Gravitational and electromagnetic perturbations of a charged black hole in a general gauge condition

We derive a set of coupled equations for the gravitational and electromagnetic perturbation in the Reissner-Nordstr\"om geometry using the Newman Penrose formalism. We show that the information of the physical gravitational signal is contained in the Weyl scalar function $\Psi_4$, as is well known, but for the electromagnetic signal the information is encoded in the function $\chi$ which relates the perturbations of the radiative Maxwell scalars $\varphi_2$ and the Weyl scalar $\Psi_3$. In deriving the perturbation equations we do not impose any gauge condition and our analysis contains as a limiting case the results obtained previously for instance in Chandrashekhar's book. In our analysis, we also include the sources for the perturbations and focus on a dust-like charged fluid distribution falling radially into the black hole. Finally, by writing the functions on a basis of spin weighted spherical harmonics and the Reissner-Nordstr\"om spacetime in Kerr-Schild type coordinates a hyperbolic system of coupled partial differential equations is presented and numerically solved. In this way, we solve completely a system which generates a gravitational signal as well as an electromagnetic/gravitational one, which sets the basis to find correlations between them and thus facilitating the gravitational wave detection via the electromagnetic signal.


I. INTRODUCTION
Gravitational wave astronomy was born in 2015 with the discovery of the first astronomical source named GW150914 [1][2][3]. This first observation of a binary black hole system was done with the gravitational wave interferometer LIGO [4], which detected the gravitational wave signal produced by the merger of two black holes in a binary system. The two black holes were not surrounded by significant amount of matter which could generate electromagnetic emission and, in this respect, the emission of the merger was purely in the gravitational channel. Unlike black hole binaries, a system involving neutron stars do possess electromagnetic counterpart, as demonstrated by the detection of the neutron star merger event in gravitational and electromagnetic messenger channels in 2017 named GW170817 [5,6].
During the formation of an accretion disk around a black hole, electrons may escape from the influence of the central object leaving a net charge in the system, these charged particles may be captured by the black hole producing a charged black hole [7]. Thus, the study of the electromagnetic and gravitational perturbations can be used to describe the scattering of both types of waves. However, for a charged black hole a gravitational perturbation of the metric inevitably accompanies a perturbation on the electromagnetic field and vice versa. The description of coupled electromagnetic and gravitational perturbations on charged black holes have been discussed in several works [8][9][10][11][12][13][14] using different techniques including dispersion of waves due to curvature potentials [15][16][17] and the Newman-Penrose formalism [18][19][20].
Chandrasekhar described the scattering of electromagnetic waves on a Reissner-Nordström black hole and the resulting generation of outgoing gravitational waves, by using the gauge freedom of the Maxwell equations in a curved background, he derived the electromagnetic equations by finding a gauge that restore the symmetry to the perturbation equations [21]. He showed that the curved spacetime produced by a black hole is sensitive to the electromagnetic field part χ of the spacetime and this awareness is manifested in the symmetry of the equations for the scalars ϕ (1) 2 and Ψ (1) 3 in a curved background. With the introduction of this particular gauge the electromagnetic and gravitational perturbation equations simplify greatly. Furthermore, using this gauge the equations for the gravitational and electromagnetic Weyl scalars decouple from the rest of the functions appearing in the system of equations. Due to the apparent cognizance of the curved geometry to the existence of the Maxwell field in which the symmetry in the equations is recovered, Chandrasekhar dubbed this gauge as the phantom gauge. Since then, the scattering of both gravitational and electromagnetic waves have been described using this gauge in a variety of works [22][23][24]. In Ref [22], Lee found a pair of equations for only two gauge invariant quantities involving electromagnetic and gravitational perturbations in a Kerr-Newman spacetime without using the phantom gauge. In this work we revisit the findings of Lee including explicitly the matter sources that may cause the perturbation in a Reissner-Nordström background.
As a direct application of our setting we consider a pressure-less charged perfect fluid (dust) as the cause of the perturbation falling radially into the black hole. We write explicitly the equations in a coordinate system and expand the functions using a spherical harmonic basis with the appropriate spin weight. With this choice, we show that the dynamics of the perturbed functions Ψ (1) 4 and χ, is completely determined by a set of partial differential equations that depend only on the radial and temporal coordinates. We numerically solve such set of equations and obtain the wave forms for several representative values of the parameters of the system. Setting the basis for a thorough comparatively analysis which might find correlations between the waveforms and thus, by a detection of one of these electromagnetic/gravitational signal, one will be able to infer the presence of a purely gravitational one. This program will be carried out in future works.
The paper is structured as follows: In section II we introduce the basics of the Newman Penrose formalism including the Bianchi and Maxwell identities. In section III we provide a detailed derivation of the the perturbation equations in a Reisnner-Nordström bakground. In section IV we describe the sources of the perturbation and show that choosing an adequate decomposition in spin weighted spherical harmonics it is possible to separate the time-spatial structure of the equations. In Section III A we introduce the tetrad and geometric quantities in the Reissner-Nördstrom background described in horizon penetrating coordinates, present a numerical scheme to solve the equations for some particular scenario of matter falling into the black hole, and show some waveforms, gravitational as well as those related with χ. Finally some concluding remarks are given in section VI. In the rest of the paper we use η to indicate the signature of the metric: η = 1 for signature (+, −, −, −), and η = −1 for the signature (−, +, +, +), and we will use geometric units where c = G = 1.

II. FOUNDATIONS: NEWMAN-PENROSE FORMALISM
The starting point in the Newman Penrose formalism is to define a tetrad of null vectors [25]. The choice of the tetrad is made to reflect symmetries of spacetime, since certain components may vanish, leading to simplification of field equations. In this work we use l µ and k µ , ingoing and outgoing null vectors respectively, which satisfy the normalization conditions k µ l µ = −1 and m µ m µ = 1. The metric tensor can be represented by g µν = −2(l (µ k ν ) − m (µ m ν) ) where m means complex-conjugate, Greek index runs from 0 to 3. The directional derivative operators are defined as D = l µ ∂ µ , ∆ = κ µ ∂ µ and δ = m µ ∂ µ . The spin coefficients are obtained from the projections where ";" stands for covariant derivative. The Weyl scalars related with the curvature Ψ 0 Ψ 1 , Ψ 2 , Ψ 3 and Ψ 4 , and the source terms Φ ij , are defined as and where C µνλτ is the Weyl tensor and T µν is the stress energy tensor of the matter content. The information of the electromagnetic fields is encoded in the scalars, where F µν is the Faraday tensor [24]. The stress energy tensor for the electromagnetic field has the form with µ 0 the magnetic permeability in vacuum. In the rest of this work we set µ 0 = 1. From the definition of the scalars Φ ij and Eqs. (4) and (5) we consider that Φ ij = 2 η ϕ i ϕ j . Latin index runs from 1 to 3. In order to obtain the electromagnetic and gravitational perturbation equations, we depart from the projected Maxwell equations, and the Bianchi identities [21,26] as explained below.

A. Maxwell equations
The dynamics of the electromagnetic fields with sources is described by the Maxwell equations F µν ;ν = J ν , where J ν is the external electric current. The Maxwell equations projected along the tetrad η pn F ap |n = J a (" | " means intrinsic derivative), written in terms of the spin coefficients and the electromagnetic scalars are [27] where J l = J µ l µ , J m = J µ m µ and J k = J µ k µ .

B. Bianchi identities
By projecting the equations R µν (λτ ;σ) = 0 on the tetrad, one gets a set of equations which include the spin coefficients and operators acting on the Weyl scalars known as the Bianchi identities. In the following we will use two of them, Eq. (321d) and Eq. (321h) from [21]: For a detailed description on the projections we refer the reader to [26]. Similarly, projecting the Weyl tensor on the tetrad, we obtain the expression: which are the three equations needed in the forthcoming derivation.

III. EQUATIONS FOR THE PERTURBATIONS
Corresponding to the six parameters of the Lorentz group of transformation, there are six degrees of freedom to rotate a chosen tetrad frame. It is usual to encode a general Lorentz transformation in terms of the basis vectors l, k and m, and classify them in three classes of rotations, each one of them leaving invariant a vector under the transformation. The effect of the basis transformation on the various Newman Penrose quantities can be found in [21]. and where the superscript denotes a perturbed quantity. For the Reissner-Nordström spacetime the background scalars Ψ 2 and ϕ 1 are the only non zero. Consequently Ψ 0 , Ψ 1 , Ψ 2 , Ψ 4 , ϕ 0 and ϕ 1 are unaffected to first order under an infinitesimal rotation. However, Ψ 3 and ϕ 2 are indeed affected since Ψ 2 and ϕ 1 are different from zero in the background. Nevertheless, the radiative combination is invariant to first order and, as it is shown below, one can get a coupled system of equations for this function, χ and the perturbed Weyl component Ψ 4 . In the following we describe in detail the general procedure to find such coupled set of equations.

A. Perturbed Maxwell Equations
The ingoing and outgoing electromagnetic radiation is given by the perturbations of the scalars ϕ 0 and ϕ 2 respectively. However, in a charged spacetime the outgoing electromagnetic perturbations couples with the perturbations of the Weyl scalar Ψ 3 that carries the so called electromagnetic part of the spacetime. A similar coupling occurs with the ingoing perturbation ϕ 0 , and Ψ 1 . In this section we will derive an equation for the perturbation of ϕ 2 .
First, consider the following identities relating the derivative operators [21]: Second, on the projected Maxwell's Eqs. (6)(7)(8)(9) operate with [∆ + η(µ − γ + γ + 2µ)] on Eq. (8) and with [δ − η(τ − α − β − 2π)] on Eq. (9), then subtract these equations and after some algebra one arrives to the following expression: where In order to describe the perturbation of ϕ 2 let us perform a first order perturbation of the form f → f + f (1) in all the functions on Eq. (19). In a Reissner-Nordström like background, considering that the spacetimes are type D, in Petrov classification, the spinor quantities ν, λ, κ, σ, are zero. Furthermore, we will consider spherical symmetry and that ϕ 0 , ϕ 2 are zero as well. These considerations imply with the perturbed current term defined as In the previous equation we have kept only first order and non-vanishing background quantities. Eq. (21) can be further simplified by using background Maxwell equations Eqs. (6); Dϕ 1 = 2 ηρ ϕ 1 , and Eq. (7); δϕ 1 = 2 ητ ϕ 1 to obtain This equation can be written in a more convenient form using the commutation expressions for the operators acting on ϕ and the Ricci identities [28]: on Eq. (23). After some algebra one arrives to the following expression relating the perturbations ϕ In order to simplify the notation let us define the derivative operators: where N takes integer values. In terms of these expressions we can rewrite the electromagnetic perturbation equation Eq. (28) as: With the notation of Eqs. (29,30) we have denoted D +1 = D+η ((+1) ρ + 4 ǫ) for N = +1, D −1 = D+η ((−1) ρ + 4 ǫ) for N = −1, and analogously for other values and operators.

B. Perturbed Bianchi identities
In order to derive the equations for the perturbations Ψ (1) 4 and Ψ 3 we start by perturbing the the Bianchi identities Eqs. (10,11). As mentioned above, in a Petrov type D space-time background, the background Weyl scalars vanish except Ψ 2 , and the spinor coefficients κ, ν, λ, σ, are equal to zero. Furthermore, we will consider spacetimes, such as the Schwarzschild or the Reissner-Nördstrom, where is always posible to chose a tetrad so that the non zero spinor coefficients are real. Finally, we will consider stress energy tensors of the form of the Reissner-Nordström one, so that the only non-zero Ricci scalar is Φ 11 .
By performing a first order perturbation in Eq. (10) and Eq. (11) one gets the following two equations: In the forthcoming analysis we consider that the external matter that causes the perturbation satisfies T 22 /4π = 0. As we will show bellow, this condition is consistent with matter falling only in the radial direction.
In an analogous manner, the perturbation of Eq. (12) gives Finally, the following Ricci identities describing the action of the operators D and ∆ on the unperturbed fields also will be useful:

C. System of equations for the perturbations
In the previous sub-section, we obtained four equations relating the perturbations Ψ 2 , ν (1) and λ (1) due to the perturbed sources T 22 (1) and J 2 . It is an under-determined system, with four equations for five unknowns. However, as we will show, one can partially solve such system using the particular combination of Ψ and χ. This remarkable combination has been related to a freedom in the rotation of the tetrad [21], although its physical meaning is not clearly understood and, to our knowledge, the physical meaning of such combination has not been discussed in the literature. In this section, we present a detailed derivation of such sub system of equations.
Acting with ∆ 5 on Eq. (32), and with δ −2 on Eq. (33), adding and using the identity A further simplification can be done using the fact that the action of the operator ∆ N on ϕ 1 , is for an arbitrary function f . Thus, using Eq. (39), Eq. (38) takes the form Substituting in the previous equation ∆ 2 λ (1) given by Eq. (34) and after collecting terms one gets Simplifying and using the definition of the operator ∆ N in Eq. (29), one obtains: Next, one can use Eqs. (32, 33) to express the perturbed spinors as: and substitute them in Eq. (42). The resulting equation is: where we have defined the operators Remarkably the operator takes the same form as the operator A 3 acting on 2 η ϕ 1 Ψ 3 . Indeed one can writte the action of A 2 and A 3 on ϕ and using this remarkable property in Eq. (45), one gets: where and In order to derive a second equation for Ψ Considering that D −2 = D +1 − 3 η ρ s we can use the Maxwell equation Eq. (31) to expand this last equation as: After some algebra the previous equation becomes From Eqs. (32, 33) one can obtain the following expressions for the terms involving the perturbed spinor coefficients: Using this expressions, in Eq. (57), and after some simplifications we obtain an equation involving only Ψ 3 , ϕ 2 , and the sources: where the operators have the form: We have collected the operators acting on Ψ 3 and on ϕ 2 in those involving D N and ∆ N , and the rest, as long as they involve more algebraic manipulation in the next steps in the derivation. Indeed, using Ψ and for the radial operators where O 1a , O 1r , and O 2r have the form: and We have written the operator O 2r as a sum of different operators, since as shown below each element of the sum is equal to the corresponding element of O 1r : thus, each of the coefficients of O 2r take the same form as the corresponding coefficients of O 1r acting on the variable χ defined in Eq. (53). As a result, we can again express the operators acting on Ψ 3 with the operators acting on ϕ We have shown that it is possible to get a couple of equations for Ψ (1) 4 and χ that are independent of λ (1) and ν (1) . However, the complete system is not solved as long as, in order to determine the perturbed spinor coefficients one should obtain Ψ independently, which is not possible within this formalism because, as mentioned above, the system is under determined.

IV. SOURCES OF THE PERTURBATION AND HARMONIC DECOMPOSITION
Since the previous section appears quite lengthy due to all the calculations, we find helpful to present in short the two important equations that describe the gravitational and electromagnetic/gravitational perturbations for Ψ (1) 4 and χ and introduce further comments on the sources that may cause these perturbations. From Eqs. (60, 78), we have: where As the source of perturbation, let us consider a charged dust like matter falling radially into the black hole with stress energy tensor T µν = ρ f u µ u ν , where u µ is the four velocity and ρ f is the rest mass density of the fluid. In our analysis we consider that the fluid is falling radially into the black hole with four velocity: When the fluid is charged it induces an electric current given by J µ(1) = ρ e u e µ where the electric density is ρ e = qρ f , and q is the charge per unit of mass of each particle.
An important property of the system of Eqs. (79, 80) with the given form of the sources, is that the system can be decoupled into an angular and radial set of equations.
First, one must notice that the different functions of the system Eqs. (79, 80), have different spin weight. For instance, the Weyl scalar Ψ (1) 4 has spin weight minus two, whereas χ has spin weight minus one, also the rest mass density ρ n , and the density of charge ρ e , are scalar functions with zero spin. Thus, the different operators acting on these functions have to be such that they finally produce quantities with the same spin weight, indeed, as we show bellow, Eq. (79) has spin weight −2 and Eq. (80) spin weight −1. With this in mind and using the fact that the spin-weighted spherical harmonics Y s (l,m) form a basis for each weight s, we write In spherical symmetry we can choose the vector k µ = (1, −1, 0, 0) such that the operators δ N = m µ ∂ µ andδ N in Eq. (30), can be written in terms of the eth and eth-bar operators, ð s = − (∂ θ + i csc θ ∂ ϕ − s cot θ) andð s = − (∂ θ − i csc θ ∂ ϕ + s cot θ) as: The action of the eth, eth-bar operators on Y s (l, m) is to rise or lower the spin weight as: Given the action of the operators on the spin-weight, one gets the following identities for the terms in the perturbation equations: Thus the angular dependence of the dynamical equations is encoded in the spin weighted spherical harmonics. Furthermore, all the terms appearing in Eq. (79) have spin weight −2, and those appearing in Eq. (80) have spin weight −1, which confirms the equations are balanced with respect the spin weight.
The angular part of the perturbation equations can be factorized using the orthogonality properties of the spherical harmonics, and each equation and can be reduced to a set of equations for each mode as follows; multiply Eq. (79) by Y −2 (l ′ ,m ′ ) , and equation (79) by Y −1 (l ′ ,m ′ ) . Then integrate the element of solid angle. Due to the orthogonality of the spherical harmonics each sum reduces and one gets an equation for each mode (l, m).
Finally, recalling the Peeling theorem [29] which states that the Weyl scalars have the asymptotic decay Ψ s ≡ 1/r 5−s , it proves convenient to rewrite the equation for the gravitational perturbation in terms of the quantity r Ψ 4 (1) which does not decay in the asymptotic region. For the electromagnetic/gravitational perturbation the product between the Weyl and Maxwell scalars forming χ decays as 1/r 4 , thus the product r 4 χ has a constant amplitude. For simplicity, let us define for each radial mode the amplitudes R l m4 = r Ψ l m4 , and R l m3 = r 4 χ l m .
We obtain the following system in terms of dimensionless quantities where for simplicity in the notation, we have drop the indices l, m on each mode.

A. Matter models
We shall consider that there are two sources of matter that cause a perturbation into the black hole. One associated to neutral matter characterized by the rest mass densityρn, and the other associated to charged particles with densityρe. We consider that neutral particles move only in the radial direction and are free falling into the black hole. The conservation of the number of particles ∇µJ µ = 0, where J µ = ρnu µ n , for the metric Eq. (101) gives: where we have defined the 3-velocity v r n =û r n u 0 n . and we have assumed that the dimensionless four velocity has componentŝ u µ n = (û 0 n ,û r n , 0, 0). For charged particles we make the same assumptions and obtain that the conservation of the number of particles implies Considering the normalization of the four velocityû µû µ = −1 for both types of particles, we can expressû 0 in terms ofû r as [28]:û where∆ = 1 − 2 σ 2 r + σ 4Q2 r 2 . As previously stated, this expression is valid for neutral and charged particles. For spherically symmetric static spacetimes one can determine the movement of test particles up to quadrature by means of the Euler Lagrange equations as follows. Let us consider the dimensionless lagrangian for neutral particleŝ using the Euler Lagrange equation for x 0 , and the metric Eq. (101), one obtains a conserved quantity associated to the energy of the particles at infinityǫn and consequentlyû 0 n =ǫ whereǫn = − ∂L ∂û 0 . Proceeding in a similar way for charged particles, the dimensionless lagrangian iŝ where the vector potential was taken as Aµ = Q ǫ 0 r , 0 = Qmax µ 0 c 2 R 0Q r , 0 and we have written q = Qmaxq. From the staticity of the space time via the Euler lagrange equation for t we get Finally, the projection of the four velocity along the null vector k µ uµ for both types of particles provides a couple of expressions that will be useful bellow:û 0 e +û r e =ǫ e − λ 2r +û r ∆ , andû 0 n +û r n =ǫ n +û r ∆ . Given the components of the velocity, for both neutral and charged particles and the evolution for the densities one can solve the perturbation equations numerically given an initial distribution of matter and compute the resultant gravitational signal due to the gravitational and electromagnetic perturbation induced into the black hole.

B. First order reduction and numerical implementation
In terms of the 3+1 decomposition of the spacetime ds 2 = −(α 2 − β r βr)dt 2 + 2 βrdrdt + γ 2 rr dr 2 + r 2 (dθ 2 + sin 2 dφ 2 ) one can determine the lapse function α, the shift vector β r and the metric component γrr, from the metric Eq. (101) which gives In order to write the system of second order differential equations Eqs. (98-99) as a first order system suitable for numerical integration, we introduce the auxiliary functionŝ From the definition ofπa one obtains the time development ofRa, while by taking the time derivative ofψa and replacing the lapse and shift one gets ∂tψa =r The equations forπ4 andπ3 are obtained by substituting first order functions into Eqs. (109-110), ∂tπ4 = σ 2 2r − σ 2Q2 ∂rπ4 +r 2 ∂rψ4 ρn(t,r) (ûn 0 +ûn r ) 2 .

C. Waveforms
For our simulations we solve the system of equations for the gravitational perturbation and the electromagnetic/gravitational perturbation with sources Eqs. (124-125). For the source we are considering non spherical shell of particles falling into the hole. The numerical code evolves the first order variables with a third order Runge Kutta integrator with a fourth order spatial stencil. For more detailed description of the code see Ref. [28]. We use as initial data a gaussian packet in the density describing a non spherical shell of particles falling into the hole ρn(0,r) = ρ0e (r−rcg ) 2 /2σ 2 g = ρe(0,r); where ρ0 = 5 × 10 −3 , rcg = 50 and σg = 0.8. For the simulation we are using Kerr-Schild type coordinates and rmin lies inside the event horizon, we choosermin = 1.5 andrmax = 1000. The gravitational and electromagnetic/gravitational waveforms produced by the infalling matter are extracted at a fixedr = 100 radius. The gravitational and electromagnetic/gravitational functions,R4,R3, initially are set to zero, as well as their time derivatives; the outer boundary was set far enough from the horizon to ensure that any possible incoming radiation has no effect on our results.
In Fig. 1 we show the radial profile for the gravitational (a) and the electromagnetic/gravitational component (b) for the mode l = 2. The simulation have been done for different values of chargeQ = 0.0, 0.5, 0.9, 0.95 and 0.97. We can observe that the waveforms are quiet similar in structure and that the amplitude changes for different values ofQ; the gravitational waveform amplitude is greater and present more oscillations that the electromagnetic/gravitational case. It is interesting to observe that the amplitude decreases for greater values ofQ in the gravitational case; on the other hand, in the electromagnetic/gravitational case the amplitude increase for greater values ofQ. The time response for both, gravitational and electromagnetic/gravitational waveforms stars and ends almost at similar times.
In Fig. 2 we plot the absolute value on a logarithmic scale to show the different stages of the signal: the initial burst due to the initial data, the quasinormal ringing and the tail. The left figure (a) for the gravitational signal and (b) for the electromagnetic/gravitational signal. We find no sign of mixing between gravitational and electromagnetic/gravitational frequencies. Each signal displays its characteristic ring down frequency and the power-law decay. The ring down frequencies has been extracted fromt ∼ 50 tot ∼ 350 for the gravitational case and fromt ∼ 50 to 400 for the electromagnetic/gravitational case. The frequency of the gravitational waves is the one associated to the quadrupolar quasinormal mode. In order to find the frequencies we fit the data with a sinusoidal waveform. The numerical values of the corresponding frequencies are shown in table I. The values we find are consistent with the values given in [21]. Although it is known that quasinormal mode frequencies are complex, we were interested on the oscillatory behaviour of the signal. For black holes of with masses between 10M⊙ < M < 10 3 M⊙ the electromagnetic frequencies are in the interval 8 Hz to 800 Hz whereas the gravitational waves produced for such a range of masses are in the interval of 12Hz to 1.2kHz [28]. As has been pointed out, quasinormal ringing can be used to determine the intrinsic properties of the black hole [30]. Electromagnetic waves with such low frequencies however could be absorbed easily by the interstellar medium during its propagation and it will be almost impossible to detect them directly.
Next we present the behavior of the energy carried by the gravitational wave for several values of the charge of the infalling     matter for the l = 2 mode (any −l ≤ m ≤ l), obtained by from the energy loss formula, that is, the power of the gravitational wave, Pgw = dEgw dt , (see [27]): From Fig. 3, we notice how the flux of energy carried by the gravitational wave Ψ 4 , reaches a constant value. This value decreases for large values of the charge of the black hole. Although the same integral can be made for the gauge invariant quantity χ, its interpretation as electromagnetic energy is not immediate, as long as it is coupled to the gravitational radiation through Ψ (1) 3 , and deserves a deeper discussion. Here we only mention that it is zero forQ = 0 and opposite in behaviour to dEgw/dt; its asymptotic value increases withQ.
Our system of equations allows to have a purely electromagnetic/gravitational response (encoded in the function χ). Indeed, for the l = 1 case, which corresponds to a dipole angular distribution, there is no gravitational response, as the gravitational waves occur starting from the quadrupolar angular distribution [31], but there might be an electromagnetic one. We see that this is the case; solving the system of equations with l = 1 we obtain only a response in χ, which we present in Figure 4.

VI. FINAL REMARKS
In this work, we revisited the gravitational and electromagnetic perturbations in a Reissner-Nordström black hole by means of the Newman Penrose formalism. In our analysis we include the sources that cause the perturbations and discuss the particular case of a charged perfect fluid falling radially into the black hole. Using both, Maxwell equations for the Maxwell scalars and the Bianchi identities for the Weyl scalars we have found a system of coupled equations for the gravitational and electromagnetic perturbations without choosing a specific gauge. A common practice to study electromagnetic and gravitational perturbations within the Newman Penrose formalism is to choose the so dubbed phantom gauge (imposing ϕ (1) 2 = 0 ) since using this gauge one can obtain a sub-system of equations for the perturbation of the Weyl scalars Ψ (1) 4 and Ψ (1) 3 . However, although convenient, this choice is not unique. In this work we have shown that a similar system of equations can be obtained for Ψ (1) 4 and χ = 2ϕ1Ψ 3 . Our results thus opens up the possibility to explore electromagnetic and gravitational perturbations without any a priori assumption on the value of any of the scalars.
It is remarkable that it is not possible to obtain a perturbation equation for Ψ without choosing a particular gauge, only the combination given by χ can be determined via this formalism. However, this fact is not related with some physical properties of the fields since one can always obtain the fields by solving numerically the Einstein-Maxwell field equations and computing all the gravitational scalars at each time step. The actual physical meaning of such constraint in the Newman Penrose formalism is an ongoing work.
We also considered a dust-like charged fluid as a source of the perturbations. We used the spin weighted spherical harmonics as a basis to expand the functions and obtain a system of partial differential equations for the temporal and radial components, leaving all the angular dependence of the functions on the respective basis of spherical harmonics. The resulting system of partial differential equations constitutes a hyperbolic system suitable to be solved numerically by standard means. In this way, we see that we have a robust procedure which sets the basis for accurately determine the simultaneous generation of gravitational and electromagnetic waveforms. A thorough study comparing amplitudes, frequencies, harmonic dependence and power between the gravitational and the electromagnetic/gravitational signals, for several fiducial values of the parameters will allow to determine correlations between these waveforms which not only will give a better understanding of the process, but in general might shed light in the multimessenger program as well, as long as it might be possible to extrapolate the correlations found in the system presented in this work, to other scenarios where signals of different interactions are generated.
Furthermore, our analysis can be used as a simple model to describe the correlation existing between electromagnetic and gravitational wave signals occurred during the accretion of charged matter around a compact object, since the frequency of the gravitational waves due to the quasinormal ringing of black holes of (10-10 3 )M⊙ with moderate charge lies within the range of sensibility of current ground base gravitational wave interferometers.
Finally, we remark that the studies made in a Reissner-Nordström spacetime frequently give valuable insight in the Kerr geometry. The resulting relations arising from the interaction of the electromagnetic field of the matter with the charge of the black hole, might have a similitude with an interaction of the angular momentum of the accreting matter with the angular momentum of the black hole. The results and derivations presented in this work, might prove to be useful in the perturbation analysis generated by accreting rotating matter in a Kerr background. Such studies are currently underway.