Horizon hair of extremal black holes and measurements at null infinity Stefanos Aretakis (joint with Yannis Angelopoulos and Dejan Gajic) University of Toronto International Congress on Mathematical Physics July 24, 2018 1 / 20
Evolution of scalar fields ◮ Scalar fields: Investigate the evolution of solutions to wave equation ✷ g ψ = 0 on black hole backgrounds (Schwarzschild, Kerr, Reissner–Nordstr¨ om) event horizon null infinity ◮ Motivation: In harmonic gauge ✷ g x µ = 0 the vacuum equations take the form ✷ g g µν = N µν ( g, ∂g ) . Hence the wave equation serves as a (necessary) toy model in studying the dynamics of Einstein equations. ◮ Goal: ◮ Upper bounds for stability considerations ◮ Lower bounds for strong cosmic censorship ◮ This talk: Emphasis on conservation laws, asymptotics and physical conse- quences 2 / 20
Conservation laws along characteristic hypersurfaces Let S v be a foliation with section of a null hypersurface H . Then roughly speaking a conservation law consists of integrals of the form � F ( ψ, D a ψ ) S v which are independent of v for all scalar fields ψ satisfying the wave equation. 3 / 20
Late-time tails on sub-extremal black holes 4 / 20
Previous mathematical works Very active research area in the past decade. ◮ Main difficulties: Low frequencies, superradiant, trapping, redshift ◮ Contributors: Dafermos, Rodnianski, Andersson, Tataru, Moschidis, Blue, Holzegel, Shlapentokh-Rothman, Dyatlov, H¨ afner, Bony, Smulevici, Klain- erman, Ionescu, Tohaneanu, Sterbenz, Soffer, Schlue, Luk, Finster, Kamran, Smoller, Yau, Donninger, Schlag, Vasy, Hintz, Metcalfe, Wald, ... ◮ Lower bounds were first proved in the work of Luk–Oh. ◮ All methods break down at the extremal case. 5 / 20
The Newman–Penrose constant ◮ The Newman–Penrose constant gives rise to a conservation law along null infinity. The constant is equal to � r →∞ r 2 · ∂ v ( rψ ) NP [ ψ ] = lim S τ ◮ However, NP [ ψ ] = 0 for compactly supported data. Then, generically, NP [ ∂ − 1 ψ ] � = 0 t where ∂ − 1 ψ is canonically defined as long as ∂ t � = 0 . t ◮ The constant NP [ ∂ − 1 ψ ] can be explicitly computed using the initial data t of ψ . ◮ Denote I (1) [ ψ ] := NP [ ∂ − 1 ψ ] . t ◮ I (1) is the unique obstruction to inverting T 2 . 6 / 20
Late-time asymptotics Theorem (Angelopoulos, A., Gajic) If ψ is a solution to the wave equation on a sub-extremal Reissner–Nordstr¨ om space-time with smooth compactly supported initial data then Asymptotics in the exterior region ψ | H ψ | r = R rψ | I − 2 I (1) [ ψ ] · τ − 2 − 8 MI (1) [ ψ ] log τ · τ − 3 8 I (1) [ ψ ] · τ − 3 8 I (1) [ ψ ] · τ − 3 Comments: ◮ I (1) [ ψ ] = M � ψ d Ω + M � 1 ∂ t ψ r 2 drd Ω . 1 − 2 M 4 π 4 π { t =0 }∩ S BF { t =0 } r ◮ Sharp lower and upper pointwise bounds. ◮ I (1) [ ψ ] related to the quantity L of Luk–Oh. ◮ Correlated asymptotics along H + ( ψ ∼ 8 I (1) [ ψ ] · τ − 3 ) and I + ( rψ ∼ − 2 I (1) [ ψ ] · τ − 2 ) . ◮ Leading order asymptotics recover work of Leaver. ◮ Precise logarithmic corrections along I + appear to be new. ◮ We further obtain (2 ℓ + 3) -asymptotics. 7 / 20
Late-time tails (or late-time tales) for extremal black holes 8 / 20
Why extremal black holes? ◮ Mass minimizers ◮ Applications in supersymmetry, quantum gravity, string theory ◮ Electromagnetic and gravitational signatures ◮ Turbulent gravitational behavior ◮ Vast astronomical evidence for near-extremal black holes. ◮ Rees et al. ( The distribution and cosmic evolution of massive black hole spins , Astrophys. J.) report that “the spin distribution is heavily skewed toward fast-rotating Kerr black holes” and that “about 70% of all stellar black holes at all epochs are maximally rotating”. Gas accretion dominant effect and spins black holes up. 9 / 20
Firstly, we have the following Proposition (A.) If ψ satisfies the wave equation on extremal Reissner–Nordstr¨ om then the integral � 1 � � H [ ψ ] = − Y ψ + 2 M ψ dvol S τ is independent of τ . Here Y is transversal to the horizon. ◮ For smooth solutions ψ we have H [ Tψ ] = 0 . Hence H is an obstruction to inverting T . 10 / 20
“Outgoing radiation” Solutions ψ with H [ ψ ] � = 0 and NP [ ψ ] = 0 11 / 20
“Initially static moment” Solutions ψ with H [ ψ ] � = 0 and NP [ ψ ] � = 0 12 / 20
“Ingoing radiation” Solutions ψ with H [ ψ ] = 0 and NP [ ψ ] = 0 13 / 20
H [ ψ ] as a “horizon hair” ◮ Outgoing perturbations and perturbations with an initially static moment ( H [ ψ ] � = 0 ) satisfy along the event horizon: 1) Non-decay : Y ψ → − 1 M H [ ψ ] 1 2) Blow-up : Y Y ψ → M 3 H [ ψ ] · τ ◮ H [ ψ ] : “horizon” “hair” since 1) Energy density measured by incoming observers: T rr [ ψ ] ∼ H [ ψ ] where T is the E-M tensor, 2) | Y k ψ | , | T rr [ ψ ] | ≤ 0 away from the horizon. ◮ Generic ingoing perturbations: | Y Y Y ψ | → ∞ , as τ → + ∞ . ◮ Later extensions/applications by: Reall, Murata, Casals, Zimmerman, Gralla, Tana- hashi, Bizon, Lucietti, Angelopoulos, Gajic, Ori, Sela, Tsukamoto, Kimura, Harada, Hadar, Dain, Dotti, Godazgar, Burko, Khanna, Bhattacharjee, Chow, Berti et al, Cardoso et al,... 14 / 20
Late-time asymptotics Theorem (Angelopoulos, A.,Gajic) The following asymptotics hold on ERN: Asymptotics along the event horizon Perturbation outgoing data ingoing data 2 H · τ − 1 − 2 H (1) · τ − 2 ψ | H − 1 M 2 · H (1) · τ − 2 2 Y ψ | H M · H 1 M 3 · H (1) 1 Y Y ψ | H M 3 · H · τ M 5 · H (1) · τ 2 M 5 · H · τ 2 3 3 Y Y Y ψ | H − − ◮ H registers in the asymptotics for ψ . ◮ Here H (1) = H [ T − 1 ψ ] . It is well-defined for ingoing perturbations. ◮ Asymptotics on H + confirm numerical results of Murata–Reall–Tanahashi and is consistent with decay rates of Blaskley–Burko, Ori–Sela and Casals– Gralla–Zimmerman. ◮ Asymptotics (with log corrections) on the event horizon are important for dynamics in the interior of extremal black holes– C 2 extendibility. (Gajic, Reall et al., Gajic–Luk). 15 / 20
Late-time asymptotics Theorem (Angelopoulos, A.,Gajic) The following asymptotics hold on ERN: Asymptotics away from the event horizon ψ | r = R rψ | I Data � 4 MH − 2 I (1) � r − M H · τ − 2 4 M · τ − 2 outgoing � � M · τ − 2 2 · NP [ ψ ] · τ − 1 4 NP + static moment r − M H � I (1) + r − M H (1) � − 2 I (1) · τ − 2 M · τ − 3 ingoing − 8 ◮ Here I (1) = NP [ T − 1 ψ ] and H (1) [ ψ ] = H [ T − 1 ψ ] . ◮ H [ ψ ] registers in the asymptotics away from H , even on null infinity I . 4 M r − M is the static solution. ◮ ◮ For outgoing perturbation ψ , T − 1 ψ is singular on H + : its local energy is infinite. ◮ Asymptotics for rψ | I were not known in physics literature. 16 / 20
Measuring the horizon hair H from null infinity ◮ In principle, precise asymptotics allow to observe/measure the horizon in- stability from afar. Let’s consider outgoing radiation. ◮ Along r = R > M : We have a slower decay rate if H � = 0 . In fact, H [ ψ ] = R − M τ →∞ τ 2 · ψ | r = R · lim 4 M ◮ Along I + : We have the same decay rate for the radiation field, but the � 4 MH − 2 I (1) � · τ − 2 . In horizon hair registers in the asymptotics rψ | I ∼ fact, it turns out that I (1) = M � I + ∩{ τ ≥ 0 } rψ d Ω dτ which yields 4 π � 1 + 1 τ 2 · ( rψ ) | I � � H [ ψ ] = 4 M lim rψ | I d Ω dτ 8 π τ →∞ I + ∩{ τ ≥ 0 } ◮ We conclude that for extremal black holes information “leaks” from the event horizon to null infinity. 17 / 20
Comparison with sub-extremal tails For ERN: 1 + 1 � τ 2 · ( rψ ) | I � � H [ ψ ] = 4 M lim rψ | I d Ω dτ 8 π τ →∞ I + ∩{ τ ≥ 0 } For sub-extremal RN we have: ◮ The RHS vanishes for sub-extremal RN! ◮ Specifically, we have � = − M τ 2 · ( rψ ) | I � � lim rψ | I d Ω dτ, 2 π τ →∞ I + ∩{ τ ≥ 0 } = 2 M � � τ 3 · ψ | r = R � lim rψ | I d Ω dτ, π τ →∞ I + ∩{ τ ≥ 0 } = 2 M � � τ 3 · ψ | H � lim rψ | I d Ω dτ, π τ →∞ I + ∩{ τ ≥ 0 } ◮ Late time tails are dictated by the weak-field dynamics, namely by dynamics at very large r . ◮ Integral of the radiation field had been used by Luk–Oh for lower bounds on sub-extremal RN. 18 / 20
Physics Literature ◮ Work by Reall, Murata and Tanahashi suggests that perturbations of initial data of extremal R–N in the context of the Cauchy problem for the Einstein– Maxwell-scalar field equations exhibit a version of the horizon instability. ◮ Work by Casals–Gralla–Zimmerman and subsequently by Hadar–Reall ob- tained that the decay rate for non-zero azimuthal frequencies along the event 1 horizon on extremal Kerr is √ τ and for the first-order transversal derivative is √ τ (amplified instability). 19 / 20
Thank you! 20 / 20
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