Gravitational waves and electrodynamics: New perspectives

# Gravitational waves and electrodynamics: New perspectives

Francisco Cabral Instituto de Astrofísica e Ciências do Espaço, Faculdade de Ciências da Universidade de Lisboa, Edifício C8, Campo Grande, P-1749-016 Lisbon, Portugal.    Francisco S. N. Lobo Instituto de Astrofísica e Ciências do Espaço, Faculdade de Ciências da Universidade de Lisboa, Edifício C8, Campo Grande, P-1749-016 Lisbon, Portugal.
July 22, 2019
###### Abstract

Given the recent direct measurement of gravitational waves (GWs) by the LIGO-VIRGO collaboration, the coupling between electromagnetic fields and gravity have a special relevance since it opens new perspectives for future GW detectors and also potentially provides information on the physics of highly energetic GW sources. We explore such couplings using the field equations of electrodynamics on (pseudo) Riemann manifolds and apply it to the background of a GW, seen as a linear perturbation of Minkowski geometry. Electric and magnetic oscillations are induced that propagate as electromagnetic waves and contain information about the GW which generates them. The most relevant results are the presence of longitudinal modes and dynamical polarization patterns of electromagnetic radiation induced by GWs. These effects might be amplified using appropriate resonators, effectively improving the signal to noise ratio around a specific frequency. We also briefly address the generation of charge density fluctuations induced by GWs and the implications for astrophysics.

###### pacs:
04.20.Cv, 04.30.-w, 04.40.-b

## I Introduction

A century after General Relativity (GR) we have celebrated the first direct measurement of gravitational waves (GWs) by the LIGO-VIRGO collaboration Abbott:2016blz (), and ESA’s LISA-Pathfinder e-LISA () science mission officially started on March, 2016. For excellent reviews on GWs see Colpi:2016fup (); Sathyaprakash:2009xs (). The waves that were measured by two detectors independently, beautifully match the expected signal following a black hole binary merger, allowing the estimation of physical and kinematic properties of these black holes. This is the expected celebration which not only confirms the existence of these waves and reinforces Einstein’s GR theory of gravity, but it also marks the very birth of GW astronomy. Simultaneously, if the general relativistic interpretation of the data is correct it gives an indirect observation of black holes and the dynamics of black hole merging in binaries.

It should be said however that GW emission from the coalescence of highly compact sources provides a test for astrophysical phenomena in the very strong gravity regime which means that a fascinating opportunity arises to study not only GR but also extended theories of gravity both classically Bogdanos:2009tn () as well as those including “quantum corrections” from quantum field theory which can predict a GW signature for the non-classical physics happening at or near the black hole’s horizon (see Cardoso:2016oxy (); Abedi:2016hgu () for recent claims on the detection of GW echoes in the late-time signals detected by LIGO, which point to physics beyond GR). The window is opened for the understanding of the physical nature of the sources but the consensus on the discovery of GWs is general. At the moment of the writing of this paper, after the first detection, the LIGO team has announced other two detections also interpreted as black hole binary merger events. The celebrated measurement of GW emission was done using laser interferometry, but other methods such as pulsar timing arrays Zhu:2015ara () will most probably provide positive detections in the near future.

However, it is crucial to keep investigating different routes towards GW measurements (see Zhu:2015ara (); Barausse:2014tra (); Hacyan:2015kra (); GWsRiles:2012yw (); Singer:2014qca ()) and one such route lies at the very heart of this work. Instead of using test masses and measuring the minute changes of their relative distances, as it is done in Laser interferometry (used in LIGO, VIRGO, GEO600, TAMA300 and will be used in KAGRA, LIGOIndia and LISA), we can also explore the effects of GWs on electromagnetic fields. For this purpose, one needs to compute the electromagnetic field equations on the spacetime background of a GW perturbation. This might not only provide models and simulations which can test the viability of such GW-electromagnetic detectors, but it might also contribute to a deeper understanding of the physical properties of astrophysical and cosmological sources of GWs, since these waves interact with the electromagnetic fields and plasmas which are expected to be common in many highly energetic GW sources (see (Brodin:2000du, )). Thus, an essential aim, in this work, will be to carefully explore the effects of GWs on electromagnetic fields.

Before approaching the issue of GW effects on electromagnetic fields, let us mention very briefly other possible routes in the quest for GW measurements. Recall that linearized gravity is also the context in which gravitoelectric and gravitomagnetic fields can be defined (GravitomagnetismMashhoon:2003ax, ). In particular, gravitomagnetism is associated to spacetime metrics with time-space components. Similarly, as we will see, the () polarization of GWs is related to space-space off-diagonal metric components, which therefore resemble gravitomagnetism. This analogy might provide a motivation to explore the dynamical effects of GWs on gyroscopes. In fact, an analogy with gravitomagnetism brings interesting perspectives regarding physical interpretations, since other analogies, in this case with electromagnetism, can be explored. In particular, gravitomagnetic effects on gyroscopes are known to be fully analogous to magnetic effects on dipoles. Now, in the case of gravitational waves these analogous (off-diagonal) effects on gyroscopes will, in general, be time dependent.

The tiny gravitomagnetic effect on gyroscopes due to Earth’s rotation, was successfully measured during the Gravity Probe B experiment Everitt:2011hp (), where the extremely small geodetic and Lens-Thirring (gravitomagnetic) deviations of the gyro’s axis were measured with the help of Super Conducting Quantum Interference Devices (SQUIDS). Analogous (time varying) effects on gyroscopes due to the passage of GWs, might be measured with SQUIDS. On the other hand, rotating superconducting matter seems to generate anomalous (stronger) gravitomagnetic fields (anomalous gravitomagnetic London moment) Tajmar:2004ww (); Tajmar:2006gh () so, if these results are robustly confirmed then superconductivity and super fluidity might somehow amplify gravitational phenomena. This hypothesis deserves further theoretical and experimental research as it could contribute for future advanced GW detectors.

Another promising route comes from the study of the coupling between electromagnetic fields and gravity, the topic of our concern in the present work. Are there measurable effects on electric and magnetic fields during the passage of a GW? Could these be used in practice to study the physics of GW production from astrophysical sources, or applied to GW detection? Although very important work has been done in the past (see for example Marklund:1999sp (); Brodin:2000du ()), it seems reasonable to say that these routes are far from being fully explored.

Regarding electromagnetic radiation, there are some studies regarding the effects of GWs (see for example Montanari:1998gd (); Rakhmanov:2009zz ()) it has been shown that gravitational waves have an important effect on the polarization of light Hacyan:2015kra (). On the other hand, lensing has been gradually more and more relevant in observational astrophysics and cosmology and it seems undoubtedly relevant to study the effects of GWs (from different types of sources) on lensing, since a GW should in principle distort any lensed image. Could lensing provide a natural amplification of the gravitational perturbation signal due to the coupling between gravity and light? These topics need careful analysis for a better understanding of the possible routes (within the reach of present technology) for gravity wave astronomy and its applications to astrophysics and cosmology.

This paper is outlined in the following manner: In Section II, we briefly review the foundations of electrodynamics and spacetime geometry and present the basic equations that will be used throughout this work. In Section III, we explore the coupling between electromagnetic fields and gravitational waves. In Section IV, we discuss our results and conclude.

## Ii Electrodynamics: General formalism

In this section, for self-consistency and self-completeness, we briefly present the general formalism that will be applied throughout the analysis. We refer the reader to FCFL ()-(Hehl:1999bt, ) for detailed descriptions of the deep relation between electromagnetic fields and spacetime geometry. We will be using a signature.

Recall that in the pre-metric formalism of electrodynamics, the charge conservation provides the inhomogeneous field equations while the magnetic flux conservation is given in the homogeneous equations HehlYuribook (); Gronwald:2005tv (); Hehl:2000pe (); Hehl:2005hu (); Hehl:1999bt (). The field equations are then given by

 dF=0,dG=J. (1)

Note that these are general, coordinate free, covariant equations and there is no need for an affine or metric structure of spacetime. is the charge current density 3-form; is the 2-form representing the electromagnetic excitation, is the Faraday 2-form, so that , where is the electromagnetic potential 1-form; the operator stands for exterior derivative.

The constitutive relations (usually assumed to be linear, local, homogeneous and isotropic) between and ,

 G⟷⋆F, (2)

provide the metric structure via the Hodge star operator , which introduce the conformal part of the metric and maps -forms to -forms, with the dimension of the spacetime manifold. On these foundations, the electromagnetic field equations on the background of a general (pseudo) Riemann spacetime manifold can be obtained. In the tensor formalism we get

 ∂μFμν+1√−g∂μ(√−g)Fμν=μ0jν,∂[αFβγ]=0, (3)

where we have used in the inhomogeneous equations the general expression for the divergence of anti-symmetric tensors in pseudo-Riemann geometry, . The homogeneous equations are independent from the metric (and connection) due to the torsionless character of Riemann geometry.

We introduce the definitions

 F0k = 1c∂tAk−∂kA0≡Ekc, (4) Fjk = ∂jAk−∂kAj≡−ϵijkBi, (5)

where is the totally antisymmetric 3-dimensional Levi-Civita (pseudo) tensor, is a vector density (a natural surface integrand) and is a co-vector (a natural line integrand). Then, the homogeneous equations are the usual Faraday and magnetic Gauss laws

 ∂tBi+ϵijk∂jEk=0,∂jBj=0, (6)

while the inhomogeneous equations can be separated into the generalized Gauss and Maxwell-Ampère laws. These are, respectively

 ∂kEj(g0kgj0−gjkg00)−∂μBkcgmμgn0ϵkmn +Ejγj−Bkcϵkmnσmn0=ρε0, (7)
 1c∂μEj(g0μgji−gjμg0i)−ϵkmn∂μBkgmμgni +Ejξji−Bkϵkmnσmni=μ0ji, (8)

where

 γj ≡ [∂k(g0kgj0−gjkg00) (9) +1√−g∂k(√−g)(g0kgj0−gjkg00)],
 σmnβ≡[∂μ(gmμgnβ)+1√−g∂μ(√−g)(gmμgnβ)], (10)
 ξji ≡ 1c[∂μ(g0μgji−gjμg0i) (11) +1√−g∂μ(√−g)(g0μgji−gjμg0i)].

One sees clearly, that new electromagnetic phenomena are expected due to the presence of extra electromagnetic couplings induced by spacetime curvature. In particular, the magnetic terms in the Gauss law are only present for non-vanishing off-diagonal time-space components , which in linearized gravity correspond to the gravitomagnetic potentials. These terms are typical of axially symmetric geometries (see FCFLapplications (); Rezzolla:2000dk ()).

For diagonal metrics, the inhomogeneous equations, the Gauss and Maxwell Ampère laws, can be recast into the following forms

 −gkkg00∂kEk+Ekγk=ρε0, (12)
 ϵijkgiigjj∂jBk+1c2g00gii∂tEi+ϵijkσjiiBk+Eiξii=μ0ji, (13)

with

 γk(x)≡ −[gkkg001√−g∂k(√−g)+∂k(gkkg00)], (14)

and

 σjii(x)≡gjjgii1√−g∂j(√−g)+∂j(gjjgii), (15)
 ξii(x)≡g00gii1c21√−g∂t(√−g)+1c2∂t(g00gii). (16)

The Einstein summation convention is applied in Eq. (13) only for and while the index is fixed by the right hand side. Also, no contraction is assumed in Eq. (14) nor in the expression for .

New electromagnetic effects induced by the spacetime geometry include an inevitable spatial variability (non-uniformity) of electric fields whenever we have non-vanishing geometric functions , electromagnetic oscillations (therefore waves) induced by gravitational radiation and also additional electric contributions to Maxwell’s displacement current in the generalized Maxwell-Ampère law. This last example becomes clearer by re-writing Eq. (13) in the form

 ϵijk∂j¯Biijjk=μ0(ȷi+ȷiD), (17)

where and

 ȷiD ≡ −ε0√−g(g00gii∂tEi+c2Eiξii), (18) = −ε0∂t(√−gg00giiEi)

is the generalized Maxwell displacement current density and

 ¯Biijjk≡giigjj√−gBk. (19)

Again no summation convention is assumed for the index in Eqs. (17) and (18). The functions vanish for stationary spacetimes but might have an important contribution for strongly varying gravitational waves (high frequencies), since they depend on the time derivatives of the metric. Analogously, Eq. (12) can be written as

 ∂k~Ek=ϱε0, (20)

where

 ~Ej≡−gjjg00√−gEj,ϱ≡√−gρ. (21)

These are physical, observable effects of spacetime geometry in electromagnetic fields expressed in terms of the extended Gauss and Maxwell-Ampère laws which help the comparison with the usual inhomogeneous equations in Minkowski spacetime, making clearer the physical interpretations of such effects.

Finally, we review the field equations in terms of the electromagnetic 4-potential which in vacuum are convenient to study electromagnetic wave phenomena. From the definition of the Faraday tensor and Eq. (3), we get

 ∇μ∇μAν−gλνRελAε−∇ν(∇μAμ)=μ0jν, (22)

where is the Ricci tensor. Using the expression for the (generalized) Laplacian in pseudo-Riemann manifolds, , and assuming the Generalized Lorenz gauge () in vacuum, we arrive at

 ∂μ∂μAν+1√−g∂μ(√−ggμλ)∂λAν−gλνRελAε=0. (23)

For a diagonal metric, we get

 ∂μ∂μAν+1√−g∂μ(√−ggμμ)∂μAν−gννRενAε=0, (24)

with no contraction assumed in . In general, and contrary to electromagnetism in Minkowski spacetime, the equations for the components of the electromagnetic 4-potential are coupled even in the (generalized) Lorenz gauge. Notice also that for Ricci-flat spacetimes, the term containing the Ricci tensor vanishes. Naturally, the vacuum solutions of GR are examples of such cases. New electromagnetic phenomena are expected to be measurable, for gravitational fields where the geometric dependent terms in Eq. (23) are significant.

This completes the main axiomatic (foundational) formalism of electrodynamics in the background of curved (pseudo-Riemann) spacetime.

## Iii Gravitational waves and electromagnetic fields

As mentioned in the Introduction, GWs have been recently detected by the LIGO team using laser interferometry Abbott:2016blz (). Another method that has been carried over the last decade to detect GWs is that of pulsar timing arrays. Nevertheless, it is crucial to keep exploring different routes towards GW detection and its applications to astrophysics and cosmology. Due to the huge distances in the Cosmos, any GW reaching Earth should have an extremely low amplitude. Therefore, the linearisation of gravity is usually applied which allows to derive the wave equations. It is a perturbative approach which is background dependent and its common to consider a Minkowski background. In any case, the GW can be seen as a manifestation of propagating spacetime geometry perturbations.

In principle, the passage of a GW in a region with electromagnetic fields will have a measurable effect. To compute this we have to consider Maxwell’s equations on the perturbed background of a GW. We shall consider a GW as a perturbation of Minkowski spacetime given by , with , so that

 ds2=c2dt2−dx2−dy2−dz2+hαβdxαdxβ, (25)

where the perturbation corresponds to a wave travelling along the axis, i.e.,

 ds2 = c2dt2−dz2−[1−f+(z−ct)]dx2 (26) − [1+f+(z−ct)]dy2+2f×(z−ct)dxdy,

and and refer to the two independent polarizations characteristic of GWs in GR. This metric is a solution of Einstein’s field equations in the linear approximation, in the so-called TT (Transverse-Traceless) Lorenz Gauge. For this metric, we get

 1√−g∂z(√−g)=f×(∂zf×)+f+(∂zf+)f2×+f2+−1, (27)
 1√−g∂t(√−g)=f×(∂tf×)+f+(∂tf+)f2×+f2+−1. (28)

These quantities will be useful further on.

### iii.1 GW effects in electric and magnetic fields

Consider an electric field in the background of a GW travelling in the direction. The general expression for Gauss’ law (7), in vacuum, is now given by

 [1−f+(z,t)]−1∂xEx+[1+f+(z,t)]−1∂yEy +∂zEz−f−1×(z,t)(∂yEx+∂xEy) +[1√−g∂z(√−g)]Ez=0, (29)

which clearly shows that physical (possibly observable) effects are induced by the propagation of GWs.

As for the Maxwell-Ampère law, Eq. (8) provides the following relations in vacuum:

 1c2[f−1×∂tEy−(1−f+)−1∂tEx]+Exξxx+Eyξyx −(1−f+)−1[(1+f+)−1∂yBz−∂zBy]−Byσzxx +Bxσzyx−f−1×(f−1×∂yBz+∂zBx)=0, (30)
 1c2[f−1×∂tEx−(1+f+)−1∂tEy]+Eyξyy+Exξxy −(1+f+)−1[(1−f+)−1∂xBz−∂zBx]+Bxσzyy −Byσzxy+f−1×(f−1×∂xBz+∂zBy)=0, (31)
 −1c2∂tEz+Ezξzz−f−1×(∂yBy−∂xBx) +[(1−f+)−1∂xBy−(1+f+)−1∂yBx]=0, (32)

with the non-vanishing geometric coefficients given by

 ξxx=1c2f×(f+−1)∂tf×−(f2×+f+−1)∂tf+(f+−1)2(f2×+f2+−1),
 ξyx=ξxy=1c2−(f2+−1)∂tf×+f×f+∂tf+f2×(f2×+f2+−1),
 ξyy=1c2−f×(f++1)∂tf×+(f2×−f+−1)∂tf+(f++1)2(f2×+f2+−1),
 ξzz=−1c2f×(∂tf×)+f+(∂tf+)f2×+f2+−1,
 σzxx=−f×(f+−1)∂zf×−(f2×+f+−1)∂zf+(f+−1)2(f2×+f2+−1),
 σzyy=−−f×(f++1)∂zf×+(f2×−f+−1)∂zf+(f++1)2(f2×+f2+−1),
 σzxy=σzyx=−−(f2+−1)∂zf×+f×f+∂zf+f2×(f2×+f2+−1).

A natural consequence of these laws is the generation of electromagnetic waves induced by gravitational radiation. Initially static electric and magnetic fields become time dependent during the passage of GWs which might be detectable in this way.

In general, the system of coupled Eqs. (7)-(8) and the homogeneous equations in (3) have to be taken as a whole. As we will see from Eq. (29), in some specific situations the electric field can be solved directly from Gauss’ law. This electric field can in turn act as a source for magnetism via the Maxwell-Ampère relations in Eqs. (3.6)-(32), where the presence of the GW induces extra terms proportional to the electric field. In this work, we will explore relatively simple situations in order to illustrate the effects of GWs in electric and magnetic fields. Let us start by considering the effects of GWs in electric fields.

### iii.2 Electric field oscillations induced by GWs

We will consider electric fields in the following three scenarios.

#### iii.2.1 Electric field aligned with the z axis

An electric field along the axis can easily be achieved by charged plane plates constituting a capacitor. In the absence of GWs the electric field thus produced is approximately uniform (neglecting boundary effects) for static uniform charge distributions or time variable if there is an alternate current (as in the case of a RLC circuit with a variable voltage signal generator). With the passage of the GW, in general the electric field is perturbed by both the () and () modes. To see this let us look at Gauss’ law when the electric field is aligned with the direction of the GW propagation. From Eq. (29), we have

 (33)

where is given by the expression in Eq. (27). We can see that even if in the absence of any GW the electric field was static and uniform, during the passage of the spacetime disturbance, the field will be time varying and non-uniform, oscillating with the same frequency of the passing GW. In fact, the general solution is

 Ez(x,y,z,t)=E0√−g=E0√1−f2+−f2×, (34)

where in the most general case, . To get the full description of the electric field one has to consider also both the Maxwell-Ampère relations in Eqs. (3.4)-(32) and the Faraday law. Nevertheless it is already clear from Eq. (34) that GWs induce propagating electric oscillations.

We will consider the most simple case in which is a constant (without any dependence on , or ). Indeed, one can easily verify that the fields , constitute a (trivial) solution of the full Maxwell equations, namely Eqs. (29)-(32) together with the homogeneous equations in (3). Notice that for zero magnetic field the Maxwell-Ampère equation (32) is

 −1c2∂tEz+Ezξzz=0, (35)

which is verified by the solution in (34) for a constant . This can easily be seen when one considers that

 ξzz=−1c21√−g∂t(√−g), (36)

in accordance with the expressions previously shown for and Eq. (28). In this case, the coupling between the electric field and the GW in the expression for the generalized Maxwell displacement current density, compensates the traditional term which depends on the time derivative of the electric field. In fact, by multiplying by , then Eq. (35) can be interpreted as the conservation of the total electric flux density. This situation is thus compatible with the experimental scenario where there are no currents producing any magnetic field and the electric field, although changing in time, due to the coupling with gravity does not give rise to any magnetic field, since the total electric flux is conserved. Naturally, this is not the general case. For example in the presence of currents along the axis, and due to the gravitational factors in the equations (3.6)-(32) the magnetic field is dynamical (time dependent). Therefore, this field necessarily affects the electric field via the Faraday law,

 ∂yEz=∂yE0√−g=−∂tBx,∂xEz=∂xE0√−g=∂tBy, (37)

which implies that in general . Since is time dependent, in such a case the electric field contributes to the magnetic field via the (non-null) generalized Maxwell displacement current, in accordance with Eq. (32), where now

 −1c2∂tEz+Ezξzz≠0. (38)

As a practical application consider the following harmonic GW perturbation

 f+(z,t) = acos(kz−wt), (39) f×(z,t) = bcos(kz−wt+α). (40)

In this case, we get the following electric oscillations

 Ez(z,t) = ~E0[1−a2cos2(kz−wt) (41) −b2cos2(kz−wt+α)]−1/2,

for , which is obeyed by the extremely low amplitude GWs reaching the Solar System. Here is an arbitrary fixed constant and is the phase difference. These electric oscillations will show distinct features sensitive to the or GW modes. It provides a window for detecting and analysing GW signals directly converted into electromagnetic information.

Notice that the electric waves produced are longitudinal, since these are propagating along the same direction of the GW, even though the electric field is aligned with this direction. To grasp the physical interpretation behind this non-intuitive result, recall that the electric energy density depends quadratically on the field and therefore it is the energy density fluctuation induced by the GW which propagates along the direction of .

In order to have an approximate idea on the energy density of the resulting electromagnetic wave we can use the usual expression (derived from Maxwell electrodynamics in Minkowski spacetime). We get

 uem∼ε0E2z(z,t) = ε0~E20[1−a2cos2(kz−wt) (42) − b2cos2(kz−wt+α)]−1,

and the energy per unit area and unit time through any surface (with a normal making an angle with the axis) is approximately expressed by

 ∥→S∥cosϑ=ε0c~E20[1−a2cos2(kz−wt) −b2cos2(kz−wt+α)]−1cosϑ, (43)

where is the Poynting vector, and .

If is the electric field in the absence of GWs, then the relevant dimensionless quantity to be measured is given by the following expression

 ∣∣∣Ez(z,t)−~E0~E0∣∣∣=∣∣[1−a2cos2(kz−wt) −b2cos2(kz−wt+α)]−1/2−1∣∣, (44)

and in terms of the energy density, we get

 −b2cos2(kz−wt+α)]−1−1∣∣, (45)

with .

Substituting in these two expressions the typical amplitudes for GWs due to binaries (), the induced electric field and corresponding energy density oscillations signal will be extremely small. Concerning GWs reaching the Solar System, the detectors which might have a response proportional to the electric field magnitude or rather to its energy (proportional to the square of the electric field magnitude), must be extremely sensitive. We emphasize the fact that, in principle, this electromagnetic wave can be confined in a cavity using very efficient reflectors for the frequency . Then, under appropriate (resonant) geometric conditions, the signal can be amplified. This might have very important practical applications for future GW detectors.

#### iii.2.2 Electric field in the xy plane

Suppose we have an electric field in the direction. The electric field could initially be uniform and confined within a plane capacitor. In these conditions, the Gauss law in vacuum becomes

 [1−f+(z,t)]−1∂Ex∂x−(f×)−1(z,t)∂Ex∂y=0. (46)

A similar expression is obtained if the electric field is aligned with the axis. Assuming a separation of variables , where and are seen as external parameters, substituting in the above equation and dividing it by , we obtain

 (1−f+)−1∂xF1F1=f−1×∂yF2F2, (47)

therefore, one arrives at the following expressions

 F1(x;z)=C1(z,t)e−(1−f+)x,F2(x;z)=C2(z,t)e−f×y. (48)

Since we can always add a constant to the solution obtained from , we can write

 Ex(x,y,z,t)=E0x[1+~C(z,t)e−[(1−f+)x+f×y]], (49)

where in general can be obtained by taking into account the other Maxwell equations. The full solution should be compatible with the limit without gravity in which we recover the uniform field . Therefore

 f+=f×=0⇒~C(z,t)=0. (50)

A natural Anszatz is

 ~C(z,t)=ηfα1++βfα2×+μfα3+fα4×, (51)

where and () are constants. But as previously said the form of this function can be studied by considering compatibility with the remaining Maxwell equations.

For the harmonic GW introduced before, the second term in the solution above, Eq. (49) is given by the following expression

 E0x~C(z,t)exp{−[(1−acos(kz−wt))x +bcos(kz−wt+α)y]}. (52)

The solution obtained is also sensitive to the existence or not of two modes in the GW, to their amplitudes and phase difference. Although this solution obeys the Gauss law, it implies a non-zero dynamical magnetic field, according to Faraday’s law. As mentioned, to get a full treatment one should then check the consistency with the other Maxwell equations, in order to derive restrictions on the mathematical form of .

Let us consider now the case where an electric field is generated by a plane capacitor oriented along the axis and a second electric field is generated by another similar capacitor oriented along the axis. In this condition, the resulting electric field in the vacuum between the charged plates, , obeys the equation

 (1−f+)−1∂xEx − (f×)−1∂yEx+(1+f+)−1∂yEy (53) −(f×)−1∂xEy=0.

A possible solution to this equation is given by

 Ex(x,y,z,t)=E0x[1+~C1(z,t)e−[(1−f+)x+f×y]],
 Ey(x,y,z,t)=E0y[1+~C2(z,t)e−[f×x+(1+f+)y]],

where for we get .

The resulting electric oscillations propagate along the axis as an electromagnetic wave with non-linear polarization. This wave results from a linear gravitational perturbation of Minkowski spacetime and therefore (in this first order approximation) it can be thought of as an electromagnetic disturbance propagating in Minkowski background with a dynamical polarization. In fact, the angle between the resulting electric field and the axis is then i.e., for

 Θ(x,y,z,t)≃arctan⎧⎪ ⎪⎨⎪ ⎪⎩[1+~C2(z,t)e−[f×x+(1+f+)y]][1+~C1(z,t)e−[(1−f+)x+f×y]]⎫⎪ ⎪⎬⎪ ⎪⎭.

Even if we had , we still necessarily get a non-linear, dynamical polarization. Such an oscillating polarization could in principle be another distinctive signature of the GW that is causing it. The solutions obtained already give sufficient information to conclude that it is possible to obtain polarization fluctuations induced by GWs, where for the strength of the effect is given by . A dynamical spatial polarization pattern is therefore expected in our detector. This contrasts with the other cases where the resulting wave was linearly polarized. This effect is shown in Figs. 1 and 2.

Nevertheless, again, the Faraday law and the Maxwell-Ampère relations can provide constraints on the functions and .

#### iii.2.3 Electric field in the background of a GW with zero (×) mode

If we consider solely the GW mode, the spacetime metric (26) becomes diagonal and the Gauss and Maxwell-Ampère equations simplify to the following expressions in vacuum

 ∂k~Ek=0,ϵijk∂j¯Biijjk=μ0ȷiD, (56)

respectively, where

 ~Ej≡−gjjg00√−gEj,¯Biijjk≡giigjj√−gBk, (57)

and the generalized Maxwell displacement current density is

 ȷiD = ε0∂t~Ei, (58)

in accordance with Eqs. (17)-(21). Let us search for a trivial electric field solution which is fully compatible with the complete system of Maxwell equations. If we consider the field

 ~E=(~Ex0(y,z,t),~Ey0(x,z,t),~Ez0(x,y,t)), (59)

the Gauss law is trivially obeyed and the electric field is given by

 E=⎛⎜ ⎜⎝1−f+√1−f2+~Ex0,1+f+√1−f2+~Ey0,~Ez0√1−f2+⎞⎟ ⎟⎠. (60)

Furthermore, if , the generalized Maxwell displacement current density is zero, therefore effectively the electric field does not contribute to the Maxwell-Ampère equations. Consequently, in the absence of electric currents, such an electric field solution seems to be compatible with the condition . Let us assume that this is the case. Regarding the remaining Maxwell equations, the Magnetic Gauss law is trivially obeyed but what about Faraday’s law? In this case, one can show that the condition , leads to a field which necessarily depends on time which contradicts the hypothesis of zero magnetic field according to the Maxwell-Ampère relations in (56) and the expression (58). In fact, one arrives at the field.

 ~E=(~Ex0(z,t),~Ey0(z,t),~Ez0), (61)

where is a constant and are given by

 ~Ex0=~Cx0exp[−∫∂z(1−f+√−g)√−g1−f+], (62)
 ~Ey0=~Cy0exp[−∫∂z(1+f+√−g)√−g1+f+], (63)

where and are constants of integration. These functions clearly depend on time and therefore the generalized Maxwell displacement current cannot be zero leading to a non-vanishing magnetic field. When considering a generic electric field with three components as in (60), one cannot assume that neither a zero magnetic field.

Therefore in the general case one needs to consider the influence of the electric field on the magnetic field, through the generalized Maxwell displacement current. An exception to this is the special case first considered in , where the electric field is aligned with the direction of the propagation of the GW.

### iii.3 Magnetic field oscillations induced by GWs

The passage of the GW can induce a non-vanishing time varying magnetic field, even for an initially static electric field. In general the full system of the Maxwell equations can be explored numerically to compute the resulting electric and magnetic oscillations. These magnetic fluctuations could be measured in principle using SQUIDS (Super Conducting Quantum Interference Devices) that are extremely sensitive to small magnetic field changes.

To get a glimpse of the gravitationally induced magnetic field fluctuations, we can consider for simplicity only the (+) GW mode and take the generalized Maxwell-Ampère law in the form of Eq. (17). We will be considering an electric field aligned with the axis given by the following solution to the Gauss law

 E=⎛⎜ ⎜⎝0,0,~Ez0(x,y,t)√1−f2+⎞⎟ ⎟⎠,∂k~Ek=0. (64)

We can also consider an electric current along the axis such that in principle, by symmetry we expect a magnetic field in the plane, . Then the Maxwell-Ampère equations (17) are

 ∇×¯B=μ0(√−gj+ε0∂t~E0), (65)

where , while the Faraday law provides the equations

 ∂tBx=−∂y~Ez0√1−f2+,∂tBy=∂x~Ez0√1−f2+. (66)

Then we can perform an integration over an “amperian” closed line coincident to a magnetic field line (in perfect analogy with the method taken in usual electromagnetism) to integrate the Maxwell-Ampère law, assuming axial symmetry, around the charge current distribution and electric flux (Maxwell displacement) current.

We obtain the following solution to Eq. (65)

 ¯B=μ0~Itot2π√x2+y2(cosϕey−sinϕex), (67)

where and . is the (constant) electric current and is the Maxwell displacement current density. We then get the magnetic field components

 Bx=−1+f+√1−f2+[μ0~Itot(x,y,z,t)2π(x2+y2)y], (68)

and

 By=1−f+√1−f2+[μ0~Itot(x,y,z,t)2π(x2+y2)x], (69)

respectively.