A comparative study of Dirac quasinormal modes of charged black holes in higher dimensions
Sayan K. Chakrabarti
备注：
21 pages, several eps figures, change in title, corrections made in the discussion of αgoing to zero limit, some references added, version to appear in Eur. Phys. J. C
In this work we study the Dirac quasinormal modes of higher dimensional charged black holes. Higher dimensional Reissner-Nördstrom type black holes as well as charged black holes in Einstein Gauss-Bonnet theories are studied for Fermionic perturbations using WKB method. A comparative study of the quasinormal modes in the two different theories of gravity has been performed. The beahviour of the frequencies with the variation of black hole parameters as well as with the variation of spacetime dimensions are done. We also study the large multipole number limit of the black hole potential in order to look for an analytic expression for the frequencies.
We connect the quasinormal modes corresponding to Dirac fermions in various black holes backgrounds to an N=2 supersymmetric quantum mechanics algebra, which can be constructed from the radial part of the fermionic solutions of the Dirac equation. In the massless fermion case, the quasinormal modes are in bijective correspondence with the zero modes of the fermionic system and this results to unbroken supersymmetry. The massive case is more evolved but as we prove, supersymmetry remains unbroken even in this case.
We study Dirac quasinormal modes of Schwarzschild-Anti-de Sitter (Schwarzschild-AdS) black holes, following the generic principle for allowed boundary conditions proposed in \cite{PhysRevD.92.124006}. After deriving the equations of motion for Dirac fields on the aforementioned background, we impose vanishing energy flux boundary conditions to solve these equations. We find a set of two Robin boundary conditions are allowed. These two boundary conditions are used to calculate Dirac normal modes on empty AdS and quasinormal modes on Schwarzschild-AdS black holes. In the former case, we recover …
We study charged Dirac quasinormal modes (QNMs) on Reissner-Nordström-Anti-de Sitter (RN-AdS) black holes with generic Robin boundary conditions, by extending our earlier work of neutral Dirac QNMs on Schwarzschild-AdS black holes. We first derive the equations of motion for charged Dirac fields on a RN-AdS background. To solve these equations we impose a requirement on the Dirac field: that its energy flux should vanish at asymptotic infinity. A set of two Robin boundary conditions compatible with QNMs is consequently found. By employing both analytic and numeric methods, we then obtain the q…
We study the quasinormal modes of fermionic perturbations for an asymptotically Lifshitz black hole in 4-dimensions with dynamical exponent z=2 and plane topology for the transverse section, and we find analytically and numerically the quasinormal modes for massless fermionic fields by using the improved asymptotic iteration method and the Horowitz-Hubeny method. The quasinormal frequencies are purely imaginary and negative, which guarantees the stability of these black holes under massless fermionic field perturbations. Remarkably, both numerical methods yield consistent results; i.e., both m…
The quasinormal modes of the Reissner-Nordström de Sitter black hole for the massless Dirac fields are studied using the Pöshl-Teller potential approximation. We find that the magnitude of the imaginary part of the quasinormal frequencies decreases as the cosmological constant or the orbital angular momentum increases, but it increases as the charge or the overtone number increases. An interesting feature is that the imaginary part is almost linearly related to the real part as the cosmological constant changes for fixed charge, and the linearity becomes better as the orbital angular momentum …
In a D-dimensional Lifshitz black hole we calculate exactly the quasinormal frequencies of a test Dirac field in the massless and zero angular eigenvalue limits. These results are an extension of the previous calculations in which the quasinormal frequencies of the Dirac field are determined, but in four dimensions. We discuss the four-dimensional limit of our expressions for the quasinormal frequencies and compare with the previous results. We also determine whether the Dirac field has unstable modes in the D-dimensional Lifshitz spacetime.
The quasinormal modes (QNMs) associated with the decay of Dirac field perturbation around a Schwarzschild black hole is investigated by using continued fraction and Hill-determinant approaches. It is shown that the fundamental quasinormal frequencies become evenly spaced for large angular quantum number and the spacing is given by $ω_{λ+1}- ω_λ=0.38490-0.00000i$. The angular quantum number has the surprising effect of increasing real part of the quasinormal frequencies, but it almost does not affect imaginary part, especially for low overtones. In addition, the quasinormal frequencies also bec…
In this paper we study the Dirac quasinormal modes of an uncharged 2 + 1 black hole proposed by Mandal et. al and referred to as MSW black hole in this work. The quasi- normal mode is studied using WKB approximation method. The study shows that the imaginary part of quasinormal frequencies increases indicating that the oscillations are damping and hence the black hole is stable against Dirac perturbations.
We investigate the quasinormal mode frequencies for the massless Dirac field in static four dimensional $AdS$ space-time. The separation of the Dirac equation is achieved for the first time in $AdS$ space. Besides the relevance that this calculation can have in the framework of the $AdS/CFT$ correspondence between M-theory on $AdS_4\times S^7$ and SU(N) super Yang-Mills theory on $M_3$, it also serves to fill in a gap in the literature, which has only been concerned with particles of integral spin $0,1,2$.
The dirac quasinormal modes of the Reissner-Nordström black hole surrounded by quintessence are investigated using the third WKB approximation. We find that the magnitude of the imaginary part of the quasinormal mode frequencies increases firstly and then decreases as the charge $Q$ increases, but it decreases as the absolute value of $w_{q}$ increases. The magnitude of the imaginary part of quasinormal complex frequencies is smaller than those with no quintessence. That is to say, the dirac field damps more slowly due to the presence of quintessence.
We study charged fermionic perturbations in the background of two-dimensional charged Dilatonic black holes, and we present the exact Dirac quasinormal modes. Also, we study the stability of these black holes under charged fermionic perturbations.
Motivated by the recent computations of the quasinormal frequencies of higher dimensional black holes we exactly calculate the quasinormal frequencies of the Dirac field propagating in D-dimensional (D > 4) massless topological black hole. From the exact values of the quasinormal frequencies for the fermion and boson fields we discuss whether the recently proposed bound on the relaxation time of a perturbed thermodynamical system is satisfied in D-dimensional massless topological black hole. Also we study the consequences of these results.
The quasinormal modes (QNMs) associated with the decay of Dirac field perturbation around a Schwarzschild-anti-de Sitter (SAdS) black hole is investigated by using Horowitz-Hubeny approach. We find that both the real and the imaginary parts of the fundamental quasinormal frequencies for large black holes are linear functions of the Hawking temperature, and the real part of the fundamental quasinormal frequencies for intermediate and small black holes approximates a temperature curve and the corresponding imaginary part is almost a linear function of the black hole radius. The angular quantum n…
Recently, the anomalous decay rate of quasinormal modes has been studied for some geometries under scalar field perturbations, which occurs when the longest-lived modes are the ones with higher angular number, as well as, the existence of a critical scalar field mass, i.e the value of scalar field mass such that the decay rate does not depend appreciably on the angular number, and beyond which the behaviour of the decay rate is inverted. Here, we consider the propagation of fermionic fields in the background of Schwarzschild de Sitter black holes, and we show that the anomalous decay rate beha…
We find exact solutions to the Dirac equation in D-dimensional de Sitter spacetime. Using these solutions we analytically calculate the de Sitter quasinormal (QN) frequencies of the Dirac field. For the massive Dirac field this computation is similar to that previously published for massive fields of half-integer spin moving in four dimensions. However to calculate the QN frequencies of the massless Dirac field we must use distinct methods in odd and even dimensions, therefore the computation is different from that already known for other massless fields of integer spin.
The quasinormal modes of a massless Dirac field in the de Rham-Gabadadze-Tolley (dRGT) massive gravity theory with asymptotically de Sitter spacetime are investigated using the Wentzel- Kramers-Brillouin (WKB) approximation. The effective potential for the massless Dirac field due to the dRGT black hole is derived. It is found that the shape of the potential depends crucially on the structure of the graviton mass and the behavior of the quasinormal modes is controlled by the graviton mass parameters. Higher potentials give stronger damping of the quasinormal modes. We compare our results to th…
We study new type black holes in three-dimensional New Massive Gravity and we calculate analytically the quasinormal modes for fermionic perturbations for some special cases. Then, we show that for these cases the new type black holes are stable under fermionic field perturbations.
While quasinormal modes of bosonic fields for the non-trivial $4$-dimensional Einstein-Gauss-Bonnet theory of gravity suggested in [D.~Glavan and C.~Lin, Phys.\ Rev.\ Lett.\ {\bf 124}, 081301 (2020)] have been recently studied, there is no such study for fermionic fields. Here we calculate quasinormal modes of the Dirac field for spherically symmetric asymptotically flat black hole in this novel $4D$ Einstein-Gauss-Bonnet theory. The values of the quasinormal frequencies, calculated by the 6th order WKB method with Padé approximants and the time-domain integration, show that the real part of t…
Using the third-order WKB approximation and monodromy methods, we investigate the influence of Lorentz violating coefficient $b$ (associated with a special axial-vector $b_μ$ field) on Dirac quasinormal modes in the Schwarzschild black hole spacetime. At fundamental overtone, the real part decreases linearly as the parameter $b$ increases. But the variation of the imaginary part with $b$ becomes more complex. For the larger multiple 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…
We consider quasinormal modes of the massive Dirac field in the background of a Schwarzschild- Tangherlini black hole. Different dimensions of the spacetime are considered, from d = 4 to d = 9. The quasinormal modes are calculated using two independent methods: WKB and continued fraction. We obtain the spectrum of quasinormal modes for different values of the overtone number and angular quantum number. An analytical approximation of the spectrum valid in the case of large values of the angular quantum number and mass is calculated. Although we don't find unstable modes in the spectrum, we show…
We evaluate quasinormal mode frequencies for RN black hole space times with cosmic string perturbed by a massless Dirac field, using Pöschl-Teller potential method. We find that only in the case of RN black hole having small charge, the effect due to cosmic string will dominate when perturbed by a negatively charged Dirac field, but if we are perturbing with positively charged Dirac field decay will be less in the case of black hole having cosmic string compared to the RN black hole without string
We have studied quasinormal modes of spinor $\frac{1}{2}$, massless Dirac field perturbations of a black hole in massive gravity. The parameters of the theory, such as the mass of the black hole, the scalar charge of the black hole, mode number and the multipole number are varied to observe how the corresponding quasinormal frequencies change. We have also used the P$\ddot{o}$schl-Teller approximation to reach analytical values for the frequencies of quasinormal modes for comparison with the numerically obtained values. Comparisons are done with the frequencies of the Schwarzschild black hole.
The quasinormal modes (QNMs) of Dirac field perturbations of a Reissner-Nordström black hole in an asymptotically Anti-de Sitter spacetime are investigated. We find that both the real and imaginary parts of the fundamental quasinormal frequencies for large black holes are the linear functions of the Hawking temperature, and the slope of the lines for the real parts decreases while that for the magnitude of the imaginary parts increases as the black hole charge increases. According to the Anti-de Sitter/Conformal Field Theory (AdS/CFT) correspondence, the fact shows that different charge presen…
The Dirac quasinormal modes (QNMs) of the Kerr-Newman black hole are investigated using continued fraction approach. It is shown that the quasinormal frequencies in the complex $ω$ plane move counterclockwise as the charge or angular momentum per unit mass of the black hole increases. They get a spiral-like shape, moving out of their Schwarzschild or Reissner-Nordström values and "looping in" towards some limiting frequencies as the charge and angular momentum per unit mass tend to their extremal values. The number of the spirals increases as the overtone number increases but decreases as the …
Using the WKB approximation, massless and massive Dirac quasinormal modes (QNMs) are studied in spherically symmetric regular spacetimes. We analyze the relationships between QNM frequencies and the parameters (angular momentum number $l$, magnetic monopole charge $β$ and the mass of the field $m$), and discuss the extreme charge of magnetic monopole $β_{e}$ for spherically symmetric regular black holes (BHs). Furthermore, we apply an expansion method to expand QNMs in inverse powers of $L=l+1/2$, and confirm good precision with $l>n$. Finally, we improve traditional finite difference method t…
We evaluate the quasinormal modes of massless Dirac perturbation in a Schwarzschild black hole surrounded by the free static spherically symmetric quintessence by using the third-order WKB approximation. The result shows that due to the presence of quintessence, the massless field damps more slowly. The real part of the quasinormal modes increases and the the absolute value of the imaginary part increases when the state parameter $w_q$ increases. In other words, the massless Dirac field decays more rapidly for the larger $w_q$. And the peak value of potential barrier gets higher as $|k|$ incre…
The quasinormal modes of charged and uncharged massive scalar fields and also of charged Dirac fields against the background of a charged spherical black hole endowed with a scalar hair have been investigated. Special emphasis has been given to the case where negative scalar charge dominates over the electric charge of the black hole which mimics an Einstein-Rosen bridge. Except for the complete monotonic behaviour of the damping (imaginary part of the quasinormal mode) against the charge of the black hole as opposed to the existence of a peak for the pure RN case, the qualitative behaviour do…
In this work, quasinormal modes (QNMs) of the Schwarzschild black hole are investigated by taking into account the quantum fluctuations. Gravitational and Dirac perturbations were considered for this case. The Regge-Wheeler gauge and the Dirac equation were used to derive the perturbation equations of the gravitational and Dirac fields respectively and the third order Wentzel-Kramers-Brillouin (WKB) approximation method is used for the computing of the quasinormal frequencies. The results show that due to the quantum fluctuations in the background of the Schwarzschild black hole, the QNMs of t…
We find the low lying quasinormal mode frequencies of the recently proposed novel four dimensional Gauss-Bonnet de Sitter black holes for scalar, electromagnetic and Dirac field perturbations using the third order WKB approximation as well as Padé approximation, as an improvement over WKB. We figure out the effect of the Gauss-Bonnet coupling $α$ and the cosmological constant $Λ$ on the real and imaginary parts of the QNM frequencies. We also study the greybody factors and eikonal limits in the above background for all three different types of perturbations.
The regularization proposed in [D.~Glavan and C.~Lin, Phys.\ Rev.\ Lett.\ {\bf 124}, 081301 (2020)] led to the black hole solutions which turned out to be the solutions of the consistent well-defined $4$-dimensional Einstein-Gauss-Bonnet theory of gravity suggested in [K.~Aoki, M.~Gorji and S.~Mukohyama, arXiv:2005.03859]. Recently the quasinormal modes of bosonic and fermionic fields for this theory were studied. Here we calculate quasinormal frequencies of the test scalar, electromagnetic and Dirac fields for the spherically symmetric black hole in the novel $4D$ Einstein-Gauss-Bonnet-de Sit…
We study Chern-Simons black holes in d-dimensions and we calculate analytically the quasinormal modes of fermionic perturbations. Also, we consider as background the five-dimensional Chern-Simons black hole with torsion and the BTZ black hole with torsion. We have found that the quasinormal modes depend on the highest power of curvature present in the Chern-Simons theory, such as occurs for the quasinormal modes of scalar perturbations. We also show that the effect of the torsion is to modify the real part of the quasinormal frequencies, which modify the oscillation frequency of the field for …
Quasinormal modes of Dirac field in the background of a non-Schwarzschild black holes in theories with higher curvature corrections are investigated in this paper. With the help of the semi-analytic WKB approximation and further using of Padé approximants as prescribed in [1] we consider quasinormal modes of a test massless Dirac field in the Einstein-dilaton-Gauss-Bonnet (EdGB) and Einstein-Weyl (EW) theories. Even though the effective potential for one of the chiralities has a negative gap we show that the Dirac field is stable in both theories. We find the dependence of the modes on the new…
Quasinormal modes of a massive Dirac field were calculated for various static black hole backgrounds with the help of the WKB formula. These estimations, however, are rough and valid only for very small values of $μM$, where $M$ and $μ$ are mass of the black hole and field respectively. Thus, no accurate calculations of massive Dirac modes are known even for the Schwarzschild black hole and this is all the more so for the Kerr solution. Here we calculate quasinormal modes of a massive Dirac field in the Kerr background. We have shown that the infinitely long-lived quasinormal modes (quasireson…
We study the quasinormal modes of massive Dirac spinors in the background of rotating black holes. In particular, we consider the Kerr geometry as well as the five dimensional Myers-Perry spacetime with equal angular momenta. We decouple the equations using the standard methods from the literature. In the five dimensional Myers-Perry black hole the angular equation is solved analytically. Using the continued fraction method, we calculate the spectrum of quasinormal modes for the ground modes and first excited modes. We analyze, in a systematic way, its dependence on the different parameters of…
In this paper, we study the quasinormal modes of the massless Dirac field for charged black holes in Rastall gravity. The spherically symmetric black hole solutions in question are characterized by the presence of a power-Maxwell field, surrounded by the quintessence fluid. The calculations are carried out by employing the WKB approximations up to the thirteenth order, as well as the matrix method. The temporal evolution of the quasinormal modes is investigated by using the finite difference method. Through numerical simulations, the properties of the quasinormal frequencies are analyzed, incl…
We discuss whether the minimally coupled massless Klein-Gordon and Dirac fields have well defined quasinormal modes in single horizon, asymptotically flat two-dimensional black holes. To get the result we solve the equations of motion in the massless limit and we also calculate the effective potentials of Schrodinger type equations. Furthermore we calculate exactly the quasinormal frequencies of the Dirac field propagating in the two-dimensional uncharged Witten black hole. We compare our results on its quasinormal frequencies with other already published.
We calculate numerically the quasinormal frequencies of the Klein-Gordon and Dirac fields moving in the two-dimensional dimensionally reduced BTZ black hole. Our work extends results previously published on the damped oscillations of this black hole. First, we compute the quasinormal frequencies of the minimally coupled Klein-Gordon field for a range of the dimensionally reduced BTZ black hole physical parameters that is not previously analyzed. Furthermore we determine, for the first time, the quasinormal frequencies of the Dirac field propagating in the dimensionally reduced BTZ black hole. …
An analytical solution representing traversable asymptotically flat and symmetric wormholes was obtained without adding exotic matter in two different theories independently: in the Einstein-Maxwell-Dirac theory and in the second Randall-Sundrum brane-world model. Further, a smooth normalizable asymmetric wormhole solution has been recently obtained numerically in the Einstein-Maxwell-Dirac theory. Using the time-domain integration method we study quasinormal ringing of all these wormholes with emphasis to the regime of mimicking the near extremal Reissner-Nordström black holes, which is chara…
So far the study of black hole perturbations has been mostly focussed upon the classical black holes with singularities at the origin and hidden by event horizon. Compared to that, the regular black holes are a completely new class of solutions arising out of modification of general theory of relativity by coupling gravity to an external form of matter. Therefore it is extremely important to study the behaviour of such regular black holes under different types of perturbations. Recently a new regular Bardeen black hole solution with a de Sitter branch has been proposed by Fernando. We compute …
We extensively study the exact solutions of the massless Dirac equation in 3D de Sitter spacetime that we published recently. Using the Newman-Penrose formalism, we find exact solutions of the equations of motion for the massless classical fields of spin s=1/2,1,2 and to the massive Dirac equation in 4D de Sitter metric. Employing these solutions, we analyze the absorption by the cosmological horizon and de Sitter quasinormal modes. We also comment on the results given by other authors.
We analyze the fermionic quasinormal modes of the BTZ black hole in the presence of space-time noncommutativity. Our analysis exploits a duality between a spinless and spinning BTZ black hole, the spin being proportional to the noncommutative deformation parameter. Using the AdS/CFT correspondence we show that the horizon temperatures obtained from the dual CFT pick up noncommutative contributions. We demonstrate the equivalence between the quasinormal and non-quasinormal modes for the noncommutative fermionic probes, which provides further evidence of holography in the noncommutative setting.…
Quasinormal modes of scalar, electromagnetic, and gravitational fields in the extreme Schwarzschild-de Sitter background are known to be expressed in analytic form as eigenvalues of the Pöschl-Teller wavelike equation. We show that perturbations of fermionic fields (given by Dirac and Rarita-Schwinger equations) do not lead to the Pöschl-Teller effective potential. Nevertheless, using the Frobenius method we find quasinormal modes analytically in this case as well. We write down the analytical formula for quasinormal frequencies of the near-extreme Schwarzschild-de Sitter black holes, which is…
Both scalar and Dirac quasinormal modes in Garfinkle-Horowitz-Strominger black hole spacetime are studied by using the WKB approximation and the Pöschl-Teller approximation. For scalar field, we find that the QNMs with higher dilatons decay more rapidly than that with lower ones. However, this is not the case for mass $m$. Fields with higher mass will decay more slowly. The similar behaviors appear in the case of Dirac field. We also find that QNM frequencies evaluated by using WKB approximation and Pöschl-Teller approximation have a good agreement with each other when mode number $n$ is small…
We find the asymptotic formula for quasinormal mode frequencies omega_n of the Dirac equation in a Schwarzschild-AdS_D background in space-time dimension D > 3, in the large black-hole limit appropriate to many applications of the AdS/CFT correspondence. By asymptotic, we mean large overtone number n with everything else held fixed, and we find the O(n^0) correction to the known leading O(n) behavior of omega_n. The result has the schematic form omega_n =~ n Delta(omega) + A ln(n) + B, where Delta(omega) and A are constants and B depends logarithmically on the (D-2)-dimensional spatial momentu…
To find what influence the charge of the black hole $Q$ will bring to the evolution of the quasinormal modes, we calculate the quasinormal frequencies of the neutrino field (charge $e=0$) perturbations and those of the massless Dirac field ($e\neq 0$) perturbations in the RN metric. The influences of $Q$, $e$, the momentum quantum number $l$, and the mode number $n$ are discussed. Among the conclusions, the most important one is that, at the stage of quasinormal ringing, the larger when the black hole and the field have the same kind of charge ($eQ>0$), the quasinormal modes of the massless ch…
We exactly calculate the quasinormal frequencies of the electromagnetic and Klein-Gordon perturbations propagating in a five-dimensional dilatonic black hole. Furthermore we exactly find the quasinormal frequencies of the massive Dirac field. Using these results we study the linear stability of this black hole. We compare our results for the quasinormal frequencies and for the linear stability of the five-dimensional black hole with those already published.
Using the Frobenius method, we find high overtones of the Dirac quasinormal spectrum for the Schwarzschild black hole. At high overtones, the spacing for imaginary part of $ω_{n}$ is equidistant and equals to $\Im{ω_{n+1}}-\Im{ω_{n}} =i/8M$, ($M$ is the black hole mass), which is twice less than that for fields of integer spin. At high overtones, the real part of $ω_{n}$ goes to zero. This supports the suggestion that the expected correspondence between quasinormal modes and Barbero-Immirzi parameter in Loop Quantum Gravity is just a numerical coincidence.
