Monodromy Approach to Pair Production of Charged Black Holes and Electric Fields Chiang-Mei Chen1 2Toshimasa Ishige3Sang Pyo Kim4 5Akitoshi Takayasu6and Chun-Yu Wei1 1Department of Physics National Central University Chungli 32001 Taiwan

2025-05-02 0 0 736.63KB 16 页 10玖币

侵权投诉

Monodromy Approach to Pair Production of Charged Black Holes and Electric Fields

Chiang-Mei Chen,1, 2, ∗Toshimasa Ishige,3, †Sang Pyo Kim,4, 5, ‡Akitoshi Takayasu,6, §and Chun-Yu Wei1, ¶

1Department of Physics, National Central University, Chungli 32001, Taiwan

2Center for High Energy and High Field Physics (CHiP),

National Central University, Chungli 32001, Taiwan

3Graduate School of Science and Engineering, Chiba University, Chiba-shi 263-8522, Japan

4Department of Physics, Kunsan National University, Kunsan 54150, Korea

5Asia Paciﬁc Center for Theoretical Physics (APCTP), Pohang 37673, Korea

6Faculty of Engineering, Information and Systems, University of Tsukuba, Ibaraki 305-8573, Japan

(Dated: October 10, 2023)

To ﬁnd the pair production, absorption cross section and quasi-normal modes in background ﬁelds,

we advance the monodromy method that makes use of the regular singular points of wave equations.

We ﬁnd the mean number of pairs produced in background ﬁelds whose mode equations belong to the

Riemann diﬀerential equation and apply the method to the three particular cases: (i) charges near

the horizon of near-extremal black holes, (ii) charges with minimal energy under the static balance

in nonextremal charged black holes, and (iii) charges in the Sauter-type electric ﬁelds. We then

compare the results from the monodromy with those from the exact wave functions in terms of the

hypergeometric functions with three regular singular points. The explicit elaboration of monodromy

and the model calculations worked out here seem to reveal evidences that the monodromy may

provide a practical technique to study the spontaneous pair production in general black holes and

electromagnetic ﬁelds.

I. INTRODUCTION

One of nonperturbative aspects of quantum ﬁeld theory is spontaneous particle production from background ﬁelds,

two of whose most prominent phenomena are the (Sauter-)Schwinger mechanism in electromagnetic ﬁelds [1, 2] and

the Hawking radiation in black holes [3]. The physical concept behind the particle production is that the background

ﬁelds change the vacuum in such a way that the out-vacuum is superposed of multiparticle states of the in-vacuum

and vice versa [4]. The Klein-Gordon equation for a scalar ﬁeld, though a linear equation, has been solved only for

a few background ﬁelds [5]. Indeed it has been a challenge for a long time either to directly ﬁnd the exact wave

functions in terms of the special functions or to develop some approximation schemes for the wave functions, such as

the WKB method [6] or Borel-summed WKB method [7].

Particle production has been an interesting topic in cosmology, in particular, in expanding universes [8] (for a recent

review, see [9]). Recently the Schwinger pair production of charged particles and antiparticles by a strong electric

ﬁeld has attracted much attention because ultra-strong lasers have been proposed to reach ﬁeld strengths near the

Schwinger ﬁeld in the near future, and spontaneous production of electrons and positrons will be a direct test of

QED in strong ﬁeld region (for a review on astrophysical applications, see [10] and for a recent review, see [11]).

Charged black holes are an arena in which both the Schwinger mechanism and the Hawking radiation intertwine to

spontaneously emit charges.

The ﬁeld equation for a charged scalar in charged black holes, such as the Reissner-Nordstr¨om (RN) and Kerr-

Newman (KN) black holes in an asymptotically ﬂat or (anti-)de Sitter space (A)dS, has not been exactly solved yet

in terms of special functions in the global covering space. A conventional wisdom has been to solve the ﬁeld equation

in the near-horizon region and the asymptotic region, and then to connect those wave functions. Another method

is to use the enhanced symmetry of background geometry in some limits. The (near-)extremal black holes have a

near-horizon geometry whose enhanced symmetry allows one to exactly solve wave functions. Two of us (CMC,

SPK) have studied spontaneous production of charged particles from (near-)extremal RN or KN black holes in the

asymptotically ﬂat or (A)dS spaces [12–18].

To understand the emission of charges from charged black holes, one has to solve the Klein-Gordon or Dirac equation

in the RN or KN black holes. However, the Klein-Gordon equation in nonextremal charged black holes is a conﬂuent

Heun equation, with three poles including a double pole at inﬁnity, which in general cannot be analytically solved [19].

∗cmchen@phy.ncu.edu.tw

†ishiget@yahoo.co.jp

‡sangkim@kunsan.ac.kr

§takitoshi@risk.tsukuba.ac.jp

¶weijuneyu@gmail.com

arXiv:2210.14792v3 [hep-th] 6 Oct 2023

In this paper we will study an alternative technique, the so-called monodromy that directly computes the scattering

coeﬃcients from the information of poles without knowing the analytic solutions [20, 21]. This technique can pass

over the formidable task of ﬁnding the wave functions in terms of the special functions but give one useful information

necessary for the mean number or absorption cross section of emitted particles from black holes. Moreover, it can

also be used to calculate the quasi-normal modes by imposing a proper boundary condition.

As the ﬁrst step in this direction, we ﬁnd the mean number of pairs produced in background ﬁelds whose mode

equations belong to the Riemann diﬀerential equation. We then consider the three special cases of the Riemann

equation: (i) charges in near-extremal charged black holes, (ii) charges with minimal energy under a static balance

in nonextremal black holes, and (iii) charges in electric ﬁelds that belong to a generalized P¨oschl-Teller potential.

All the three cases have wave functions in terms of the hypergeometric functions with three regular singular points

and with transformation formulas between diﬀerent regions, and will provide one with models to test the monodromy

approach. The phase-integral method that also makes use of the regular singular points [22] gives the leading term

for the pair production from near-extremal charged black holes [12–14] and explains the Stokes phenomenon [23]. On

the other hand, the monodromy method recovers the exact results for those models studied in this paper. Thus we

argue that the monodromy seems a completion of the phase-integral method and may work for more general cases.

In section II, we introduce the deﬁnition and properties of monodromy and impose the boundary conditions for

pair production and absorption cross section. And we ﬁnd the mean number of pair production in background ﬁelds

whose mode equations are described by the Riemann diﬀerential function. In section III, we derive, case by case, the

pair production from monodromy when the governing equations are the hypergeometric function with three regular

singular points. One speciﬁc case is the emission of scalar charges from the near-horizon region of near-extremal RN

black holes [12], and the other case is the scalar charges with minimal energy under a static balance in nonextremal

RN black holes [19], which is also given by the Riemann diﬀerential equation. In section IV, we apply the monodromy

to charges in electric ﬁelds of the generalized P¨oschl-Teller potential, one of which is the Sauter-type electric ﬁeld.

The monodromy conﬁrms the pair production formula from the exact solutions of the wave equation. In section V,

we discuss the features and applications of the monodromy to more general cases.

II. MONODROMY AND BOUNDARY CONDITIONS

In this section, we are going to give a concise introduction to monodromy and to explain how to apply this

technique to study pair production mechanisms. Such ideas had been discussed in [21], but the explanations here

are more rigorous and complete, and, in particular, several obscure issues during the physical applications will be

clariﬁed. Thus the application will turn out to be just a straightforward process.

A. Monodromy Representation of Fundamental Group

Consider a second order ordinary diﬀerential equation (ODE) of the form

dr2R(r) + U(r)d

dr R(r) + V(r)R(r) = 0,(1)

where U(r) and V(r) are meromorphic functions in Cwith ﬁnite poles. Let ri(i= 1,2, . . . , m) be the whole set of

poles of either U(r) or V(r). If both (r−ri)U(r) and (r−ri)2V(r) are holomorphic at ri, then the singularity riis

called regular, otherwise irregular.

The ODE (1) can be extended to the Riemann sphere, i.e. P=C∪ {∞}. Setting s= 1/r and S(s) = R(r), the

ODE (1) is transformed into

ds2S(s) + U∞(s)d

dsS(s) + V∞(s)S(s)=0,

U∞(s)=2s−1−s−2U(1/s), V∞(s) = s−4V(1/s).

(2)

Note that ri∈C\ {0}is a pole of U(r) or V(r) if and only if si= 1/ri∈C\ {0}is a pole of U∞(s) or V∞(s). If

s= 0 is a pole, then rm+1 =∞is a singularity of (1) and the number of singularities in Pbecomes n=m+ 1,

otherwise n=m. If all of the singularities r1, . . . , rnare regular, then the ODE (1) is called a Fuchsian diﬀerential

equation. Let X=P\ {r1, r2, . . . , rn}. It is clear that the solutions of the ODE (1) are holomorphic in X, since they

are diﬀerentiable everywhere in X. However, since Xis not simply connected, a solution of the ODE (1) may be a

multivalued holomorphic function in X. Such multivaluedness of the solutions is characterized by the monodromy

which reﬂects the topological property of X.

Γ2

Γ3

Γ1

FIG. 1. A sketch of the loops on the Riemann sphere P: The loops Γ1,Γ2, and Γ3from the base point r0are enclosing the

singular points r1, r2, and r3, respectively. Γ1·Γ2·Γ3continuously deforms to r0and yields the global relation (3).

Fix a base point r0∈X. Let Γi(i= 1,2, . . . , n) be a loop in Xinitiated at r0enclosing only ricounterclockwise.

The product of loops Γj·Γiis deﬁned to be a loop going ﬁrstly along Γithen Γj. Moreover, π1(X, r0), the fundamental

group of Xwith the base point r0, is generated by [Γ1],...,[Γn] where [Γi] denotes the homotopy class of loops for r0

containing Γi. Since the base point r0is a constant loop (i.e., the image of the constant map [0,1] → {r0}), [r0] is the

identity element of π1(X, r0) denoted by 1. We can construct Γ1,...,Γnto be arranged clockwise in this order. Then,

the composite loop Γ1·Γ2···Γn, which is homotopic to r0in X(i.e., it continuously deforms to r0in Xas illustrated

in Figure 1 for the case of n= 3), gives the following global relation of π1(X, r0):

[Γ1]·[Γ2]···[Γn]=1.(3)

Let Dbe a simply connected open set in Xwith r0∈D. Since Dis simply connected, the monodromy theorem

in the complex analysis implies that all of the solutions of the ODE (1) are single-valued in D. In virtue of the

existence and uniqueness of solutions for the initial value problem, the map from the space of initial values at r0, say

[R(r0), R′(r0)]T=R0where R′=dR/dr, to the solution space of the ODE (1) over D

Φ : C2→Sol(D),R07→ R(r),(4)

is a C-linear isomorphism. Let R1

0and R2

0be C-linearly independent vectors in C2, then Φ(R1

0) and Φ(R2

0) are

C-linearly independent solutions in Sol(D). Suppose that Γ is the image of the smooth map

rΓ: [0,1] →X, t 7→ rΓ(t) with rΓ(0) = rΓ(1) = r0.(5)

There exist 0 < t1< t2<1 such that rΓ([0, t1]) ⊂Dand rΓ([t2,1]) ⊂D. As for rΓ([t1, t2]), it is assumed to be

any path in Xconnecting rΓ(t1) and rΓ(t2). In the course of the analytic continuation along Γ, the solutions Φ(R1

and Φ(R2

0) stay as they are while in rΓ([0, t1]), and after the analytic continuation along rΓ([t1, t2]) when ﬁnally

going into rΓ([t2,1]), they consequently become respective C-linear sums of Φ(R1

0) and Φ(R2

0). Thus, the monodromy

matrix MΓalong the loop Γ with respect to [Φ(R1

0),Φ(R2

0)] is deﬁned by a nonsingular matrix such that

Φ(R1

0),Φ(R2

0)MΓ= Γ∗Φ(R1

0),Φ(R2

0),(6)

where Γ∗denotes the analytic continuation along Γ.

Let us transform (1) into the Pfaﬃan form

dr R

R′=0 1

−V−UR

R′⇐⇒ d

dr R=AR.(7)

Using the discretization of (7), the path-ordered-exponential of Aalong Γ is deﬁned by

LΓ=Pexp ZΓ

A(r)dr=Pexp Z1

A(rΓ(t))r′

Γ(t)dt

= lim

N→∞ (I+A(rΓ(tN−1)) r′

Γ(tN−1)∆t)···(I+A(rΓ(t0)) r′

Γ(t0)∆t),

(8)

where ∆t= 1/N,tk=k/N . It is clear that

LΓ′·Γ=LΓ′LΓ(9)

holds. Applying Φ−1to the both sides of (6), we have

R1

0,R2

0MΓ=LΓR1

0,R2

0⇐⇒ MΓ=R1

0,R2

0−1LΓR1

0,R2

0.(10)

Thus, the monodromy matrix is similar to the path-ordered-exponential, which is denoted by MΓ∼LΓ.

The monodromy representation of the fundamental group is given by

ρ:π1(X, r0)→GL(2,C),[Γ] 7→ MΓ.(11)

From (9) and (10), it holds that

ρ([Γ′]·[Γ]) = MΓ′·Γ=MΓ′MΓ=ρ([Γ′])ρ([Γ]).(12)

Therefore, ρis a homomorphism of groups. Here the image of ρis called the monodromy group. From the global

relation of the fundamental group (3), we obtain the global relation of the monodromy group such that

MΓ1MΓ2···MΓn=ρ([Γ1])ρ([Γ2]) ···ρ([Γn]) = ρ([Γ1]·[Γ2]···[Γn]) = ρ(1) = I.(13)

In particular, LΓ1LΓ2···LΓn=Iholds with respect to [R1

0,R2

0] = I.

Set ˆ

Ui(r) = (r−ri)U(r) and ˆ

Vi(r) = (r−ri)2V(r) for i= 1,2, . . . , m. If rm+1 =∞is a singularity, set

U∞(r) = 2 −rU (r) and ˆ

V∞(r) = r2V(r). Then the ODE (1) is transformed as follows;

ODE (1) ⇐⇒ d2R

dr2+ˆ

(r−ri)

dr +ˆ

(r−ri)2R= 0

⇐⇒ (r−ri)2d2R

dr2+ˆ

Ui(r−ri)dR

dr +ˆ

ViR= 0.

(14)

We introduce the Euler operator δi= (r−ri)d/dr. Using (r−ri)2d2/dr2=δ2

i−δi, as a sequel to the transformation

in (14) we obtain

δ2

iR+ ( ˆ

Ui−1)δiR+ˆ

V R = 0.(15)

Thus, we have another Pfaﬃan form:

δiR

δiR=0 1

−ˆ

Vi1−ˆ

Ui R

δiR⇐⇒ δiˆ

R=ˆ

Aiˆ

R.(16)

Using the discretization of (16), the path-ordered-exponential of ˆ

Aialong Γ is deﬁned by

(ˆ

Li)Γ=Pexp ZΓ

Ai(r)dlog(r−ri)=Pexp Z1

Ai(rΓ(t)) r′

Γ(t)

rΓ(t)−ri

dt

= lim

N→∞ I+ˆ

Ai(rΓ(tN−1)) r′

Γ(tN−1)∆t

rΓ(tN−1)−ri···I+ˆ

Ai(rΓ(t0)) r′

Γ(t0)∆t

rΓ(t0)−ri,

(17)

where ∆t= 1/N, and tk=k/N .

In virtue of the existence and uniqueness of the solutions for the initial value problem with the initial value

(R, δiR)T=ˆ

R0for (15), the map

Φi:C2→Sol(D),ˆ

R07→ R(r),(18)

is a C-linear isomorphism. Here we suppose [Φ(R1

0),Φ(R2

0)] = [ˆ

Φi(ˆ

0),ˆ

Φi(ˆ

0)]. Then, applying ˆ

Φ−1

ion the both

sides of (6), we have

hˆ

0,ˆ

0iMΓ= (ˆ

Li)Γhˆ

0,ˆ

0i⇐⇒ MΓ=hˆ

0,ˆ

0i−1(ˆ

Li)Γhˆ

0,ˆ

0i.(19)

文档加载中……请稍候！
如果长时间未打开，您也可以点击刷新试试。

下载文档到电脑，查找使用更方便

10 玖币 0人已下载

立即下载

摘要：

MonodromyApproachtoPairProductionofChargedBlackHolesandElectricFieldsChiang-MeiChen,1,2,∗ToshimasaIshige,3,†SangPyoKim,4,5,‡AkitoshiTakayasu,6,§andChun-YuWei1,¶1DepartmentofPhysics,NationalCentralUniversity,Chungli32001,Taiwan2CenterforHighEnergyandHighFieldPhysics(CHiP),NationalCentralUniversity,Ch...

展开>> 收起<<

Monodromy Approach to Pair Production of Charged Black Holes and Electric Fields Chiang-Mei Chen1 2Toshimasa Ishige3Sang Pyo Kim4 5Akitoshi Takayasu6and Chun-Yu Wei1 1Department of Physics National Central University Chungli 32001 Taiwan.pdf

共16页,预览4页

还剩页未读，继续阅读

声明：本站为文档C2C交易模式，即用户上传的文档直接被用户下载，本站只是中间服务平台，本站所有文档下载所得的收益归上传人(含作者)所有。玖贝云文库仅提供信息存储空间，仅对用户上传内容的表现方式做保护处理，对上载内容本身不做任何修改或编辑。若文档所含内容侵犯了您的版权或隐私，请立即通知玖贝云文库，我们立即给予删除！

Monodromy Approach to Pair Production of Charged Black Holes and Electric Fields Chiang-Mei Chen1 2Toshimasa Ishige3Sang Pyo Kim4 5Akitoshi Takayasu6and Chun-Yu Wei1 1Department of Physics National Central University Chungli 32001 Taiwan

相关推荐

开通VIP享超值会员特权

作者详情

相关内容

热门标签

举报选择: