呂 嫣, 花 巍

(沈陽師范大學(xué) 物理科學(xué)與技術(shù)學(xué)院, 沈陽 110034)

?

準(zhǔn)極端Reissner-Nordstr?m黑洞對(duì)Dirac粒子的散射

呂 嫣, 花 巍

(沈陽師范大學(xué) 物理科學(xué)與技術(shù)學(xué)院, 沈陽 110034)

在Reissner-Nordstr?m時(shí)空下,給出分離變量后的Dirac方程的徑向方程和角向方程。按照Chandrasekhar的方法,通過定義烏龜坐標(biāo)變換,將徑向耦合方程退耦成獨(dú)立的一維波動(dòng)方程,并給出方程中勢(shì)的表達(dá)式?？紤]到極端Reissner-Nordstr?m黑洞內(nèi)外視界是重合的,這將導(dǎo)致黑洞的質(zhì)量和電荷量相同。在準(zhǔn)極端Reissner-Nordstr?m背景時(shí)空下,討論了當(dāng)粒子的能量大于勢(shì)壘高度時(shí),Dirac粒子被黑洞的散射情況。首先,將勢(shì)函數(shù)用一組階梯函數(shù)代替;之后,利用量子力學(xué)的方法給出透射反射系數(shù)以及波函數(shù)的數(shù)值變化曲線;最后,比較分析不同質(zhì)量的粒子被散射的情況。從結(jié)果來看,不同質(zhì)量Dirac粒子的散射是明顯不同的,質(zhì)量越大的粒子散射越強(qiáng)。從這個(gè)角度來講,黑洞也可以看做是一個(gè)質(zhì)量攝譜儀。

準(zhǔn)極端Reissner-Nordstr?m黑洞; Dirac粒子; 散射; 烏龜坐標(biāo)

0 引 言

自從霍金發(fā)現(xiàn)黑洞存在熱輻射之后[1],黑洞的量子特性越來越受到人們的關(guān)注。目前為止,已經(jīng)有各種各樣的方法和手段被應(yīng)用于研究各種背景時(shí)空下黑洞的輻射和散射問題[2-13]。本文所研究的準(zhǔn)極端Reissner-Nordstr?m(RN)黑洞對(duì)Dirac粒子的散射問題,實(shí)際上是求解該背景時(shí)空下的Dirac方程。彎曲時(shí)空中Dirac方程的解一方面反映了高度彎曲時(shí)空附近天體的自身演化圖像,另一方面也可以解釋并預(yù)言Dirac粒子在彎曲時(shí)空中的行為,后者對(duì)于進(jìn)一步研究黑洞的熱性質(zhì)又具有極其重要的理論意義。

1 波動(dòng)方程

RN時(shí)空中的線元形式為

其中:M是黑洞質(zhì)量;Q是黑洞電荷量。根據(jù)文獻(xiàn)[14],RN時(shí)空中的Dirac方程可分離為徑向和角向方程,徑向方程為

且有

這里:n是整數(shù)或半整數(shù);σ代表入射波的頻率;m是粒子的靜止質(zhì)量;λ是Dirac方程的本征值;Δ可稱作視界函數(shù)。

分離變量后的角向方程為

定義烏龜坐標(biāo)

其中

并且

上式中:r±為黑洞兩視界的視界半徑;κ±為相應(yīng)的表面引力。

按照Chandrasekhar的方法[14],徑向耦合方程(2)、(3)可以被退耦為一維波動(dòng)方程

(14)式中勢(shì)壘的表達(dá)式為

2 波動(dòng)方程的解

在每一階梯上局域反射透射系數(shù)有如下形式

在邊界處采用無反射的內(nèi)邊界條件,即Vn→0,Rn→0。圖2給出了不同質(zhì)量粒子的即時(shí)透射反射系數(shù),參數(shù)選擇情況是:σ=0.8,m=0.8(實(shí)線),m=0.75(虛線),m=0.7(點(diǎn)劃線)。在邊界處,勢(shì)為0,所有粒子反射系數(shù)均為0,隨著遠(yuǎn)離邊界,所有粒子的反射系數(shù)均在增加,而透射系數(shù)相應(yīng)減小?？梢钥闯?質(zhì)量越大的粒子反射系數(shù)越大而透射系數(shù)越小,這意味著質(zhì)量小的粒子更容易穿越勢(shì)壘。圖3給出了相應(yīng)粒子的波函數(shù)變化曲線,圖3a代表波函數(shù)的實(shí)部,圖3b代表波函數(shù)的虛部。在靠近邊界處,不同質(zhì)量粒子的波長、振幅近似相同,隨著遠(yuǎn)離邊界,質(zhì)量較大的粒子波長更長,振幅更大,其散射更為明顯。

圖1 勢(shì)與階梯勢(shì)

圖2 反射系數(shù)與透射系數(shù)

圖3 波函數(shù)

3 結(jié) 論

本文研究了準(zhǔn)極端RN黑洞對(duì)Dirac粒子的散射,包括透射反射系數(shù)和波函數(shù)。采用階梯勢(shì)的方法得到全空間的透射反射系數(shù)以及波函數(shù)的變化情況。透射反射系數(shù)是逐點(diǎn)定義的,因此也可以稱作即時(shí)透射反射系數(shù)。在所討論的情況中,不同質(zhì)量的粒子散射情況明顯不同,質(zhì)量越大的粒子散射越強(qiáng),可以推斷不同能量的粒子入射黑洞,散射情況也會(huì)明顯不同,所以黑洞可以看作是一個(gè)質(zhì)量(或能量)攝譜儀。以上討論的是入射粒子的散射,若根據(jù)霍金輻射考慮事件視界處產(chǎn)生的出射粒子,則在無窮遠(yuǎn)處的平均粒子數(shù)應(yīng)為霍金輻射譜與勢(shì)壘穿透系數(shù)的乘積。

[ 1 ]HAWKING S W. Black hole explosions?[J]. Nature, 1974,248:30-31.

[ 2 ]PARIKH M K, WILCZEKil F. Hawking radiation as tunneling[J]. Phys Rev Lett, 2000,85(24): 5042-5045.

[ 3 ]ZHANG Jingyi, ZHAO Zheng. Massive particles’ black hole tunneling and de sitter tunneling[J]. Nucl Phys B, 2005,725(1/2):173-180.

[ 4 ]GUO Guanghai, GUI Yuanxing, TIAN Jianxiang. The real scalar field in extreme RNdS space[J]. Gen Relativ Gravit, 2005,37(7):1323-1330.

[ 5 ]GUO Guanghai. Real scalar field scattering in the nearly extremal Schwarzschild desitter space[J]. Chin Phys B, 2010,19(11):110403.

[ 6 ]LYU Yan, HUA Wei. Solution of Dirac equation around a charged rotating black hole[J]. Chin Phys B, 2014,23(4):040403.

[ 7 ]SHAO Jianzhou, WANG Yongjiu. Scattering and absorption of particles by a black hole involving a global monopole[J]. Chin Phys B, 2012,21(4):040404.

[ 8 ]CHAKRABARTI S K, MUKHOPADHYAY B. Scattering of Dirac waves off Kerr black holes[J]. Mon Not R Astron Soc, 2000,317(4):979-984.

[ 9 ]MUKHOPADHYAY B,CHAKRABARTI S K. Semi-analytical solution of Dirac equation in Schwarzschild geometry[J]. Class Quant Grav, 1999,16(10):3165-3181.

[10]LYU Yan, GUI Yuanxing. Numerical solution of Dirac equation in Schwarzschild-de sitter spacetime [J]. Phys Scr. 2007,75(2):152-156.

[11]LYU Yan, CUI Song, LIU Ling. The solution of Dirac equation in quasi-extreme Reissner-Nordstr?m de sitter space[J]. Mod Phys Lett A, 2009,24(30):2433-2443.

[12]呂嫣,方戈亮,徐琦. Schwarzschild-de Sitter度規(guī)下旋量場(chǎng)方程的解[J]. 沈陽師范大學(xué)學(xué)報(bào):自然科學(xué)版, 2006,24(4):426-429.

[13]BREVIK I, SIMONSEN B. The scalar field equation in Schwarzschild-de Sitter space[J]. Gen Relativ Gravit, 2001,33(10):1839-1861.

[14]CHANDRASEKHAR S. The mathematical theory of black holes[M]. New York: Oxford University press, 1983.

[15]CHAKRABARTI S K. On mass-dependent spheroidal harmonics of spin one-half[J]. Proc R Soc Lond A, 1984,391:27-38.

Scattering of Dirac particles off quasi-extreme Reissner-Nordstr?mblack holes

LYUYan,HUAWei

(College of Physical Science and Technology, Shenyang Normal University, Shenyang 110034, China)

Dirac equations are separated into radial and angular equations in Reissner-Nordstr?m (RN) geometry. By tortoise coordinate transformation, the radial parts of Dirac equations are decoupled into independent one-dimensional wave equations according to Chandrasekhar’s method. The form of the potential in the wave equations is also given. For extreme RN black holes, the two black hole horizons are coincidence, which leads to the equal mass and the electric charge of the black hole. In quasi-extreme Reissner-Nordstr?m geometry, the case that the energies of the waves are greater than the height of the potential barrier is considered. First, the potential is replaced by a collection of step functions. Second, the reflection and transmission coefficients and the wave functions are computed by quantum mechanics method. Last, the solutions to Dirac equations with different masses are compared. The results show that Dirac particles with different mass are scattered obviously different, the greater the mass is, the stronger the scattering is. Therefore, the black hole can act as a mass spectrograph.

quasi-extreme Reissner-Nordstr?m black holes; Dirac particles; scattering; tortoise coordinate

2015-03-08。

國家自然科學(xué)基金青年科學(xué)基金資助項(xiàng)目(11301350)。

呂 嫣(1978-),女,遼寧沈陽人,沈陽師范大學(xué)副教授,碩士。

1673-5862(2015)03-0405-04

O412.1

A

10.3969/ j.issn.1673-5862.2015.03.019