Noncommutative information is revealed from the static detector outside the black hole

We investigate the transition behavior of the two-level atom as the Unruh-DeWitt detector outside a noncommutative black hole. When the mass of the black hole is small enough, the difference between the commutative and noncommutative black hole can be distinguished. In particular, an evident fluctuation appearing at a far distance from the horizon by calculating the quantum Fisher information of the transition rate with regard to the local Hawking temperature provides a novel and interesting result about the information extraction of the noncommutativity for a small-mass black hole.


I. INTRODUCTION
The Unruh-Dewitt (UDW) detector [1] is modeled from a two-level atom with a fixed energy gap.The detector works by interacting with an external field that has been prepared in a given initial state [2].Since the response of the detector is only based on the particle content of the field, it can be used to study the quantum field in any spacetime without relying on spacetime symmetry.Based on this, the detector model is widely applied in many different situations (see the review [3] and references therein).However, the field contents only for the commutative spacetime (irrespective of the spacetime is flat or curved) were studied in the past several years as far as we know, so in this paper, we will investigate the response of the detector outside a noncommutative Schwarzschild black hole [4].
The study of noncommutative geometry is probably stemmed from the singularity problem in General Relativity and it is believed that the spacetime noncommutativity can cure the divergence in the singular points that exists in the commutative spacetime [5,6].The special interest in the noncommutativity is sparked by the prediction of string theory along with the brane-world scenario [9][10][11].The noncommutativity was also introduced into the black holes since it can remove the socalled Hawking paradox where the temperature diverges as the radius of a commutative black hole shrinks to zero [12,13], and adoption of spacetime noncommutativity leads to different black holes [4,[14][15][16][17][18][19] and their associated thermodynamics [20][21][22][23].Since the black hole images [24][25][26][27][28][29][30] can be observed, it is possible to glean some information about the structure of the black hole [31].If the noncommutativity is the fundamental property of nature, the general black holes should carry the information about the spacetime noncommutativity.So it is significant to study the influence of noncommutative black holes on the physics outside the black hole.* Electronic address: zhangbaocheng@cug.edu.cn In this paper, we numerically investigate the UDW detector coupled to a massless scalar field in the noncommutative Schwarzschild spacetime under the Hartle-Hawking state by extending the method of Ref. [32] to the noncommutative case.We calculate the transition rate of a static two-level atom at a fixed position outside the black hole for some different situations, and compare the results with those of the commutative Schwarzschild black hole.We also use the quantum Fisher information (QFI) [33] to study the possibility of extracting the information about noncommutativity from the transition rate of the atom.In this paper, we use units with = c = G = 1.

II. NONCOMMUTATIVE BLACK HOLE AND HARTLE-HAWKING STATE
In this paper, we adopt the idea assumed in Ref. [4] where the spatial noncommutative effect is attributed to the modified energy-momentum tensor as a source while the Einstein tensor is not changed, since the direct application of noncommutative coordinates to black holes is inconvenient.Along the line, we can change the mass of a gravitational body to include the noncommutative effect in gravity.Thus, the smearing mass could be taken by redefining its density through a Gaussian distribution of minimal width √ θ (θ is the noncommutative parameter with the dimension length squared) instead of the Dirac delta function for the usual definition of mass density in commutative space.With the mass density for a smearing mass, ρ θ (r) = M (4πθ) 3/2 exp −r 2 /4θ , one can solve the Einstein field equation to obtain the metric for the noncommutative black hole as where ), and the lower incomplete gamma function γ(3/2, r 2 /4θ) = r 2 /4θ 0 t 1/2 e −t dt which approaches to √ π/2 as r → ∞.In the θ → 0 limit, the incomplete γ-function becomes the usual gamma function and the noncommutative metric in Eq. (1) becomes the commutative Schwarzschild metric.From the condition of f (r) = 0, the event horizon can be found as r H = 4M √ π γ(3/2, r 2 H /4θ) ≡ 4M √ π γ H .The temperature is obtained for the noncommutative metric (1) as The modal solutions of the Klein-Gordon equation, where ω > 0, and Y ℓm is a spherical harmonic function.Then, we separate the radial equation, which is obtained by using the tortoise coordinate r * = dr f (r) as where . To solve the equation, the boundary condition must be known.It is not hard to obtain that which has the same form as the commutative case as given in Ref. [32].Thus, we use the same method as in Ref. [32] to numerically solve the scalar field Ψ.
For the noncommutative black hole, the Hartle-Hawking state is also regular across the both the past and future horizon, and it reduces to a thermal field at spatial infinity with the temperature T H for the noncommutative black hole.For the Hartle-Hawking state, the basic modes have the properties of positive-frequency plane waves with respect to the horizon generators and are expressed as [34,35], where "in" represents the ingoing modes at infinity, "up" represents the outgoing modes at the horizon, and they have the same meaning as that in Ref. [32] but the mass is modified by the noncommutative effect.
We calculate the Wightman function of the quantum field in the Hartle-Hawking vacuum state |0 H as where Φ ωℓ (r) is the normalized radial function ρ ωl (r) according to the Wronskian relation [32], R is the position of the detector, ∆τ = dt f (r) is the proper time difference along the time-like geodesic trajectory, and Φ up,in ωℓ (R) is given by solving the Eq. ( 2) and makes the normalization.

III. TRANSITION RATE
Consider a point-like two-level atom (detector) with the ground state |g and the excited state |e and it is placed outside the noncommutative black hole and holds still there.The atom will interact with the field φ in the Hartle-Hawking state and evolves according to the Hamiltonian, Ĥint = λχ(τ )µ(τ )φ(x(τ )) where λ ≪ 1 is the coupling constant, χ(τ ) = exp(−τ 2 /2σ 2 ) is a timevarying switching function, and µ(τ ) is a SU(2) ladder operator describing the change of the atom, defined as µ(τ ) = |e g|e iEτ + |g e|e −iEτ (E is the energy difference between two levels of the atom).The atom moves along a trajectory x(τ ) in the noncommutative spacetime where τ is the proper time of the detector.In the interaction picture, the time evolution operator is expressed as Û = T exp −i dτ Ĥint (τ ) where T represents the time order.Expanding the operator to the first order, we obtain where Û (0) = I and Û (1) = −i dτ Ĥint .
For the initial state of the atom with the ground state |e and the field with the state |0 H , the density matrix is written as ρi = |e e| ⊗ |0 H 0 H |. Through the evolution of unitary operators, we get the final state's density where the field state has been traced out.
The transition probability of the detector is read from the second term of the density operator (7), Since the Wightman function satisfies the time translation invariance along the static trajectory, one can for-mally drop the τ ′ -integral of Eq. ( 8) with a change of variables.Then, by first factoring out the total effective interaction time and subsequently taking the infinite interaction time limit σ → ∞, the transition rate is obtained as where s = τ − τ ′ and λ = 1 for convenience.
Using the result of Eq. ( 5), the transition rate is given with an explicit form as where ω = E f (R), and T loc refers to the local Hawking temperature, It is not hard to confirm that this result satisfies Kubo-Martin-Schwinger (KMS) condition [36,37] of thermal balance with the local temperature T loc .The upper panel of Fig. 1 presents the change of the local temperature at the distance R outside the noncommutative black hole.It is found that the local temperature also reduces to zero when the mass approaches to the minimum value M 0 , similar to the behavior of T H .This is interesting that the local temperature is higher than T H when M > M 0 , but both of them decay to zero when M → M 0 .The lower panel of Fig. 1 presents the transition rate for the case in which the detector is fixed at the position outside the black hole and the mass of the black hole is changed.As expected, the transition rate approaches to zero when M → M 0 where the local temperature is also zero.
We also study the change of local temperature and the transition rate for the case in which the mass of the black hole is fixed at a small value and the distance between the detector and the horizon is changed.The upper panel of Fig. 2 presents the results for the local temperature.It is seen that the local temperature for the noncommutative black hole is lower than that for the commutative black hole and both of them will diverge when the detector approaches to the horizon since the red shift factor exists in the denominator.The lower panel of Fig. 2 presents the results for the transition rate.It shows that the farther the distance, the smaller the transition rate, which is consisten with the situation as in the lower panel of Fig. 1 where it is seen that the transition rate decreases for the noncommutative black hole but it increases for the commutative black hole when the mass of the black hole become small enough.

IV. FISHER INFORMATION
To analyze the possibility of measuring the noncommutativity that exists in the black hole spacetime, we use QFI to estimate it.The QFI is a central quantity in quantum metrology and the quantum analogue of the classical Fisher information.
According to the quantum Cramér-Rao theorem, for a given observable temperature T loc , the measurement precision is determined by [33,38] V ar(T loc ) ≥ where V ar represents the covariant variance, and n represents the number of repeated measurements.F Q is just the QFI defined by [39]  ).Therefore, we obtain Figure 3 presents the change of QFI with the mass of the black hole for a static detector fixed at a position outside the black hole, as that in Fig. 1.It can be observed that the QFI is nearly the same for both commutative and noncommutative black holes.However, a difference emerges after the mass decreases, where the QFI for the noncommutative black hole surpasses that for the commutative black hole due to the influence of spatial noncommutativity.In particular, there is a divergent peak at M d ≃ 16.67.This can be obtained by rewriting the QFI as The QFI as a function of R for the case in which the detector is fixed at a position outside the black hole.The inserted figure presents the behaviors of QFI for the case that the distance between the detector and the horizon is short.The model parameters are the same as in Fig. 2. firmed by taking the numerical value to the forty units after the decimal point.
Figure 4 gives the change of QFI with the distance between the detector and the horizon for a black hole with a fixed small mass.It is interesting to find that the QFI present a fluctuating behavior, and the amplitude of fluctuation for the noncommutative black hole is larger than that for the commutative black hole.The reason for the fluctuation is that the transition rate presents a fluctuating behavior when the detector is placed at a far away location, as in the lower panel of Fig. 2. But the amplitude of the fluctuation is very small for the transition rate, so it is not easy to be seen.The QFI enlarges the fluctuation which is helpful for the possible observation.

V. CONCLUSION
In this paper, we investigate the behavior of UDW detectors under the background of noncommutative black holes.We study two cases: one is that the position of the detector is fixed and the mass of the black hole is changed, in which the local Hawking temperature and the transition rate of the detector present the different phenomena when the mass becomes small enough; the other one is that the position of the detector is changed for a fixed small mass of the black hole, in which the local Hawking temperature and the transition rate of the detector present the similar change but with different numerical values.We also calculate the QFI for these two cases.A novel and interesting phenomenon is found that the noncommutativity can enhance the fluctuation of the QFI with the distance between the detector and the horizon for a fixed small mass of the black hole.In particular, the fluctuation is still strong enough even for a far enough distance, which is helpful for the possible future observation.

FIG. 1 :
FIG.1:The local temperature (upper panel) and the transition rate (lower panel) as a function of M for the case in which the detector is fixed at a position outside the black hole.The model parameters employed are E = 0.01 and R = 50. operator

FIG. 2 :
FIG. 2: The local temperature (upper panel) and the transition rate (lower panel) as a function of R for the case in which the mass of the black hole is fixed.The model parameters employed are E = 0.01 and M/ √ θ = 2.

FIG. 3 :
FIG.3:The QFI as a function of M for the case in which the detector is fixed at a position outside the black hole.The inserted figure presents the behaviors of QFI for the case that the mass of the black hole is small.The model parameters are the same as in Fig.1.

2
FIG.4:The QFI as a function of R for the case in which the detector is fixed at a position outside the black hole.The inserted figure presents the behaviors of QFI for the case that the distance between the detector and the horizon is short.The model parameters are the same as in Fig.2.
The function u is the normalized form of the scalar field Ψ, and the function υ is analogous to u on the second exterior of the Kruskal manifold.Thus, the quantum field under the Hartle-Hawking state can be expanded as ψ = ∞