Gravitino Perturbations in Schwarzschild Black Holes
We consider the time evolution of massless gravitino perturbations in Schwarzschild black holes, and show that as in the case of fields of other values of spin, the evolution comes in three stages, after an initial outburst as a first stage, we observe the damped oscillations characteristic of the quasinormal ringing stage, followed by long time tails. Using the sixth order WKB method and Prony fitting of time domain data we determine the quasinormal frequencies. There is a good correspondence between the results obtained by the above two methods, and we obtain a considerable improvement with respect to the previously obtained third order WKB results. We also show that the response of a black hole depends crucially on the spin class of the perturbing field: the quality factor becomes a decreasing function of the spin for boson perturbations , whereas the opposite situation appears for fermion ones.
The study of the evolution of small perturbations in black hole backgrounds is a very interesting subject. Actually, we know that this evolution, at intermediate times, is dominated by damped single frequency oscillations. These characteristic oscillations have been termed quasinormal modes and the associated frequencies quasinormal frequencies, and depend only on the parameters characterizing the black hole, as it mass, electric charge and angular momentum [1, 2, 3]. In this sense we can say that black holes have a characteristic sound, resembling for example the familiar sound produced bay the ringing of a bell or the strum of a guitar.
Once one realizes the importance of black holes in fundamental physics, one can grasp the meaning and significance of their characteristic vibrations [1, 2, 3]. Black holes have been called the hydrogen atom of general relativity, perfect comparison because like the hydrogen atom in quantum mechanics,a black hole, as a solution of the Einstein’s field equations has all the general relativistic properties embodied in it, but still is simple enough to be a model for starting a complete understanding of all the physics that go with Einstein’s gravitation theory.
There are different contexts in which the study of quasinormal modes of black holes appears to be motivated: the estimation of astrophysical black hole parameters [3, 4] , the study of the stability of these solutions under small perturbations, semiclassical ways to quantize the black hole area and the estimation of thermalization timescales in connection with the ADS/CFT correspondence, in which a large static black hole in asymptotically AdS spacetime corresponds to a thermal state in a Conformal Field Theory ( CFT ) at its boundary, and the decay of the test field in the black hole spacetime corresponds to the decay of the perturbed state in the CFT. In this sense, we can determine the timescale for the return to thermal equilibrium studying its dynamics in AdS spactime, and then translating it onto the CFT, using the AdS/CFT conjecture  .
Since the discovery of quasinormal oscillations of black holes, many studies have been done on quasinormal modes of various spin fields and a considerable variety of analytical, semi-analytical and numerical methods have been developed to determine them [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 20].
An interesting problem to consider is the study of quasinormal modes of spin fields, called Rarita-Schwinger fields. In a previous paper Shu and Shen  determined the quasinormal modes of this fields in Schwarzschild spacetime using the third order WKB method developed by Iyer and Will .
In this paper we investigate for the first time the complete time evolution of gravitino perturbations in the Schwarzschild background and determine with better accuracy the quasinormal frequencies by fitting time domain data by a superposition of damping exponents, using the Prony method. We also extend previous results by Shu and Zheng beyond the third order WKB, using the sixth order approach developed by Konoplya [17, 18]. The results obtained by this two approaches match very well, ensuring the validity of the determined quasinormal frequencies. We also studied the complete time evolution of scalar, electromagnetic, gravitational and Dirac perturbations in the mentioned spacetime, and compute numerically some quasinormal frequencies by the two methods mentioned above.
The structure of this paper is as follows. In Section II we use the Newmann-Penrose formalism to obtain the equations for gravitino test perturbations in Schwarzschild spacetime, and use separation of variables to obtain decoupled equations for the radial and angular variables involved. The radial equation is written in the familiar way useful to study the quasinormal spectrum of the perturbations, obtaining an analytical expression for the effective potential related to it. Section III is devoted to determine the time domain evolution and to compute some quasinormal frequencies of gravitino perturbations, and to compare them with similar calculations for other boson and fermion fields. The last section contains our conclusions and presents some lines for future research on this subject.
2 Rarita-Schwinger equation in Schwarzschild spacetime
In supergravity theory the gravitational field is coupled to the massless spin 3/2 Rarita-Schwinger field, that acts as a source of torsion and curvature. if we denote by the spin 3/2 field, then the supergravity fields equations are invariant under supergravity transformations , where is an arbitrary spinor field.
When the Rarita-Schwinger field vanishes, the supergravity equations becomes Einstein’s vacuum equations. Therefore, since the torsion and the curvature produced by the Rarita-Schwinger field depend quadratically on this field, then in the linear approximation about a solution with , the supergravity field equations reduces to the Rarita-Schwinger field equations together with the vacuum Einstein’s field equations [22, 23].
The Rarita-Schwinger equations for the field in a curved background can be written as
where we have defined and (the parenthesis denotes symmetrization on the indices enclosed and the indices between bars are excluded from the symmetrization). It is possible to show that in an algebraically special vacuum spacetime, the contraction of defined above, with a multiple principal spinor of the conformal curvature satisfies a decoupled equation .
Using the Newmann-Penrose notation, we can show that in a frame such that is the only nonvanishing component of the Weyl spinor, the Rarita-Schwinger reduces to only two equations for the components and , that can be solved by separation of variables in all the type D vacuum backgrounds. In terms of the differential operators , , and that contains the usual directional derivatives of the Newmann-Penrose formalism, we can put the Rarita-Schwinger equations in the above mentioned frame as
We are interested in the specific case of Schwarzschild background, where the metric can be expressed as
where is the frequency, is a half integer, and the functions and satisfy the ordinary differential equations
where is a separation constant that takes the values with . In the above equations we have used the operators defined by
Now introducing the tortoise coordinate defined by and making use of the variable we can put the second equation in (7) in the standard form
where , and
As explained in Reference , it suffices to consider the equation for only, and an entirely similar reduction can be made with the equation for . Finally. It is possible to reduce (11) to a, Schrodinger-type equation in the form
where the effective potential for the Rarita-Schwinger field is given by
with the function defined by , and .
In Figure (1) we show the effective potential for gravitino perturbations with two different multipole numbers , for a Schwarzschild background with . As we can see, has the form of a definite positive potential barrier, i.e, it is a well behaved function that goes to zero at spatial infinity and gets a maximum value near the event horizon.
3 Time domain evolution of perturbations and quasinormal frequencies
The most direct approach to study the gravitino perturbations is to solve numerically the evolution equation associated with (14), that is
where the function results from the factorization of in (6) as . In order to integrate numerically the equation (16) we use the technique developed by Gundlach, Price and Pulling , and the result can be observed as the time-domain profile showed in Figure (2).
As we can see, the temporal evolution of gravitino perturbation can be divided in three stages. The first depends on the initial conditions and the point where we observe the profile. After the initial outburst at the first stage, we observe the exponential damping of the perturbations called quasinormal ringing, that can be split to the superposition of exponentially damping oscillations, represented by a set of complex frequencies, called quasinormal frequencies, whose real parts describe the actual frequency of the oscillation, while the imaginary part is the damping rate.
The quasinormal modes are solutions of the wave equation (14) with the specific boundary conditions requiring pure out-going waves at spatial infinity and pure in-coming waves on the event horizon. Thus no waves come from infinity or the event horizon. As we can see from the figure, the quasinormal ringing stage is followed by the so-called tails at asymptotically late times. Thus, the time evolution of gravitino perturbations in Schwarzschild spacetime follows the same stages as well known boson and fermion perturbations.
To show this, we also integrated numerically the perturbation equations corresponding to scalar, electromagnetic, gravitational and Dirac perturbations in this spacetime. The resulting profiles are observed in Figure (3). Note that in all of the cases considered, the time evolution is similar, then the gravitino is not the exception.
In order to evaluate the quasinormal modes we used two different methods. The first is a semianalytical method to solve equation (14) with the required boundary conditions, based in a WKB-type approximation, that can give accurate values of the lowest ( that is longer lived ) quasinormal frequencies, and was used in several papers for the determination of quasinormal frequencies in a variety of systems .
The WKB technique was applied up to first order to finding quasinormal modes for the first time by Shutz and Will . Latter this approach was extended to the third order beyond the eikonal approximation by Iyer and Will  and to the sixth order by Konoplya [17, 18]. We use in our numerical calculation of quasinormal modes this sixth order WKB expansion. The sixth order WKB expansion gives a relative error which is about two order less than the third WKB order, and allows us to determine the quasinormal frequencies through the formula
where is the value of the potential at its maximum as a function of the tortoise coordinate, and represents the second derivative of the potential with respect to the tortoise coordinate at its peak. The correction terms depend on the value of the effective potential and its derivatives ( up to the 2i-th order) in the maximum, see  and references therein.
Assuming that the quasinormal ringing stage begins at and ends at , where , then the expession (18) is satisfied for each value in the time profile data
From the above expression, we can determine, as we know , the quasinormal frequencies once we have determined as functions of . The Prony method allows to find the as roots of the polinomial function defined as
It is possible to show that the unknown coefficients of the polinomial function satisfy
Solving the linear equations (21) for we can determine numerically the roots and then the quasinormal frequencies.
It is important to mention the fact that with the Prony method we can obtain very accurate results for the quasinormal frequencies, but the practical application of the method is limited because we need to know with precision the duration of the quasinormal ringing epoch. As this stage is not a precisely defined time interval, in practice, it is difficult to determine when the quasinormal ringing begins. Therefore, we are able to calculate with high accuracy only two or, sometimes three dominant frequencies.
In Table 3 we show the results for the evaluation of the first four fundamental quasinormal modes for gravitino perturbations in Schwarzschild black holes, using the two methods mentioned above.
The parameter entering in the calculation is the black hole mass , that we take to be a unit mass. We also show for comparison the results obtained using a third order WKB method, that were obtained in reference . With sixth order WKB approximation we obtain an improve of the results for the quasinormal frequencies, as we can easily see after comparison with the numerical results obtaining using the Prony fitting of the time domain data.
The obtained results are showed in Figure (4). For a given angular number, higher overtones becomes less oscilatory, i.e, the real part of the quasinormal frequencies decreases whereas the imaginary part increases. In this situation, the quality factor of modes, defined as the ratio , decreases. On the contrary, modes with higher multipole number and the same overtone number have higher frequencies and the damping shows only little increments, with an effective increase of the quality factors.
To test the accuracy of the Prony method to finding quasinormal frequencies by fitting time domain data, we also
calculated the quasinormal frequencies of scalar, electromagnetic, gravitational and dirac perturbations in the Schwarzschild spacetime.
To comparative purposes, we also calculated the above frequencies using sixth order WKB method, and correct the results presented in reference  for the quasinormal frequencies of Dirac and gravitino perturbations, where apparently the author uses an improper generalization of the formula for the effective potential due to boson fields in Schwarzshild spacetime to the case of fermion fields. The equation for the radial part of boson perturbations have the form (14) with the effective potential:
where and the quantity is related to the spin-weight of the perturbing field by . For scalar, electromagnetic and axial gravitational perturbations has the integer value 1, 0, -3, respectively.
The polar perturbations corresponding to a gravitational perturbing field are governed by the so called Zerilli potential, that is different from (22), but as was rigorously shown by Chandrasekar in , the quasinormal frequencies belonging to axial and polar perturbations are identical.
For the case of Dirac perturbing field, we use the form of the effective potential obtained by Cho in reference , given by:
where the quantity . Tables 3 to 3 shows the results obtained. We also show the third order WKB results for all the perturbations. In the case of scalar, electromagnetic and gravitational perturbations, we listed the numerical results given in the paper of Leaver .
As we can see, the Prony method gives values that are in perfect agreement with previous numerical results existing for scalar, electromagnetic and gravitational perturbations. Also we obtain a improvement of the third order WKB results using the sixth order semianalytic formula.
The dependance of the quality factor with the multipole and overtone number is similar as for the case of gravitino test field perturbations. In Figure (5) we show this dependance for gravitino and gravitational perturbations. For a fixed multipole number, higher overtones shows a decreasing in the quality factor, and the modes are more damping. In contrast, increasing the multipole number for a given overtone, the quality factor increases due to the increasing of the real oscillating frequency in a more pronounced way that the damping rate. At this point it is interesting to note that fermion perturbations in the black hole background shows higher quality factors that boson perturbations, as Figure (6) shows. As the spin weight of a boson perturbation increases, the quality factor decreases, and the black holes perturbed by this fields becomes more poor oscillators. Then, the lowest quality factors corresponds to gravitational perturbations, and the higher to scalar test perturbations.
The situation for fermion fields is opposite. In this case the quality factor increases with the increment of the field spin, and as a consequence the highest quality factor belongs to the gravitino perturbations. Then, as a general fact, black holes perturbed by fermion fields are better oscillators than those perturbed by boson fields.
4 Concluding remarks
We have studied the evolution of Rarita-Schwinger field perturbations in a Schwarzschild background. Solving numerically the time evolution equation for this perturbations, we find similar time domain profiles as in the case of fields of other spins: the usual three stages in the time evolution dominated at intermediary times by quasinormal ringing. We determined the quasinormal frequencies by two different approaches, 6th order WKB and time domain integration with Prony fitting of the numerical data. The two methods give close values of QNMs for well pronounced potential barriers.
Also we apply the Prony method to numerically calculated the quasinormal frequencies of scalar, electromagnetic, gravitational and Dirac perturbations in the Schwarzschild background, with results in perfect agreement with previous numerical calculations by other authors.
The time domain evolution of gravitino perturbations are similar to that corresponding to other fields of different spin. The behaviour of the quasinormal frequencies with respect to the black holes masses is also similar for all spin perturbations. However, an interesting difference occur in the oscillating behaviour of black holes perturbed by fields of different spin classes: if the perturbing field has integer spin, then the system becomes a poor oscillator as the spin increases. The opposite situation occurs in the presence of fermion perturbations, black holes perturbed by higher spin fields are better oscillators. In general, the highest quality factor is associated with gravitino perturbations and the smallest with gravitational ones.
There are extensions of this work that are interesting to consider, first the generalization to the case of charged black holes, and second, the determination of the changes of the gravitino quasinormal spectrum for semiclassical solutions, due to the vacuum polarization of quantized fields. The solution of the above problems will be presented in future reports.
We are grateful to Dr. Alexander Zhidenko from USP, Brazil for providing me with his MATHEMATICA code with the implementation of the Prony method and valuable information about numerical methods usually employed in the calculation of quasinormal frequencies. Also we would like to acknowledge helpful discussions with Elcio Abdalla and Jeferson de Oliveira from Department of Mathematical-Physics at the University of São Paulo, where this work was completed .
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