A Stringy Effect on Hawking Radiation

In string theories, interactions are exponentially suppressed for trans-Planckian space-like external momenta. We study a class of quantum field theories that exhibit this feature modeled after Witten's bosonic open string field theory, and discover a Lorentz-invariant UV/IR relation that leads to the spacetime uncertainty principle proposed by Yoneya. Application to a dynamical black hole background suggests that Hawking radiation is turned off around the scrambling time.

However, a recent study [30] has shown that more general UV dispersions could significantly modify Hawking radiation.In any case, there is growing evidence suggesting that the trans-Planckian problem associated with Hawking radiation cannot be dismissed as a mere artifact of the choice of the coordinate system.Through this work, we aim to understand the behavior of Hawking radiation in string theory, which is a prominent candidate for a theory of quantum gravity.Notable related works include Refs.[31,32].
A common characteristic of string theories is the exponential suppression of interactions beyond the Planck scale, and this work will focus specifically on its effect.Motivated by superstring field theories (as quantum field theories of spacetime fields) [33][34][35][36][37], we consider a class of scalar field theories with the action 3 in Minkowski space, where all of the fields ϕ a appear in interactions only in the nonlocal form 4  φa ≡ e ℓ 2 □/2 ϕ a . (1.2) We have adopted the mostly plus signature (−, +, The model (1.1) mimics the trans-Planckian suppression in string field theories, and has been utilized as a tool for studying such theories in the literature [34,36,42].
Henceforth, we will refer to it as the stringy model.Nevertheless, it is not known how closely (or poorly) this string-theory-inspired model resembles the string theory.It is a toy model without a controlled approximation to string theory.Furthermore, as the stringy model (1.1) is nonlocal, its quantization is potentially problematic.While the unitarity of this model in the path integral formulation has already been checked in Refs.[34,36,[40][41][42], there can still be various pathologies that will only be uncovered after a thorough examination.
Naively, since ϕ a satisfies (□ − m 2 a )ϕ a = 0 at zeroth order in the perturbation theory, we have φa ≃ e ℓ 2 m 2 a /2 ϕ a at the leading order [43,44].If this were the case, Hawking radiation would remain largely unchanged compared to the low-energy effective theory [8].However, this interpretation only holds in the low-energy limit of the model (1.1),where the perturbative treatment is valid [43,44].
Since string theory is proposed as a theory of everything, both the radiation field describing Hawking particles and the apparatus used for their detection should be governed by the same action (1.1).Therefore, the detection of a field ϕ a is actually the detection of φa , as all interactions depend exclusively on φa .For instance, an Unruh-DeWitt detector [45,46] 3 More generally, finite derivatives of the fields φa are allowed in the interaction terms, and the discussions below remain essentially the same with this generalization. 4This is of the same form as Witten's bosonic open string field theory [38,39], in which the length parameter is given by ℓ 2 ≡ 2α ′ log 3 √ 3/4 .In superstring field theories, the interaction vertex e −ℓ 2 k 2 in the abovementioned theory is in general the exponential of a quadratic function of momenta.
would couple to one of the fields ϕ a in the form of where τ represents the proper time along the trajectory x(τ ) of the detector, M (τ ) is an operator acting on the Hilbert space of the detector, and φa is the corresponding field operator in the Hamiltonian formulation.Hence, the Unruh effect [45,47,48] as the response of a uniformly accelerating detector to the Minkowski vacuum |0⟩ is determined by the Wightman function ⟨0| φa (x) φa (x ′ )|0⟩, rather than ⟨0| φa (x) φa (x ′ )|0⟩.By the same token, Hawking radiation is determined by the Wightman function of φa instead of φa .In fact, all physical observables directly probed by measurements should be expressed in terms of φa .
In view of this, it is natural to treat φa as the fundamental fields and express everything in the action (1.1) in terms of φa as where the infinite derivatives in the interaction terms have been transferred to the kinetic terms through the field redefinition (1.2).This then allows us to treat the interaction terms as perturbations when studying the effect of the infinite derivatives on the field quantization as well as the correlation functions of φa .
As in Witten's bosonic open string field theory, the action (1.4) is used in the path integral formulation, which needs to be evaluated via analytic continuation, typically in the Euclideanized momentum space [34,36,42].To compute Hawking radiation, all we need is the Wightman function ⟨0| φa (x) φa (x ′ )|0⟩.However, it is unclear how to derive the Wightman function in the path integral formulation of a nonlocal theory.In this work, we find that the Wightman function can be derived in the light-cone frame by analytically continuing the string length parameter ℓ 2 → ℓ 2 E = −iℓ 2 , which corresponds to a complexification of the string worldsheet modular parameter [49].
Interestingly, it turns out that the Wightman function is subject to a Lorentz-invariant UV cutoff on the light-cone momentum, leading to the uncertainty relation which aligns with the light-cone version of the spacetime uncertainty principle ∆T ∆X ≳ ℓ 2 proposed by Yoneya [50][51][52][53] as a fundamental feature of string theory.We will show that, as a result of eq.(1.5), the magnitude of Hawking radiation tends towards zero after the scrambling time ∼ 4a log(a/ℓ), although the Hawking temperature remains unchanged.This conclusion is in line with the understanding that a UV cutoff at the Planck scale shuts down Hawking radiation around the scrambling time [19].It also resonates with a recent study [54] regarding the fate of Hawking radiation under the generalized uncertainty principle which is designed to capture a nonlocal aspect of string theory.This study [54] revealed that the GUP also results in the termination of Hawking radiation around the scrambling time.
This paper is structured as follows.In Sec. 2, we derive the Wightman function of the stringy model and establish the algebra obeyed by the creation and annihilation operators in the decomposition of the field operator φa .The emergence of a UV/IR connection is explored, and its associated physical interpretations are also discussed.In Sec. 3, we calculate the Hawking radiation in the stringy model, and find that its amplitude diminishes to zero around the scrambling time.Finally, we present our conclusions and comment on the implications of our results in Sec. 4.

Stringy Model
In this section, we start by illustrating the necessity of employing analytic continuation in the stringy model, and then introduce the analytic continuation of the string length parameter ℓ in the light-cone frame.This extension enables us to derive the Wightman function, which showcases a UV/IR connection that ultimately gives rise to the spacetime uncertainty principle (1.5) with a minimum length.As we will see, the same property also implies a weaker interaction between the field and background fluctuations.

The Feynman propagator
From now on, we shall focus on a massless scalar field denoted simply as φ.Extensions of our discussion to massive scalar fields are straightforward.In this subsection, we explain the importance of analytic continuation for the stringy model (1.4).We also illustrate how the nonlocality inherent in the theory hinders the derivation of the Wightman function.
The Feynman propagator of φ can be read off from the action (1.4) as It differs from the standard propagator of the low-energy effective theory by an exponential factor The factor e ℓ 2 (k 0 ) 2 is unphysical, as it is exponentially large for time-like momenta, which further leads to UV divergences in integrals over the momentum space.For this reason, Feynman-diagram calculations are typically carried out in the Euclideanized momentum space [34,36,42].
In the position space, the Feynman propagator is formally given by where in the propagator (2.1) suppresses the contributions from trans-Planckian modes, and the position-space representation (2.3) can then be evaluated to give where There is again an unphysical factor e −x 2 /4ℓ 2 that grows exponentially in time-like directions.Consequently, whether we consider the Feynman propagator of the stringy model as defined in eq.(2.1) or eq.( 2.4), it cannot be directly interpreted in the Lorentzian signature.Analytic continuation is inevitably required.
In the context of Feynman diagrams, there is a prescription for selecting appropriate integration contours for loop energies in the Euclideanized momentum space which ensures that the Cutkosky rules hold and that unitarity is preserved [34,36,42].In this sense, the stringy model (1.4) is a good candidate for investigating UV physics in terms of S-matrix calculations.
Our first task is to construct the Wightman function.It is tempting to interpret the Feynman propagator (2.3) as the time-ordered product where x = (T, X), Θ(T ) is the step function, and is the Wightman function.However, this naive expectation fails for the stringy model.Take for example in two dimensions (D = 2) where x = (T, X) and k = (k 0 , K). Upon Wick the stringy propagator (2.3) can be evaluated explicitly as The complementary error function erfc(z) can be approximated by the step function 2Θ(−z) for large |z|, thus eq.(2.8) reduces to the form of the low-energy effective theory in the limit ℓ → 0. However, the fact that the step function Θ(T E ) is now replaced by a smooth function erfc(|K|ℓ−T E /2ℓ) is a reflection of the nonlocality in the stringy model, and it is incompatible with eq.(2.5) at finite ℓ.This prevents us from a direct deduction of the Wightman function from the Feynman propagator.

Wightman function in the light-cone frame
Witten's bosonic open string field theory can be formulated as a local theory only in one light-cone direction [63].To derive the Wightman function, we adopt the light-cone frame and perform an analytic continuation of the string length parameter ℓ.
Let us begin by reviewing the standard low-energy effective theory in the light-cone frame, working with just two spacetime dimensions for simplicity.The generalization to higher dimensions is straightforward.In terms of the light-cone coordinates defined by the Feynman propagator takes the form where Ω ≡ (k 0 + K)/2 and Ω ≡ (k 0 − K)/2 are the light-cone momenta.We carry out the contour integral over Ω while assuming that Ω ̸ = 0 (ignoring contributions from the ingoing modes with Ω = 0).This yields the propagator for the outgoing modes: By interpreting this propagator as a time-ordered product analogous to eq. (2.5) but adapted for the light-cone frame, i.e.
we can identify the Wightman function for the outgoing modes as We now turn to the 2D propagator in the stringy model.Due to the unphysical feature of exponential growth in time-like directions in the Lorentzian signature, we anticipate that eq.(2.12) holds only through analytic continuation.Recall the Schwinger parametrization of the stringy Feynman propagator (2.1): which is similar to the expression for the ordinary Feynman propagator, with differences being the truncation of the integral and the use of an imaginary (Euclideanized) Schwinger proper time.Transforming α into the real proper time τ corresponds to a complexification of the string worldsheet modular parameter [49], after which eq. (2.14) becomes where Notice that the exponential factor e −iτ k 2 does not lead to UV divergences for both space-like and time-like momenta, removing the need for Wick rotation.
We propose a prescription for analytic continuation in the light-cone quantization as follows: All integrals over the Lorentzian momentum space are initially evaluated with ℓ 2 E > 0, and subsequently, the obtained results are analytically continued as functions of ℓ 2 E onto the imaginary axis ℓ 2 E = −iℓ 2 .This approach preserves the Lorentzian signature of spacetime.Based on this implementation, the 2D stringy Feynman propagator (2.3) is thus (2.17) Ignoring ingoing modes with Ω = 0, we arrive at As expected from the findings in Ref. [63], the expression above manifests locality in one lightcone coordinate (in this case V ) but exhibits nonlocality in the other light-cone coordinate (U ), since the conjugate momentum Ω of U is subjected to a UV cutoff at |V |/4ℓ 2 E .Through eq.(2.12), we can now identify the stringy Wightman function as for V > 0. Imposing the reality condition on the field leads to thus implying that eq.(2.19) also applies when V < 0. The stringy Wightman function differs from that of the low-energy effective theory (2.13) merely by the cutoff in the light-cone momentum Ω.Given two spacetime points (U, V ) and (U ′ , V ′ ) separated by (∆U, ∆V ) ≡ ) indicates that the outgoing modes contributing to the twopoint correlation function W out (∆U, ∆V ) are those with momenta below Ω max = |∆V |/4ℓ 2 E .As this cutoff value is proportional to |∆V |, eq.(2.19) demonstrates a UV/IR connection featured in the stringy model.In particular, to probe the UV limit in the U -direction, the IR limit should be taken in the V -direction.We will explore this aspect further in section 2.4.

Algebra of creation and annihilation operators
While the calculation of Hawking radiation only requires knowledge of the Wightman function (2.6), the latter assumes the existence of a Hamiltonian formulation. 5To gain a deeper insight into the stringy model, we shall construct a consistent mode expansion for the field operator φout within the framework of the stringy free-field theory, including the operator algebra of the associated creation and annihilation operators.
Let us derive the algebra of the creation and annihilation operators from the stringy Wightman function (2.19), which should be interpreted as in the Hamiltonian formulation of the quantum stringy model.Decomposing the field operator into Fourier modes as Defining the vacuum state by a Ω (V )|0⟩ = 0 , ( we can evaluate ⟨0|[a Ω (V ), a † Ω ′ (V ′ )] |0⟩, and similarly the vacuum expectation values of other commutators.We deduce that These commutation relaions have the correct low-energy limit ℓ 2 E → 0. It may come as a surprise to the reader that the creation and annihilation operators are dependent on V .It implies that the wave equation e −ℓ 2 □ □ φ = 0 derived from the action (1.4) via the principle of least action is not satisfied by the field operator φ.This is a consequence of the nonlocal nature of the stringy model.For the same reason, the Wightman function (2.19) does not satisfy the wave equation as well; that is, An interesting implication of eqs.(2.24) and (2.25) is that the norm of the one-particle state a † Ω (V )|0⟩ vanishes for any Ω > 0. To define a state with a non-vanishing norm, we need to superpose a † Ω (V ) at different values of V .More generally, a one-particle state is typically a superposition of Ω-eigenstates: where |Ψ Ω ⟩ is taken to be a † Ω |0⟩ in the low-energy effective theory.However, since a † Ω (V ) is now V -dependent, a generic Ω-eigenstate can have the form It has a norm proportional to which vanishes if f Ω (V ) has a finite support whose width is less than 4ℓ 2 E Ω.In other words, in the stringy model, an outgoing mode with frequency Ω must be defined over a range of E Ω.In contrast, in the low-energy effective theory, an outgoing mode can be defined at an initial instant of time V (with ∆V = 0).

Spacetime uncertainty principle
We established in section 1 that measuring the field ϕ is essentially a detection of φ, as all measurements rely on interactions.Therefore, the Wightman function (2.19) for φ reveals the need for a significant time interval |∆V | in order to detect a particle with a large frequency Ω in the stringy model.We emphasize, though, that the discussions around eqs. (2.27)- (2.29) suggest that the mere existence of a non-zero norm state with a large Ω already necessitates a substantial |∆V |, regardless of how or whether a measurement is carried out.This constraint is in the form of a UV/IR connection: In a region of Minkowski space restricted to a finite range |∆V | in the V -direction, the frequency Ω of a quantum mode has a UV cutoff with a large extension ∆V in the V -direction.From an intuitive standpoint, one can imagine that for an outgoing mode with a large Ω, the background geometry is effectively probed by an extended string, thereby reducing its resolution.This conceptual picture is jusified by an explicit calculation presented in appendix B, where it is demonstrated perturbatively that the coupling between the field φ and a background fluctuation with a characteristic length scale λ is suppressed when ∆V ≥ 4ℓ 2 E Ω ≫ λ.Therefore, eqs.(2.30) and (2.31) showcase a pivotal aspect of the stringy model, namely that not only do long-wavelength modes perceive a smeared background, but this holds true for high-frequency modes as well.
As an extreme scenario (which will be relevant later), consider a quantum mode with an extraordinarily high frequency, to the extent that it requires to be defined on a scale as large as the entire universe.In this case, the mode becomes insensitive to the precise configurations of individual black holes or galaxies.A more appropriate approach for accurately describing the evolution of such a mode is to adopt the Friedmann-Robertson-Walker (FRW) metric to represent the background geometry, essentially rendering the black holes and galaxies akin to negligible specks of dust.In particular, this high-frequency quantum mode cannot be localized near the horizon of a black hole.Rather, it is more accurate to state that it is defined in the asymptotically flat region of the black hole.Hence, such a mode is not expected to contribute to Hawking radiation.This insight turns out to have significant implications for the magnitude of late-time Hawking radiation, a subject we will delve into in detail in the next section.Nevertheless, it is worth mentioning that, a priori, there could be an alternative interpretation of the stringy model, in which one considers ϕ as the fundamental field, with φ playing a role solely in interactions (see eq. (1.1)). 7In this perspective, an outgoing mode of ϕ can be defined at any given moment of V (with ∆V = 0), just as is the case in the low-energy effective theory, without the necessity of imposing any constraints on ∆V . 8The requirement for a bound on ∆V only arises when it comes to the detection of this outgoing mode.As a result, the UV/IR relation (2.30) applies exclusively to measurement processes, and we would not be able to make the argument that Hawking radiation is turned off around the scrambling time in the next section.
That said, we should not take this viewpoint if we do not believe in hidden degrees of freedom that can never be detected.For example, after imposing the periodic boundary condition of period 2πR on V , ∆V is bounded from above by 2πR.As a result, only propagating modes with Ω ≤ πR/2ℓ 2 can be detected.But, according to this alternative interpretation, all modes with Ω > πR/2ℓ 2 still exist in the theory.On the other hand, if we quantize φ as the fundamental field and impose eqs.(2.25) and (2.26), what is non-detectable is also non-existent.

Hawking Radiation
It has been previously argued in Ref. [19] that the spacetime uncertainty principle (2.32) leads to the termination of Hawking radiation around the scrambling time.Roughly speaking, Hawking radiation arises primarily due to the exponential blueshift, a geometric effect that occurs as the Hawking modes are traced backwards in time close to the horizon.A substantial uncertainty in space would effectively dilute the effects of this blueshift, resulting in the suppression of Hawking radiation.In this section, we will reach the same conclusion for the stringy model through an explicit calculation, offering a more comprehensive explanation to substantiate the argument presented in Ref. [19].

Background geometry
In the conventional model of an evaporating black hole, it is assumed that the quantum correction to the energy-momentum tensor is sufficiently weak so that the Schwarzschild metric is a good approximation. 9The four-dimensional Schwarzschild metric is where dΩ 2 2 = dθ 2 + sin 2 θ dφ 2 is the metric of a unit 2-sphere, and the Eddington-Finkelstein light-cone coordinates (u, v) are defined by with r * being the tortoise coordinate 10 In terms of the Kruskal light-cone coordinates defined by 9 For our purpose, the back reaction on the Schwarzschild geometry is mild assuming that the energymomentum tensor is bounded from above by ⟨T µ µ ⟩, ⟨T θ θ ⟩ ≲ O(1/ℓ 2 p a 2 ) [69], where ℓ p is the Planck length. 10Our definition of the tortoise coordinate differs slightly from the usual one by a shift of a in order to simplify some equations that follow.
the Schwarzschild metric (3.1) is equivalent to with the horizon located at r = a (i.e.U = 0).In the near-horizon region where 0 ≤ r − a ≪ a, the metric is approximately Upon the s-wave reduction, this metric reduces to that of Minkowski space in two dimensions, with the Kruskal coordinates playing the role of the light-cone coordinates in flat spacetime.
For simplicity, we assume that the collapsing matter forming the black hole is a thin shell falling at the speed of light.Without loss of generality, we set the trajectory of the null shell through a translation of the coordinate t.Consequently, t = 0 (or u = 0) corresponds to the moment when the collapsing shell reaches r * = 0, where r ≃ 1.567a.
The region inside the collapsing null shell is Minkowski space, in which we can define the light-cone coordinates where T is the Minkowski time coordinate and r is the radial coordinate.This coordinate system inside the shell can be smoothly patched with the Kruskal coordinates in the nearhorizon region outside the shell by shifting T such that the shell intersects the horizon at T = r = a.That way, the horizon is situated at U (T, r) = 0, and the null shell coincides with V (T, r) = 2a, aligning with the setup (3.7) discussed earlier.
So far, the stringy model has only been formulated in flat spacetime.Therefore, before delving into the examination of Hawking radiation in this model, it is crucial to note the range of the Kruskal coordinates U and V within which the background remains approximately flat.From the line element (3.5), it is clear that this requires both r − a ≪ a as well as r ≃ (V − U )/2.According to eqs.(3.2)-(3.4), the former condition is equivalent to whereas the latter condition demands that The region relevant to the evolution of late-time (large u) Hawking quanta is |U | ≪ a.In this case, the second condition on the areal radius further constrains that V − 2a ≪ a.
Hence, only a small neighborhood just outside the shell where |U | ≪ a and ∆V ≪ a, along with the region inside the shell, can be treated as Minkowski space.The maximum range ∆V of V near the horizon, beyond which the approximation loses its validity, is roughly ∆V ∼ O(a) . (3.11)

Hawking radiation in low-energy effective theory
In this subsection, we will revisit the derivation of Hawking radiation in the low-energy effective theory.This review will serve as a foundation for our subsequent calculation regarding the stringy model in the following subsection.
Hawking radiation is characterized by the expectation value of the number operator associated with Hawking particles described by a given wave function ψ.For an outgoing Hawking particle localized around the retarded time u = u 0 and with central frequency ω 0 , it can be represented by a wave function of the form where the profile function f ω 0 (ω) is concentrated around ω 0 and has a width of ∆ω in the frequency domain.The annihilation operator corresponding to the wave function ψ is defined as where In the low-energy effective theory, we do not distinguish ϕ from φ.With f ω 0 (ω) appropriately normalized according to it follows that [b ψ , b † ψ ] = 1.Although we have written ϕ = ϕ(u, v) in the definition (3.15) of b ω , since the outgoing sector of the field is purely a function of u, b ψ does not depend on v. Thus, in principle, one can imagine measuring the observable n ψ in the near-horizon region or even inside the collapsing shell.In this sense, the detection of Hawking radiation can be achieved by coupling a detector to the field operator, analogous to the Unruh effect.The reason why Hawking radiation is typically associated with a detection performed at large distances is because the operator n ψ agrees with the notion of particle number in the asymptotically flat region.
To calculate ⟨0|n ψ |0⟩, we first derive where the Kruskal coordinates U (u) and V (v) are defined in eq.(3.4), and Φ(U (u), V (v)) = ϕ(u, v) is the same field expressed in terms of U and V .Using the Wightman function (2.13) for outgoing modes in the low-energy effective theory, we find that where the Bogoliubov coefficient is Let us assume that the profile function f ω 0 (ω) possesses a narrow width ∆ω ≪ ω 0 ∼ 1/a such that functions of ω in the integrand can be approximated by their values at ω 0 , provided they change slowly with ω over scales not shorter than 1/a.This allows us to further rewrite eq.(3.21) as (3.22) Next, we replace the integration over Ω with an integration over u via the change of variable As a result, eq.(3.22) becomes where we have utilized the normalization condition (3.16).Thus, we arrive at the conventional result of a stationary Hawking radiation, featuring a Planck distribution at the Hawking temperature T H = 1/(4πa).
In the derivation above, we have neglected interactions in the low-energy effective theory.In general, renormalizable interactions result in only minor modifications to Hawking radiation [70,71].However, non-renormalizable, higher-derivative interactions can lead to exponentially large corrections to Hawking radiation [16][17][18][19].Nevertheless, this effect can be safely disregarded in the stringy model, thanks to the exponential suppression of trans-Planckian interactions.For this reason, our subsequent discussion will center solely on the stringy free-field theory.

Hawking radiation in stringy model
In this subsection, we will derive the Hawking radiation within the context of the stringy model.Contrasting with the derivation presented in the previous subsection, the Wightman function (2.13) of the low-energy effective theory should be replaced by the stringy Wightman function (2.19).Additionally, due to the nonlocal nature of the stringy model, particle detection can only be achieved over a certain time interval.
A particle detector that operates within a finite range V in time v can be described by a switching function s(v), which has the property that s(v) ≃ 0 for v / ∈ V and s(v) ≃ 1 for v ∈ V.The expectation value of the number of Hawking particles detected in the state ψ is then where the number operator is Following the calculation procedure outlined in the preceding subsection, we now incorporate the v-dependence of the operators {b ω , b † ω } and utilize the stringy Wightman function W out (U, V ) (2.19).This leads to the expression where The Kruskal coordinate U (u) is defined in eq.(3.4), and we will comment more on the function V (v) in the next subsection.The sole distinction between eq.(3.29) and its counterpart (3.19) in the low-energy effective theory is the presence of a UV cutoff Ω max = ∆V /4ℓ 2 E in the integration over Ω.Consequently, instead of eq.(3.21), we obtain Due to the translation symmetry in V in the near-horizon region, ⟨0|n ψ (v, v ′ )|0⟩ depends on v and v ′ only through ∆V (3.30), so we shall denote it as ⟨0|n ψ (∆V )|0⟩ going forward.
The quantity ⟨n ψ (V)⟩ defined in eq.
where we have reintroduced the wave function ψ(u) (3.13) of the Hawking particle into the expression.The result closely resembles eq.(3.24) but with the addition of a cutoff at In the calculation above, we performed the analytic continuation ℓ 2 E → ℓ 2 = iℓ 2 E (2.16), while also applying the change of variable u → u + iπa.This introduced an extra factor e (ω−ω ′ )πa , which is approximately 1 due to the assumption that f ω 0 (ω) has a narrow width ∆ω ≪ ω 0 .With only minor adjustments, this calculation aligns closely with the one presented in Ref. [19], where it was demonstrated that a UV cutoff in Ω results in the termination of Hawking radiation around the scrambling time.
According to eq. (3.27), a finite detection range V in the V -direction places an upper bound ∆V max on ∆V .On the other hand, the expectation value (3.32) vanishes when the wave packet ψ(u) (3.13), which is centered at u = u 0 with a width of ∆u, lies outside the range of integration u ∈ −∞, u cut (∆V ) .As a result, the number of Hawking particles ⟨n ψ (V)⟩ (3.27) vanishes when For typical Hawking radiation with dominant frequency ω 0 ∼ 1/a, we need ∆u ≫ a to ensure precise resolution of the frequency (i.e.∆ω ≪ ω 0 ).For instance, for ω 0 = 1/a and ∆ω = ω 0 /100, we have ∆u = 100a.Therefore, we conclude that Hawking radiation can no longer be detected when Put differently, this signifies that to detect Hawking radiation at a late time (large u), a large ∆V max (an extended duration of detection V) is necessary in order to define the quantity Strictly speaking, the applicability of the stringy model is confined to the near-horizon region where the 4D background geometry is approximately flat (see Sec. 3.1).Hence, our derivation of Hawking radiation in the stringy model remains reliable only up to a detection range ∆V max ∼ O(a), which corresponds to the detection of Hawking quanta before the scrambling time: u ≲ u cut (a) ≃ 4a log(a/ℓ).On the other hand, it is often assumed that, even when the 4D geometry is curved, the local Minkowski vacuum approximation holds as long as the 2D geometry (the radial part) is approximately flat (i.e. when eq. (3.9) is valid).Naively, this would imply that the condition (3.35) can be easily bypassed, allowing for the successful detection of Hawking radiation at all times.More explicitly, suppose that the range of detection is set to be V ∈ (v i = 0, v f ), then using eq.(3.4), we have As a result, to detect Hawking radiation at time u, it would suffice to choose v f > u, which is easily attainable.However, there are additional subtleties in the stringy model arising from the UV/IR connection (2.30).As we will elucidate below, not only do ∆V max and v f become roughly linearly related when dealing with highfrequency modes that require a range ∆V max ∼ v f ≫ O(a), rendering it practically impossible to detect late-time Hawking quanta, but Hawking radiation itself ceases to exist after the scrambling time.

Hawking radiation after scrambling time
In the conventional description of Hawking radiation in the low-energy effective theory, Hawking quanta detected after the scrambling time originate from fluctuations with trans-Planckian frequencies Ω ≫ O(a/ℓ 2 ) in the Unruh vacuum (i.e. the local Minkowski vacuum) confined within a Planckian distance from the horizon.In the case of the stringy model, the UV/IR connection (2.30) between the frequency Ω and the associated minimal scale ∆V applies not only to detection processes but also to the very existence and definition of quantum modes, as elaborated in Sec.2.3.As a consequence, if there were to be late-time Hawking quanta, their trans-Planckian ancestors in the distant past would have spanned an interval ∆V ≫ O(a) in the V -direction. 11  11 For a solar-mass black hole (a ≃ 3 km), a Hawking particle later than merely u ≃ 5 ms would have had a central frequency in the remote past corresponding to a ∆V that is already larger than the size of the universe!To put this into perspective, the conventional estimate of the black hole's lifetime is O(a 3 /ℓ 2 ) ∼ 10 64 years.matter matter collapse Figure 1.An outgoing wave packet with a width of ∆U can only be defined over a range of ∆V > 4ℓ 2 /∆U .The vertical line represents a region considerably smaller than ∆V , consisting of matter with low density that eventually collapses into a black hole.Such a wave packet cannot be localized in the near-horizon region of the black hole, and thus will not contribute to Hawking radiation To illustrate this further, suppose that a Hawking particle described by a wave packet with central frequency ω ∼ 1/a is detected at a retarded time u = 2(n + 1)a log(a/ℓ).Let us trace this wave packet backward in time to the Minkowski region inside the shell, and then following it all the way to the Minkowski spacetime in the far past.During this backward evolution, the wave packet would have experienced a significant blueshift in the near-horizon region such that its central frequency Ω in the distant past would have been However, according to eq. (2.30), such a high-frequency quantum mode demands to be defined at the scale For a macroscopic black hole (a ≫ ℓ), this length scale vastly exceeds the size a of the black hole for n > 1.Consequently, in the Minkowski space of the far past, the black hole appears to be no more than a tiny speck of dust to this mode.This implies that the black hole background would have had little to no impact on the evolution of the mode in the first place, as illustrated in Fig. 1.Given this, it becomes unlikely that this mode would eventually give rise to Hawking radiation.
In fact, for a solar-mass black hole, ∆V (3.37) is of the scale of the observable universe for a value of n as small as 1.26!It is shown in Appendix B (and was also argued in Sec.2.4) that the effects of background fluctuations with length scales much shorter than ∆V ≥ 4ℓ 2 Ω are suppressed for high-frequency modes Ω.Therefore, on such an enormous scale, all geometric structures, including black holes and galaxies, can be effectively treated as if they have been smoothed out.It is then natural to define the vacuum state of the high-frequency modes on the Friedmann-Robertson-Walker (FRW) background, which is better approximated by the vacuum in the asymptotically flat region of a black hole, as opposed to that in the nearhorizon region.Since this vacuum state is insensitive to the time-dependent background of black hole formation that occurs on a minuscule scale compared to the extent of these modes, it remains practically unchanged both in the far past and the far future.As a result, these modes cannot contribute to the particle creation process, and we conclude that Hawking radiation no longer exists after the scrambling time when u ≳ 4a log(a/ℓ). 12ast but not least, let us briefly address the function V (v) used in the derivation of Hawking radiation in Sec.3.3.In principle, this function should correspond to the Kruskal coordinate V (v) as defined in eq.(3.4) for the Schwarzschild metric.However, since the geometry is probe-dependent, and high-frequency modes experience a smoother background, the exponential relation between V and v would be effectively smeared to an approximately linear relation for high-frequency modes defined on immense scales ∆V ≫ a.In our setup, we have assumed a negligible back reaction of the vacuum energy-momentum tensor, as in the conventional model of black hole evaporation.Thus, our findings may not apply to models with a large vacuum energy-momentum tensor around the horizon (e.g.

Conclusion and Discussion
the KMY model [13,14]).Additionally, it is worth noting that our conclusion could potentially be influenced by stringy effects other than the suppression of high-energy interactions.
Nonetheless, it suffices to say that the conventional model of black hole evaporation is, if not entirely ruled out, far from being confirmed by string theory.
As Hawking radiation lasts for only approximately the scrambling time, the mass of the black hole decreases by merely a fraction of order O (ℓ/a) 2 log(a/ℓ) relative to its initial mass.Consequently, the black hole remains essentially classical with just small quantum corrections, and the information loss paradox is absent in this scenario.
Our conclusion stands in contrast to some common beliefs regarding Hawking radiation.
It is often asserted that the "nice-slice" argument [21] suggests that Hawking radiation persists as an adiabatic process well described by the low-energy effective theory until the black hole becomes microscopic.However, there exists a loophole in this argument [18]: the time evolution of a quantum mode with a trans-Planckian center-of-mass energy in a scattering process with the background curvature depends on UV physics, even when highenergy excitations are not present.In the context of the stringy model discussed in this work, even though the high-energy interactions are suppressed, the nice-slice argument is still inapplicable due to nonlocality inherent in the stringy model.
It was pointed out in Ref. [73] that an effective UV cutoff that shuts down Hawking radiation around the scrambling time is in accordance with the Trans-Planckian Censorship Conjecture (TCC) [74] in the context of inflationary cosmology.For reliable predictions about primordial perturbations on an inflationary background, the phase of accelerated expansion should not last beyond the TCC time scale, analogous to the scrambling time for black holes.Otherwise, the fluctuations we observe today would have been smaller than the Planck length in the early stages of inflation, rendering the low-energy effective theory invalid [75].Similarly, if Hawking radiation originates from trans-Planckian modes, it cannot be reliably described within the framework of effective field theory.One way to circumvent this issue is by restricting Hawking radiation to occur within the scrambling time.
Notice that the final result (3.32) of Hawking radiation presented in this work still displays a Planck spectrum characterized by the Hawking temperature T H = (4πa) −1 , despite the decrease in the magnitude of the radiation over time.The relation dS BH = (T H ) −1 dM then yields the same Bekenstein-Hawking entropy S BH [76], thereby respecting the universality of black hole thermodynamics [77].Our result is also not in direct contradiction to recent developments in the computation of black hole entanglement entropy using the prescription of quantum extremal surfaces [78][79][80].This prescription allows us to calculate the entanglement entropy of Hawking radiation, but is unable to unveil the ultimate fate of a specific initial state, which is a dynamical issue.In this work, we have highlighted a potential source of nonlocality required by the notion of the islands.
Our investigation has revealed that the stringy model leads to a Lorentz-invariant uncertainty relation (2.31) with a minimum length scale, which is the light-cone version of the spacetime uncertainty principle (2.32) [50][51][52][53].This is the essential reason leading to the peculiar result that Hawking radiation terminates around the scrambling time.However, strictly speaking, this UV/IR connection, as well as other peculiar features, could have their origins in certain pathologies of the model.Our claims remain to be fully justified.Nevertheless, we believe that this hardly discussed alternative resolution to the information loss paradox deserves more exploration.

B Interaction With the Background
Let us consider a background interaction term where h(V ) is a background field assumed to be constant in U for simplicity, and we have used the mode expansion (2.22) of φ.The total action for the field φ is now S = S 0 + S I , where S 0 is the free-field action for the stringy model: with the analytic continuation ℓ 2 → ℓ 2 E = −iℓ 2 performed.As an illustrative example, we examine the impact of the interaction (B.1) between the field φ and a background fluctuation described by with a small amplitude h 0 .In the lowest-order approximation with respect to h 0 , we obtain where ϵ(z) ≡ Θ(z) − 1/2.The question is whether the second term in this expression, arising from the background interaction, becomes effectively suppressed at large Ω when ℓ E ̸ = 0.
In a physical scenario, we typically have a wave packet of the field φ characterized E Ω ≫ a, as indicated by the UV/IR relation (2.30).This is the case for Hawking particles, as their corresponding wave packets not only exhibit blueshifted central frequencies Ω in the past but also encompass a broad spectrum ∆Ω ≳ Ω [18].Therefore, the deviation from flat spacetime caused by the black hole background is highly suppressed for Hawking modes.
(3.27)  then represents the number of Hawking particles detected in the near-horizon region.If ⟨n ψ (V)⟩ vanishes in this region, it would imply the absence of Hawking particles at large distances as well.Finally, by taking the same steps in eqs.(3.22)-(3.24)and adopting the same approximation scheme as in the previous subsection, we end up with

Using a class of
non-local quantum field theories, we have analyzed how Hawking radiation is affected by the stringy effect characterized by the exponential suppression of interactions in the ultraviolet regime.Owing to the UV/IR relation (2.30) constraining the frequency and spatial extent of quantum modes, the behavior of the trans-Planckian ancestors of late-time Hawking radiation differs significantly from that in the standard low-energy effective theory, where high-frequency modes are confined in the vicinity of the black hole horizon.We argued based on this distinctive feature that the specific stringy effect under consideration results in the termination of Hawking radiation beyond the scrambling time.
by a finite width ∆Ω in frequency on a background consisting of a superposition of fluctuations (B.3) with a finite width ∆λ in wavelength.The presence of the oscillatory factors e ±i4λ −1 ℓ 2 E Ω in eq.(B.4) then suppresses the effect of the background field on φ when 4(∆λ) −1 ℓ 2 E Ω ≫ 1 or when 4λ −1 ℓ 2 E (∆Ω) ≫ 1.For instance, on a Schwarzschild background, where the characteristic length scale is the Schwarzschild radius, we have λ ∼ ∆λ ∼ O(a) for the background fluctuations.Consequently, the coupling of the field to the Schwarzschild background becomes significantly suppressed when Ω ≫ a/4ℓ 2 E , or equivalently, when |∆V | ≥ 4ℓ 2 [50][51][52][53]][52][53]. It is shown in appendix A that eq.(2.32) (with ℓ 2 replaced by 2 ℓ 2 E ) is also directly reflected in the stringy Feynman propagator (2.3).A physical interpretation of the spacetime uncertainty principle (2.32) [50-53] is that high-energy strings are long, hence a large spatial uncertainty.Similarly, eqs.(2.30) and (2.31) indicate that a substantial light-cone momentum P U = Ω can only be carried by a string 30)such that approaching the UV limit where Ω → ∞ demands the IR limit |∆V | → ∞.Since a UV cutoff in Ω is equivalent to having a minimal uncertainty ∆U in the light-cone coordinate U , eq. (2.30) can be interpreted as giving rise to an uncertainty relation 66It may appear that the uncertainty relation (2.31) is potentially problematic as we analytically continue ℓ 2 E to −iℓ 2 at the end of each calculation.We will see in the next section that, in the calculation of Hawking radiation, there is only a negligible difference between writing ℓ 2 E or ℓ 2 in the Wightman function(2.19).proposed