Quasinormal modes of Dirac field in the Einstein-dilaton-Gauss-Bonnet and Einstein-Weyl gravities
aa r X i v : . [ g r- q c ] S e p Quasinormal modes of Dirac field in the Einstein-dilaton-Gauss-Bonnet andEinstein-Weyl gravities
A. F. Zinhailo ∗ Institute of Physics and Research Centre of Theoretical Physics and Astrophysics,Faculty of Philosophy and Science, Silesian University in Opava, CZ-746 01 Opava, Czech Republic
Quasinormal modes of Dirac field in the background of a non-Schwarzschild black holes in theorieswith higher curvature corrections are investigated in this paper. With the help of the semi-analyticWKB approximation and further using of Pad´e approximants as prescribed in [1] we consider quasi-normal modes of a test massless Dirac field in the Einstein-dilaton-Gauss-Bonnet (EdGB) andEinstein-Weyl (EW) theories. Even though the effective potential for one of the chiralities has anegative gap we show that the Dirac field is stable in both theories. We find the dependence of themodes on the new dimensionless parameter p (related to the coupling constant in each theory) fordifferent values of the angular parameter ℓ and show that the frequencies tend to linear dependenceon p . The allowed deviations of qausinormal modes from their Schwarzschild limit are one orderlarger for the Einstein-Weyl theory than for the Einstein-dilaton-Gauss-Bonnet one, achieving theorder of tens of percents. In addition, we test the Hod conjecture which suggests the upper boundfor the imaginary part of the frequency of the longest lived quasinormal modes by the Hawkingtemperature multiplied by a factor. We show that in both non-Schwarzschild metrics the Dirac fieldobeys the above conjecture for the whole range of black-hole parameters. PACS numbers: 04.50.Kd,04.70.-s
I. INTRODUCTION
Recently, the interest in studying new alternative theories of gravity has been increasing see, for instance, [2]. In spiteof its efficiency, the unmodified general theory of relativity is not able to answer some fundamental questions. Someof the most important problems are the construction of non-contradictory quantum gravity, the singularity problem,the problems of dark matter and dark energy. The problem of the construction of non-contradictory quantum gravityis connected with the non-renormalizability of General Relativity. This can usually be solved by adding the higherorder terms in curvature to the theory [3]. In this paper, we will consider two different approaches: the first approachis related to adding of the Gauss Bonnet term coupled to a dilaton [4–6], while the second theory consists of theWeyl term [7] added to the Einstein action. Both theories are inspired by the low energy limit of string theory[8], which contain quadratic corrections in curvature, but the Gauss-Bonnet term alone leads to the full divergenceand does not contribute to the equations of motions, so that there remaining only two options for adding highercurvature corrections: either coupling of the Gauss-Bonnet term to other fields or choosing essentially non-Gauss-Bonnet quadratic corrections. Thus, here we will consider example of the both options.The Lagrangian of the Einstein-dilaton-Gauss-Bonnet gravity is: L EdGB = 12 R − ∂ µ φ∂ µ φ (1)+ α ′ g e φ (cid:0) R µνρσ R µνρσ − R µν R µν + R (cid:1) , where α ′ is the Regge slope, g is the gauge coupling constant and φ is the dilaton field function. Black holes in theEinstein-dilaton-Gauss-Bonnet gravity has been recently investigated in number of papers [6], [9–17].For the Einstein-Weyl gravity the Lagrangian can be written as follows: L EW = √− g ( γR − αC µνρσ C µνρσ + βR ) , (2)where α , β and γ are coupling constants, C µνρσ is the Weyl tensor. For spherically symmetric and asymptotically flatsolutions we can choose γ = 1 and β = 0 [18], so that the only new coupling constant is α . The condition R = 0 isevidently satisfied in this case, so that the Schwarzschild solution is also the solution of the above theory. The static ∗ Electronic address: [email protected] spherically symmetric and asymptotically flat black holes in the Einstein-Weyl theory represent the generic class ofblack hole solutions in the quadratic theories of gravity if no other matter fields are added. They have been recentlystudied in [19–22].Recently black holes in the both theories have been extensively studied. In particular, quasinormal modes werefound for test scalar and electromagnetic fields [17, 20]. Although quasinormal modes of a Dirac field around blackholes in the Einstein gravity were studied in detail in a number of papers (see [23–27] and reference therein), tothe best of our knowledge there are no works devoted to Dirac quasinormal modes in theories with higher curvaturecorrections. When considering the neutrino field, the special attention must be paid to the presence of a negativeregion of a potential curve with negative chirality. The positive definite effective potential guarantees dynamicalstability of perturbations, that is, absence of unboundedly growing modes. For the one of the chiralities of Dirac fieldin the Schwarzschild background the negative gap does not lead to the instability because the other chirality providespositive definite potential and the both chiralities are proved to be iso-spectral. However, the iso-spectrality has neverbeen proved for the considered non-Einsteinian theories, so that the instability cannot be excluded a priori. Becauseof this, it would be interesting to study the quasinormal spectrum of the Dirac field in the above non-Einsteiniantheories of gravity and see whether there is an instability. After all, the test of stability is extremely importantfor higher curvature corrected theories because of the so called eikonal instability which occurs in a abroad class oftheories with various higher curvature corrections, spacetime dimensions and asymptotics [28–33], and not only forgravitational, but also for test fields [34].We will analyze values of modes at the low angular parameter ℓ and in the eikonal regime. We will find dependenciesof the complex frequency on the new dimensionless parameter p (related to the coupling constant in each theory). Inaddition, we will compare quasinormal modes of both theories between each other and with modes of other fields ineach theory. In addition here we will test the quasinormal modes of Dirac field in the above two theories as to theHod’s conjecture [35] who claims that there must always be a minimal mode whose damping rate is limited by theHawking temperature multiplied by some factor.This work is organized as follows. In Sec. II we introduce a metric and a general wave equation and considerthe effective potential for the Dirac field for a spherically symmetric black hole. For this case we prove that theDirac perturbations are linearly stable in both theories. In Sec. III, the basic principles of the WKB method arebriefly considered, an analytical approximation in the eikonal regime is analysed Sec. III A, the quasinormal modesfor test massless Dirac field are found, a comparative analysis is made for the our result with the results for otherfields in these theories of gravity Sec. III B. In Sec. III C we will check the Hod’s conjecture for the Dirac field inEinstein-dilaton-Gauss-Bonnet and Einstein-Weyl gravities. II. BLACK HOLE METRIC AND ANALYTICS FOR THE WAVE EQUATION
In the general case the metric for a spherically symmetric black hole can be written in the form: ds = − e µ ( r ) dt + e ν ( r ) dr + r (sin θdφ + dθ ) , (3)where e µ ( r ) and e ν ( r ) are the metric coefficients. The explicit expression for the metric coefficients were obtainednumerically in [36] for Einstein-dilaton-Gauss-Bonnet gravity and in [18] for Einstein-Weyl gravity. The approximateanalytical expressions (which will be used here) were obtained in [37] for the Einstein-dilaton-Gauss-Bonnet metric,in [38] for the Einstein-Weyl metric. They are also written down in Appendixs A, B.We parameterize the both black-hole solutions in theories (1, 2) via the following dimensionless parameter p up tothe rescaling: p EdGB ≡ e φ = 6 α ′ g r e φ − φ ∞ ) ( Einstein − dilaton − Gauss − Bonnet ) (4a) p EW = r √ α ( Einstein − W eyl ) . (4b)For convenience we fix radius of the black-hole event horizon to be r = 1. For all p the Schwarzschild metric isthe exact solution of the Einstein-Weyl equations as well, but only at some minimal nonzero p min , in addition tothe Schwarzschild solution, there appears the non-Schwarzschild branch which describes the asymptotically flat blackhole, whose mass is decreasing, when p grows. The approximate maximal and minimal values of p are: p min,EdGB ≥ , p max,EdGB ≤ . , (5a) p min,EW ≈ / ≈ . , p max,EW ≈ . . (5b)The general covariant Dirac equation has the form [39]: γ α (cid:18) ∂∂x α − Γ α (cid:19) Ψ = 0 , (6)where γ α are noncommutative gamma matrices and Γ α are spin connections in the tetrad formalism. We separate ofangular variables in equation (6) and rewrite the wave equation in the following general master form in terms of the“tortoise coordinate” r ∗ [39]: d Ψ dr ∗ + ( ω − V ( r ))Ψ = 0 , dr ∗ = p e ν ( r ) − µ ( r ) dr. (7)The effective potentials of test Dirac ( s = ± /
2) field in the general background (3) can be written as follows: V ± ( r ) = kr e µ ( r ) kr ∓ e µ ( r ) √ e ν ( r ) r ± p e µ ( r ) − ν ( r ) ( p e µ ( r ) ) ′ ! , (8)where the prime designates the differentiation with respect to the “tortoise coordinate” r ∗ . (a) (b) FIG. 1: The effective potential V ( r ) for the EdGB gravity for ℓ = 1; the blue line corresponds to p = 0, the red line correspondsto p = 0 . p = 0 .
97: (a) V − ( r ); (b) V + ( r ). (a) (b) FIG. 2: The effective potential V ( r ) for the EW gravity for ℓ = 1; the blue line corresponds to p = 0 . p = 0 . p = 1 .
14: (a) V − ( r ); (b) V + ( r ). In the both cases for the “plus” (“minus”) potential of the Dirac field k = ℓ + 1 ( k = ℓ ). As can be seen from figs.(1(a), 2(a)) the potential V − ( r ) has a negative gap near the event horizon. The same behavior is appropriate to thepotential V − ( r ) in the Schwarzschild case. However, as it was shown earlier for black holes for which both metriccoefficients are equal (like for the Schwarzschild case e µ ( r ) = e − ν ( r ) [40]), the potentials of opposite chiralities can betransformed into each another with help of the Darboux transformation. This means that from both potentials we getthe same quasinormal spectrum. It allows us to ignore the negative gap of the “minus” potential an and talk aboutoverall stability for the Schwarzschild case. When both metric coefficients are not the same anymore, to the best ofour knowledge the iso-spectrality of both chiralities was not shown. Here we can see that the following replacements:Ψ + = q ( W + ddr ∗ )Ψ − , W = p e µ ( r ) , q = const ; (9)provides the Darboux transformation of equations (7, 9) for transition between “minus” (given by V − ( r ) and “plus”( V + ( r )) perturbations. As the potential for one of the chiralities is positive definite, this immediately guaranteesthe stability of the Dirac field for the other chirality in both considered theories. Therefore, we can use only stablepotential. Later in the work, we will use the potential V + ( r ). III. QUASINORMAL MODES OF MASSLESS DIRAC FIELD FOREINSTEIN-DILATON-GAUSS-BONNET AND EINSTEIN-WEYL GRAVITIES
For finding quasinormal modes, it is necessary to solve the spectral problem with the appropriate boundary con-ditions: for a functions Ψ there are only incoming waves at the horizon ( r ∗ → −∞ ) and only the outgoing wavesat the infinity ( r ∗ → + ∞ ). Quite effectively this problem can be solved using the WKB-method [1, 41–43]. Theadvantages of this method over numerical methods is the ability to obtain low-lying quasinormal modes with sufficientaccuracy automatically for a broad class of effective potentials, and, thereby, not to tailor the method for each case.The method gives good accuracy when n ≤ ℓ , where n = 0 , , , .. is a overtone number. The general formula for them-order of the WKB approach can be written in form: i ( ω − V ) p − V ′′ − m X i =2 Λ i = n + 12 . (10)Here, the Λ i are the correction term of the i-th order and Λ i depend on the value of the potentials V ( r ) and itsderivative at the maximum, V is a value of V ( r ) in r max and V ′′ is a second derivative in r max . But the WKB seriesconverges only asymptotically, there is no strict criterium for evaluation of an error. The higher accuracy of the WKBapproach can be achieved the averaging of the Pad´e approximation [1]. We will use the fourth-order of the WKBapproximation and apply further Pad´e expansion of the order which provides the best accuracy in the Schwarzschildlimit [1, 43]. A. An analytical approximation in the eikonal regime
In the regime of high multipole numbers ℓ (eikonal regime) it is sufficient to use the first order WKB formula: ω = s V − i (cid:18) n + 12 (cid:19) p − V ′′ . (11)When the multipole numbers ℓ is high the behavior of test fields of different spin obey the same law in the dominantorder and the expression for ω for the Dirac field will be identical to the formulas for other spin. For Einstein-dilaton-Gauss-Bonnet case it was found for electromagnetic field [17] for small 1 /ℓ : ω EdGB = 23 √ r (cid:18)(cid:18) ℓ + 12 (cid:19) (1 − . p ) − i (cid:18) n + 12 (cid:19) (1 − . p ) (cid:19) + O ( p , ℓ − ) , (12a) r max = 3 r , r p + O ( p , ℓ − ) , (12b)where r max is the position of peak of the effective potential.For the Einstein-Weyl gravity the values of ω was found in [38] for small 1 /ℓ , where t = 1054 − p is a deviationsfrom the Schwarzschild branch: ω EW = 23 √ r (cid:18)(cid:18) ℓ + 12 (cid:19) (1 − . t ) − i (cid:18) n + 12 (cid:19) (1 − . t ) (cid:19) + O ( t , ℓ − ) , (13a) r max = 3 r . t ) + O ( t , ℓ − ) . (13b)When p = 0 in the formula (12a) and t = 0 in (13a) these formulas go over into the well-known eikonal formula forthe Schwarzschild black hole. A general approach to finding eikonal quasinormal modes for static asymptotically flatand spherically symmetric black holes has been recently suggested in [44]. It is worthwhile mentioning that the realand imaginary parts of the above eikonal formulas for test fields will coincide with the oscillation frequency and theLyapunov exponents of the null geodesics in the background of the Einstein-dilaton-Gauss-Bonnet and Einstein-Weylblack holes [45]. However, this is not expected for the gravitational or other non-test (non-minimally coupled) fields[46–48]. B. Quasinormal modes for low ℓ For obtaining accurate values of quasinormal modes at low numbers ℓ we will use the fourth-order of the WKBapproximation (10) and apply further Pad´e expansion of the order. In the figs. (3, 4) we construct the real (oscillationfrequency) and imaginary (damping rate of oscillation) parts of the frequency ω on the values of the parameter p forvarious multipole numbers ℓ . As can be seen, the function ω ( p ) tends to be linear for all cases. This behavior is alsocharacteristic of other test fields that were previously considered [17], [20]. Comparing figs. (3, 4), we can see thatthe deviations from Schwarzschild branch by the Weyl correction are much larger than the Einstein-dilaton-Gauss-Bonnet gravity. For the Einstein-dilaton-Gauss-Bonnet case values of the modes are decreasing when increasing thedimensionless parameter p . On the contrary, for Weyl case we see, that the oscillation frequency and the dampingrate of oscillations are increasing with increasing p . This also follows from the form of curves for potentials figs. (1,2). With an increase the dimensionless parameter p for the Einstein-dilaton-Gauss-Bonnet gravity, the height of thepotential barrier decreases, which is on the favor of lower bound states. For the Einstein-Weyl gravity, the maximumof potential V ( r ) increases with increasing p . It means, that with increasing p the corresponding frequencies arehigher. (a) (b) (c) FIG. 3: The fundamental quasinormal mode of EdGB ( n = 0) for the Dirac field ( s = 1 / ℓ = 1; (b) ℓ = 2; (c) ℓ = 3. (a) (b) (c) FIG. 4: The fundamental quasinormal mode of EW ( n = 0) for the Dirac field ( s = 1 / ℓ = 1; (b) ℓ = 2; (c) ℓ = 3. Approximate calculation formulas for the complex frequency were found from the obtained data for different values ℓ . For Einstein-dilaton-Gauss-Bonnet it is (14): Re ( ω s =0 . ,ℓ =1 ) ≈ . − . p,Im ( ω s =0 . ,ℓ =1 ) ≈ − .
284 + 0 . p ; (14a) Re ( ω s =0 . ,ℓ =2 ) ≈ . − . p,Im ( ω s =0 . ,ℓ =2 ) ≈ − .
269 + 0 . p ; (14b) Re ( ω s =0 . ,ℓ =3 ) ≈ . − . p,Im ( ω s =0 . ,ℓ =3 ) ≈ : − .
266 + 0 . p. (14c)Accordingly, for Einstein-Weyl ℓ = 1, ℓ = 2, ℓ = 3, we have (15): Re ( ω s =0 . ,ℓ =1 ) ≈ .
011 + 0 . p,Im ( ω s =0 . ,ℓ =1 ) ≈ . − . p ; (15a) Re ( ω s =0 . ,ℓ =2 ) ≈ − .
227 + 1 . p,Im ( ω s =0 . ,ℓ =2 ) ≈ . − . p ; (15b) Re ( ω s =0 . ,ℓ =3 ) ≈ − .
407 + 1 . p,Im ( ω s =0 . ,ℓ =3 ) ≈ . − . p. (15c)Approximate dependencies for parts of the complex frequency in p were obtained in formulas (14) and (15) for low ℓ . For all options, we have a reasonable linear approximation, which is clearly visible in figs. (3) and (4). C. The checking of Hod’s conjecture
In work [35], Hod put forward the statement for damping rate of the fundamental oscillation. In other words inthe spectrum of quasinormal modes there always must exist a frequency whose absolute value of the imaginary partis smaller than π times Hawking temperature of the black hole. According to this statement, for asymptotically flatblack holes as well as for nonasymptotically flat ones, the following inequality holds: | Im ( ω ) | ≤ πT H , (16)where T H is the Hawking temperature. The Hawking temperature T H for Einstein-dilaton-Gauss-Bonnet andEinstein-Weyl gravities can be written in the next form: T H = 14 π r de µ ( r ) dr de − ν ( r ) dr (cid:12)(cid:12)(cid:12)(cid:12) r = r . (17) (a) (b) FIG. 5: The dependencies of Hod’s conjecture on p : (a) EdGB ; (b) EW . From the fig. (5) it can be seen that for the whole interval of values of parameter p for Einstein-dilaton-Gauss-Bonnet and Einstein-Weyl metrics | Im ( ω ) | πT H ≤
1. It means, that for the Hod’s conjecture also holds for the casesconsidered.
IV. CONCLUSIONS
In this work we considered test massless Dirac field in Einstein-dilaton-Gauss-Bonnet and Einstein-Weyl gravities.It was shown that although the potential V − ( r ) of the Dirac field has a negative gap near the event horizon, wehave proved that the Dirac field is stable in both considered theories. This is possible because of the stability of thesecond potential V + ( r ) and the iso-spectrality of both potentials. Quasinormal modes were obtained for both metricsfor different values of the angular parameter ℓ . The dependence of the complex frequency on the new parameter p was constructed. The Einstein-Weyl gravity allows for much stronger deviations form the Schwarzschild geometry.Therefore, quasinormal modes of Einstein-Weyl black hole are more different from the Schwazrschild case than thoseof Einstein-dilaton-Gauss-Bonnet black hole, achieving the effect of tens of percents. In the last part of this work weshown, that the Hod’s conjecture holds for Einstein-dilaton-Gauss-Bonnet and Einstein-Weyl gravities for the Diracfield.In the future, it would be interesting to investigate the Dirac field including the massive term and check thepossibility of the existence of the arbitrarily long-lived quasinormal modes, called quasiresonances [49], for this case.In [50] it has recently been shown that the quasiresonances exist for the massive Dirac field in the Einstein theory, butno such study was performed in the higher curvature corrected theories. Our approach could also be extended to thecase of Einstein-Gauss-Bonnet black holes with other types of coupling of the scalar field [51] as well as to scalarizedblack holes for whose metrics analytical approximations are known [52]. Acknowledgments
The author acknowledges the support of the grant 19-03950S of Czech Science Foundation ( GA ˇ CR ) and acknowl-edges the SU grant SGS/12/2019. The author would like to acknowledge Roman Konoplya for useful discussions. Appendix A: Analytical form of the metric functions of Einstein-dilaton-Gauss-Bonnet metric
The analytical approximations for the metric functions e µ ( r ) and e ν ( r ) have the forms: e µ ( r ) = [( r − r )(11528( − p + 937132 p − p + 325377 p ) r + 8(263522875+497564855 p − p + 1833700801 p − p − p + 3579147 p ) r r − − p ) p ( − p − p + 33 p ) r r + p (283646440 − p +1868830098 p − p + 470844780 p − p ) rr + 1441 p ( − p − p − p + 11115 p ) r )] / [11528( − p )( − p ) r (( − − p +108459 p ) r + (36575 + 121424 p − p ) r )] , (A1a) e ν ( r ) = [2882( − p )( − p ) r (( − − p + 108459 p ) r + (36575 + 121424 p − p ) r )(18( − p − p + 24795 p ) r + 18(223782 − p + 110455 p +16245 p ) rr − p ( − p − p + 1404 p ) r ) ] / [81(13 − p ) ( r − r )((22914 − p + 2755 p ) r + ( − p + 1805 p ) r ) (11528( − p + 937132 p − p + 325377 p ) r + 8(263522875 + 497564855 p − p + 1833700801 p − p − p + 3579147 p ) r r − − p ) p ( − p − p +33 p ) r r + p (283646440 − p + 1868830098 p − p + 470844780 p − p ) rr + 1441 p ( − p − p − p + 11115 p ) r )] . (A1b) Appendix B: Analytical form of the metric functions of Einstein-Weyl metric
The metric coefficients are determined as follows: e µ ( r ) = (cid:16) − r r (cid:17) A ( r ) , e ν ( r ) = B ( r ) (cid:0) − r r (cid:1) A ( r ) , (B1)where A ( r ) = " (cid:0) p − p + 806771764 p − p + 604749333 (cid:1) r (B2a)+78279 (cid:0) p − p + 393815823540843 p + 2680050514097926 p − p + 10978748485369369 p − (cid:1) r r − (cid:0) p + 180905642811 p + 417682197141 p − p − p + 3382539200269 p − (cid:1) r r − (cid:0) p − p − p + 2389090517292988952355 p − p + 2186684376605688462974 p − (cid:1) rr +31 (cid:0) p + 410672271594465801 p − p + 51431640078486304191 p − p + 43250367615320791700 p − (cid:1) r / " r (cid:0) p − p + 806771764 p − p + 604749333 (cid:1) r − (cid:0) p − p + 386036980 p − p + 341153481 (cid:1) rr +899 (cid:0) p + 115659 p − p − p + 1101579 (cid:1) r ! , B ( r ) = " (cid:0) p − p + 20865434326 p + 23094914865 (cid:1) r (B2b) − (cid:0) p − p + 100077612184 p − (cid:1) r r − (cid:0) p + 17950319416564777 p − p + 5097428297648940 (cid:1) rr +635371 (cid:0) p − p + 90235778452 p − (cid:1) r / " r (cid:0) p − p + 20865434326 p + 23094914865 (cid:1) r − (cid:0) p − p + 100077612184 p − (cid:1) rr +6 (cid:0) p − p + 13202029643 p + 2626009760 (cid:1) r ! . 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