Effects of gravitational waves on electromagnetic fields
Matteo Luca Ruggiero













 









 




















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        <ref>jnlProc. SPIE====matteoluca.ruggiero@unito.itDipartimento di Matematica “G.Peano”, Università degli studi di Torino, Via Carlo Alberto 10, 10123 Torino, ItalyINFN - LNL , Viale dell'Università 2, 35020 Legnaro (PD), Italy
        
        
        We focus on the interaction of a plane gravitational wave with electromagnetic fields and we describe this interaction in the proper detector frame where, thanks to the introduction of Fermi coordinates, it is possible to refer to directly measurable quantities. The presence of a gravitational field can be addressed in terms of an effective electromagnetic medium and, within this framework, we show that the coupling of pre-existing electromagnetic fields with the gravitational field of the wave gives rise to new effective currents.
        To assess the impact of these effects, we solve Maxwell's equations for some standard configurations of the electric and magnetic fields.
        Effects of gravitational waves on electromagnetic fields
            Matteo Luca Ruggiero
            April 29, 2024
        ========================================================
        
        
        
        
        § INTRODUCTION
        
        
        After their first direct interferometric detection  <cit.>, gravitational waves  have opened a new window for research in astrophysics and cosmology: in fact, gravitational waves are important not only because they constitute yet another test of Einstein's theory <cit.>, but also because  they serve as crucial instruments for exploring the universe in the age of multi-messenger astronomy. Advancements in technology and specialized missions are poised to significantly enhance the wealth of information accessible through this avenue (for further details, see, e.g., <cit.> and related references). 
        
        Furthermore, experiments with gravitational waves are crucial for testing gravitational theories beyond General Relativity. Indeed, challenges to Einstein's theory emerge from observations of the universe on a large scale, with the problems of dark matter and dark energy  <cit.>, and at a more fundamental level we still don't know how to reconcile General Relativity with the Standard Model  of particle physics. In order to  try to cope with these issues, theories with additional fields (such as scalar ones) or provided with a richer geometrical structure were proposed and, in these cases, gravitational waves  might have different features with respect to the general relativistic ones, such as for instance longitudinal effects (see e.g. <cit.>). These motivations drive enormous efforts to further develop current interferometers, such as LIGO and VIRGO, as well as to design and create brand new ones, such as LISA.
        
        
        More generally, investigations on gravitational waves  do not solely rely on interferometers, where changes in relative distances between test masses are measured using lasers. Other Earth-based experimental setups have been proposed, including  accelerators <cit.>, storage rings <cit.>, two-level system resonance <cit.> and microcavities  <cit.>. In the latter case, particularly, researchers study the effects of gravitational waves on electromagnetic fields in resonant cavity experiments. So, the interaction between gravitational waves and electromagnetic fields, which has extensively been studied (see e.g. <cit.> and references therein) presents potential for exploring new experimental paths. 
        
        
        
        The purpose of this paper is to develop a formalism to study the effect of a plane gravitational wave on electromagnetic fields. In order to do this, we solve Maxwell's equation in the curved spacetime of a plane gravitational wave and we do this in the proper detector frame. Indeed, gravitational waves are typically studied using the transverse and traceless (TT)  tensor, which allows to introduce the so-called TT coordinates or TT frame. The proper detector frame is an alternative approach <cit.> which is based on the use of Fermi coordinates. The latter  are a quasi-Cartesian coordinates system that can be build in the neighbourhood of the world-line of an observer, and their definition depends both on the background field where the observer is moving and, also, on the kind of motion. Fermi coordinates are  defined, by construction, as scalar invariants <cit.>; they have a concrete meaning, since they are the coordinates an observer would naturally use to make space and time measurements in the vicinity of his/her world-line. 
        
        Recently, the interaction of gravitational waves and electromagnetic fields has been studied in the TT frame by <cit.>; here we use the alternative approach described above and, in addition, we use a formalism that allows Maxwell's equations in curved spacetime to be written as flat spacetime equations in the presence of an anisotropic medium. Subsequently, some simplified interaction models will be discussed to understand and evaluate the impact of a plane gravitational wave on the electromagnetic fields already present before its passage.
        
        The paper is organized as follows: in Section <ref> we introduce the formalism to study Maxwell's equations in curved spacetime, while in Section <ref> we focus on the construction of the local spacetime metric in the laboratory frame, with the use of Fermi coordinates, and we apply  it to the field of a plane gravitational wave. Next, in Section <ref>, we present the fundamental equations necessary for investigating the influence of gravitational waves on electromagnetic fields, followed by their solution in select prototypical scenarios. Subsequently, discussion and conclusions are provided in Section <ref>.
        
        
        
        
        
        
        
        § MAXWELL'S EQUATIONS IN CURVED SPACETIME
        
        
        We aim to investigate the influence of gravitational waves on electromagnetic fields. To achieve this, we must solve Maxwell's equations in curved spacetime. If F_μν is the antisymmetric tensor of the electromagnetic field, the equations that determine the electromagnetic field in curved spacetime are[Latin indices refer to space coordinates, while Greek indices to spacetime ones. We will use bold-face symbols like  W to refer to space vectors; the spacetime signature is +2 in our convention.]<cit.>:
        
            F_μν;α+F_να;μ+F_αμ;ν    =     0, 
            ε_0F^μν_     ;ν    =     j^μ,
        
        where “;” indicates covariant derivative, and j^μ is the four-current. The above equations, due to the antisymmetry of the electromagnetic tensor, can be also written in the form 
        
            F_μν,α+F_να,μ+F_αμ,ν    =     0, 
            ε_0/√(-g)∂_v(√(-g) F^μν)     =     j^μ,
         
        where “,” stands for partial derivative, and g is the determinant of the spacetime metric. The covariant electromagnetic tensor F_μν is usually defined in terms of the three-dimensional electric and magnetic fields[We use Cartesian coordinates here and henceforth.]E and B
            F_μν=  [     0   E_x   E_y   E_z;  -E_x     0 -cB_z  cB_y;  -E_y  cB_z     0 -cB_x;  -E_z -cB_y  cB_x     0 ].
        
        
        The above equations (<ref>)-(<ref>) can be simplified and written like Maxwell's equations in flat spacetime in a  medium (see e.g. <cit.> and references therein). To do this, we introduce  H^μν such that
        
        H^μν=ε_0√(-g)F^μν=
        ε_0√(-g)g^μλg^νρF_λρ, 
        which contains the fields D and H since
        
            H^μν=  [      0   -D^x   -D^y   -D^z;    D^x      0 -H^z/c  H^y/c;    D^y  H^z/c      0 -H^x/c;    D^z -H^y/c  H^x/c      0 ].
        
        Now, if we set
        
        J^μ=√(-g)j^μ, 
        the source equations (<ref>) can be written in the form
        ∂_νH^μν=J^μ.  
        In other words, Maxwell's source equations in curved spacetime are written in the form of electromagnetic equations in flat spacetime in a material medium, with current J^μ, with the constitutive equations given by (<ref>). The homogenous equations (<ref>) are instead expressed in terms of the fields E,  B as in flat spacetime. 
        In three dimensional notation we have
        
            ∇·𝐃 = ρ ,  ∇×𝐇 = ∂𝐃/∂ t
            + 𝐣 ,  ∇·𝐁 = 0   ,  ∇×𝐄 = -∂𝐁/∂ t .
        
        
        
        
        
        Using Eq. (<ref>), it is possible to obtain the relation between E,  D,  H. In fact, we get
        
        D^j=-ϵ_0√(-g)/g_00g^jiE_i-ϵ_jik w_kH_i/c, 
        where w_i=g_0i/g_00; the above equation can be written in three-dimensional form
        𝐃 = ε_0ε 𝐄
        + 𝐰/c×𝐇, 
        whith
        ε  = -√(-g)/g_00 g^ij .
        To obtain the other constitutive equations between B,  H,  E, we introduce the dual tensors
        ^*F^μν and ^*H_μν 
        defined by  <cit.>
        ^*F^μν=
        1/2e^μνλρF_λρ , 
        ^*H_μν= 
        1/2e_μνλρH^λρ,  
        where 
        
        e_μνλρ=√(-g) ϵ_μνλρ,   
        e^μνλρ=
        -1/√(-g)ϵ_μνλρ. 
        In the above equations ϵ_μνλρ is the four-dimensional Levi-Civita tensor, with ϵ_0123=1, 
        Re-expressed in terms of the dual tensors,  the constitutive equations (<ref>) are
        
        ^*H_μν=
        ε_0√(-g)g_μλg_νρ ^*F^λρ.  
        
        Accordingly, we get
        
            B_i=-√(-g)/ε_0c^2g_00g^ijH_j-
            1/ε_0cg_00ϵ_ijk g_j0E_k,
        
        which, in three-dimensional notation becomes
        𝐁 = μ/ε_0 c^2 𝐇
        - 𝐰/c×𝐄
        where μ=ε, as defined in Eq. (<ref>). 
        
        
        
        We notice that when g_μν=η_μν, where η_μν is the Minkowski tensor, then μ=ε=δ_ij; for instance, this is true for spacetimes that are flat at space infinity and, in these cases, we have D=ε_0 E, B =μ_0 E, where ε_0μ_0=c^-2. 
        
        In summary, Maxwell's equations in curved spacetime are equivalent to the flat spacetime equations (<ref>) in a medium with the constitutive equations (<ref>) and (<ref>).  The above description shows that the interaction between electromagnetic fields and the gravitational field is determined by ε, which describes the features of an equivalent  medium, and by the vector w, which originates from the off-diagonal elements of the spacetime metric, which are usually related to the so-called gravitomagnetic effects  <cit.>. In this framework, it is relevant to point out that both ε and w are independent of a confarmal factor: this fact reflects the conformal invariance of Maxwell's equations <cit.>. 
        
        
        
        
         
        
        
        § THE SPACETIME METRIC IN THE LABORATORY FRAME
        
        
        
        
        
        The spacetime metric in the laboratory frame - or proper detector frame - can be obtained using Fermi coordinates; more generally speaking, the latter  are used to express the spacetime metric in the vicinity of a reference world-line, which can be thought of as the world-line of the laboratory frame where a measurement device (the “detector”) is placed. As showed for instance by <cit.>, <cit.>  this expression depends both on how the laboratory frame moves in spacetime (i.e. on its acceleration and the rotation of its axes) and on the spacetime curvature through the Riemann  tensor.  Here, we are concerned with the effects produced by gravitational waves, hence we neglect the inertial effects in the definition of the metric elements: in other words we  consider a geodesic and non rotating frame. Accordingly, using  Fermi coordinates (cT,X,Y,Z), up to quadratic displacements |X^i| from the reference world-line, the line element turns out to be (see e.g.  <cit.>)
        
        ds^2=-(1+R_0i0jX^iX^j )c^2dT^2-4/3 R_0jikX^jX^k cdT dX^i+(δ_ij-1/3R_ikjlX^kX^l )dX^idX^j. 
        Here  R_αβγδ=R_αβγδ(T) is the projection of the 
        Riemann curvature tensor on the orthonormal tetrad e^μ_(α)(τ) of the
        reference observer, parameterized by the proper time[In e^μ_(α) tetrad indices like (α) are within parentheses, while  μ is a  background spacetime index; however, for the sake of simplicity, we drop here and henceforth parentheses to refer to tetrad indices, which are the only ones used.]τ: R_αβγδ(T) = R_αβγδ(τ)=R_μνρσe^μ_(α)(τ)e^ν_(β)(τ)e^ρ_(γ)(τ)e^σ 
        _(δ)(τ) and it is evaluated along the reference geodesic, where T=τ and X^i=0.
        The line element (<ref>) can be written in a compact form by introducing Φ, 𝒜_i, Ψ_ij defined as
        
            Φ/c^2=g_00+1/2  Ψ_ij/c^2=g_ij-δ_ij/2  𝒜_i/c^2=-g_0i/2,
        
        Accordingly, we get
        d s^2= -c^2 (1-2Φ/c^2) T^2 -4/c𝒜_i X^i T  +
         (δ_ij+2Ψ_ij/c^2) X^i  X^j , 
        where, explicitly:
        
            Φ (T,  X^i)   =   -c^2/2R_0i0j(T )X^iX^j, 
            𝒜^_i(T ,X^i)   =   c^2/3R_0jik(T )X^jX^k, 
            Ψ_ij (T, X^i)     =     -c^2/6R_ikjl(T)X^kX^l.
        
        In the above definitions, Φ and 𝒜_i are, respectively, the gravitoelectric and gravitomagnetic potential, and Ψ_ij is the perturbation of the spatial metric <cit.>. Notice that the line element (<ref>) is a perturbation of flat Minkowski spacetime; in other words |Φ/c^2| ≪ 1, |Ψ_ij/c^2| ≪ 1, |𝒜_i/c^2| ≪ 1.  
        
        As for our purposes, we are interested in describing the field of a plane gravitational wave using this formalism. Before doing that, we briefly recall the standard approach to the description of plane gravitational waves. 
        Starting from Einstein's equations
        
        G_μν=8π G/c^4T_μν, 
        we suppose that the spacetime metric g_μν is in the form g_μν=η_μν+h_μν, where |h_μν|≪ 1 is a small perturbation of the Minkowski tensor η_μν of flat spacetime. Setting h̅_μν=h_μν-1/2η_μνh, with h=h^μ_μ, Einstein's field equations (<ref>) in the Lorentz gauge ∂_μh̅^μν=0 turn out to be
        □h̅_μν=-16π G/c^4T_μν, 
        where □ = ∂_μ∂^μ=∇^2-1/c^2∂/∂ t^2. The vacuum (i.e. T_μν=0) solutions of Eq. (<ref>) are gravitational waves propagating in empty space. Imposing  the transverse and traceless (TT) gauge and using the corresponding coordinates (ct_TT,x_TT,y_TT,z_TT) <cit.> the solutions for a plane wave propagating along the x direction are given by 
        h̅_μν=-(h^+e^+_μν+h^×e^×_μν), with
        
        h^+=A^+cos(ω t_TT-kx_TT +ϕ^+),    h^×=A^×cos(ω t_TT-kx_TT +ϕ^×), 
        where ϕ^+, ϕ^× are constants, and the linear polarization tensors of the wave are
        
        e^+_μν=[[  0  0  0  0;  0  0  0  0;  0  0  1  0;  0  0  0 -1 ]],    e^×_μν=[[ 0 0 0 0; 0 0 0 0; 0 0 0 1; 0 0 1 0 ]] .  
        
        Here A^+, A^× are the amplitude of the wave in the two polarization states, ϕ^+, ϕ^× the corresponding phases,  while ω is the frequency and k the wave number, so that the wave four-vector is k^μ=(ω/c, k, 0, 0 ), with k^μk_μ=0.  The two linear polarizations states can be added with phase difference of ±π/2 to get circularly polarized waves. We will use 
        
        h^+=A^+sin(ω t_TT-kx_TT),    h^×=A^×cos(ω t_TT-kx_TT), 
        thus fixing the phase difference: accordingly, circular polarization corresponds to the condition A^+=± A_×. In conclusion, in TT coordinates the spacetime element is given by
        
        ds^2= -c^2dt_TT^2+dx_TT^2 +(1-h^+)dy_TT^2 +(1+h^+)dz_TT^2 -2h_× dy_TT dz_TT . 
        
        To express the metric (<ref>) where the curvature tensor appears,  we remember that up to linear order in the perturbation h_μν, we can write the following expressions for the Riemann tensor <cit.>:
        
        R_ikjl=1/2(h_il,jk+h_kj,li-h_kl,ji-h_ij,lk) 
        and
        
        R_ij0l=1/2( h_il,j0-h_jl,i0). 
        Accordingly, we exploit the gauge invariance in linear approximation <cit.>  and use the above expressions  to calculate the Riemann tensor in Fermi coordinates. 
        
        In particular,  in the metric (<ref>) the Riemann tensor is evaluated along the reference world-line: so, after calculating the components of Riemann tensor using Eq. (<ref>), we set X^i=0. We point out that the expression of the metric tensor is obtained in the large wavelength limit, which means that the typical dimension L of the frame  is negligible with respect to the wavelength λ; more accurate expressions can be obtained, which contains higher order terms in the small parameter ϵ = L/λ  (see e.g. <cit.> and references therein). Actually, the series expansion can be exactly summed to obtain a compact form <cit.>.
        
        As a consequence, if we define the functions
        
        f^+=1/2 A^+,    f^×=1/2 A^×
        the gravitoelectric potential (<ref>) is written as
        Φ=ω^2/2[ f^+ Y^2+2f^× YZ-f^+ Z^2], 
        
        
        
        
        
        while the components of the gravitomagnetic potential (<ref>)  are
        
        
            𝒜_X   =   ω^2/3[f^+ (Y^2-Z^2)+2f^× ZY ], 
            𝒜_Y   =   ω^2/3[-f^+ YX-f^× ZX ], 
            𝒜_Z   =   ω^2/3[-f^× YX+f^+ XZ ].
        
        
        
        
        In addition, starting from the definition
        Ψ_ij (T, X^i)  =  -c^2/6R_ikjl(T)X^kX^l, 
        we explicitly calculate the remaining metric components:
        
            Ψ_XX    =     -1/6ω^2(- Y^2+ Z^2-2 YZ ), 
            Ψ_XY    =     -1/6ω^2( YX+ ZX ), 
            Ψ_XZ    =     -1/6ω^2( YX- ZX ), 
            Ψ_YY    =     -1/6ω^2(- X^2), 
            Ψ_YZ    =     -1/6ω^2(- X^2), 
            Ψ_ZZ    =     -1/6ω^2( X^2).
        
        
        
        § MODIFIED ELECTROMAGNETIC FIELDS
        
        
        
        
        We showed that it is possible to study electromagnetic fields in curved spacetime in complete analogy with the formulation of Maxwell's equations in flat spacetime in presence of an equivalent (generally non isotropic) medium. It is important to emphasize the limitations of this approach. Specifically, we assume that the electromagnetic field doesn't impact the background spacetime; in essence, we focus on how the background spacetime affects the electromagnetic fields while disregarding any reciprocal influence of these fields on the spacetime itself.  Of course, this is not the general case, in fact in extreme astrophysical events the electromagnetic field can be source of the gravitational field, and the coupled Maxwell's and Einstein's equations must be solved. However, this approach is adequate for our purposes. 
        
        In what follows, we suppose to have and electric field or a magnetic field before the passage of the wave: we are interested in understanding how these fields are perturbed by the passage of the wave. As we discussed above, the natural framework to study  this interaction is the laboratory metric (<ref>), with the definitions (<ref>)-(<ref>). 
        
        Let us start by the case when, before the passage of the gravitational wave, an electric field E^0 is present. Hence, we consider  the first of Eqs. (<ref>) with the constitutive equation (<ref>); notice that a magnetic field eventually produced by the gravitational wave, does not enter this equation, since both w_i and H_i are first order terms in the wave perturbation, so if we work at linear order their contribution can be neglected. As a consequence, the constitutive equation reads
        
        D^j=-ϵ_0√(-g)/g_00g^jiE_i. 
        
        We work at linear order with respect to the wave amplitude, so we may write
        
        -√(-g)/g_00≃ 1+f, 
        g^ji≃δ^ji+f^ji, √(-g)≃ 1+ p, 
        where  f, f^ji and p are proportional to the wave amplitude, i.e. they are linear functions of A^+, A^×. In addition, we look for a solution for the electric field in the form  E_i=E^0_i+e_i where, again, e_i is a linear function of   A^+, A^×. So, up to first order in the wave amplitude, we may write Eq. (<ref>) in the form
        
        D^j=[δ^ij(1+f )+f^ji]E_i^0+δ^ji e_i. 
        As a consequence:
        ∂_jD^j= ∂_j(δ^jiE_i^0)+∂_j[(δ^jif+f^ji) E^0_i]+∂_j(δ^jie_i). 
        According to Eq. (<ref>), we have  ρ=(1+p)ρ_0 where ρ_0 is the “true” charge density, i.e. the source of the unperturbed electric field E^0:
        ∂_jE^0  j=ρ_0/ϵ_0. 
        Then, the  first of Eqs.  (<ref>) can be written as
        ∂_j(δ^jiE_i^0)+∂_j[(δ^jif+f^ji) E^0_i]+∂_j(δ^jie_i)=(1+p)ρ_0/ϵ_0. 
        Taking into account Eq. (<ref>), we finally have
        ∂_j(δ^jie_i)=pρ_0/ϵ_0- ∂_j[(δ^jif+f^ji) E^0_i]. 
        We see that the effective charge density
        ρ_eff/ϵ_0=pρ_0/ϵ_0- ∂_j[(δ^jif+f^ji) E^0_i] 
        becomes source for the perturbation of the electric field
        ∂_ie^i =ρ_eff/ϵ_0.  
        From Eq. (<ref>), we observe that in vacuum, where ρ_0=0, gravitational waves interact with the unperturbed electric field, determining  an effective charge density. This leads to a perturbed electric field similar to that found in a medium rather than in a vacuum.
        
        
        Now, let us consider the case of the presence of a magnetic field B^0 before the passage of the wave. To this end, we consider the second of the Maxwell's equations (<ref>)  with the constitutive equation (<ref>): exactly as before, an electric field eventually produced by the gravitational wave does not enter this constitutive equation, since both w_i and E_i are first order terms in the wave perturbation. 
        
        
        
        From Eq. (<ref>) we may write
        
        H_i=-ϵ_0c^2g_00/√(-g)γ_ij B^j, 
        where γ_ij=g_ij-g_i0jg_j0/g_00 is the spatial metric <cit.>;  at linear order we have γ_ij=g_ij. As before, we write
        
        -g_00/√(-g)=1+ℓ
        where ℓ=-f,
        
        g_ij=δ_ij+s_ij,  
        B_j=B^0_j+b_j. 
        Again, we observe that ℓ, s_ij and b_j are linear functions of the gravitational wave amplitudes A^+,A^×.  
        
        
        Accordingly, at linear order in the perturbation due to the gravitational wave, we may write the lhs member of the second Maxwell's equation (<ref>) ϵ_kmi∂_m H_i  in the form
        ϵ_kmi∂_m[ (δ_ij+s_ij+δ_ijf )B^0_j+δ_ij b_j]. 
        Then,  the source equation for the magnetic field reads
        ϵ_kmi∂_mδ_ijB^0_j+ϵ_kmi∂_m[(s_ij+δ_ijf  )B^0_j]+ϵ_kmi∂_m(δ_ij b_j)=μ_0(1+p )j^0_k. 
        where j_k^0 is the “true” current which appears in the equation for the unperturbed magnetic field
        ϵ_kmi∂_m B^0_i=μ_0 j^0_k. 
        Consequently, we may write Eq. (<ref>) in the form
        ϵ_kmi∂_m(δ_ij b_j)=μ_0 p j^0_k-ϵ_kmi∂_m[(s_ij+δ_ijf  )B^0_j]. 
        
        If we define the effective current
        μ_0 j_k^eff=μ_0 p j^0_k-ϵ_kmi∂_m[(s_ij+δ_ijf  )B^0_j] 
         we finally may write
        ϵ_kmi∂_mb_i=μ_0 j_k^eff. 
        Once more, we observe that the effective current, originating from the interaction between the gravitational field and the initial magnetic field, acts as a source for the perturbation b_i.
        
        In summary, at linear order the gravitational wave amplitude, we obtained the source equations  for the perturbed electric (<ref>) and magnetic (<ref>) field.
        
        
        
        
        
        
        
        § SOME EXAMPLES
        
        
        Below, we will apply the formalism described so far to solve Maxwell's equations in some particular cases. We point out that Eqs. (<ref>) and (<ref>) correspond to the perturbed Maxwell's equations for the sources of the electric and magnetic field. However, in order to fulfill the whole set of Maxwell's equations, we need to consider also the homogenous ones. To this end, as it customary, we will introduce a scalar and vector potential.
        
        
        
        
         §.§ A Uniform Electric Field
        
        
        We suppose that before the passage of the wave we have a uniform electric field E^0=E^0 u_Z, where E^0=const. To fix ideas,  we can consider the field within a capacitor.
        
         
        We need to solve Eq. (<ref>), with ρ_0=0. In order to solve also the homogenous Maxwell equations, we set e=-∇ϕ. Before the passage of the wave, we have ϕ_0(Z)=V_0(1-Z/d), where V_0 is the difference of potential  between the two armatures, and d the distance between them. In this case, E^0=V_0/d is the unperturbed constant electric field, and ϕ_0 satisfies the Laplace equation. 
        
        Accordingly,  Eq. (<ref>) becomes
        ∇^2ϕ=-ρ_eff/ϵ_0. 
        Taking into account the expression of the line element (<ref>), with the definitions (<ref>)-(<ref>), we obtain the following effective charge density:
        ρ_eff=-ϵ_0ω^2/c^25    (A^×cos(ω  T ) Y -A^+sin(ω  T ) Z ) 𝐸^0/6.  
        
        
        The solution of Eq. (<ref>) with the conditions ϕ(Y,0)=V_0=E^0d, ϕ(Y,d)=0 turns out to be:
        ϕ(Y,Z)=E^0(d-Z)[1+5/36ω^2/c^2A^+sin(ω  T ) (d+Z )Z-5/12ω^2/c^2A^×sin(ω  T )YZ   ]. 
        
        Accordingly,  the electric field is in the form E= E_Y u_Y+E_Z u_Z, where
        
        E_Y=5/12E^0(d-Z)Z ω^2/c^2A^×cos(ω  T ),  
        E_Z=E^0[1+5/12(d-2Z)Y ω^2/c^2 A^×cos(ω  T )-5/36(d^2-3Z^2)ω^2/c^2A^+sin(ω  T )   ] ,
        
        
        A sketch of his electric field is plotted in Figure <ref>.
        
        
        
        
         §.§ A Uniform Magnetic Field
        
        
        Let us consider the case when a uniform magnetic field B^0=B^0 u_Z is present before the passage of the wave. To this end, we need to solve Eq. (<ref>) and, in addition, in order to solve also the homogenous Maxwell's equation for the magnetic field, we set b= ∇× A, where A is the vector potential. Accordingly, exploiting the gauge freedom and imposing the Coulomb gauge (∇· A=0), we may write Eq.   (<ref>)  in the form
        ∇^2 A=-μ_0 j^eff,  
        where, in this case, from Eqs. (<ref>), (<ref>)-(<ref>), we obtain
        
            j^eff_X    =    ω^2/c^2(2 A^+sin(ω  T ) Y +2 A^×cos(ω  T ) Z ) /3 B^0, 
            
            j^eff_Y    =     - ω^2/c^2 A^+sin(ω  T ) X /2 B^0, 
            
            j^eff_Z    =     -ω^2/c^2 A^×cos(ω  T ) X /2 B^0.
        
        On solving Eq. (<ref>), taking into account the vacuum unperturbed solution A^0 such that B^0=∇× A^0 where
        
        A^0_X=-B^0Y/2,    A^0_Y=B^0X/2,    A^0_Z=0 ,
        
        we have
        
            A_X    =     -ω^2/c^2 A^+sin(ω  T )   Y^3/9B^0-ω^2/c^2  A^×cos(ω  T ) Z^3/9B^0-Y 𝐵^0/2, 
            
            A_Y    =    ω^2/c^2  A^+sin(ω  T ) X   Z^2/4B^0+X 𝐵^0/2,  
            
            A_Z    =    ω^2/c^2   A^×cos(ω  T ) X   Z^2/4B^0,
        
        and, consequenty the magnetic field B has the following components
        
            B_X    =     -ω^2/c^2  A^+sin(ω  T ) XZ/2 B^0
            
            B_Y    =      -7ω^2/c^2  A^×cos(ω  T ) Z^2/12B^0
            
            B_Z    =      B^0[1+ω^2/c^2  A^+sin(ω  T )   Z^2/4 +ω^2/c^2 A^+sin(ω  T )   Y^2/3]
        
        
        
        This magnetic field in the plane YZ is plotted in Figure <ref>.
        
        
        
        
        
        
        § DISCUSSION AND CONCLUSIONS
        
        
        We studied the interaction of a plane gravitational wave with electromagnetic fields. For this purpose we used a formalism that allows Maxwell's equations in curved spacetime to be written in the form of flat spacetime equations in an effective medium, with appropriate constitutive equations. In particular, we  focused on the perturbation induced by a gravitational wave on  electromagnetic fields that are present before its passage. Working at first order in the wave amplitude, we showed that the perturbed electromagnetic fields satisfy equations Eqs.  (<ref>) and (<ref>), where new terms are present which originate  from the coupling of the gravitational field of the wave with the sources of the unperturbed fields and with the unperturbed fields themselves; similar results were obtained by <cit.>. 
        
        To give an idea of the impact of gravitational waves on electromagnetic fields, we considered two simple situations:  the case where static and uniform electric and magnetic fields are present. These situations ideally correspond to the field within a capacitor and a within a solenoid.  In both cases (see Eqs. (<ref>)-(<ref>) and (<ref>)-(<ref>)) we showed not only that the preexisting fields are perturbed along their initial direction, but also that components in new directions arise. In particular, the perturbations  are in the order of
        Δ E/E^0≃ h L^2/λ^2,   Δ B/B^0≃ h L^2/λ^2, 
        where h ≃ A^+≃ A^× is the wave amplitude, λ≃(ω/c)^-1 is the wavelength and L is a typical dimension of the measurement device.  
        
        We remark that we used a reference frame adapted to the direction of propagation of the wave: namely the X axis is the direction of propagation, and YZ is the plane orthogonal to it. In general, the laboratory frame has an arbitrary orientation with respect to the direction of propagation of the wave so we should introduce a set of coordinates that are adapted to the laboratory: this is usually accomplished by using the  Euler angles (see e.g. <cit.>). However, this would not change the order of magnitude and the physical meaning of the effects that we are studying.  
        
        What is the physical interpretation of the modified electric (<ref>)-(<ref>) and magnetic  (<ref>)-(<ref>)) fields? We must remember that gravitational waves produce tidal effects, as it is manifest since their impact in the laboratory metric (<ref>) enters through the components of the Riemann tensor. As a consequence, these effects cannot be detected locally, e.g. along the reference world-line, but only by means of a comparison of what happens at different locations in the laboratory frame. To fix ideas, if we consider two identical capacitors, one at the origin of our frame, and the other at the location (X=X_0,Y=Y_0,Z=Z_0) what we expect is a relative variation of the electric field as described by (<ref>)-(<ref>); the same happens for the magnetic field. As a matter of fact,  the passage  of gravitational waves also in presence of static electric and magnetic field produce time-varying electromagnetic fields with the same frequency of the waves.
        
        The electromagnetic perturbations determined by the passage of a gravitational wave have the order of magnitude expressed by Eq. (<ref>).  To estimate them it is important to remember that  the laboratory metric (<ref>) is approximated up to second order in L/λ.  That being said, and considering that the typical amplitude of gravitational waves reaching the Earth is h ∼ 10^-21, we see that relative magnitude of the perturbations can be of the order of ∼ 10^-23.  Notice that previous estimates such as those reported by  <cit.>  overestimated the impact of the effect, since they are developed in the TT frame where, however, quantities have not a direct meaning in terms of observables. In any case, as discussed by <cit.> (see also references therein), one possibility to detect very small magnetic fields is the use of SQUIDS which can measure magnetic field ranging from 10^-15 T to 10^-18 T <cit.>; however, the order of magnitude of the effects we are considering suggests that even using such devices huge unperturbed magnetic field are needed, which are really out of our current technological possibilities. Similar considerations apply to electric fields.
        
        Another kind of approach is the use of resonant cavities (see e.g. <cit.>): the basic idea is that the passage of a gravitational wave affects the shape of a cavity and alters the magnetic flux passing through it, resulting in an electric signal that can be measured. In addition, as we showed, the passage of the wave produces an effective current in Maxwell’s equations, causing an electromagnetic field to oscillate at the same frequency as the gravitational wave. Recent proposals of experimental setups seem quite promising for the detection of gravitational waves <cit.>. 
        
        The general feeling is that, even if current technology does not allow these effects to be measured in terrestrial laboratories, this possibility is not excluded in the future.  A completely different scenario is offered by astrophysical events, where the combination of enormous electromagnetic fields and more intense gravitational waves can open up new interesting possibilities; however, this would require a different analysis than the one used in this work as it requires the simultaneous solution of Einstein's and Maxwell's equations.
        
        In conclusion, our approach lends itself to a very simple description of the interaction of electromagnetic devices with gravitational waves in terms of observable quantities, and can be used for further studies in this field; the possibility of developing a real experimental proposal is in any case guided by technological developments.
        
        
        
        
        
        § ACKNOWLEDGEMENTS
        
        
        The author expresses gratitude to Dr. Antonello Ortolan for engaging discussions and valuable insights. Special appreciation is extended to the local research project "Modelli gravitazionali per lo studio dell'universo" (2022) from the Department of Mathematics "G. Peano," Università degli Studi di Torino, as well as to the Gruppo Nazionale per la Fisica Matematica (GNFM) for their contributions.
