Lifshitz scaling, ringing black holes, and superradiance
YYITP-21-07, IPMU21-0006
Lifshitz scaling, ringing black holes, andsuperradiance
Naritaka Oshita a Niayesh Afshordi a,b
Shinji Mukohyama c,da
Perimeter Institute, 31 Caroline St, Waterloo, Ontario N2L 2Y5, Canada b Waterloo Centre for Astrophysics, University of Waterloo, Waterloo, ON, N2L 3G1,Canada c Center for Gravitational Physics, Yukawa Institute for Theoretical Physics, Kyoto Uni-versity, 606-8502, Kyoto, Japan d Kavli Institute for the Physics and Mathematics of the Universe (WPI), The Universityof Tokyo, Kashiwa, Chiba 277-8583, Japan
E-mail: [email protected] , [email protected] , [email protected] Abstract:
We investigate the ringdown waveform and reflectivity of a Lifshitzscalar field around a fixed Schwarzschild black hole. The radial wave equation ismodified due to the Lorentz breaking terms, which leads to a diversity of ringdownwaveforms. Also, it turns out that Lifshitz waves scattered by the Schwarzschildblack hole exhibits superradiance. The Lorentz breaking terms lead to superlumi-nal propagation and high-frequency modes can enter and leave the interior of theKilling horizon where negativity of energy is not prohibited. This allows the Lif-shitz waves to carry out additional positive energy to infinity while leaving negativeenergy inside the Killing horizon, similar to the Penrose process in the ergosphereof a Kerr spacetime. Another interesting phenomenon is emergence of long-livedquasinormal modes, associated with roton-type dispersion relations. These effectsdrastically modify the greybody factor of a microscopic black hole, whose Hawkingtemperature is comparable with or higher than the Lifshitz energy scale. a r X i v : . [ g r- q c ] F e b ontents The Hořava-Lifshitz (HL) gravity theory [1] is one of the most promising candidatesfor a quantum theory of gravity. In this theory, space ( x ) and time ( t ) are anisotropicand they follow the Lifshitz scaling x → b x , t → b z t, (1.1)where b is a constant and z is the dynamical critical exponent. In order for thetheory to be renormalizable (at least in a power-counting level ), z = 3 is required .The theory of HL gravity has advantage not only in the context of quantum gravitybut also in cosmology. For example, the Hamiltonian constraint in the projectableHL gravity is not a local equation but an integrated equation, which allows theemergence of dark matter as integration constant [3, 4]. Also, the superluminalitycaused by the anisotropy of the z = 3 HL gravity can solve the horizon problem andlead to scale-invariant cosmological perturbations [5, 6]. This means that the HL It was rigorously shown in [2] that the projectable HL theory is renormalizable. For z > the theory is super-renormalizable at least in the sense of power-counting. – 1 – igure 1 . A schematic picture showing the superradiant scattering around (a) a Kerrblack hole and (b) Schwarzschild black hole with a Lifshitz field. The negativity of energyis allowed in the region where spacetime is superluminally dragged such as the ergosphere ofa Kerr black hole or the interior of Killing horizon of a Schwarzschild black hole. Therefore,modes leaving from such a region can carry out additional positive energy to infinity whileleaving negative energy there. This is nothing but the superradiance effect. gravity provides an alternative to inflationary cosmology . In this sense, the theoryof HL gravity is well-motivated by cosmological considerations while being one ofthe promising candidates for the theory of quantum gravity.However, the Lifshitz scaling breaks the Lorentz symmetry of gravity and theLorentz breaking is significant at short-length scales of (cid:46) /M HL . Could any novelphenomena happen if spacetime has such a microscopic length scale caused by theanisotropy between space and time? In this paper, we will consider this interestingissue by focusing on the perturbation of a static black hole. Black hole perturbationtheory demonstrates many non-trivial features, such as superradiance [9, 10] andthe universality of the late-time ringdown waveform [11, 12]. Although the ringingbehavior has been investigated in the framework of the HL gravity with a covariant(no-higher-derivative) scalar field [13] , to the best of our knowledge, it has not yetbeen investigated how the spatial higher-derivative terms affect the ringing behaviorand reflectivity of black holes.By numerically solving the Lifshitz field equation, we find that the standard be-haviour of ringing black hole, i.e. the exponential suppression and constant-frequencyoscillation (see [12] for a review), is not guaranteed. Depending on the parameterscharacterizing the Lifshitz theory, we find novel features such as a power-law tailwithout ringing oscillations or long-lived quasinormal modes. We also compute thereflectivity of scattered Lifshitz waves by a Schwarzschild black hole and find that The flatness problem may also be addressed by the Lifshitz scaling [6, 7]. For a review of the cosmological aspect of the HL gravity as well as the "Vainshtein screening"for the scalar graviton, see Ref. [8] Note, however, that ref. [13] considers a non-projectable HL theory without terms dependingon the spatial derivatives of the lapse function and such a setup is known to be inconsistent. – 2 –t exhibits the superradiance even though the black hole has no angular momen-tum. This is not surprising since the Killing horizon is no longer a causal boundarydue to the superluminality of the Lifshitz field, and high-frequency modes couldenter and leave the interior of the Killing horizon while low-frequency modes arestill trapped. This is very similar to the nature of ergosphere of a Kerr black holewhere co-rotating modes can enter and leave the ergosphere and the counter-rotatingmodes are trapped. The (superluminal) spacetime dragging inside the ergosphere al-lows negative energy to exist, which allows co-rotating modes to carry out someadditional positive energy to infinity while leaving negative energy in the ergosphere.Similarly, negative energy can exist inside the Killing horizon of a Schwarzschildblack hole, and superluminal modes could extract some positive energy out of thehorizon (FIG. 1). In contrast, if the superluminal propagation is prohibited due toLorentz invariance, the negative energy inside the horizon is causally disconnectedfrom outside, which is why the superradiance never occurs for standard Schwarzschildblack hole. We expect that the superradiance would be more significant for a smallerblack hole whose Hawking temperature is higher than the Lifshitz energy scale M HL .This may drastically change the greybody factor and evaporation rate at the finalstage of black hole evaporation [14, 15], which might be testable via the observationof stochastic gravitational waves (GWs) since some small primordial black holes (ifexisted) would have evaporated at the early stage of the Universe and may havecaused the sudden reheating process, which results in inducing stochastic GWs [16].The superradiance is closely related to Penrose process and the latter is expectedto occur in theories with spontaneous breaking of the Lorentz symmetry [17]. Theargument is that when particles of different species interact with the ghost condensate[18] and their propagation speeds are different, two apparent horizons appear withdifferent radii and the Penrose process is made possible in the region between the twohorizons. In Ref. [19], the apparent violation of the generalized second law (GSL)was studied in that setup, and their gedanken experiment showed that a perpetuummobile involving a black hole and two thermal shells could be realized. However,this is not the case [20, 21], at least in the original ghost condensation scenario, sincethe accretion rate of the ghost condensate onto the black hole [22], which increasesthe black hole entropy, overwhelms the effect of the perpetuum mobile . In oursituation, unlike the case with ghost condensate [20, 21], the GSL may be violateddue to the Penrose process, and so, one might wonder if it allows construction of aperpetuum mobile of the second kind. The hierarchy between two different Hawkingtemperatures is essential in the above gedanken experiment [19]. On the other hand,in our situation the universal horizon is the unique horizon and the temperature See also [23] for the compatibility of the ghost condensate with the de Sitter entropy boundintroduced in [24] as a closely related issue. It was pointed out that the universal horizon is unstable against the perturbations [25]. How-ever, it takes infinite time to form the universal horizon in the preferred frame while the evaporation – 3 –ssociated with the universal horizon is also uniquely determined [26, 28]. Therefore,the perpetuum mobile would not be allowed in our case, at least in the same manneras Ref. [19]. On a separate note, it is even clear if the violation of the GSL isproblematic. For example, the Hawking-Moss transition [29] also violates the GSL[30–33] where the cosmological horizon shrinks. Moreover, the Jarzynski equality[34, 35] in the non-equilibrium statistical mechanics implies that the second law ofthermodynamics can be violated.In the next section, we introduce a simplified model of the HL gravity, wherethe tensor perturbation is modeled by a massless scalar field ψ , and briefly reviewthe appearance of a preferred frame and a universal horizon due to the extra scalardegree of freedom ϕ , often called Khronon. We also explain the methodology ofour numerical computation. In section 3, we will show our results of the black holeringing at late time. Also, the reflectivity of scattered waves around the black holeis investigated, and it is found out that the superradiance occurs due to the Lifshitzscaling. In section 4, we summarize our achievements and discuss the possibility ofa perpetuum mobile in our case. We will use the notation ( − , + , + , +) throughoutthe manuscript. We will investigate the following simplified model to see how the universal featuresof the ringdown and reflectivity of a static black hole are affected by the Lifshitzscaling: L = (cid:90) d x √− g [ L EH + L SG + L GW ] , (2.1) L EH ≡ πG R, (2.2) L SG ≡ πG (cid:8) α ( u µ ∇ µ u ν ) − β ∇ µ u ν ∇ ν u µ − γ ( ∇ µ u µ ) (cid:9) , (2.3) L GW ≡ − ψ ( F (∆) + (cid:50) ) ψ, (2.4)where ψ is a scalar field modeling the tensor perturbation, F (∆) ≡ ∆ /M − ν ∆ /M , ∆ is the Laplacian on the constant- ϕ hypersurfaces (the definition of time is finite. Therefore, the stability of a pparent universal horizon originating from a gravitationalcollapse is still open question. Here we implicitly assume that all matters have the same power of momentum in their dispersionrelation at high energies. Otherwise, as shown in [26], the Hawking temperature of the universalhorizon is not unique. Also, the temperature associated with the universal horizon depends onvacuum choice. According to [26], the inconsistency between the results in [27] and [28] can beexplained by the difference of vacuum choice. However, in either case, the Hawking temperaturecan be unique for the universal horizon. – 4 –hich will be given in subsection 2.2), and ν is a constant of the order of unity. Theunit normal vector u µ is expressed in terms of the khronon field ϕu µ ≡ ∂ µ ϕ √∇ ν ϕ ∇ ν ϕ . (2.5)This theory models a situation where the background is given by a solution of theEinstein equations in general relativity whereas gravitational perturbations (mod-eled by a scalar field ψ ) follows the Lifshitz scaling at short-length scales. To discusswhat situations can be covered by this simple model, we will come back to thispoint in the discussion section. In this paper, we will consider the Schwarzschildbackground, whose line element is ds = − (cid:16) − r s r (cid:17) dt + (cid:16) − r s r (cid:17) − dr + r d Ω , (2.6)and investigate how the Lifshitz scaling affects the scattering process around thestatic black hole. The theory in (2.1) has a preferred direction given by u µ , which stems from thepreferred frame ( ϕ = const. ) one should respect. We here briefly review the preferredframe and the universal horizon based on Ref. [25]. The dynamics of the khrononfield induces the preferred frame. The khronon field equation in the Schwarzschildbackground is given by [25] ∂ ξ UU − c χ ∂ ξ VV + 2 c χ ξ = 0 , (2.7)where c χ ≡ (cid:112) ( β + γ ) /α and U ≡ u t , V ≡ u r , ξ ≡ r s r = 1 r . (2.8)Here, we have set r s = 1 . The unit normal vector u µ satisfies ( u t ) − ( u r ) = 1 − ξ, (2.9)and so the relation between U and V is given by U − V = 1 − ξ. (2.10) The contribution of the scalar-graviton to the background spacetime can be negligible becausethe parameters α , β , and γ are assumed to be much smaller than unity. Indeed, α and β are requiredto be small by the observational constraints. On the other hand, either | γ | (cid:28) or γ = O (1) iscompatible with the constraints. See [36] and (2.17)-(2.19) below. – 5 –hoosing the branch with in-going u µ (i.e. V = u r < and thus plugging V = − (cid:112) U − ξ into (2.7), one obtains ∂ ξ U + c χ UU (1 − c χ ) − ξ (cid:20) − ( ∂ ξ U ) + ( U ∂ ξ U + 1 / U − ξ + 2( U − ξ ) ξ (cid:21) = 0 . (2.11)One can also rewrite the background metric as ds = − (1 − ξ ) dτ − V dτ dr ∗ + dr ∗ + r d Ω , (2.12)where dτ = dt − V − ξ dr ∗ and dr ∗ = dr/U . Note that r ∗ differs from the standarddefinition of tortoise coordinate in general relativistic black holes, as r ∗ → −∞ refersto the universal (not Killing) horizon where U → .When U = 1 , the metric (2.12) reduces to the one in the Gullstrand-Painlevécoordinates ds = − dτ + ( dr + (cid:112) ξdτ ) + r d Ω . (2.13)The sound horizon appears at ξ = ξ c that satisfies U ( ξ c )(1 − c χ ) = 1 − ξ c . (2.14)In order for the second term in (2.11) to be regular, one has to impose ∂ ξ U ( ξ c ) = 12(1 − c χ ) U ( ξ c ) − c χ (cid:115) − c χ (1 − c χ ) U ( ξ c ) ξ c . (2.15)Now imposing the boundary condition of U (0) = 1 and using the shooting method,one can numerically solve (2.11). When c χ → or c χ → ∞ , (2.11) has analyticsolutions U ( ξ ) = − ξ ( c χ → ) , (cid:113) − ξ + ξ ( c χ → ∞ ) . (2.16)In the next section, we will investigate the perturbations of the Lifshitz scalar fieldin the both limits: c χ → ∞ and c χ → . The two limits can be compatible with theobservational and theoretical constraints on the parameters obtained in Ref. [36].Most of the constraints are satisfied for α , β , γ (cid:28) . The non-trivial constraints arethe constraints on the parametrized post-Newtonian (ppN) parameters quantifyingpreferred-frame effects, which translate to [36] (cid:12)(cid:12)(cid:12)(cid:12) α − β )1 − β (cid:12)(cid:12)(cid:12)(cid:12) (cid:46) − , (2.17) (cid:12)(cid:12)(cid:12)(cid:12)(cid:18) α − β − α (cid:19) (cid:18) − ( α − β )(1 + β + 2 γ )(1 − β )( β + γ ) (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) (cid:46) − . (2.18) The vacuum Cherenkov constraint from the scalar graviton was not considered in [36] and thistreatment seems consistent with the decoupling limit implied by α , β , γ (cid:28) . This may not havebeen the case if γ = O (1) (see footnote 9). – 6 –lso, the observation of gravitational waves emitted from the event GW170817 withthe gamma ray emission put a stringent constraint on β | β | (cid:46) − . (2.19)Assuming β = 0 , for example, the ppN constraints become | α | (cid:46) − and ( | α | / | − c − χ | (cid:46) − . In the two cases, c χ (cid:29) and c χ (cid:28) , the latter constraint reducesrespectively to | α | (cid:46) × − ( c χ (cid:29) , (2.20) c χ (cid:38) | α | × × ( c χ (cid:28) , (2.21)Therefore, it turns out that the limit of c χ → ∞ is compatible with the constraintsonce taking a small value of α so that (2.20) is satisfied. The other limit c χ → maybe also compatible with them when taking the infinitesimal value of α . We will investigate the dynamics of incoming scalar waves around a static black holebased on the following wave equation: (cid:20) ∆ M − ν ∆ M + (cid:50) (cid:21) ψ ( t, r, θ, φ ) = 0 , (2.22)where ∆ ≡ D a D a and D a is the covariant derivative on the khronon surfaces, M HL is the Lifshitz energy scale, and ν is a dimensionless parameter. In order to obtainthe explicit form of D a , let us decompose the metric as ds = − N dτ + h ij ( dx i + N i dτ )( dx j + N j dτ ) . (2.23)Comparing it with (2.12), one can read N = (1 − ξ ) + V , N r = − V, h ij = diag (1 , r , r sin θ ) . (2.24)The definition of D i v j is D i v j = ∂ i v j + (3) Γ jik v k , (2.25)and (3) Γ jik is the Levi-Civita connections w.r.t. h ij : (3) Γ jik = 12 h jl ( ∂ i h lk + ∂ k h il − ∂ l h ik ) . (2.26)The explicit form of the Levi-Civita connections in three-space is presented in Ap-pendix A. Therefore, the Laplacian on the khronon surface is ∆ ψ = D i D i ψ = D i ∂ i ψ = ∂ i ∂ i ψ + (3) Γ iik ∂ k ψ = ∂ i h ij ∂ j ψ + (3) Γ iik ∂ k ψ. (2.27) But note that c χ → limit may lead to development of non-perturbative behavior due tocaustic formation. – 7 –et us explicitly write down the d’Alembertian by using the metric (2.12). Thecovariant part of the equation of motion is (cid:50) ψ = g µν ∂ µ ∂ ν ψ − g µν Γ αµν ∂ α ψ = 0 , (2.28)and the inverse metric g µν is g µν = − /U − V /U − V /U (1 − /r ) /U /r
00 0 0 1 / ( r sin θ ) . (2.29)Then the Lorentz breaking equation of motion (2.22) can be explicitly written as U (cid:18) − ∆ M + κ ∆ M (cid:19) Ψ+ (cid:20) ∂ τ − (cid:18) − r (cid:19) ∂ r ∗ + 2 V ∂ τ ∂ r ∗ + U (cid:96) ( (cid:96) + 1) r (cid:21) Ψ+ 1 U (cid:20) V (cid:48) (cid:18) − r (cid:19) − U V r (cid:21) ( ∂ τ + V ∂ r ∗ ) Ψ − Ur (cid:20) − r (cid:21) ∂ r ∗ Ψ + 2
U Vr ∂ τ Ψ = 0 , (2.30)where ψ = Ψ( t, r ) Y (cid:96)m ( θ, φ ) , (2.31) ∆ = ∂ r ∗ + 2 Ur ∂ r ∗ − (cid:96) ( (cid:96) + 1) r . (2.32)The explicit form of the quadratic and cubic Laplacians are shown in Appendix A. We numerically solve the wave equation (2.30) with the 4th-order Runge-Kuttamethod. First, we decompose it into two first-order differential equations with re-spect to τ d Ψ dτ = Π( τ, r ) , (2.33) d Π dτ = − U (cid:18) − ∆ M + κ ∆ M (cid:19) Ψ (2.34) − (cid:20) − (cid:18) − r (cid:19) ∂ r ∗ + U (cid:96) ( (cid:96) + 1) r (cid:21) Ψ − V ∂ r ∗ Π (2.35) − U (cid:20) V (cid:48) (cid:18) − r (cid:19) − U V r (cid:21) (Π + V ∂ r ∗ Ψ) (2.36) + 2 Ur (cid:20) − r (cid:21) ∂ r ∗ Ψ − U Vr Π , (2.37)– 8 –here we introduced a new function Π ≡ d Ψ /dτ , and we should solve Ψ( τ, r ∗ ) and Π( τ, r ∗ ) simultaneously. We compute the spatial derivative terms ∂ nr ∗ X ( X is Ψ or Π ) with the Mathematica ’s function
NDSolve‘FiniteDifferenceDerivative . Inthis function, the derivatives at the spatial boundaries are calculated with one-sidedformulas.In the following, we use two parameters Ξ and Ξ defined as Ξ ≡ ν M r s , Ξ ≡ M r s . (2.38)In this definition, the dispersion relation at infinity becomes ω = Ξ k + Ξ k + k , (2.39)where ω and k is a frequency and wavenumber with respect to τ and r ∗ , respectively.The ratio of the time step ∆ τ to the spatial step ∆ r ∗ ( λ ≡ ∆ τ / ∆ r ∗ ) should befixed with a value smaller than unity, in order to satisfy an approximate Courantcondition for the superluminal modes that propagate outside the light cone. InAppendix B, we simulate the wave propagation with λ = 0 . , . , and . , andfound out that the results converge for λ (cid:46) . when | Ξ | (cid:46) . and Ξ (cid:46) . ,provided that the typical value of the wavenumber k is of order unity. Therefore,we will use λ = 0 . in the following analysis. However, note that a computationinvolving stronger superluminal modes (e.g., Ξ (cid:29) . or Ξ (cid:29) . for typicalwavenumber of order unity) may need a smaller value of λ . In the Appendix B,we also confirm the consistency of our simulations with the known results (i.e., thefundamental quasinormal mode and reflectivity) of the Lorentz invariant case ( Ξ =Ξ = 0 ). We control numerical high-frequency unstable modes with the Kreiss-Oligerdissipation[37] of an amplitude of / .As an initial configuration of the Lifshitz scalar field, we assume wave packetcentered at r = r w with the form of Ψ( τ = 0 , r ∗ ) = exp (cid:20) − ( r ∗ − r w ) s (cid:21) cos ˜ ωr ∗ , (2.40) ˙Ψ( τ = 0 , r ∗ ) = − (cid:18) r ∗ − r w ) s cos ˜ ωr ∗ + ˜ ω sin ˜ ωr ∗ (cid:19) exp (cid:20) − ( r ∗ − r w ) s (cid:21) . (2.41)We use s = 2 , ˜ ω = 1 throughout the analysis, meaning that the typical value ofthe wavenumber k in the dispersion relation (2.39) is of order unity. Although thisis purely ingoing waves at infinity for Ξ = Ξ = 0 , this leads to partial outgoingmodes when the dispersion relation is modified due to the Lifshitz scaling. Thereare two types of modified dispersion relation: Ξ > and Ξ < . In analogy tosuperfluid perturbations, we call the latter case a roton dispersion relation, as itcould lead to backward propagation in a range of frequency/distance from the blackhole. The sign of the sixth-derivative terms should be positive in order to guaranteethe renormalizability and the UV stability.– 9 – Results
With the initial condition (2.40) and (2.41), we numerically solve for Ψ( t, r ∗ ) , follow-ing the prescription outlined in Section 2.3. Some snapshots of these solutions areshown in Appendix C. In this section, we present and analyze the detailed resultsfor the ringdown and reflectivity of the black hole. In order to see the late-time behavior of ψ with the Lifshitz scaling, we calculate Ψ( t, r ∗ o ) , where r ∗ o is the position of an observer. The results are shown in FIG. 2.For the case of Ξ > (left panel of FIG. 2), the quasinormal ringing are suppressedand the power-law tail appears earlier than the Lorentz invariant case (grey line inFIG. 2). Also, high-frequency modes appear earlier than the power-law tail. Theform of the tail is universal and independent of the parameters of Lifshitz scaling.This is consistent with the fact that the late-time tail is caused by back-scattering offthe background curvature, and therefore, its power depends only on the asymptoticbackground spacetime [12]. On the other hand, the ringdown lasts longer for theroton dispersion relation, Ξ < . In this cases, as we show below, the group velocity v g ≡ dω/dk is suppressed (enhanced) at the intermediate (high) frequencies in thiscase, and therefore wavepackets are dispersed significantly in space/time. We believethis is the origin of the long-lived ringing at late time. We computed the spectrumof the long-lived modes for (Ξ , Ξ ) = ( − . , . and found that the dominantmodes are around ω ≈ . (black dashed line in FIG. 3-b), which can be explainedusing the following simple analytic estimate: To see this, let us simplify the modifieddispersion relation as ω = (cid:18) − r (cid:19) k + 2 V ( r ) kω + U ( r ) (cid:0) Ξ k + Ξ k (cid:1) . (3.1)For the case of c χ = 0 , the group velocities of ingoing and outgoing modes are v g ( r, k ) = − r − (cid:18) − r (cid:19) (cid:112) k + 3Ξ k < , (3.2) v g ( r, k ) = − r + (cid:18) − r (cid:19) (cid:112) k + 3Ξ k > , (3.3)respectively, and the position r = r ( ω, k ) is obtained by solving (3.1) r ( ω, k ) = ω flat ( k ) + k ω flat ( k ) − ω ) , (3.4) The first two terms in (3.1) represent the inward frame dragging and the last term gives theLifshitz scaling at high frequencies. We assume that the simplified dispersion relation captures theessence of wave propagation with the frame dragging and Lifshitz scaling. We also confirmed thatthe numerical result (FIG. 3) is well consistent with the analysis based on the simplified dispersionrelation (FIG. 4). – 10 – igure 2 . Time domain functions of | Ψ | with r ∗ o = 60 and r ∗ w = 80 . The cases of Lifshitzscaling with Ξ = 0 . (left) and Ξ = − . (right) are shown. The Lorentz invariant case(gray-dashed) is also shown for comparison. Figure 3 . (a) The time domain function for Ξ = − . and Ξ = 0 . . (b) The absolutevalue of the spectrum for the time domain function in the range of ≤ τ ≤ (red), ≤ τ ≤ (blue), and ≤ τ ≤ (black-dashed). with ω flat ( k ) ≡ √ k + Ξ k + Ξ k . The incoming and outgoing trajectories in thephase diagram ( v g − r plane) are shown in FIG. 4. Note that the ingoing andoutgoing trajectories are separated and do not describe the reflection at the angularmomentum potential as the modified dispersion relation shown in (3.1) does notinclude the potential term. Therefore, the trajectories around r (cid:46) (cid:96)/ω shown inFIG. 4 is not reliable. As shown in FIG. 4, the group velocity is indeed suppressedfor intermediate frequencies around ω (cid:39) . . This is consistent with the fact that thelate-time ringing within ≤ τ ≤ are dominated by the modes of ω ∼ . (seethe blue line in FIG. 3-(b)). On the other hand, the neighbouring modes ω ∼ . and ω ∼ . get out earlier (see the red line in FIG. 3-(b)), which is also consistentwith the analytically obtained trajectories in the phase space (FIG. 4) as the groupvelocities for ω = 1 . and ω = 1 . are higher than that for ω = 1 . .– 11 – igure 4 . The trajectories in the phase space obtained from (3.1). We use the sameparameters as in FIG. 3. The red and blue lines represent the outgoing and ingoing modes,respectively. The gray lines show the trajectories for the Lorentz invariant case ( Ξ = Ξ =0 ). As the next exercise, we numerically calculate the reflectivity of a black hole withthe (non-roton) Lifshitz scaling of Ξ > . Here, we implement the Fourier trans-formation for the ingoing and outgoing wavepackets, measured by the observer at r ∗ = r ∗ o , and calculate the reflectivity defined by the absolute value of the ratio be-tween the ingoing and outgoing Fourier coefficients. The result (FIG. 5) shows thatthe superradiance (i.e. Reflectivity larger than unity) occurs for Ξ > . One mightwonder why the superradiance occurs even though the black hole has no angularmomentum. In our situation, the superluminal propagation is allowed due to theLifshitz scaling, and the superluminal modes can enter and leave the interior of theKilling horizon where negative energy can exist as in the ergosphere of a Kerr blackhole. Therefore, superluminal modes of the Lifshitz scalar can access the interiorto carry out additional positive energy to infinity while leaving the negative energyinside the Killing horizon. One can also understand the superradiance effect due tothe Lifshitz scaling from the negativity of the angular momentum potential term.Let us show how the potential term is modified due to the Lorentz breaking terms F (∆) . We here define the potential term as the term which does not involve thederivative in the wave equation. Hence the modified potential term V ang ( r ) is V ang ( r ) = U (cid:20) (cid:96) ( (cid:96) + 1) r + Ξ D + Ξ (cid:18) D (cid:48)(cid:48) + 2 Ur D (cid:48) − (cid:96) ( (cid:96) + 1) r D (cid:19)(cid:21) , (3.5)where the definition of D = D ( r ) is given in (A.8). In FIG. 6, we plot the potentialterm including the Lorentz breaking effect and one can see that the negative energy– 12 – igure 5 . The reflectivity of the static black hole with the Lifshitz scalar field of Ξ = 0 . and Ξ = 0 . . The left and right panels show the frequency-dependence of reflectivity for c χ = 0 and c χ → ∞ , respectively. For (cid:96) = 2 and , the reflectivity exceeds unity, whichmeans that the non-spinning black hole exhibits the superradiance effect. Figure 6 . The angular momentum potentials for the various parameters. The negativeenergy region inside the potential barrier is deeper for larger values of Ξ ( > and Ξ . region locally appears inside the potential barrier, which can lead to superradiance.Therefore, we conclude that Lifshitz scaling could lead to superradiant scattering,even without angular momentum of the background black hole.Even if the energy scale of the Lifshitz scaling M HL is higher than the typicalfrequency of the ringing black hole, it eventually reaches M HL due to the evapora-tion of (an isolated) black hole. At the stage where the Hawking temperature iscomparable with M HL , the greybody factor would be drastically modified due to thesuperradiance effect. Therefore, our result implies that the final stage of the blackhole evaporation can be drastically different from the standard picture, provided thatthe Lifshitz scaling is ubiquitous at high energy scales.We compute the maximum values of the reflectivity as a function of the mass– 13 – igure 7 . The maximum values of the amplification factor (=(reflectivity) − ) for ν = 1 .The superradiance is observed for smaller values of r s . of black hole. Note that the non-dimensional parameters Ξ and Ξ increase as theblack hole shrinks and r s becomes smaller (see Eq. (2.38)). In FIG. 7, we showthe r s -dependence of the maximum value of reflectivity for ν = 1 . It is challengingto extend our computation to the case of M HL r s (cid:28) ( Ξ (cid:29) . ) since higher-frequency (highly superluminal) modes are involved and a smaller value of λ ( = time step/ spatial grid) is required to numerically resolve those highly superluminalmodes. Nevertheless, we expect that the trend would continue even for M HL r s (cid:28) because the negative-energy region inside the angular momentum potential becomesdeeper for a larger value of Ξ (smaller value of black hole mass) as is shown in FIG.6. On the other hand, the negativity of ν (i.e. the roton dispersion relation) makesthe negative-energy region inside the potential small, which results in quenching thesuperradiance. In this paper, we have investigated the effect of the Lifshitz scaling on the late-timeringing and reflectivity of a Schwarzschild black hole with the simplified model (2.1).We have considered a situation where the background is given by a Schwarzschildsolution whereas gravitational perturbations (modeled by a scalar field ψ ) followsthe Lifshitz scaling at short-length scales. Such a situation can be realized, forexample, by considering the following minimal theory of the HL gravity with staticand spherical symmetry of background L = (cid:90) d x √− g (cid:20) κ (cid:0) K ij K ij − λK + R (cid:1) + κ w C ij C ij (cid:21) , (4.1)where K ij is the extrinsic curvature, C ij is the Cotton tensor, R is the three-dimensional Ricci scalar, K ≡ tr [ K ] , and κ , λ , w are constants. The first three– 14 –erms reduce to L EH and L SG with c χ → ∞ , and the quadratic action for tensorialgravitational waves is modeled by the Lorentz invariant part of L GW of our simplifiedmodel (2.1). The second term including the Cotton tensor leads to the Lorentz break-ing terms corresponding to F (∆) in L GW . Let us note that if the scalar-gravitonbecomes dynamical under the renormalization group flow beyond some energy scale, M S , then the action (4.1) should contain other higher-order derivative terms (withrespect to scalar-graviton) for the theory to be renormalizable. Therefore, the min-imal theory (4.1) can be a low-energy effective theory of quantum gravity, providedthat the degree of freedom of the scalar-gravition can be traced out up to the in-termediate energy scales ∼ M HL (cid:28) M S . Our work would be applicable not only tosome specific situations in the HL gravity but also to the scattering problem of ablack hole in other higher-derivative gravity theories. For example, the consistenttheory of D → Einstein-Gauss-Bonnet gravity [38], that amends ambiguities andfatal problems in the proposal of [39], leads to spatial higher-derivative terms in thedispersion relation .We found out that the black hole ringing at late time disappears when the quarticderivative term is dominant with Ξ > . On the other hand, the black hole ringingexhibits long-lived modulation when Ξ < . We also showed that the Lifshitz wavesscattered around a static black hole exhibits superradiance. The superradiance isstronger for a smaller black hole as its quasinormal frequency becomes comparablewith or higher than M HL . This superradiance may significantly affect the evaporationprocess of a primordial black hole since it would change their greybody factor. Ifthe energy flux of Hawking radiation is enhanced at the final stage of black holeevaporation, it could cause stronger reheating than expected before and may induceamplified stochastic gravitational waves [16] that could be observable with the futuregravitational-wave detectors such as DECIGO [40], BBO [41], and LISA [42].The Lifshitz scaling leads to the modifications to dispersion relation. The coeffi-cients of the modifications have been constrained by the observations of gravitationalwave by the LIGO and Virgo collaboration [43]. Based on the latest observationalconstraint [43], the Lifshitz scaling is less important at least for the typical fre-quency (quasinormal frequency) of a black hole with M (cid:29) − M (cid:12) . Therefore, thenovel phenomena investigated here, at least for ν > , could be important only forasteroid-mass or smaller primordial black holes. For ν < , one may imagine highenergy excitations (e.g., ultra high energy cosmic rays) that could excite long-livedroton modes, even in the vicinity of black hole horizons. Acknowledgments
We thank Sergey Sibiryakov for the feedback on a draft of the manuscript. The workof NO was supported in part by the JSPS Overseas Research Fellowships and bythe Perimeter Institute for Theoretical Physics. The work of SM was supported in– 15 –art by Japan Society for the Promotion of Science Grants-in-Aid for Scientific Re-search No. 17H02890, No. 17H06359, and by World Premier International ResearchCenter Initiative, MEXT, Japan. Research at Perimeter Institute is supported bythe Government of Canada through the Department of Innovation, Science and Eco-nomic Development Canada and by the Province of Ontario through the Ministry ofResearch, Innovation and Science.
A Levi-Civita connections, quadratic, and cubic Laplacians
In this appendix, we show explicit forms of Levi-Civita connections and quadratic/cubicLaplacians that appear in the radial wave equation (2.30). The non-zero Levi-Civitaconnections in three-space are (3) Γ r ∗ θθ = − rU, (3) Γ r ∗ φφ = − rU sin θ, (3) Γ θr ∗ θ = U/r, (3) Γ θθr ∗ = U/r, (3) Γ θφφ = − cos θ sin θ, (3) Γ φr ∗ φ = U/r, (3) Γ φθφ = cot θ, (3) Γ φφr ∗ = U/r, (3) Γ φφθ = cot θ, (A.1)and the non-zero Levi-Civita connections of the metric (2.12) are Γ τττ = − V r U , Γ ττr ∗ = 12 r U , Γ τr ∗ τ = 12 r U , Γ τr ∗ r ∗ = V (cid:48) U , Γ τθθ = rVU , Γ τφφ = r sin θVU , Γ r ∗ ττ = 1 − /r r U , Γ r ∗ τr ∗ = V r U , Γ r ∗ r ∗ τ = V r U , Γ r ∗ r ∗ r ∗ = V V (cid:48) U , Γ r ∗ θθ = 1 − rU , Γ r ∗ φφ = (1 − r ) sin θU , Γ θr ∗ θ = Ur , Γ θθr ∗ = Ur , Γ θφφ = − cos θ sin θ, Γ φr ∗ φ = Ur , Γ φθφ = cot θ, Γ φφr ∗ = Ur , Γ φφθ = cot θ, (A.2)where a prime denotes the derivative with respect to r ∗ . The quadratic and cubiclaplacians can be computed directly from (2.32) ∆ = ∂ r ∗ + A ( r ) ∂ r ∗ + B ( r ) ∂ r ∗ + C ( r ) ∂ r + D ( r ) , (A.3) ∆ = ∂ r ∗ + (cid:18) A + 2 Ur (cid:19) ∂ r ∗ + (cid:18) A (cid:48) + B + 2 Ur A − (cid:96) ( (cid:96) + 1) r (cid:19) ∂ r ∗ + (cid:18) B (cid:48) + C + 2 Ur A (cid:48) + 2
Ur B − (cid:96) ( (cid:96) + 1) r A + A (cid:48)(cid:48) (cid:19) ∂ r ∗ + (cid:18) B (cid:48)(cid:48) + 2 C (cid:48) + 2 Ur B (cid:48) + 2
Ur C − (cid:96) ( (cid:96) + 1) r B + D (cid:19) ∂ r ∗ + (cid:18) C (cid:48)(cid:48) + 2 Ur C (cid:48) − (cid:96) ( (cid:96) + 1) r C + 2 D (cid:48) + 2 U Dr (cid:19) ∂ r ∗ + D (cid:48)(cid:48) + 2 Ur D (cid:48) − (cid:96) ( (cid:96) + 1) r D, (A.4)– 16 –here A = 4 Ur , (A.5) B = 4 U (cid:48) r − (cid:96) ( (cid:96) + 1) r , (A.6) C = − r (cid:18) U U (cid:48) r − U (cid:48)(cid:48) (cid:19) , (A.7) D = (cid:96) ( (cid:96) + 1) r (cid:18) (cid:96) ( (cid:96) + 1) r − U r + 2 U (cid:48) r (cid:19) . (A.8) B The convergence and consistency of numerical solutions
The convergence of our simulations is tested by changing the resolution. We per-formed the numerical simulations with (∆ τ, ∆ r ∗ , λ ) = (0 . , . , . , (0 . , . , . ,and (0 . , . , . and one can find that the waveform converges well (FIG. 8).We also confirmed the Kreiss-Oliger dissipation does not affect the numerical resultby performing our numerical simulation with different coefficients (FIG. 9). As a Figure 8 . Comparison among the results with (∆ τ, ∆ r ∗ ) = (0 . , . , (0 . , . , and (0 . , . . The ratio λ ≡ ∆ τ / ∆ r ∗ is . , . , and . , respectively. The coefficientof the Kreiss-Oliger dissipation is / and we use Ξ = 0 . , Ξ = 0 . , and c χ = 0 . consistency check, we check that our numerical simulation reproduces the fundamen-tal quasinormal mode at a late time when M HL → ∞ (FIG. 10). The fundamentalmode for a massless scalar field with (cid:96) = 2 is ω qnm (cid:39) . − i . , and our resultis well consistent with the fundamental mode. We performed the simulation with (cid:96) = 2 mode. Also, the reflectivity we obtained from the simulation for Ξ = Ξ = 0 isconsistent with the solution of the Regge-Wheeler equation [44] for a massless scalarfield (FIG. 11). The list of quasinormal modes is presented in https://pages.jh.edu/ eberti2/ringdown/ andhttps://centra.tecnico.ulisboa.pt/network/grit/files/ringdown/. – 17 – igure 9 . The numerical simulation with the Kreiss-Oliger coefficient of / and / .We use Ξ = 0 . , Ξ = 0 . , and c χ = 0 . Figure 10 . The ringdown waveform of Ξ = Ξ = 0 computed by our numerical com-putation with c χ → ∞ (top), and c χ = 0 (bottom). The red solid lines are the ringdownwaveform obtained from the massless scalar fundamental quasinormal mode for (cid:96) = 2 . C Snap shots of the Lifshitz scalar waves
Here we show some snap shots of the perturbations of the Lifshitz scalar wavesfor three parameter sets: (Ξ , Ξ ) = (0 , (FIG. 12), (0 . , . (FIG. 13), and– 18 – igure 11 . The reflectivity of Ξ = Ξ = 0 computed by our numerical computationwith c χ → ∞ (cross) and c χ = 0 (plus). The red solid line is obtained from the numericalsolution of the Regge-Wheeler equation. ( − . , . (FIG. 14). Although we present the snap shots only for c χ → ∞ , thetrend does not change for the case of c χ = 0 .– 19 – igure 12 . Snap shots of the propagating scalar waves without the Lifshitz scaling. Thered lines represent the position of the Killing horizon and we use r ∗ w = 80 and r ∗ o = 60 . Theuniversal horizon is located at r ∗ → −∞ . – 20 – igure 13 . Snap shots of the propagating Lifshitz scalar waves with Ξ = 0 . , Ξ = 0 . ,and c χ → ∞ . We use r ∗ w = 80 and r ∗ o = 60 . – 21 – igure 14 . Snap shots of the propagating Lifshitz scalar waves with Ξ = − . , Ξ = 0 . ,and c χ → ∞ . We use r ∗ w = 80 and r ∗ o = 60 . – 22 – eferences [1] P. Horava, Phys. Rev. D , 084008 (2009), arXiv:0901.3775 [hep-th] .[2] A. O. Barvinsky, D. Blas, M. Herrero-Valea, S. M. Sibiryakov, and C. F. Steinwachs,Phys. Rev. D , 064022 (2016), arXiv:1512.02250 [hep-th] .[3] S. Mukohyama, Phys. Rev. D , 064005 (2009), arXiv:0905.3563 [hep-th] .[4] S. Mukohyama, JCAP , 005 (2009), arXiv:0906.5069 [hep-th] .[5] S. Mukohyama, JCAP , 001 (2009), arXiv:0904.2190 [hep-th] .[6] E. Kiritsis and G. Kofinas, Nucl. Phys. B , 467 (2009), arXiv:0904.1334 [hep-th] .[7] S. F. Bramberger, A. Coates, J. a. Magueijo, S. Mukohyama, R. Namba, andY. Watanabe, Phys. Rev. D , 043512 (2018), arXiv:1709.07084 [hep-th] .[8] S. Mukohyama, Class. Quant. Grav. , 223101 (2010), arXiv:1007.5199 [hep-th] .[9] S. A. Teukolsky and W. H. Press, Astrophys. J. , 443 (1974).[10] R. Brito, V. Cardoso, and P. Pani, Superradiance: New Frontiers in Black HolePhysics , Vol. 906 (Springer, 2015) arXiv:1501.06570 [gr-qc] .[11] S. Chandrasekhar and S. L. Detweiler, Proc. Roy. Soc. Lond. A , 441 (1975).[12] E. Berti, V. Cardoso, and A. O. Starinets, Class. Quant. Grav. , 163001 (2009),arXiv:0905.2975 [gr-qc] .[13] S. Chen and J. Jing, Phys. Lett. B , 124 (2010), arXiv:0905.1409 [gr-qc] .[14] S. Hawking, Commun. Math. Phys. , 199 (1975), [Erratum: Commun.Math.Phys.46, 206 (1976)].[15] S. Hawking, Nature , 30 (1974).[16] K. Inomata, M. Kawasaki, K. Mukaida, T. Terada, and T. T. Yanagida, Phys. Rev.D , 123533 (2020), arXiv:2003.10455 [astro-ph.CO] .[17] C. Eling, B. Z. Foster, T. Jacobson, and A. C. Wall, Phys. Rev. D , 101502(2007), arXiv:hep-th/0702124 .[18] N. Arkani-Hamed, H.-C. Cheng, M. A. Luty, and S. Mukohyama, JHEP , 074(2004), arXiv:hep-th/0312099 .[19] S. Dubovsky and S. Sibiryakov, Phys. Lett. B , 509 (2006), arXiv:hep-th/0603158.[20] S. Mukohyama, Open Astron. J. , 30 (2010), arXiv:0908.4123 [hep-th] .[21] S. Mukohyama, JHEP , 070 (2009), arXiv:0901.3595 [hep-th] .[22] S. Mukohyama, Phys. Rev. D , 104019 (2005), arXiv:hep-th/0502189 .[23] S. Jazayeri, S. Mukohyama, R. Saitou, and Y. Watanabe, JCAP , 002 (2016),arXiv:1602.06511 [hep-th] . – 23 –
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