Snowmass2021 - Letter of Interest

Peering inside Black Holes with Gravitational Waves

Thematic Areas:  (TF01) String theory, quantum gravity, black holes  (TF2) Effective field theory techniques  (TF3) CFT and formal QFT  (TF4) Scattering amplitudes  (TF5) Lattice gauge theory  (TF6) Theory techniques for precision physics  (TF7) Collider phenomenology  (TF8) BSM model building  (TF9) Astro-particle physics & cosmology  (TF10) Quantum Information Science  (TF11) Theory of neutrino physics  (CF6) Dark Energy and Cosmic Acceleration: Complementarity of Probes and New Facilities  (CF7) Cosmic Probes of Fundamental Physics

Contact Information: Emil Mottola (Los Alamos National Laboratory & Perimeter Institute)[[email protected]]

Author: Emil Mottola

Abstract: In classical General Relativity (GR) the interiors of Black Holes (BHs) are not only singular but, if rotating, also admit closed timelike curves, violating causality.1 This feature occurs at macroscopic distance scales, far larger than the microscopic Planck scale LP l. When quantum effects are considered, severe conflicts with statistical thermodynamics, conservation of probability and an enormous BH entropy arise also at the macroscopic horizon scale.2,3 This suggests that a low energy semi-classical Effective Field Theory (EFT) approach should be applicable. In this LOI such an approach based the conformal anomaly is proposed, which leads to a non-singular horizonless, but ultra-compact object called a gravitational condensate star. The gravastar hypothesis can be tested by searching for discrete surface modes and GW echoes emitted after binary merger events. In this new era of GW and multi-messenger astronomy with additional GW detectors coming online in this decade, the time is now ripe for a full-fledged effort to confront these theoretical ideas with the observational data that hold the promise of resolving the conundrum that BHs pose, and potentially point to a new path to ultimate synthesis of gravitation and quantum theory.

1 The presumption that gravitational collapse of stars above a certain mass leads inevitably to an event horizon and a BH singularity is based upon an essentially classical view of the collapsing matter as a collection of uncorrelated pointlike particles, with pressure p ≥ 0 and Equation of State (EoS) obeying classical energy conditions, neglecting quantum coherence. These assumptions remain widely held, despite the experimental evidence of the wavelike properties of matter, quantum violation of classical energy conditions and macroscopic quantum coherence in a wide variety of sufficiently cold, condensed systems, as well as the quantum degeneracy pressure responsible for highly compact stars such as White Dwarfs and Neutron Stars. The fact that BHs and the problems they pose arise at macroscopic scales strongly suggests that there is a missing element in the low energy content of classical gravity. A first principles Effective Field Theory (EFT) approach to including quantum effects in gravity has been developed, based on the conformal or trace anomaly of the energy-momentum tensor of massless quantum fields,7,8 and the one-loop local effective action corresponding to it. This identifies the missing element in GR to be the long range massless scalar degree of freedom this effective action implies.9–13 The conformal anomaly action9,10,13 expressed in local form is b0Z √ n o 1Z S [ϕ] = d4x −g − ( ϕ)2 + 2 Rαβ − 1 Rgαβ
(∂ ϕ)(∂ ϕ) + d4x A ϕ (1) A 2 3 α β 2 where A is the conformal trace anomaly, made up of curvature invariants as well as matter invariants such as the gluonic contribution of QCD in the SM.7,8 The conformal anomaly is a quantum effect with no intrinsic length scale, and in particular does not involve the ultrashort Planck scale LP l. The new scalar field ϕ is the Goldstone-like boson of conformal symmetry breaking, and an additional propagating massless scalar degree of freedom in the low energy EFT of gravity, not present in classical GR, that is induced by the quantum fluctuations of SM matter/radiation. When added to the usual Einstein-Hilbert term of classical GR, the Wess-Zumino action SA amounts to a well-defined modification of Einstein’s classical theory fully consistent with, and in fact required by first principles of QFT and general covariance, with no additional assumptions.14 αβ Since the stress-energy tensor TA [ϕ] derived from (1) is the source of the gravitational metric field through Einstein’s equations, the macroscopic effects of the conformal anomaly are transferred to the gravitational 9–12,15 αβ field. Qualitatively new phenomena are then predicted. In particular, TA [ϕ] can dominate the classical terms in the vicinity of the apparent horizon of a forming BH. Indeed the linear eq. for ϕ resulting from variation of (1) may be solved in static spherical backgrounds that possess a time translation symmetry and α Killing vector K = ∂/∂t, with scalar invariant norm K Kα = gtt = −f(r). The result is that α
  ϕ(r) = cH ln −K Kα + ··· = cH ln f(r) + ... (2) 9,11 generically diverges as f(r) → 0, where cH is a dimensionless state-dependent integration constant. The α condition f(r) = −K Kα = 0 of the Killing vector becoming null is the condition for the location of the Schwarzschild BH horizon at r = rS = 2GM. Because of the logarithmically divergent behavior (2) at the arbitrarily large ∝ c2 /f 2 r horizon the stress tensor derived from (1) also generically grows H as approaches 9,11 the horizon, in any state for which cH 6= 0. This shows that it is possible for coherent quantum effects to become large enough to affect the classical geometry of BHs no matter how small the local curvature is. That quantum effects arise at BH horizons is supported by a variety of other considerations, arising from QFT, string theory, AdS/CFT, gauge/gravity duality, supergravity, and loop quantum gravity4,19–23. Thus αβ TA affects the classical geometry and also causes a change in the vacuum energy in the interior, so that the final endpoint of collapse may not be that of a classical BH at all, but a gravitational condensate star, free of singularities and all BH paradoxes.6,15–17. By taking a positive value in the interior of a fully collapsed star, the effective cosmological term with p=−ρ EoS removes any singularity, replacing it with a smooth dark energy de Sitter interior. The quantum boundary layer where the effective value of the gravitational vacuum energy changes rapidly is determined by the anomaly stress tensor, and gives rise to a finite surface tension.18 With a definite EFT Lagrangian this LOI is to find the self-consistent spherically symmetric condensate star solution and derive its linear perturbations and normal modes of free oscillation.24. With no BH singularity

2 GRAVITATIONAL-WAVE SIGNATURES OF EXOTIC … PHYSICAL REVIEW D 94, 084031 (2016) 0.15 black hole On the other hand, when the Schwarzschild horizon is 0.10 replaced by a surface (as, e.g., in the gravastar case) or by a throat (as in the wormhole case), the potential also develops 0.05 outgoing at infinity ingoing at horizon a minimum (i.e., an innermost stable PS) which can trap 0.00 low-frequency modes [12,15,28–30] (cf. Fig. 1). This inner 0.15 wormhole PS can also be thought of as being caused by the centrifugal 2 0.10 barrier, and it may become nonlinearly unstable [12]. These ) M * modes make their way to the waveforms in Fig. 2 in the 0.05 outgoing at infinity V(r trapped outgoing at infinity form of “echoes” of the initial PS modes after they leak 0.00 0.15 through the potential barrier: the radiation pulse generated centrifugal barrier star-like ECO at the potential barrier peak (the PS modes) is then trapped 0.10 in a semipermeable cavity bounded between the two PSs.

0.05 outgoing at infinity Indeed, the time delay between two consecutive echoes is regular at the center trapped roughly the time that light takes for a round trip between the 0.00 -50 -40 -30 -20 -10 0 10 20 30 40 50 potential barrier. In general, this delay time reads r /M * Z 3M dr FIG. 1. Qualitative features of the effective potential felt by Δt ∼ 2 pffiffiffiffiffiffiffi ; ð5Þ perturbations of a Schwarzschild BH compared to the case of rmin FB wormholes [12] and of starlike ECOs with a regular center [22]. The precise location of the center of the star is model dependent where r is the location of the minimum of the potential and was chosen for visual clarity. The maximum and minimum of min shown in Fig. 1. If we consider a microscopic correction at the potential corresponds approximately to the location of the l ≪ unstable and stable PS, and the correspondence is exact in the the horizon scale ( M), then the main contribution to eikonal limit of large angular number l. In the wormhole case, the time delay comes near the radius of the star and modes can be trapped between the PSs in the two “universes.” In therefore, to which perturbations can be lost, the spectrum is expectedthe starlike to case, be real modes, and arediscrete trapped, classified between the by its PS angular and the   – l momentum eigenvalue. The discrete frequency spectrumcentrifugal of barrier normal near modes the center of the of thesurface star [28 oscillations30]. In all cases will Δt ∼−nM log ; l ≪ M; ð6Þ provide a clear basis for distinguishing gravitationalthe potential condensate is of stars finite from height, BH and ringdown the modes modes leak away, in GW with M higher-frequency modes leaking on shorter timescales. signatures. Predicting these characteristic frequencies of the surface prior to possible detection in present and where n is a factor of order unity that takes into account the planned GW antennas, such as aLIGO II will be a clean and powerful test of the theory. structure of the objects. For wormholes, n ¼ 8 to account for the fact that the signal is reflected by the two maxima in Another striking observational signature of a non-singular horizonless ‘BH’ are ‘echoes’ after a merger 25 25–27 event. a characteristic time ∆t ' rS ln (1/) after the coalescence and main burst of GWs. The echo GRAVITATIONAL-WAVEsignal delay isthe result SIGNATURES of (possibly OF multiple) EXOTIC … reflection(s) from the interiorPHYSICAL centrifugal REVIEW barrier: D 94, Fig.0840311. (2016) 6 0.15 0.4 black hole 4 On the other hand, when the Schwarzschildwormhole horizon is wormhole 0.10 replaced2 by a surface (as, e.g., in the gravastar case) or by a 0.2 throat0 (as in the wormhole case), the potential also develops 0.0 0.05 outgoing at infinity -0.2 ingoing at horizon a-2 minimum (i.e., an innermost stable PS) which can trap 0.00 low-frequency-4 modes [12,15,28–30] (cf. Fig. 1). This inner -0.4 0.15 wormhole PS4 can also be thought of as being caused by theempty centrifugal shell 0.5 empty shell 2 2 (t) 0.10 barrier, and it may become nonlinearly unstable [12]. These (t) ) M 10 10 * 0 0.0 Ψ Ψ modes make their way to the waveforms in Fig. 2 in the 0.05 outgoing at infinity V(r trapped outgoing at infinity form-2 of “echoes” of the initial PS modes after they leak -0.5 0.00 -4 0.15 through the potential barrier: the radiation pulse generated centrifugal barrier star-like ECO gravastar gravastar at4 the potential barrier peak (the PS modes) is then trapped 0.5 0.10 in2 a semipermeable cavity bounded between the two PSs. 0 0.0 0.05 outgoing at infinity Indeed, the time delay between two consecutive echoes is regular at the center -2 -0.5 trapped roughly the time that light takes for a round trip between the 0.00 -4 -50 -40 -30 -20 -10 0 10 20 30 40 50 potential-100 barrier. 0 In general, 100 this 200 delay time 300 reads 400 100 200 300 400 r /M t/M t/M Figure 1: LeftPanel: Effective* potentials experienced by GW perturbations for a Schwarzschild BH, wormhole, and 3M dr −6 FIG.star-like 1. Qualitative Extreme featuresCompact of Object the effective (ECO) potential such as felta gravastar byFIG. 2. respectively, Left: A dipolar showing (Δl ¼t ∼ the12, m inner¼ 0 centrifugal) scalar; wave barrier packet in scattered the5 off a Schwarzschild BH and off different ECOs with l ¼ 10 M 2 000001 r pFB ð Þ perturbationslast two cases. of a SchwarzschildRight Panel: The BH characteristic compared to the GW case echoes of(r0 ¼ produced. inM the). The latter right ECO panel cases,Z min shows emerging the late-time characteristic behavior of the waveform. The result for a wormhole, a gravastar, and a simple wormholes [12] and of starlike ECOs with a regular center [22]empty. shell of matter are26 qualitatively similar and display a series of “echoes” which are modulated in amplitude and distorted in times ∆t, 2∆t, . . . later after multiple refelections from the centrigugal barrier. ffiffiffiffiffiffiffi The precise location of the center of the star is model dependentfrequency.where Forr thisis compactness, the location of the the delay minimum time in Eq. of the(6) potentialreads Δt ≈ 110M for wormholes, Δt ≈ 82M for gravastars, and Δt ≈ 55M for andSince was chosen∆t is for logarithmic visual clarity. in The, maximum an echo and signal minimum should of emerge withinmin 90 to 100 r or a few to tens of msec emptyshown shells, in respectively. Fig. 1. If we considerS a microscopic correction at theafter potential the coalescencecorresponds approximately and formation to the of location a gravastar. of the The form of the reflecting barrier in the non-rotating unstable and stable PS, and the correspondence is exact in the the horizon scale (l ≪ M), then the main contribution to eikonalcondensate limit ofstar large final angular state number is completelyl. In the wormhole determined case, by thethe interior time delay de Sitter comes geometry. near the With radius the of surface the star layer and modesalso can determined be trapped between we can the compute PSs in the both two “ theuniverses. time delay” In andtherefore, amplitude of the GW echo signal, allowing also084031-3 thefor starlike a general case, modesfrequency are trapped dependent between scattering the PS and loss the and attenuation due to non-linear SM interactions at the l centrifugalsurface. barrier This near will the provide center of a thedetailed star [28 signature–30]. In all template cases for GW detectorsΔt ∼−nM andlog GW data; analysis.l ≪ M; 6 the potential is of finite height, and the modes leak away, with M ð Þ   higher-frequencyAny detection modes of an leaking echo on signal shorter would timescales. be clear observational evidence of a non-singular interior to a ‘BH’ where n is a factor of order unity that takes into account the and physics beyond Einstein’s GR in fully collapsed stars.structure The of startling the objects. possibility For wormholes, thus presentsn 8 to itself account of testing quantum effects in gravitational physics, peeringfor into the the fact interior that the of signal BH-like is reflected objects by for the the¼ two first maxima time. in Even more strikingly, if EM waves are produced in the collision and coalescence event, they will also be trapped by the same gravitational potential, and also emerge after the same characteristic echo delay time, in sharp6 contrast to the behavior expected of a BH. Either or both of these multi-messenger signals would be wormhole 0.4 wormhole smoking4 guns for new physics at and beyond the ‘BH’ horizon, and confirmation of the effective theory. 2 0.2 An0 in-depth analysis of the gravitational condensate star0.0 alternative to BHs is particularly timely now, as -2 -0.2 the-4 possibility of gravitational echo signals being emitted-0.4 from a ‘BH’ merger is being actively considered, and claims of their detection have been made,27,while a subsequent analysis found just a 1.5 σ significance 4 empty shell 0.5 empty shell 2 28 (t) for the echo signal . This situation is poised to change(t) with more GW data. The limitations of low S/N 10 10 0 0.0 Ψ shouldΨ be gradually ameliorated by improved sensitivity of an expanded GW detector network, multiple -2 -0.5 detections,-4 making stacked analysis with improved statistics possible, and the potential for EM counterpart

observations. In order to take full advantagegravastar of these improved observations, they must be matchedgravastar by more 4 0.5 accurate2 predictive theory based on the well-motivated EFT of (1). Any positive detection of a GW echo 0.0 signal0 would revolutionize BH physics and reverberate to the foundations of GR vis-a-vis´ quantum theory. -2 -0.5 -4 -100 0 100 200 300 400 100 200 300 400 t/M 3 t/M

FIG. 2. Left: A dipolar (l 1, m 0) scalar wave packet scattered off a Schwarzschild BH and off different ECOs with l 10−6M ¼ ¼ ¼ (r0 2.000001M). The right panel shows the late-time behavior of the waveform. The result for a wormhole, a gravastar, and a simple empty¼ shell of matter are qualitatively similar and display a series of “echoes” which are modulated in amplitude and distorted in frequency. For this compactness, the delay time in Eq. (6) reads Δt ≈ 110M for wormholes, Δt ≈ 82M for gravastars, and Δt ≈ 55M for empty shells, respectively.

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