Influence of Lorentz violation on Dirac quasinormal modes in the Schwarzschild black hole spacetime

0.17 MB·English

by  Songbai Chen

Influence of Lorentz violation on Dirac quasinormal modes in the Schwarzschild
black hole spacetime
Songbai Chen
∗
Department of Physics, Fudan University, Shanghai 200433, P. R. China
Institute of Physics and Department of Physics,
Hunan Normal University, Changsha, Hunan 410081, P. R. China
Bin Wang
†
Department of Physics, Fudan University, Shanghai 200433, P. R. China
7
0
0 Rukeng Su‡
2 China Center of Advanced Science and Technology (World Laboratory),
P.B.Box 8730, Beijing 100080, Peoples Republic of China
n
Department of Physics, Fudan University, Shanghai 200433, P. R. China
a
J

6
Abstract
1

1 Usingthethird-orderWKBapproximationandmonodromymethods,weinvestigatetheinfluence
v
9 ofLorentzviolatingcoefficientb(associatedwithaspecialaxial-vectorbµfield)onDiracquasinormal
8
modesin theSchwarzschildblack holespacetime. Atfundamentalovertone,therealpart decreases
0
1 linearly astheparameter bincreases. Butthevariation of theimaginary part with bbecomesmore
0
7 complex. For the larger multiple moment k, the magnitude of imaginary part increases with the
0
increaseofb,whichmeansthatpresenceofLorentzviolationmakesDiracfielddampsmorerapidly.
/
c
q Athighovertones,itisfoundthattherealpartofhigh-dampedquasinormalfrequencydoesnottend
-
r to zero, which is quite a different from the asymptotic Dirac quasinormal modes without Lorentz
g
: violation.
v
i
X PACSnumbers: 04.30.-w,04.62.+v,97.60.Lf
r
a
∗Electronicaddress: [email protected]
†Electronicaddress: [email protected]
‡Electronicaddress: [email protected]
2
I. INTRODUCTION
Since Lorentz invariance was discovered, it has been great of importance in many fields of the fundamental
physics,suchasthe Einstein’s specialrelativitytheory,particle physicsandhigh energyphysics. However,the
recentdevelopmentofunifiedgaugetheoriesandtheobservationofhighenergycosmicrays[1][2][3]implythat
Lorentz symmetry is only an approximate symmetry of nature and may be spontaneously broken in the more
fundamentalphysicsdefinedinahigherscaleofenergy. Obviously,theexistenceofLorentzviolationwillmake
a greatinfluence on the fundamental physics and lead to many subjects need to be reconsidered. Therefore, a
great deal of effort has been attracted to study Lorentz violation in the different fields [4]-[22].
OneofinterestingtheorymodelswithLorentzviolationistheStandardModelExtension[8][9][10]. Itoffersa
consistent theoretical framework which includes the standard model and allows for small violations of Lorentz
and CPT symmetry. This small spontaneous breaking of Lorentz symmetry maybe arise from the presence
of nonzero vacuum expectation values for Lorentz tensors defined in an underlying theory. A straightforward
method of implementing Lorentz violation in the curve spacetime is to imagine the existence of a tensor field
with a non-vanishing expectation value and couple this tensor to gravity or matter fields. In gravitational
theories,vierbeinformalismis usedwidely becauseit canbuild a link betweenthe covariantcomponents T
µν
···
of a tensor field in a coordinate basis and the corresponding covariant components T of the tensor field in
ab
···
a local Lorentz frame. The link can be described by T =ea eb T , where vierbein ea is defined by
µν··· µ ν··· ab··· µ
g =η ea ea . Moreover,itisalsofoundthatvierbeinformalismcandealwithlocalLorentztransformation
µν ab µ ν
anddiffeomorphisms, whichare two basic types of spacetime transformationsin gravitationaltheories. Bluhm
and Kostelecky[11] make use of this tool and find that any violation of diffeomorphism invariance via vacuum
values of vectors or tensors breaks local Lorentz invariance. The converse is also true. For example, if a
spacetime vector b acquires a fixed vacuum expectation value b , which breaks diffeomorphisms, then the
µ µ
h i
associated local vector b as given by contraction with the inverse of vierbein also acquires a fixed vacuum
a
expectation value b . The presence of quantity b breaks local Lorentz symmetry. Therefore, it is natural
a a
h i h i
for us to adopt vierbein formalism to study of Lorentz violation in the curve spacetime.
Adopting vierbein formalism, the fermion partial action S in Standard Model Extension can be explicitly
ψ
expressed as
1
Sψ = d4x√−g(2ieµaψΓa←D→µψ−ψM∗ψ), (1)
Z
3
where eµ is the inverse of the vierbein ea . The symbols Γa and M are
a µ ∗
1
Γa γa c eνaeµ γb d eνaeµ γ γb e eµa if eµaγ g eνaeλ eµ σbc, (2)
≡ − µν b − µν b 5 − µ − µ 5− 2 λµν b c
and
1
M m+im γ +a eµ γa+b eµ γ γa+ H eµ eν σab. (3)
∗ ≡ 5 5 µ a µ a 5 2 µν a b
The first terms of Eqs.(2) and (3) lead to the usual Lorentz invariant kinetic term and mass for the Dirac
field. The parameters a , b , c , d , e , f , g , H are Lorentz violating coefficients which arise from
µ µ µν µν µ µ λµν µν
nonzerovacuumexpectationvaluesoftensorquantitiesandcomprehensivedescribeeffectsofLorentzviolation
on the behavior of particles coupling to these tensor fields. All of coefficients can be constrained as the real
numbers if the action (1) is hermitian. In generally, they are functions of position. The two terms involving
the couplings a and b are CPT odd, which have been extensively studied in connection with Lorentz- and
µ µ
CPT-violating probing experiments including comparative studies of cyclotron frequencies of trapped-atoms
[17],clockcomparisontests[18],spectroscopiccomparisonofhydrogenandantihydrogen[19],analysisofmuon
anomalous magnetic moment[20], study of macroscopic samples of spin-polarized solids [21], and so on.
Ontheotherhand,blackholeisanotherinterestingobjectinthemodernfundamentalphysics. Itisbelieved
widely that the study of black hole may lead to a deeper understanding of the relationship among the general
relative theory, quantum mechanics, thermodynamics and statistics. This means that black hole physics play
an important role in the fundamental physics. However, at present whether black holes exist in our universe
or not is still unclear. A recent investigation shows that quasinormal modes can provide a direct way to
identify black hole existence in our universe because that they carry the characteristic information of black
holes [23][24]. Moreover, it is also found that quasinormal modes have a close connection with the AdS/CFT
correspondence [25][26][27] and the loop quantum gravity [28][29]. Thus, the study of quasinormal modes in
black hole spacetimes has become appealing in recent years [30]-[44].
Since both Lorentzviolationandthe quasinormalmodes arehottopics inphysics atpresent,it is naturalto
raise a question whether Lorentz violation affects the quasinormal modes of black holes. From the action (1),
we canobtain that due to presence of Lorentz violating coefficients Dirac equationmust be modified and then
its quasinormalfrequencies in the black hole spacetimes shouldbe changed. However,to my best knowledgeit
is still an open question how Lorentz violation affects the properties of quasinormal modes in the background
of black hole spacetimes. Obviously, different Lorentz violating coefficients (i.e. different types of breaking
of Lorentz symmetries) have different effects on quasinormal modes. In this paper, we just consider Dirac
equationwitha modifiedtermcontainingLorentzviolatingcoefficientbwhicharisesfromthe nonzerovacuum
4
expectation value of a special axial vector field b [9][10] and calculate quasinormal modes of massless Dirac
µ
fields inthe Schwarzschildblackholespacetime. Ourresultshowsthatinthis casebothfundamentalandhigh
overtones quasinormal frequencies depend on the Lorentz violating coefficient b.
The organizationofthis paper is as follows. InSec.II, we derive the equationof masslessDirac fieldcoupled
with the special axial vector field b in the Schwarzschild black hole spacetime. In Sec.III, we evaluate the
µ
fundamentalovertonesquasinormalfrequenciesoftheDiracperturbationalfieldbyusingthethird-orderWKB
approximation[45][46][47]. In Sec.IV, we adopt the monodromy technique [48][49] and study the high-damped
quasinormal frequencies. The last section is devoted to a summary.
II. THE DIRAC EQUATION WITH LORENTZ VIOLATION ASSOCIATED WITH AN
AXIAL-VECTOR b FIELD
µ
According to the variation of ψ in the action (1), we find that the massless Dirac equation only containing
the CPT and Lorentz covariance breaking kinetic term associated with an axial-vector b field in the curve
µ
spacetime can be expressed as
[iγae µ(∂ +Γ ) b e µγ γa]Ψ=0, (4)
a µ µ − µ a 5
where
I 0 0 σi 0 I
γ0 = , γi = , γ =iγ0γ1γ2γ3 = . (5)
0 I σi 0 5 I 0
(cid:18) − (cid:19) (cid:18) − (cid:19) (cid:18) (cid:19)
SincetheLorentzviolationisverysmall,itisreasonableforustoassumethattheaxial-vectorb fielddoesnot
µ
change the background metric. For convenience, we take b as a non-zero timelike vector ( b ,0,0,0), where b
µ r2
is a constant. In the Schwarzschildspacetime, the vierbein can be defined as
2M 1
ea =( 1 , , r, rsinθ). (6)
µ r − r 1 2M
− r
q
SettingΨ=(1−2Mr )−14(sinθ)−21ΦandsubstitutingEq.(6)intoEq.(4),theDiracequation(4)canbesimplified
as
γ0 ∂ 2M ∂ 1 γ2 ∂ γ3 ∂ b
+ 1 γ1( + )+ + + γ γ0 Φ=0. (7)
 1 2M ∂t r − r ∂r r r ∂θ rsinθ∂ϕ r2 1 2M 5 
− r − r
q q 
If we define a tortoise coordinate
r
r =r+2Mln( 1), (8)
∗ 2M −
and the ansatz
iG(±r)(r)φ±jm(θ,ϕ)
Φ= e iωt (9)
  −
 F(±r)(r)φ∓jm(θ,ϕ) 
 
5
with
j+mYm−1/2
2j l 1 1
φ+ =  q , k =j+ , j =l+ ,
jm 2 2
j mYm+1/2
 −2j l 
 
 q  (10)
j−2jm++21Ylm−1/2 1 1
φ−jm =  q , k =−(j+ 2), j =l− 2,
j+m+1Ym+1/2
 − 2j+2 l 
 q 
we find that the case for (+) and ( ) in the functions F and G can be put together and Eq.(7) can be
± ±
−
rewritten as
d G k 2M b I 0 G 0 ω G
+ 1 = . (11)
dr∗ (cid:18)F (cid:19) rr − r − r2!(cid:18)0 −I (cid:19)(cid:18)F (cid:19) (cid:18)−ω 0(cid:19)(cid:18)F (cid:19)
It is very easy to find the decoupled equations for variables F and G can be expressed as
d2F
+(ω2 V )F =0, (12)
dr2 − 1
∗
d2G
+(ω2 V )G=0, (13)
dr2 − 2
∗
with
dW
V = +W2
1,2 ±dr
∗
= |k|∆12[k ∆21 (r 3M) 2b]+ b2±2b(r−2M), (14)
r4 | | ∓ − − r4
where W = k 1 2M b and ∆ = r(r 2M). It is obvious that the potentials V and V are related
r − r − r2 − 1 2
q
to the coefficient b of Lorentz violations, which means that the Dirac quasinormal modes should depend on
Lorentz violation. If we set b = 0, Eqs.(12)-(14) give the results of general Dirac fields in the Schwarzschild
black hole spacetime[37]. Moreover,it is well knownthat the potentials V and V possess the same spectra of
1 2
quasinormalfrequencies because that they are supersymmetric partners derived from the same superpotential
W. In the following, we therefore just make use of Eq.(12) to evaluate the Dirac quasinormal frequencies and
write V as V.
1
III. THE FUNDAMENTAL DIRAC QUASINORMAL MODES WITH LORENTZ VIOLATING
COEFFICIENT b
µ
Inordertostudytherelationshipbetweenquasinormalfrequenciesandthecoefficientb,wecantakeM =1.
Fig. 1 shows the variation of the effective potential with the coefficient b of Lorentz violations for fixed k =3.
From this figure we can find that as b increases, the peak value of the potential barrier gets lower and the
location of the peak (r =r ) moves along the right.
p
6
0.35
V
0.3
0.25
0.2
0.15 b=0.9 b=0
0.1
0.05
0 r
0 2 4 6 8 10 12 14
FIG.1: VariationoftheeffectivepotentialforthemasslessDiracfield(k=3)withthecoefficientbofLorentzviolations
Let us now evaluate the fundamental quasinormal frequencies for the massless Dirac field with Lorentz
violations by using the third-order WKB potential approximation, a numerical method devised by Schutz,
Will and Iyer [45][46][47]. Due to its considerable accuracy for lower-lying modes, this method has been used
extensively in evaluating quasinormal frequencies of various black holes. In this approximate method, the
formula for the complex quasinormal frequencies ω is
ω2 =[V +( 2V′′)1/2Λ] i(n+ 1)( 2V′′)1/2(1+Ω), (15)
0 − 0 − 2 − 0
where
1 1 V(4) 1 1 V′′′ 2
Λ = 0 +α2 0 (7+60α2) ,
(−2V0′′)1/2 8 V0′′ !(cid:18)4 (cid:19)− 288 V0′′ ! 
′′′ 4
1 5 V 
Ω = 0 (77+188α2)
(−2V0′′)(cid:26)6912 V0′′ !
1 V′′′2V(4) 1 V(4) 2
0 0 (51+100α2)+ 0 (67+68α2)
− 384 V0′′3 ! 2304 V0′′ !
1 V′′′V(5) 1 V(6)
+ 0 0 (19+28α2) 0 (5+4α2) , (16)
288 V0′′2 ! − 288 V0′′ ! (cid:27)
and
1 dsV
α=n+ , V(s) = ,
2 0 drs
∗ (cid:12)(cid:12)r∗=r∗(rp)
(cid:12)
n is overtone number. (cid:12)
Substituting the effective potential V into the formulaabove,we canobtain the quasinormalfrequencies for
theDiracfieldswithLorentzviolations. Thefundamentmodesfrequenciesfork =1 5arelistintheTable1.
∼
From Fig.2, we find that for fixed k the real part almost decrease linearly with the increase with b. Moreover,
Fig. 3 tells us that the relationshipbetween the magnitude of the imaginaryparts andb is more complex. For
largerk, wefind thatthey increaseas bincreases,whichmeans thatpresence ofLorentzviolationmakesDirac
oscillation damps more rapidly.
7
b ω (k=1) ω (k=2) ω (k=3) ω (k=4) ω (k=5)
0 0.176452-0.100109i 0.378627-0.096542i 0.573685-0.096320i 0.767194-0.096276i 0.960215-0.096256i
0.1 0.166462-0.100956i 0.367984-0.096909i 0.562847-0.096596i 0.756264-0.096489i 0.949237-0.096431i
0.2 0.157373-0.101275i 0.357770-0.097175i 0.552294-0.096829i 0.745548-0.096681i 0.938430-0.096592i
0.3 0.149036-0.101037i 0.347972-0.097342i 0.542020-0.097021i 0.735043-0.096851i 0.927792-0.096739i
0.4 0.141230-0.100325i 0.338572-0.097408i 0.532020-0.097174i 0.724745-0.097000i 0.917321-0.096873i
0.5 0.133666-0.099621i 0.329548-0.097380i 0.522288-0.097288i 0.714652-0.097128i 0.907015-0.096994i
0.6 0.126319-0.101000i 0.320878-0.097266i 0.512819-0.097363i 0.704761-0.097234i 0.896874-0.097101i
0.7 0.122890-0.114215i 0.312532-0.097083i 0.503605-0.097402i 0.695069-0.097319i 0.886894-0.097194i
0.8 ---- 0.304478-0.096864i 0.494639-0.097404i 0.685573-0.097384i 0.877074-0.097275i
0.9 ---- 0.296676-0.096668i 0.485914-0.097372i 0.676269-0.097428i 0.867413-0.097342i
1.0 ---- 0.289080-0.096612i 0.477423-0.097310i 0.667154-0.097453i 0.857907-0.097397i
TABLEI:Thefundamentalovertones(n=0)quasinormalfrequenciesofDiracfieldwithLorentzviolatingkineticterm
associated with an axial-vector bµ in theSchwarzschild black hole spacetime for k=1∼5.
1.2 Re
k=6
1
k=5
0.8
k=4
0.6
k=3
0.4 k=2
0.2 k=1
0 b
0 0.2 0.4 0.6 0.8 1
FIG. 2: Variation of thereal parts of Dirac quasinormal frequencies with thecoefficient b of Lorentz violations.
IV. THE HIGH-DAMPED DIRAC QUASINORMAL MODES WITH LORENTZ VIOLATING
COEFFICIENT b
µ
Motivated by Hod conjecture[28], a great deal of effort has been devoted to the study of the high-damped
quasinormal modes in the different black hole spacetimes because that Hod’s conjecture suggests that there
maybe exist a connection between the high-damped quasinormal frequencies and quantum gravity. In this
section,weadopttothemonodromymethod[48][49]andinvestigatethehigh-dampedDiracquasinormalmodes
with Lorentz violations in the Schwarzschild black hole spacetime. Our purpose is to probe whether Lorentz
violations affects Hod conjecture. As in Ref.[48], after selecting the contour L as shown in Fig.4, we can
calculate the global monodromy aroundthe contour L. In the neighborhoodof the event horizonr =2M, the
effective potential V and the solution of Eq.(12) can be approximated as
b2 i ω2 b2 z
V , φ(r) e −(2M)4 . (17)
∼ (2M)4 ∼ r
Sincetheonlysingularityofφ(r)ore iωz insidethecontouroccursatthepointr =2M. Afterafullclockwise
−
round trip, φ(r) acquires a phase eπsω2−κ(2Mb2)4 , while e−iωz acquires a phase e−πκω. So the coefficient of e−iωz
π(ω+sω2−(2Mb2)4)
in the asymptotic of φ(r) must be multiplied by e κ .
8
0.114 -Im 0.0974 -Im
0.112
0.0972
0.11 k=1
0.108 0.097 k=2
0.106
0.104 0.0968
0.102
0.0966
0.1 b b
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0 0.2 0.4 0.6 0.8 1
0.0974 -Im -Im k=4
0.0974 k=5
k=6
0.0972
0.0972
0.097 k=3 0.097
0.0968 0.0968
0.0966
0.0966
0.0964
0.0964
b 0.0962 b
0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1
FIG. 3: Variation of theimaginary parts of Dirac quasinormal frequencies with the coefficient b of Lorentz violations.
B(cid:13) Im(cid:13)(r(cid:13)(cid:13))(cid:13)(cid:13) L(cid:13)
Re(cid:13)(x(cid:13)(cid:13))=(cid:13)(cid:13)0(cid:13)(cid:13)
Re(cid:13)(x(cid:13)(cid:13))<(cid:13)(cid:13)0(cid:13)(cid:13)
Re(cid:13)(r(cid:13)(cid:13))(cid:13)(cid:13)
A(cid:13) r=(cid:13)2(cid:13)M(cid:13)(cid:13)
co(cid:13)m(cid:13)p(cid:13)le(cid:13)(cid:13)x(cid:13) r(cid:13)(cid:13)- p(cid:13)(cid:13)(cid:13)la(cid:13)(cid:13)n(cid:13)e(cid:13)(cid:13)
FIG. 4: The complex r-planeand thecontourL. The regions with thehachures denotethearea Re(x)<0.
Moreover,wefindthatthe behaviorsofthe tortoisecoordinater andofthe effective potentialV inEq.(12)
∗
near the singular point r =0 are
r2
r ,
∗ ∼ −4M
b2 4Mb 1 j2
V − = − , (18)
∼ r4 − 4r2
∗
wherej =1 b . AsinRef.[48],accordingtothelocalmonodromyaroundthesingularpointr =2M,wefind
−2M
that the monodromy around the contour L must multiply the coefficient of e iωz by a factor (1+2cosπj).
−
−
Comparingthelocalandglobalmonodromy,wecanobtaindirectlythehigh-dampedquasinormalfrequencies
formula for Dirac fields with Lorentz violations in the Schwarzschild black hole spacetime
π ω+sω2−(2Mb2)4 b
e (cid:20) κ (cid:21) = [1+2cosπ(1 )], n . (19)
− − 2M →∞
Where κ is the surface gravity constant of the event horizon of the black hole. Comparing with mass M of
9
black hole,parameterb is verysmall,thus the term b2 inEq.(19) canbe neglected. The frequency formula
(2M)4
for high-damped quasinormalmodes can be further simplified as
b
ω =T ln [1+2cosπ(1 )] i2nπT , n , (20)
H | − 2M |− H →∞
where T is the Hawking temperature of Schwarzschildblack hole. It is obvious that Lorentz violating coeffi-
H
cient b affects the high-damped Dirac quasinormal frequencies.
In the higher dimensional (D > 4, D is the dimensions of spacetime ) Schwarzschild black hole spacetime,
the axial vector field can be taken as D-component form ( b ,0, , ,0). Similarly, we find that in this
rD−2 ··· ···
D 1
case the behaviors of the tortoise coordinate r and of the effective po−tential V near the singular point r = 0
∗ | {z }
are
rD 2
r − ,
∗ ∼ −2(D 2)M
−
b2 2(D 2)Mb 1 j2
V − − = − , (21)
∼ r2(D 2) − 4r2
−
∗
where j = 1 b . Repeating above operations, we can obtain that the high-damped Dirac quasinormal
− (D 2)M
−
frequencies in the D dimensional Schwarzschildblack hole spacetime satisfy
b
ω =T ln 1+2cosπ[1 ] i2nπT , n . (22)
H | − (D 2)M |− H →∞
−
It is shown that the asymptotic frequency formula of Dirac quasinormal modes can be also affected by the
coefficient b of Lorentz violation in the D dimensional Schwarzschild black hole spacetime. When b 0, we
→
findthatthe realpartsofhigh-dampedDiracquasinormalfrequenciesinbothcasesbecomezero,whichagrees
with the result of Dirac field without Lorentz violations[43][44]. According to Hod’s idea [50], one can obtain
that classical ringing frequencies with an asymptotically vanishing real part correspond to virtual quanta and
the correspondingDiractransitionsinLorentzinvariancearequantummechanicallyforbidden. However,from
the formulas(20) and(22), we find that Dirac quantum transitionis allowablein the LorentzviolationFrame.
It implies that the emergence of Lorentz violation may change the quantum property of Dirac field.
V. SUMMARY
Adopting the third-order WKB approximation and monodromy methods, we investigated the quasinormal
modesofDiracfieldswithLorentzviolatingtermassociatedwithaspecialaxial-vectorb intheSchwarzschild
µ
black hole spacetime. We find that the coefficient b of Lorentz violationaffects Dirac quasionrmalfrequencies.
Atfundamentalovertone,therealpartdecreaseslinearlyastheparameterbincreases. Butthevariationofthe
10
imaginary part with b becomes more complex. For the largermultiple moment k, the magnitude of imaginary
part increases with the increase of b, which means that presence of Lorentz violation makes Dirac field damps
more rapidly. Since the imaginary part of quasinormal frequencies for large multiple number k can be well
approximated as [36] ω i (n+ 1), the possible reason for our result is that the presence of the axial
I ∼ 3√−3GM 2
timelike vector field b with nonzero vacuum expectation values may lead to the decrease of Newton’s gravity
µ
constant G, which is possible in the Lorentz violating theories. Moreover, Dirac also supports that Newton’s
gravity constant G decrease as increase of the age of universe. At high overtones, the real parts depend on
Lorentz violating coefficient b. As b 0, we find the real part of high-damped Dirac quasinormal frequencies
→
becomes zero, which consists with the result of Dirac field without Lorentz violations. Moreover, our result
also shows that the emergence of Lorentz violation may change the quantum property of Dirac field. The
effects of other Lorentz violating coefficients in the action (1) on quasinormal modes of black holes need to be
investigated in the future.
Acknowledgments
We thank the referee for his/her quite useful and helpful comments and suggestions, which help deepen our
understanding of Lorentz violation and quasinormal modes. This work was partially supported by NNSF of
China, Ministry of Education of China and Shanghai Education Commission. R. K. Su’s work was partially
supported by the National Basic Research Project of China. S. B. Chen’s work was partially supported by
the HunanProvincialNaturalScienceFoundationofChinaunderGrantNo.05JJ40012andScientific Research
Fund of Hunan Normal University under Grant No.22040639.