(In)Stability of Kerr Black Holes with Proca Hair 3 J

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Towards Nonlinear (In)stability of Kerr Black Holes with Proca Hair 3 J. M. Miller and W. East z Introduction The Classic No Hair Theorems A New Horizon-Penetrating Coordinate System

I Our goal is to understand the nonlinear stability and dynamics of Kerr black holes with massive I Original no-hair theorems due to Bekenstein: I Define new mapping which regularizes the HR ansatz and respects the unknown ansatz functions: vector hair. All asymptotically flat, stationary black hole solutions that (1) possess no naked singularities and (2) 0 ! 0 ! 00 ! 00 ! ∂xµ ∂xν ∂xµ ∂xν ∂xµ ∂xν I Our tools are: share symmetries between matter and metric fields must be described by mass, spin, and charge. (g ) 000 000 = (g ) new µ ν old µν µ0 ν0 µ00 ν00 µ000 ν000 I Fully nonlinear simulations of the Einstein-Proca system I Basic idea—if you wait long enough, one of several things happens: ∂x ∂x ∂x ∂x ∂x ∂x 0

I Initial data generated via Herdeiro, Radu, and Runarsson’s construction [1]. I Stable orbits decay due to gravitational I Initial data generated by simulating the superradiant instability to ﬁnd the end state. 0.6 radiation Matter falls in Light, Massive Boson Fields I 2.4

I Light massive Boson fields can emerge from high-energy theory via compactification or as axion-like fields. I Matter radiates away I These fields are a candidate for dark matter.

I The coupling of massive Bosons to black holes is a potential source of gravitational waves. 1

I We study massive vector fields, called Proca fields. + I The Einstein-Proca Action: Z √ 1 1 1 S = d4x −g R − F F¯αβ − µ2A A¯α 16π 4 αβ 2 α Example lapse in new, regular coordinates Example volume form in new, regular coordinates I Mass can be constrained via various types of observations [2]: Evading the No Hair Theorems I Perform 3+1 split of Proca field

I Complex ﬁelds allow for matter and metric to have diﬀerent symmetries.

I Can suppress gravitational waves via interference. Exact same as Boson star construction.

I Bound states exist at threshold of superradiance 1 Superradiance × Can be used to mine energy from a spinning Superradiance is the wave-analogue of the I I black hole. Example imaginary vector potential in new Penrose process. Example real vector potential in new coordinates 3.2 Emerges simply from black hole coordinates Works only for boson fields. I I thermodynamics: ( A time-varying complex field...... Can have a time-invariant norm, and thus Tµν Preliminary Calculations ΩH ( k δM = δAH + ΩHδJ Stationary, Hairy, Kerr Black Holes 8π I Use cartoon method to perform axisymmetric simulations δA ≥ 0 I Use horizon penetrating coordinates, with compactification and excision h I The end point of superradiance? Preliminary calculations in-progress. Ingredients Potential Instabilities I I Multiple winding numbers ωk δAH I Threshold of superradiance ⇒ δM = ω = mΩ Lφ = im1A1 + im2A2 8π ω − mΩH H

/ I Frequency perturbation 0.4 I Complex, harmonic azimuthal and time L m dependence ω → ω + ψ(t, r, θ, φ) = ei(mφ−ωt)ψ(r, θ) E = ω δM < 0 ⇔ ω − mΩH < 0 I De-phasing

or hRe(A), Im(A)i= 6 0 Σ LφA = imA, LtA = −iωA I Stable light rings The Superradiant Instability The Herdeiro-Radu Ansatz I Superradiance with confinement implies I Growth rate (for test-fields) given by z instability—the black hole bomb. hydrogen-like spectrum [3]: I Metric ansatz: Mass provides clear confinement mechanism ! I τ ∼ (Mµ)−(4m+3) dr2 ds2 = −e2F0Ndt2 + e2F1 + r2dθ2 + e2F2r2 sin2 θ (dφ − Wdt)2 N spacetime diagram of the BH-proca system. SKSL is the new coordinate system, HRR is the old one. m

eV -10 I Field ansatz: 4.0 10 i(mφ−ωt) A = e (iVdt + H1dr + H2dθ + iH3 sin(θ)dφ) For I r References N = 1 − h r [1] C. Herdeiro, E. Radu, and H. R´unarsson. -18 with Fi, W , V , Hj, i = 0, 1, 2 and j = 1, 2, 3 subject to regularity conditions at r = rh and Kerr black holes with Proca hair. Classical and Quantum Gravity, 33(15):154001, 2016.

axion Mass in log asymptotically ﬂat conditions at r = ∞ [1].

0 8 I Solve constraint equations for spacetime and Helmholtz-like equation for Proca field to find [2] A. Arvanitaki, S. Dimopoulos, S. Dubovsky, N. Kaloper, and J. March-Russell. String axiverse. black hole mass in log M stationary solution. 10 M Phys. Rev. D, 81:123530, 2010. I Ansatz is not horizon-penetrating. Shaded region indicates BH/Boson mass where [3] R. Brito, V. Cardoso, and P. Pani. Black hole surrounded by mirrors, providing I Coordinate singularities at the horizon mean it cannot directly applied as initial data to a timescale observable [2]. Superradiance: Energy Extraction, Black-Hole Bombs and Implications for Astrophysics and Particle Physics. confinement. time-evolution. Lecture Notes in Physics. Springer International Publishing, 2015.

This research was funded by DOE SciDAC and the NNSA and used computing facilities at the Perimeter Institute and the Extreme Science and Engineering Discovery Environment (XSEDE) (TG-PHY100033). This work was approved for unlimited public release as per 0.2 LA-UR-17-30948 4.8 5.6 0.0 0.0 0.2 0.4 0.6 0.8 1.0 x/(1 + x)