Quantum Love and ringdown near extremality

Introduction to the show: ident: High Energy Physics: Theory. The day in high-energy theory: gravity, strings, fields and the questions left open.

Kai: We're looking at the paper: "Quantum Love and ringdown near extremality".

Mira: Here's the gist of the paper. In this paper we discuss the quasinormal modes and the tidal responses of four-dimensional asymptotically flat Reissner-Nordström black holes near extremality and investigate their nontrivial near-horizon quantum corrections Near extremal black holes are governed by an effective Schwarzian action coupled to boundary gauge modes, which is then coupled with a massless scalar field outside The near-horizon dynamics of near extremal black holes are generally governed by an effective Schwarzian action coupled to boundary gauge modes This setup allows for the computation of a 2 → 2 scattering process between a black hole and a scalar particle, yielding the quasinormal mode spectrum and recovering predicted zero damped modes at leading order near extremality The study reveals that static Love numbers are non-zero in contrast to classical expectations, with logarithmic dependence on energy as the near-extremal energy becomes sufficiently large Near-horizon quantum effects non-trivially change the tidal response of near-extremal black holes Effective Model and Setup The low energy near-horizon dynamics of near-extremal black holes are effectively governed by a one-dimensional theory on the boundary of the near-horizon region, which emerges from explicit and slight breaking of asymptotic symmetries of AdS2 A universal sector of this theory is the Schwarzian action arising from the breaking of time reparametrizations, coupled with some gauge fields corresponding to other isometries and gauge fields in four dimensional spacetime The effective Hamiltonian describing the near-horizon dynamics is given by H0 ≡ HSch + HSU(2) = E−10hSch(f, t) + Tr g−1∂tg2i, where f(t) is the time reparametrization mode and Sch(f, t) = f′′′f′ − 3/2f′′f’/2 The states in this theory are given as H0 E, L, M⟩ = (E + E0L)E, L, M⟩ The density of states around fixed (E, L, M) ≡ E is given by ρ(E) = E−10(2L + 1)eS0ρS(E) = eS02π2E0(2L + 1)sinh 2πr 2E/E0!Θ(E)> The near-horizon dynamics is described by the Hamiltonian

42: Decay Rate Analysis The decay rate of a black hole with available energy E and angular momentum L is computed using first-order perturbation theory induced by the interaction (2.6) The amplitude of this process is given by A(E → Ef, ω) = −iZ∞−∞dtXl′m′Nl′Zdω'O∆'(Ef, E)eitEf +E0Lf−E−E0L Taking the squared amplitude and dividing by the infinite time integral yields the decay rate γ(E) = 2πZdωZdEfρ(Ef)N2l uωO∆(Ef, E)2δ Ef + E0Lf + ω − E − E0L The Schwarzian part of the decay rate is given by the diagram EfE O∆ O∆ = Γ∆EfE22∆−1eS0Γ(2∆)(Lf L lMf M m)2 In the semiclassical limit, the decay rate simplifies to γsc(E, L) ≃ 4ζ(3)β−3Er0 = 4ζ(3)r20E02π/2 The quantum decay rate in the quantum regime is found to be γqu(E) ≃ 4√2E015π(r0E0) 2/2 In the quantum limit, the effective temperature can again be defined from the density of states as βE = ρ'S(E)/ρS(E) ≃ 1/2E The decay rates of near-extremal black holes are suppressed in both the semiclassical and quantum regimes by factors of (r0E0) squared Scattering Process and QNMs The 2 → 2 scattering amplitude is given by A(i) = −iZ∞−∞dt1Zt1−∞dt2X(lm)(l′m')Nl1Nl′AF⟨E2e iH0t1O∆e−iH0t1+iH0t2O∆'e−iH0t2E1⟩ The amplitude takes the form A(E → Ef, ω) = −2πiNlu∗ωO∆(Ef, E)δ Ef + E0Lf + ω − E − E0L The poles of the integrand give the quasinormal modes, and in leading order they recover known results of zero damped quasinormal modes

7, 41: for near-extremal black holes in the semiclassical regime In the quantum regime, we find that in leading order the quasinormal modes of this system stay the same but their functional form in terms of temperature changes The damping modes are given by ω = l(l + 1) − n2E02−i p2E0En, n ≥ l + 1 The leading behaviour of the low-lying quasinormal modes have the same structures in both semiclassical and quantum regimes In the semiclassical regime E1 ≫ E0, the imaginary part dominates over the real part in E1 > E0 The semiclassical result can also be written in a familiar form of zero-damped modes in the literature

7, 41: Static Love Numbers The static Love number is characterized by finding the real part of the amplitude (4.8) and matching it with the tidal-part of the scattering amplitude associated with static Love numbers in worldline-EFT to fix them In the zero threshold case, we find that for both classical and quantum black holes, l = 0 Love numbers vanish This is consistent with the shift symmetry of the massless scalar

11: For non-zero threshold energy, we find non-trivial response in both semiclassical and quantum black hole backgrounds with generally divergent pieces The required operator is given by λˆl(H0) = −4/24l+4πΓ(2l + 2)Γ(l + 3/2)2P 2H0E0 + λˆren l(H0) The quantum-corrected static Love numbers for arbitrary k1 in the near-extremal regime are given by λ(0)l,near−ext = (2l + 1)!!

Kai: (GN /r20) 2l+1/24l+5π l!

Mira: Γ(2l + 2) Γ(l + 3/2)2 × QqE−10E1 − 2πaP(k 21) sin(2πµ)cosh(2πk1 − cos2µ)+ λˆren l2E−10E1 The leading order static Love numbers close to extremality are given by λ(0)l,ext = GN/r20 2l+1 hbl + λˆren l2E−10E1 Discussions In this paper we have studied the ringdown and tidal deformation effects of near-extremal black holes which are related to scattering processes on the black hole backgrounds We consider a non-rotating extremal background with horizon size r0 and consider near-extremal solutions as deviation of this geometry which are parameterised by the inverse temperature β ≫ r0 The effective theory consists of Schwarzian and SU(2) modes with a coupling constant E0 We coupled this is effective quantum mechanical theory to an external massless scalar field The coupling is such that a particular mode in the far-horizon quantum field theory acts as a source from the near-horizon perspective and generates a source-operator coupled term in the effective Hamiltonian The energy E of the Schwarzian states parametrise the mass above extremality and angular momentum corresponds to rotation above extremality The real shift in the internal energy is suppressed and it can be neglected for the purposes of this paper This choice fixes the operator at leading order and shows that as energy increases, the Love numbers are definitely nonzero Near-horizon quantum gravitational fluctuations will generically induce static Love numbers for non-rotating near-extremal black holes Counterterms and Renormalization The divergent piece of order δ−1 in the amplitude (6.

Title and authors: Kai: So we’re diving into "Quantum Love and ringdown near extremality" to see what the authors actually built and measured. They start by setting up this effective model, which is based on a Schwarzian action coupled to boundary gauge modes, which they say describes the near-horizon dynamics of these black holes.

Mira: That setup is key because it lets them compute a two-to-two scattering process between the black hole and some massless scalar field that lives outside. This coupling is what allows them to find those quasinormal modes, which are essentially the characteristic frequencies of how the black hole vibrates after a disturbance.

Lev: From an engineering standpoint, that setup tells us how you'd need to model this on real hardware; you’re dealing with this effective Hamiltonian H0 which includes the Schwarzian part and some SU(two) modes, and they define states by those energy L, M labels <ref:2610.03708#pg1>.

Kai: And they give us a density of states around a fixed energy E, which is described by something like eS02pi2E0(2L + one)sinh 2pi r 2E/E0 <ref:2610.03708#pg1>!Theta(E), and that tells us how many modes are available at a given energy.

Mira: They use this to compute the decay rate, gamma(E), and they find that in the semiclassical limit, this decay rate simplifies to something like four zeta(three) beta-3Er0, which is suppressed by factors involving r0E0 for both semiclassical and quantum regimes.

Lev: When you look at the scattering amplitude calculation, they use a formula that involves terms like X(lm)(l'm')Nl1Nl'AFlangle E2e iH0t1O e-iH0t1+iH0t2O 'e-iH0t2E1, which is a second-order perturbation theory approach.

Kai: And the poles of that scattering amplitude at complex frequencies are what give us the quasinormal modes, and they mention that in leading order, these recover those known results of zero damped modes near extremality in the semiclassical regime.

Mira: But then they go into the quantum regime, and they say that while the quasinormal modes stay essentially the same in leading order, their functional form changes based on temperature.

The paper's summary: Kai: So to summarize what this paper is doing, it’s taking those near-horizon dynamics governed by the effective Schwarzian action and coupling it to a scalar field, and using that to study two things: the quasinormal modes and the tidal responses of four-dimensional asymptotically flat Reissner-Nordström black holes near extremality.

Mira: The main finding they point out is that static Love numbers are non-zero, which goes against what you might expect from classical physics, especially when we consider the logarithmic dependence on energy as the near-extremal energy gets quite large.

Lev: That non-zero response suggests a physical mechanism beyond the simplest semiclassical picture of black hole perturbations. It implies that quantum effects are definitely playing a role in how these black holes deform when they're tidally disturbed.

Kai: They also look at the decay rate, and they find it is suppressed in both the semiclassical and quantum regimes by factors of (r0E0) squared <ref:2610.03708#pg1>. That suppression is significant because it tells us that these processes are quite slow in both limits.

Mira: And they specifically address the improvements suggested for this work, which involves developing a dual representation of the interaction Hamiltonian for scattering processes, which lets them compute the amplitude using those terms like X(lm), (l'm')Nl1Nl2.

Lev: That dual representation is crucial because it’s what allows you to properly incorporate those complex energy shifts in the perturbation theory calculation, which is necessary for getting the full scattering result.

Kai: So they aren't just looking at one thing; they are trying to improve the mathematical machinery used to calculate these observables by going deeper into the interaction structure of the system.

The paper's improvements: Mira: The authors suggest that a major improvement is modeling the static Love numbers, which they characterize by matching the real part of their scattering amplitude with the tidal-part of a worldline effective field theory calculation to fix those numbers.

Kai: They found that in the zero threshold case, both for classical and quantum black holes, these Love numbers vanish, which aligns with what you expect from shift symmetry for the massless scalar field.

Lev: But when you look at non-zero threshold energy, they find non-trivial response in both semiclassical and quantum backgrounds, including generally divergent pieces that require a specific operator lambdaˆl(H0) to fix them.

Mira: For the quantum-corrected static Love numbers for arbitrary k one in the near-extremal regime, they give a formula where the leading order close to extremality is given by lambda(zero)l, ext = GN/r0 squared two l+one h bl plus that renormalization term <ref:2610.03708#pg2>.

Kai: This means that even when you're very close to extremality, the response isn't just zero; it has this specific dependence on l and the black hole parameters, which is what we need to measure.

Lev: And they also mention how they can characterize energy-dependent tidal response by showing that the exact Love number depends on your renormalization scheme, and it shows a (E1) dependence which doesn't vanish for arbitrary black holes due to a choice of renormalisation.

Conclusion: Kai: So wrapping up the paper "Quantum Love and ringdown near extremality," we see that even in the quantum regime, leading order quasinormal modes stay the same but their temperature dependence changes, and that static Love numbers are definitely non-zero when you look at non-zero threshold energy.

Mira: The implication here is that near-horizon quantum gravitational fluctuations generically induce static Love numbers for non-rotating near-extremal black holes, which is a key result they highlight.

Lev: For me, what’s important is that the decay rates are suppressed by (r0E0) squared in both semiclassical and quantum regimes, meaning the black hole ringdown is quite slow compared to what you might expect classically <ref:2610.03708#pg1>.

Kai: Yeah, and the whole point of studying this setup was to see how these near-horizon corrections change things—whether it's the QNMs or those tidal responses.

Mira: This work gives us a concrete way to predict how quantum gravity effects will manifest in the deformation of black holes right at their most extreme configurations, and that’s a big piece of information for testing general relativity predictions.

Lev: It sets up the framework for what you need to run on real hardware, showing where the overhead lands when you try to simulate these effects using error correction techniques.

Kai: So that's it for this paper on "Quantum Love and ringdown near extremality." We’ll be looking at other papers soon.