王春艷，馬 媛，董 亮

(1. 渤海大學(xué) 數(shù)理學(xué)院, 遼寧 錦州 121013； 2. 大連理工大學(xué) 物理與光電工程學(xué)院, 遼寧 大連 116024)

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Horava引力下Park黑洞的電磁擾動(dòng)

王春艷*,1,2，馬 媛1，董 亮1

(1. 渤海大學(xué) 數(shù)理學(xué)院, 遼寧 錦州 121013； 2. 大連理工大學(xué) 物理與光電工程學(xué)院, 遼寧 大連 116024)

利用三階WKB近似方法,計(jì)算Horava引力中Park黑洞電磁擾動(dòng)的似正規(guī)模頻率. 研究發(fā)現(xiàn),似正規(guī)模是復(fù)頻率且與Park黑洞的時(shí)空背景參數(shù)ΛW,w以及l(fā)和n有關(guān).復(fù)頻率的實(shí)部隨著ΛW的減小而減小,虛部絕對(duì)值是先增加后減小.態(tài)參數(shù)w對(duì)似正規(guī)模頻率的影響非常小.與標(biāo)準(zhǔn)SdS黑洞電磁擾動(dòng)的數(shù)值結(jié)果比較發(fā)現(xiàn),電磁擾動(dòng)在Horava引力下Park黑洞時(shí)空中衰減的較慢.

電磁擾動(dòng);WKB近似;Horava引力;Park黑洞

0 引言

Horava引力是一個(gè)新的四維引力理論模型〔1-3〕,在紅外低能區(qū)域,該理論自然流向愛因斯坦廣義相對(duì)論,但在紫外高能區(qū)域中,該理論是非相對(duì)論的,并且由于高階空間曲率項(xiàng)的引入破壞了洛倫茲對(duì)稱性,從而可以避免廣義相對(duì)論的重整化困難.該理論在宇宙學(xué)和黑洞方面已經(jīng)被廣泛研究〔4-8〕.

黑洞時(shí)空中的似正規(guī)模因?yàn)閿y帶有黑洞的特征信息(質(zhì)量、電荷量和角動(dòng)量),而被譽(yù)為黑洞的“特征聲音”.此時(shí),波的頻率為復(fù)數(shù)且不依賴于微擾源的初始條件,只與黑洞背景時(shí)空的參量有關(guān)〔9-11〕.因此,對(duì)黑洞在各種微擾場(chǎng)下似正規(guī)模的研究〔12-18〕,將為判斷黑洞是否存在提供有利的證據(jù).

本文利用三階WKB近似方法計(jì)算Horava引力下Park黑洞的電磁擾動(dòng)的似正規(guī)模頻率.

1 Park黑洞和電磁擾動(dòng)

在Horava引力下,引入宇宙學(xué)常數(shù)和細(xì)致平衡條件,作用量為〔8〕

(1)

其中Kij是外曲率張量,Cij是Cotton張量,κ,v,λ,μ,ΛW,w,則分別是常參數(shù).最后一項(xiàng)是細(xì)致平衡條件.

Mu-InPark〔8〕得到了一組靜態(tài)球?qū)ΨQ的一般解.當(dāng)考慮漸進(jìn)deSitter時(shí)空時(shí)，有ΛW>0且w<0，則

Horava引力下的Park黑洞解的度規(guī)線元形式如下:

(2)

其中

(3)

M是Park黑洞的質(zhì)量.

在真空中,電磁場(chǎng)擾動(dòng)滿足的Maxwell方程為〔9〕

(4)

其中,Fμv是黑洞時(shí)空背景下電磁場(chǎng)逆變張量,其協(xié)變張量Fμv滿足Fμv=Av,μ-Aμ，v.電磁勢(shì)Aμ可以展開成四維矢量球諧函數(shù)〔20〕.

(5)

等式右側(cè)第一列宇稱為(-1)l+1,第二列宇稱為(-1)l,其中l(wèi)是角量子數(shù),m是磁量子數(shù).

(6)

上式中ψ(r)=alm(當(dāng)宇稱為(-1)l+1),ψ(r)=r2/l(l+1)(-iωhlm-dflm/dr)(當(dāng)宇稱為(-1)l)〔20〕.

有效勢(shì)函數(shù)V(r)滿足

(7)

從(7)式可以看出電磁場(chǎng)的有效勢(shì)與宇宙學(xué)參數(shù)ΛW,態(tài)參數(shù)w和微擾場(chǎng)的角量子數(shù)l有關(guān),并且在r的取值范圍內(nèi)將一直是以勢(shì)壘的形式存在.圖1和圖2分別給出了有效勢(shì)V(r)隨著l和ΛW的變化關(guān)系.從圖中可見,隨著l的增加勢(shì)壘的峰值增加,峰值所在的位置沿著x軸向右邊移動(dòng).有效勢(shì)的勢(shì)壘峰值隨著ΛW,的增加而增加,但峰值所在位置則向左移動(dòng).

圖1 電磁場(chǎng)勢(shì)能隨l的變化關(guān)系,比較l=4時(shí)Park黑洞與SdS黑洞的勢(shì)能曲線 圖2 電磁場(chǎng)勢(shì)能隨r的變化關(guān)系,ΛW分別取0.1,0.01,0.0001

2 數(shù)值方法和結(jié)論

利用三階WKB近似〔21〕計(jì)算Horava引力下的Park黑洞的電磁擾動(dòng)的似正規(guī)模頻率,這里我們令M=1.基本計(jì)算公式為

(8)

(9)

(10)

表1給出了Horava引力下Park黑洞電磁擾動(dòng)的似正規(guī)模頻率的數(shù)值結(jié)果.我們發(fā)現(xiàn)這些復(fù)頻率的虛部均是負(fù)值,這意味著似正規(guī)模隨時(shí)間作指數(shù)衰減,并且在很晚的時(shí)候,指數(shù)衰減將變成冪率衰減.這現(xiàn)象說明該黑洞在電磁擾動(dòng)下隨時(shí)間的演化是趨于穩(wěn)定的.表2給出了Park黑洞隨著參數(shù)w的變化關(guān)系.另外,為了方便比較,表3給出了SdS黑洞電磁擾動(dòng)的似正規(guī)模頻率.圖3直觀的給出了電磁擾動(dòng)似正規(guī)模頻率的實(shí)部與虛部變化曲線.

表1 Horava引力下Park黑洞電磁擾動(dòng)的似正規(guī)模頻率,w=-2

表2 Horava引力下Park黑洞電磁擾動(dòng)的似正規(guī)模頻率,ΛW=0.0001,n=0

表3 SdS黑洞電磁擾動(dòng)的似正規(guī)模頻率,ΛW=0.0001

3 結(jié)論

本文利用三階WKB近似方法計(jì)算了在Horava引力下的Park黑洞電磁擾動(dòng)的似正規(guī)模頻率.從所得的數(shù)值結(jié)果我們發(fā)現(xiàn),對(duì)于固定的w,l和n,隨著ΛW的減小,電磁擾動(dòng)似正規(guī)模頻率的實(shí)部也減小,但是虛部絕對(duì)值是先增加后減小,并且當(dāng)ΛW足夠小的時(shí)候,似正規(guī)模頻率將幾乎不受影響.其次,從表2可以看出,態(tài)參數(shù)w對(duì)似正規(guī)模頻率的影響非常小.另外,當(dāng)其它參數(shù)固定時(shí),似正規(guī)模實(shí)部和虛部絕對(duì)值隨著角量子數(shù)l的增加而增大.最后,與經(jīng)典的SdS黑洞電磁擾動(dòng)數(shù)值結(jié)果比較發(fā)現(xiàn),在Park黑洞中電磁擾動(dòng)似正規(guī)模的實(shí)部比較大,而虛部絕對(duì)值比較小,這一現(xiàn)象表明,Horava引力下Park黑洞背景時(shí)空中電磁擾動(dòng)振蕩的比較快而衰減的較慢.

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Electromagnetic perturbation of Park black hole in Horava gravity

Wang Chun-yan1,2, MA Yuan1, DONG Liang1

(1. College of Mathematics and Physics, Bohai University, Jinzhou 121013, China; 2. School of physics and optoelectronic technology, Dalian University of Technology, Dalian 116024, China)

In this paper we have evaluated the quasinormal modes of the electromagnetic perturbation of Park black hole in the Horava gravity, using the third order WKB approximation method. One can find that the quasinormal modes frequencies relate to parameters ΛW、wandl、nof Park black hole .The real parts of complex frequencies decrease with ΛWdecreasing, while the magnitude of the imaginary parts decrease after increase. The influence of parameterswon the quasinormal modes frequency is so little. Moreover, by comparing the numerical results of electromagnetic perturbation with the case of SdS black hole, we find that the electromagnetic field oscillates more faster but damps more slower in Horava gravity.

electromagnetic perturbation; WKB approximation; Horava gravity; Park black hole

2016-04-08.

國家自然科學(xué)基金項(xiàng)目(No：11271055).

王春艷(1980-)，女，博士，副教授，主要從事黑洞物理方面的研究.

loveyan-zi@126.com.

P145. 8

A

1673-0569(2016)04-0313-05