstein-vacuum equations : Ric g ≡ 0. Arick Shao (University of Toronto) Null Cones to Infinity May 22, 2013 2 / 1
Introduction Elements of Mathematical GR Null Cones Wave equation , � g φ = g αβ ∇ α ∇ β φ ≡ 0. Can be thought of as linearized model for vacuum equations. Null hypersurfaces : induced metric is degenerate Characteristics of the wave equation. Generated by null geodesics. Null cone : null hypersurface N beginning from 2-sphere or point. Curvature flux : L 2 -norm on N of certain components of R . Important quantity in energy estimates. Truncated null cone. Arick Shao (University of Toronto) Null Cones to Infinity May 22, 2013 3 / 1
Introduction Explicit and Near-Explicit Solutions Schwarzschild Spacetimes Schwarzschild spacetime : spherically symmetric, black hole spacetimes m ≥ 0: “mass”. Satisfies Einstein-vacuum equations. In the outer region r > 2 m , metric can be expressed as � − 1 � � � 1 − 2 m 1 − 2 m dt 2 + dr 2 + r 2 ˚ g = − γ . r r m = 0: Minkowski spacetime ( − dt 2 + dr 2 + r 2 ˚ γ ). Infinity : represents faraway observer. In these spacetimes, timelike/null/spacelike infinity can be explicitly constructed via conformal compactification . Arick Shao (University of Toronto) Null Cones to Infinity May 22, 2013 4 / 1
Introduction Explicit and Near-Explicit Solutions Schwarzschild Spacetimes i + i + r = 0 I + I + r = 2 m i 0 i 0 r = 0 r = 2 m I − I − i − i − r = 0 Schwarzschild spacetime. Minkowski spacetime. Arick Shao (University of Toronto) Null Cones to Infinity May 22, 2013 5 / 1
Introduction Explicit and Near-Explicit Solutions Near-Minkowski Spacetimes i + I + Christodoulou-Klainerman: asymptotic stability of Minkowski spacetimes. Can recover similar structure at infinity i 0 T as Minkowski spacetime. Stability of Schwarzschild, Kerr I − spacetimes: open problem. i − Near-Minkowski, at infinity. Arick Shao (University of Toronto) Null Cones to Infinity May 22, 2013 6 / 1
Introduction ADM and Bondi Mass Mass In asymptotically flat spacetimes, with similar structures “at infinity”, there exist notions of total mass. ADM mass : applicable to spacelike hypersurfaces Computed as limit at spacelike infinity. Represents, e.g., total mass of initial data. Bondi mass : applicable to null cones Computed as limit at a cut of null infinity. Represents mass remaining in system after some has radiated away. Arick Shao (University of Toronto) Null Cones to Infinity May 22, 2013 7 / 1
Introduction ADM and Bondi Mass Mass Schwarzschild: static solution i + m ADM ( init. ) = m . m Bondi ≡ m m Bondi ≡ m on I + . I + i 0 m ADM = m Near-Minkowski: not static m Bondi ց 0 Positive mass thm. : m ADM ( init. ) ≥ 0. i + Mass loss: 0 ≤ m Bondi ≤ m ADM ( init. ) . m Bondi decreasing m Bondi ց 0 in along I + . I + i 0 m ADM ≥ 0 Arick Shao (University of Toronto) Null Cones to Infinity May 22, 2013 8 / 1
Introduction The Main Problem Main Goals Consider “near-Schwarzschild spacetime”. “Eliminate all assumptions except at single infinite null cone.” ( M , g ) : vacuum spacetime. N : future outgoing infinite null cone in ( M , g ) . N is “close to Schwarzschild null cone”. ∞ m Bondi ≈ m ? m Bondi ≈ m ? I + N N Assumed setting Intuitive picture Arick Shao (University of Toronto) Null Cones to Infinity May 22, 2013 9 / 1
Introduction The Main Problem Main Goals Assume: N is “near-Schwarzschild”. “Weighted curvature flux” of N close to Schwarzschild. “Initial data” of N close to Schwarzschild. Objective 1: control geometry of N . Quantitative bounds (for connection coefficients). Asymptotic limits for coefficients at infinity. Objective 2: connection to physical quantities. Control Bondi mass for N . Can also consider angular momentum, rate of mass loss. Arick Shao (University of Toronto) Null Cones to Infinity May 22, 2013 10 / 1
Introduction The Main Problem Main Features No global assumptions on spacetime. All assumptions on single null cone N . Low-regularity quantitative assumptions. At the level of curvature flux ( L 2 -norm of curvature on N ). Physical motivation. What controls Bondi mass, etc.? Requires finding “correct” foliation, i.e., approach to infinity. Arick Shao (University of Toronto) Null Cones to Infinity May 22, 2013 11 / 1
Geometry of Null Cones Geodesic Foliations Geodesic Foliations Geodesic foliation: express N as one-parameter family of spheres. Spheres determined by affine parameters of the null geodesics s = s 0 + 1 . 5 generating N . s = s 0 + 1 Algebraically simplest foliation. s = s 0 Write N ≃ [ s 0 , ∞ ) × S 2 . Geodesic foliation. s : affine parameter of null geodesics (starting from s 0 ). s 0 : radius of the initial sphere of N . Arick Shao (University of Toronto) Null Cones to Infinity May 22, 2013 12 / 1
Geometry of Null Cones Geodesic Foliations Geodesic Foliations S τ : level set s = τ . / γ : induced metrics on the S τ ’s. Consider adapted null frames : L L 2 spacelike directions e 1 , e 2 ( S τ ,/ γ ) e 1 , e 2 tangent to S τ . 2 null directions normal to S τ . e 1 , e 2 L tangent to N (and satisfies Ls ≡ 1) Null frame. L transverse to N (and satisfies g ( L , L ) ≡ − 2). Arick Shao (University of Toronto) Null Cones to Infinity May 22, 2013 13 / 1
Geometry of Null Cones Connection and Curvature Formulation of Null Geometry Decompose spacetime curvature and connection quantities: Spacetime curvature R : β a = 1 ρ = 1 α ab = R ( L , e a , L , e b ) , 2 R ( L , L , L , e a ) , 4 R ( L , L , L , L ) , β a = 1 σ = 1 α ab = R ( L , e a , L , e b ) , 2 R ( L , L , L , e a ) , ⋆ R ( L , L , L , L ) . 4 Connection coefficients: ζ a = 1 χ ab = g ( D e a L , e b ) , χ ab = g ( D e a L , e b ) , 2 g ( D e a L , L ) . Mass aspect function (related to Hawking and Bondi mass): γ ab ∇ a ζ b − ρ + 1 γ bd ^ γ ac / χ cd . µ = − / 2 / χ ab ^ Arick Shao (University of Toronto) Null Cones to Infinity May 22, 2013 14 / 1
Geometry of Null Cones Connection and Curvature The Null Structure Equations The connection and curvature coefficients are related via a system of geometric PDE, called the null structure equations . Evolution equations : ∇ L χ ≃ − χ · χ + α , ∇ L ζ ≃ χ · ζ + β , ∇ L χ ≃ ρ + ∇ ζ + l.o., etc. Elliptic equations : D ^ χ ≃ β + l.o., D ζ ≃ ( ρ + µ, σ ) + l.o., K ≃ − ρ + χ · χ , etc. Null Bianchi equations : ∇ L β ≃ D α + χ · β + ζ · α , etc. The vacuum equations are encoded within the structure equations. Arick Shao (University of Toronto) Null Cones to Infinity May 22, 2013 15 / 1
Geometry of Null Cones Some Important Quantities Curvature Flux Define the weighted curvature flux for N to be F ( N ) = � s 2 α � L 2 ( N ) + � s 2 β � L 2 ( N ) + � s ρ � L 2 ( N ) + � s σ � L 2 ( N ) + � β � L 2 ( N ) . Generated as a local energy quantity from Bel-Robinson tensor. Bel-Robinson tensor : “energy density” for spacetime curvature. Weights analogous to those found in C-K and K-N. Note: s will be comparable to radii r of level spheres. Arick Shao (University of Toronto) Null Cones to Infinity May 22, 2013 16 / 1
Geometry of Null Cones Some Important Quantities Hawking and Bondi Mass Hawking mass of S τ : � � � � m( τ ) = r 1 = r 1 + tr χ tr χ µ . 2 16 π 8 π S τ S τ r : area radius of S τ . If r − 2 / γ is asymptotically round : m( τ ) converges to Bondi energy as τ ր ∞ . Arick Shao (University of Toronto) Null Cones to Infinity May 22, 2013 17 / 1
Geometry of Null Cones Schwarzschild Spacetimes The Schwarzschild Case Standard outgoing shear-free null cones are: N = { t − r ∗ = c , r ≥ r 0 } , r ∗ is the “tortoise coordinate” � r r ∗ = r + 2 m log � 2 m − 1 . The affine parameter s on N is simply r . The null vector fields are � − 1 � 1 − 2 m � 1 − 2 m � L = ∂ t + ∂ r , L = ∂ t − ∂ r . r r Arick Shao (University of Toronto) Null Cones to Infinity May 22, 2013 18 / 1
Geometry of Null Cones Schwarzschild Spacetimes The Schwarzschild Case Ricci coefficients: � 1 − 2 m � χ = r − 1 / χ = − r − 1 γ , γ , ζ ≡ 0. / r Nonvanishing curvature coefficients: ρ = − 2 m r 3 . Mass aspect function: µ = 2 m r 3 . Arick Shao (University of Toronto) Null Cones to Infinity May 22, 2013 19 / 1
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