Gravitational Waves and Electrodynamics: New Perspectives

Eur. Phys. J. C (2017) 77:237 DOI 10.1140/epjc/s10052-017-4791-z

Regular Article - Theoretical Physics

Gravitational waves and electrodynamics: new perspectives

Francisco Cabrala, Francisco S. N. Lobob Instituto de Astrofísica e Ciências do Espaço, Faculdade de Ciências da Universidade de Lisboa, Edifício C8, Campo Grande, 1749-016 Lisbon, Portugal

Received: 1 February 2017 / Accepted: 26 March 2017 © The Author(s) 2017. This article is an open access publication

Abstract Given the recent direct measurement of gravita- if the general relativistic interpretation of the data is correct it tional waves (GWs) by the LIGO–VIRGO collaboration, the gives an indirect observation of black holes and the dynamics coupling between electromagnetic fields and gravity have a of black hole merging in binaries. special relevance since it opens new perspectives for future It should be said, however, that GW emission from the GW detectors and also potentially provides information on coalescence of highly compact sources provides a test for the physics of highly energetic GW sources. We explore such astrophysical phenomena in the very strong gravity regime couplings using the field equations of electrodynamics on which means that a fascinating opportunity arises to study (pseudo) Riemann manifolds and apply it to the background not only GR but also extended theories of gravity both clas- of a GW, seen as a linear perturbation of Minkowski geome- sically [5] as well as those including “quantum corrections” try. Electric and magnetic oscillations are induced that prop- from quantum field theory which can predict a GW signature agate as electromagnetic waves and contain information as of the non-classical physics happening at or near the black regards the GW which generates them. The most relevant hole’s horizon (see [6,7] for recent claims on the detection of results are the presence of longitudinal modes and dynami- GW echoes in the late-time signals detected by LIGO, which cal polarization patterns of electromagnetic radiation induced point to physics beyond GR). The window is opened for the by GWs. These effects might be amplified using appropri- understanding of the physical nature of the sources but the ate resonators, effectively improving the signal to noise ratio consensus on the discovery of GWs is general. At the moment around a specific frequency. We also briefly address the gen- of the writing of this paper, after the first detection, the LIGO eration of charge density fluctuations induced by GWs and team has announced two other possible detections, also inter- the implications for astrophysics. preted 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 [8] 1 Introduction will most probably provide positive detections in the near future. A century after general relativity (GR) we have celebrated However, it is crucial to keep investigating different routes the first direct measurement of gravitational waves (GWs) toward GW measurements (see [8–12]) and one such route by the LIGO–VIRGO collaboration [1], and ESA’s LISA- lies at the very heart of this work. Instead of using test masses Pathfinder [2] science mission which officially started on and measuring the minute changes of their relative distances, March, 2016. For excellent reviews on GWs see [3,4]. The as in laser interferometry (used in LIGO, VIRGO, GEO600, waves that were measured by two detectors independently, TAMA300 and will be used in KAGRA, LIGOIndia and beautifully match the expected signal following a black hole LISA), we can also explore the effects of GWs on electro- binary merger, allowing the estimation of physical and kine- magnetic fields. For this purpose, one needs to compute the matic properties of these black holes. This is the expected electromagnetic field equations on the spacetime background celebration which not only confirms the existence of these of a GW perturbation. This might not only provide models waves and reinforces Einstein’s GR theory of gravity, but it and simulations which can test the viability of such GW- also marks the very birth of GW astronomy. Simultaneously, electromagnetic detectors, but it might also contribute to a deeper understanding of the physical properties of astrophys- a e-mail: [email protected] ical and cosmological sources of GWs, since these waves b e-mail: [email protected] interact with the electromagnetic fields and plasmas which 123 237 Page 2 of 14 Eur. Phys. J. C (2017) 77:237 are expected to be common in many highly energetic GW undoubtedly relevant to study the effects of GWs (from dif- sources (see [13]). Thus, an essential aim, in this work, will ferent types of sources) on lensing, since a GW should in be to carefully explore the effects of GWs on electromagnetic principle distort any lensed image. Could lensing provide a fields. natural amplification of the gravitational perturbation signal Before approaching the issue of GW effects on electrody- due to the coupling between gravity and light? These top- namics, let us mention very briefly other possible routes in the ics need careful analysis for a better understanding of the quest for GW measurements. Recall that linearized gravity possible routes (within the reach of present technology) for is also the context in which gravitoelectric and gravitomag- gravity wave astronomy and its applications to astrophysics netic fields can be defined [14]. In particular, gravitomag- and cosmology. netism is associated to spacetime metrics with time-space This paper is outlined in the following manner: in Sect. components. Similarly, as we will see, the (×) polarization 2, we briefly review the foundations of electrodynamics and of GWs is related to space-space off-diagonal metric com- spacetime geometry and present the basic equations that will ponents, which therefore resemble gravitomagnetism. This be used throughout this work. In Sect. 3, we explore the analogy might provide a motivation to explore the dynami- coupling between electromagnetic fields and gravitational cal effects of GWs on gyroscopes. In fact, an analogy with waves. In Sect. 4, we discuss our results and conclude. gravitomagnetism shows interesting perspectives regarding physical interpretations, since other analogies, in this case with electromagnetism, can be explored. In particular, grav- 2 Electrodynamics: general formalism itomagnetic 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 In this section, for self-consistency and self-completeness, gyroscopes will, in general, be time dependent. we briefly present the general formalism that will be applied The tiny gravitomagnetic effect on gyroscopes due to throughout the analysis. We refer the reader to [21–27]for Earth’s rotation, was successfully measured during the Grav- detailed descriptions of the deep relation between electro- ity Probe B experiment [15], where the extremely small magnetic fields and spacetime geometry. We will be using a +−−− geodetic and Lens–Thirring (gravitomagnetic) deviations of ( ) signature. the gyro’s axis were measured with the help of super con- Recall that in the pre-metric formalism of electrodynam- ducting quantum interference devices (SQUIDS). Analo- ics, the charge conservation provides the inhomogeneous gous (time varying) effects on gyroscopes due to the pas- field equations while the magnetic flux conservation is given sage of GWs, might be measured with SQUIDS. On the in the homogeneous equations [23–27]. The field equations other hand, rotating superconducting matter seems to gen- are then given by erate anomalous (stronger) gravitomagnetic fields (anoma- = , = . lous gravitomagnetic London moment) [16,17] so, if these dF 0 dG J (2.1) results are robustly confirmed then superconductivity and superfluidity might somehow amplify gravitational phenom- Note that these are general, coordinate free, covariant equa- ena. This hypothesis deserves further theoretical and exper- tions and there is no need for an affine or metric structure imental research as it could contribute for future advanced of spacetime. J is the charge current density 3-form; G is GW detectors. the 2-form representing the electromagnetic excitation, F is Another promising route comes from the study of the cou- the Faraday 2-form, so that F = dA, where A is the electro- pling between electromagnetic fields and gravity, the topic magnetic potential 1-form; the operator d stands for exterior of our concern in the present work. Are there measurable derivative. effects on electric and magnetic fields during the passage of The constitutive relations (usually assumed to be linear, a GW? Could these be used in practice to study the physics local, homogeneous and isotropic) between G and F, of GW production from astrophysical sources, or applied to GW detection? Although very important work has been done G ←→ F, (2.2) in the past (see for example [13,18]), it seems reasonable to say that these routes are far from being fully explored. provide the metric structure via the Hodge star operator , Regarding electromagnetic radiation, there are some stud- which introduce the conformal part of the metric and maps k- ies regarding the effects of GWs (see for example [19,20]). forms to (n−k)-forms, with n the dimension of the spacetime It has been shown that gravitational waves have an impor- manifold. On these foundations, the electromagnetic field tant effect on the polarization of light [12]. On the other equations on the background of a general (pseudo) Riemann hand, lensing has been gradually more and more relevant spacetime manifold can be obtained. In the tensor formalism in observational astrophysics and cosmology and it seems we get 123 Eur. Phys. J. C (2017) 77:237 Page 3 of 14 237

√ μν 1 μν ν linearized gravity correspond to the gravitomagnetic poten- ∂μ F + √ ∂μ( −g)F = μ0 j ,∂[α Fβγ] = 0, −g tials. These terms are typical of axially symmetric geometries (2.3) (see [22,28]). For diagonal metrics, the inhomogeneous equations, the where we have used in the inhomogeneous equations the Gauss and Maxwell–Ampère laws, can be recast into the general expression for the divergence of anti-symmetric ten- following forms: ∇ μν = √1 ∂ sors in pseudo-Riemann geometry, μ − μ ρ √ g − kk 00∂ + γ k = , − μν g g k Ek Ek (2.12) g . The homogeneous equations are independent ε0 from the metric (and connection) due to the torsionless char- acter of Riemann geometry. 1 giig jj∂ Bk + g00gii∂ E + σ jii Bk + E ξ ii We introduce the definitions ijk j c2 t i ijk i 1 Ek = μ i , F = ∂ A − ∂ A ≡ , (2.4) 0 j (2.13) 0k c t k k 0 c F = ∂ A − ∂ A ≡− Bi , (2.5) with jk j k k j ijk √ where is the totally anti-symmetric three-dimensional k kk 00 1 kk 00 ijk γ (x) ≡− g g √ ∂k( −g) + ∂k g g , Levi-Civita (pseudo) tensor, Bi is a vector density (a natu- −g ral surface integrand) and Ek is a co-vector (a natural line (2.14) integrand). Then the homogeneous equations are the usual and Faraday and magnetic Gauss laws 1 √ ∂ Bi + ijk∂ E = 0,∂B j = 0, (2.6) σ jii(x) ≡ g jjgii √ ∂ ( −g) + ∂ (g jjgii), (2.15) t j k j −g j j while the inhomogeneous equations can be separated into the generalized Gauss’ and Maxwell–Ampère laws. These are, 1 1 √ 1 ξ ii(x) ≡ g00gii √ ∂ ( −g) + ∂ (g00gii). (2.16) respectively, 2 − t 2 t c g c 0k j0 jk 00 k mμ n0 ∂k E j g g − g g − ∂μ B cg g kmn The Einstein summation convention is applied in Eq. (2.13) ρ j k mn0 only for j and k, while the index i is fixed by the right hand + E j γ − B c kmnσ = , (2.7) ε0 side. Also, no contraction is assumed in Eq. (2.14) nor in the expression for σ jii. 1 0μ ji jμ 0i k mμ ni New electromagnetic effects induced by the spacetime ∂μ E g g − g g − ∂μ B g g c j kmn geometry include an inevitable spatial variability (non- ji k mni i + E j ξ − B kmnσ = μ0 j , (2.8) uniformity) of electric fields whenever we have non-vanishing geometric functions γ k, electromagnetic oscillations (there- where fore waves) induced by gravitational radiation and also addi- j 0k j0 jk 00 γ ≡ ∂k g g − g g tional electric contributions to Maxwell’s displacement current in the generalized Maxwell–Ampère law. This last exam- 1 √ +√ ∂ ( −g) g0k g j0 − g jkg00 , (2.9) ple becomes clearer by re-writing Eq. (2.13) in the form −g k ¯ iijjk i i ijk∂ j B = μ0(j + j ), (2.17) √ D mnβ mμ nβ 1 mμ nβ √ σ ≡ ∂μ(g g ) + √ ∂μ( −g)(g g ) , j i ≡ − i −g where gj and √ (2.10) j i ≡−ε − 00 ii∂ + 2 ξ ii , D 0 g g g t Ei c Ei √ ji 1 0μ ji jμ 0i =−ε ∂ ( − 00 ii ) ξ ≡ ∂μ g g − g g 0 t gg g Ei (2.18) c √ 1 0μ ji jμ 0i is the generalized Maxwell displacement current density and +√ ∂μ( −g) g g − g g . (2.11) −g √ B¯ iijjk ≡ giig jj −gBk. (2.19) One sees clearly that new electromagnetic phenomena are expected due to the presence of extra electromagnetic Again no summation convention is assumed for the index couplings induced by spacetime curvature. In particular, the i in Eqs. (2.17) and (2.18). The functions ξ ii vanish for sta- magnetic terms in the Gauss law are only present for non- tionary spacetimes but might have an important contribution vanishing off-diagonal time-space components g0 j , which in for strongly varying gravitational waves (high frequencies), 123 237 Page 4 of 14 Eur. Phys. J. C (2017) 77:237 since they depend on the time derivatives of the metric. Anal- to detect GWs is that of pulsar timing arrays. Nevertheless, ogously, Eq. (2.12) can be written as it is crucial to keep exploring different routes toward GW detection and its applications to astrophysics and cosmology. ∂ ˜ k = , k E (2.20) Due to the huge distances in the Cosmos, any GW reaching ε0 Earth should have an extremely low amplitude. Therefore, where √ √ the linearization of gravity is usually applied which allows ˜ j jj 00 E ≡−g g −gEj ,≡ −gρ. (2.21) one to derive the wave equations. It is a perturbative approach which is background dependent and its common to consider These are physical, observable effects of spacetime geom- a Minkowski background. In any case, the GW can be seen etry in electromagnetic fields expressed in terms of the as a manifestation of propagating spacetime geometry per- extended Gauss and Maxwell–Ampère laws which help turbations. the comparison with the usual inhomogeneous equations in In principle, the passage of a GW in a region with elec- Minkowski spacetime, making clearer the physical interpre- tromagnetic fields will have a measurable effect. To compute tations of such effects. this we have to consider Maxwell’s equations on the per- Finally, we review the field equations in terms of the elec- turbed background of a GW. We shall consider a GW as a per- tromagnetic 4-potential which in vacuum are convenient to turbation of Minkowski spacetime given by gαβ = ηαβ +hαβ , study electromagnetic wave phenomena. From the definition with |hαβ |1, so that of the Faraday tensor and Eq. (2.3), we get μ ν λν ε ν μ ν 2 = 2 2 − 2 − 2 − 2 + α β , ∇μ∇ A − g Rελ A −∇ ∇μ A = μ0 j , (2.22) ds c dt dx dy dz hαβ dx dx (3.1) where Rελ is the Ricci tensor. Using the expression for where the perturbation corresponds to a wave traveling along the (generalized) Laplacian in pseudo-Riemann manifolds, the z axis, i.e., √ ∇ ∇μψ = √1 ∂ − μλ∂ ψ μ μ gg λ , and assuming the gen- 2 2 2 2 2 −g ds = c dt − dz −[1 − f+(z − ct)]dx μ eralized Lorenz gauge (∇μ A = 0) in vacuum, we arrive 2 −[1 + f+(z − ct)]dy + 2 f×(z − ct)dxdy, (3.2) at √ μ ν 1 μλ ν λν ε and (+) and (×) refer to the two independent polarizations ∂μ∂ A + √ ∂μ −gg ∂λ A − g Rελ A = 0. −g characteristic of GWs in GR. This metric is a solution of (2.23) Einstein’s field equations in the linear approximation, in the so-called TT (Transverse-Traceless) Lorenz gauge. For this For a diagonal metric, we get metric, we get √ ∂ ∂μ ν + √1 ∂ − μμ ∂ ν − νν ε = , μ A μ gg μ A g Rεν A 0 1 √ f× (∂ f×) + f+ (∂ f+) −g √ ∂ ( −g) = z z , (3.3) − z 2 2 (2.24) g f× + f+ − 1 with no contraction assumed in ν. In general, and contrary √ f× (∂ f×) + f+ (∂ f+) to electromagnetism in Minkowski spacetime, the equations √1 ∂ ( − ) = t t . t g 2 2 (3.4) for the components of the electromagnetic 4-potential are −g f× + f+ − 1 coupled even in the (generalized) Lorenz gauge. Notice also that, for Ricci-flat spacetimes, the term containing the Ricci These quantities will be useful further on. tensor vanishes. Naturally, the vacuum solutions of GR are examples of such cases. New electromagnetic phenomena 3.1 GW effects in electric and magnetic fields are expected to be measurable, for gravitational fields where the geometric dependent terms in Eq. (2.23) are significant. Consider an electric field in the background of a GW traveling This completes the main axiomatic (foundational) formal- in the z direction. The general expression for Gauss’ law ism of electrodynamics in the background of curved (pseudo- (2.7), in vacuum, is now given by Riemann) spacetime. −1 −1 [1 − f+(z, t)] ∂x Ex +[1 + f+(z, t)] ∂y Ey − + ∂ E − f 1(z, t) ∂ E + ∂ E z z × y x x y 3 Gravitational waves and electromagnetic fields 1 √ + √ ∂ ( −g) E = 0, (3.5) −g z z As mentioned in the Introduction, GWs have been recently detected by the LIGO team using laser interferometry [1]. which clearly shows that physical (possibly observable) Another method that has been carried over the last decade effects are induced by the propagation of GWs. 123 Eur. Phys. J. C (2017) 77:237 Page 5 of 14 237

As for the Maxwell–Ampère law, Eq. (2.8) provides the and magnetic fields. Let us start by considering the effects of following relations in vacuum: GWs in electric fields. 1 −1 −1 xx yx f× ∂t Ey − (1 − f+) ∂t Ex + Ex ξ + Eyξ 3.2 Electric field oscillations induced by GWs c2 − ( − )−1 ( + )−1∂ z − ∂ y − yσ zxx 1 f+ 1 f+ y B z B B We will consider electric fields in the following scenarios. x zyx −1 −1 z x + B σ − f× f× ∂y B + ∂z B = 0, (3.6) 3.2.1 Electric field aligned with the z axis 1 −1 −1 yy xy An electric field along the z axis can easily be achieved by f× ∂t Ex − (1 + f+) ∂t Ey + Eyξ + Ex ξ c2 charged plane plates constituting a capacitor. In the absence −1 −1 z x x zyy − (1 + f+) (1 − f+) ∂x B − ∂z B + B σ of GWs the electric field thus produced is approximately uniform (neglecting boundary effects) for static uniform charge y zxy −1 −1 z y − B σ + f× f× ∂x B + ∂z B = 0, (3.7) distributions. The field can also be 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 1 zz −1 y x − ∂ E + E ξ − f× ∂ B − ∂ B GW, in general the electric field is perturbed by both the (+) c2 t z z y x and the (×) modes. To see this let us look at Gauss’ law −1 y −1 x + (1 − f+) ∂x B − (1 + f+) ∂y B = 0, (3.8) when the electric field is aligned with the direction of the GW propagation. From Eq. (3.5), we have with the non-vanishing geometric coefficients given by √ ∂ + √1 ∂ ( − ) = , f×( f+ − 1)∂ f× − ( f 2 + f+ − 1)∂ f+ z Ez Ez z g 0 (3.9) ξ xx = 1 t × t , −g 2 2 2 2 c ( f+ − 1) ( f× + f+ − 1) 1 √ 1 −( f 2 − 1)∂ f× + f× f+∂ f+ where √ ∂z( −g) is given by the expression in Eq. (3.3). ξ yx = ξ xy = + t t , −g 2 2 2 2 c f×( f× + f+ − 1) We can see that even if in the absence of any GW the elec- − f×( f+ + 1)∂ f× + ( f 2 − f+ − 1)∂ f+ tric field was static and uniform, during the passage of the ξ yy = 1 t × t , 2 2 2 2 spacetime disturbance, the field will be time varying and non- c ( f+ + 1) ( f× + f+ − 1) uniform, oscillating with the same frequency of the passing 1 f× (∂ f×) + f+ (∂ f+) ξ zz =− t t , GW. In fact, the general solution is 2 2 2 c f× + f+ − 1 E E ( − )∂ − ( 2 + − )∂ ( , , , ) = √ 0 = 0 , zxx f× f+ 1 z f× f× f+ 1 z f+ Ez x y z t (3.10) σ =− , −g 2 2 2 2 2 1 − f+ − f× ( f+ − 1) ( f× + f+ − 1) − f×( f+ + 1)∂ f× + ( f 2 − f+ − 1)∂ f+ = ( , ; ) σ zyy =− z × z , where in the most general case, E0 E0 x y t . To get 2 2 2 ( f+ + 1) ( f× + f+ − 1) the full description of the electric field one has to consider 2 also both the Maxwell–Ampère relations in Eqs. (3.4)–(3.8) −( f+ − 1)∂z f× + f× f+∂z f+ σ zxy = σ zyx =− . and the Faraday law. Nevertheless it is already clear from Eq. f 2( f 2 + f 2 − 1) × × + (3.10) that GWs induce propagating electric oscillations. A natural consequence of these laws is the generation We will consider the most simple case in which E0 is a of electromagnetic waves induced by gravitational radia- constant (without any dependence on x, y or t). Indeed, one tion. Initially static electric and magnetic fields become can easily verify that the fields E = (0, 0, Ez), B = (0, 0, 0) time dependent during the passage of GWs, which might constitute a (trivial) solution of the full Maxwell equations, be detectable in this way. namely Eqs. (3.5)–(3.8) together with the homogeneous In general, the system of coupled Eqs. (2.7), (2.8) and the equations in (2.3). Notice that for zero magnetic field the homogeneous equations in (2.3) have to be taken as a whole. z Maxwell–Ampère equation (3.8)is As we will see from Eq. (3.5), in some specific situations the 1 − ∂ E + E ξ zz = 0, (3.11) electric field can be solved directly from Gauss’ law. This c2 t z z electric field can in turn act as a source for magnetism via the Maxwell–Ampère relations in Eqs. (3.6)–(3.8), where the which is verified by the solution in (3.10) for a constant E0. presence of the GW induces extra terms proportional to the This can easily be seen when one considers that electric field. In this work, we will explore relatively simple 1 1 √ ξ zz =− √ ∂ ( −g), (3.12) situations in order to illustrate the effects of GWs in electric c2 −g t 123 237 Page 6 of 14 Eur. Phys. J. C (2017) 77:237 in accordance with the expressions previously shown for ξ zz In order to have an approximate idea on the energy density and Eq. (3.4). In this case, the coupling between the elec- uem of the resulting electromagnetic wave we can use the tric field and the GW in the expression for the generalized usual expression (derived from Maxwell electrodynamics in Maxwell displacement current density, compensates the tra- Minkowski spacetime). We get ditional term which depends on the time derivative of the em ∼ ε 2( , ) = ε ˜ 2 − 2 2 ( − w ) electric field. In fact, by multiplying by c2, then Eq. (3.11) u 0 Ez z t 0 E0 1 a cos kz t − can be interpreted as the conservation of the total electric − b2 cos2 (kz − wt + α) 1, (3.18) flux density. This situation is thus compatible with the exper- imental scenario where there are no currents producing any and the energy per unit area and unit time through any sur- magnetic field and the electric field, although changing in face (with a normal making an angle ϑ with the z axis) is time, due to the coupling with gravity does not give rise to approximately expressed by any magnetic field, since the total electric flux is conserved. ϑ = ε ˜ 2 − 2 2 ( − w ) Naturally, this is not the general case. For example in the pres- S cos 0cE0 1 a cos kz t

x , y = −1 ence of currents along the z axis, B B 0 and due to the − b2 cos2 (kz − wt + α) cos ϑ, (3.19) gravitational factors in Eqs. (3.6)–(3.8) the magnetic field is dynamical (time dependent). Therefore, this field necessarily where S is the Poynting vector, and S ≡ uemc. ˜ affects the electric field via the Faraday law, If E0 is the electric field in the absence of GWs, then the relevant dimensionless quantity to be measured is given by ∂y E0 ∂x E0 ∂ E = √ =−∂ Bx ,∂E = √ = ∂ B y, (3.13) the following expression: y z −g t x z −g t

E (z, t) − E˜ which implies that in general E0 = E0(x, y, t). Since E0 is z 0 = − 2 2 ( − w ) ˜ 1 a cos kz t time dependent, in such a case the electric field contributes E0 − / to the magnetic field via the (non-null) generalized Maxwell − b2 cos2 (kz − wt + α) 1 2 − 1 , displacement current, in accordance with Eq. (3.8), where (3.20) now 1 and in terms of the energy density, we get − ∂ E + E ξ zz = 0. (3.14) c2 t z z em em u (z, t) − u As a practical application consider the following harmonic 0 = 1 − a2 cos2 (kz − wt) uem GW perturbation: 0 − − b2 cos2 (kz − wt + α) 1 − 1 , f+(z, t) = a cos (kz − wt) , (3.15) (3.21) f×(z, t) = b cos (kz − wt + α) . (3.16) with uem = ε E˜ 2. In this case, we get the following electric oscillations: 0 0 0 Substituting in these two expressions the typical ampli- ˜ 2 2 −25− −21 Ez(z, t) = E0 1 − a cos (kz − wt) tudes for GWs due to binaries (10 10 ), the induced electric field and corresponding energy density oscillations − 2 2 ( − w + α) −1/2, b cos kz t (3.17) signal will be extremely small. Concerning GWs reaching for a2 +b2 ≤ 1, which is obeyed by the extremely low ampli- the solar system, the detectors which might have a response ˜ proportional to the electric field magnitude or rather to its tude GWs reaching the solar system. Here E0 is an arbitrary fixed constant and α is the phase difference. These electric energy (proportional to the square of the electric field mag- oscillations will show distinct features sensitive to the (+) or nitude), must be extremely sensitive. We emphasize the fact (×) GW modes. It provides a window for detecting and ana- that, in principle, this electromagnetic wave can be confined lyzing GW signals directly converted into electromagnetic in a cavity using very efficient reflectors for the frequency w information. . Then, under appropriate (resonant) geometric conditions, Notice that the electric waves produced are longitudinal, the signal can be amplified. This might have very important since these are propagating along the same direction of the practical applications for future GW detectors. GW, even though the electric field is aligned with this direction. To grasp the physical interpretation behind this non- 3.2.2 Electric field in the xy plane intuitive result, recall that the electric energy density depends quadratically on the field and therefore it is the energy den- Suppose we have an electric field in the x direction. The sity fluctuation induced by the GW which propagates along electric field could initially be uniform and confined within z the direction of k = k ez. a plane capacitor. Under these conditions, the Gauss law in 123 Eur. Phys. J. C (2017) 77:237 Page 7 of 14 237 vacuum becomes another similar capacitor oriented along the y axis. Under this ∂ ∂ condition, the resulting electric field in the vacuum between −1 Ex −1 Ex [1 − f+(z, t)] − ( f×) (z, t) = 0. (3.22) = + = ( , ) ∂x ∂y the charged plates, E E1 E2 Ex Ey , obeys the equation A similar expression is obtained if the electric field is −1 −1 −1 aligned with the y axis. Assuming a separation of variables (1 − f+) ∂x Ex − ( f×) ∂y Ex + (1 + f+) ∂y Ey Ex (x, y, z, t) = F1(x, z, t)F2(y, z, t), where z and t are seen −1 − ( f×) ∂x Ey = 0. (3.29) as external parameters, substituting in the above equation and dividing it by Ex , we obtain A possible solution to this equation is given by ∂ ∂ −1 x F1 −1 y F2 ˜ −[(1− f+)x+ f× y] (1 − f+) = f× , (3.23) Ex (x, y, z, t) = E0x 1 + C1(z, t)e , F1 F2 (3.30) therefore, one arrives at the following expressions: − +( + ) E (x, y, z, t) = E 1 + C˜ (z, t)e [ f×x 1 f+ y] , −(1− f+)x − f× y y 0y 2 F1(x; z) = C1(z, t)e , F2(x; z) = C2(z, t)e . (3.31) (3.24) = = ˜ ( , ) = ˜ ( , ) = Since we can always add a constant to the solution obtained where for f+ f× 0 we get C1 z t C2 z t 0. The resulting electric oscillations propagate along the z from F1(x, z, t)F2(y, z, t), we can write axis as an electromagnetic wave with non-linear polariza- ˜ −[(1− f+)x+ f× y] Ex (x, y, z, t) = E0x 1 + C(z, t)e , tion. This wave results from a linear gravitational perturbation of Minkowski spacetime and therefore (in this first order (3.25) approximation) it can be thought of as an electromagnetic ˜ ˜ where in general C(z, t) = C f+(z, t), f×(z, t) can be disturbance propagating in Minkowski background with a obtained by taking into account the other Maxwell equations. dynamical polarization. In fact, the angle between the result- The full solution should be compatible with the limit without ing electric field and the x axis is then arctan Ey/Ex gravity in which we recover the uniform field Ex = E0x . i.e., for E0x = E0y Therefore ⎧ ⎫ ⎨ + ˜ ( , ) −[ f×x+(1+ f+)y] ⎬ ˜ 1 C2 z t e f+ = f× = 0 ⇒ C(z, t) = 0. (3.26) ( , , , ) . x y z t arctan ⎩ ⎭ 1 + C˜ (z, t)e−[(1− f+)x+ f× y] A natural Anszatz is 1

˜ α1 α2 α3 α4 ˜ ˜ C(z, t) = η f+ + β f× + μf+ f× , (3.27) Even if we had C1 = C2, we still necessarily get a non- linear, dynamical polarization. Such an oscillating polariza- where η, β, μ and α (i = 1, 2, 3, 4) are constants. But, as i tion could in principle be another distinctive signature of previously said, the form of this function can be studied by the GW that is causing it. The solutions obtained already considering compatibility with the remaining Maxwell equa- give sufficient information to conclude that it is possible tions. to obtain polarization fluctuations induced by GWs, where For the harmonic GW introduced before, the second term for E x = E y the strength of the effect is given by in the solution above, Eq. (3.25) is given by the following 0 0 |π/2 − (x, y, z, t)|/(π/2). A dynamical spatial polariza- expression: tion pattern is therefore expected in our detector. This con- E C˜ (z, t) exp − (1 − a cos (kz − wt)) x trasts with the other cases where the resulting wave was lin- 0x + b cos (kz − wt + α) y . (3.28) early polarized. This effect is shown in Figs. 1 and 2. Nevertheless, again, the Faraday law and the Maxwell– The solution obtained is also sensitive to the existence or Ampère relations can provide constraints on the functions ˜ ˜ not of two modes in the GW, to their amplitudes and phase C1 and C2. difference. Although this solution obeys the Gauss law, it implies a non-zero dynamical magnetic field, according to 3.2.3 Electric field in the background of a GW with zero Faraday’s law. As mentioned, to get a full treatment one (×) mode should then check the consistency with the other Maxwell equations, in order to derive restrictions on the mathematical If we consider solely the (+) GW mode, the spacetime metric ˜ ( , ) form of C z t . (3.2) becomes diagonal and the Gauss and Maxwell–Ampère = Let us consider now the case where an electric field E1 equations simplify to the following expressions in vacuum: (Ex , 0) is generated by a plane capacitor oriented along the x = ( , ) ∂ ˜ k = , ∂ ¯ iijjk = μ j i , axis and a second electric field E2 0 Ey is generated by k E 0 ijk j B 0 D (3.32) 123 237 Page 8 of 14 Eur. Phys. J. C (2017) 77:237

in accordance with Eqs. (2.17)–(2.21). Let us search for a 3.0 trivial electric ﬁeld solution which is fully compatible with the complete system of Maxwell equations. If we consider

2.5 the ﬁeld ˜ = ( ˜ x ( , , ), ˜ y( , , ), ˜ z( , , )), E E0 y z t E0 x z t E0 x y t (3.35) 2.0 the Gauss law is trivially obeyed and the electric ﬁeld is given by y 1.5 ⎛ ⎞ ˜ z − + + + E = ⎝ 1 f ˜ x , 1 f ˜ y, 0 ⎠ . E E0 E0 (3.36) 2 2 2 1.0 1 − f+ 1 − f+ 1 − f+

∂ ˜ k = ( = , , ) 0.5 Furthermore, if t E0 0 k 1 2 3 , the general- j i ized Maxwell displacement current density D is zero, therefore effectively the electric field does not contribute to the 0.0 Maxwell–Ampère equations. Consequently, in the absence of electric currents, such an electric field solution seems to 0.0 0.5 1.0 1.5 2.0 2.5 3.0 be compatible with the condition B = 0. Let us assume x that this is the case. Regarding the remaining Maxwell equa- i Fig. 1 This vector plot illustrates the spatial (non-linear) polarization tions, the magnetic Gauss law ∂i B = 0 is trivially obeyed pattern on the (x, y) plane for the electric field at a given instant. This but what about Faraday’s law? In this case, one can show that pattern is exclusively induced by the GW. The GW parameters are a = ˜ the condition ∂t B =−curl E = 0, leads to a field E which 0.036, b = 0.766, w/2π = 89.81 Hz, α = 0.11π. We have used ˜ necessarily depends on time which contradicts the hypothe- Eqs. (3.30)and(3.31), where for simplicity we assumed C1(z, t) = ˜ −3 C2(z, t) = 1 and electric field magnitudes E0x = E0y = 10 V/m sis of zero magnetic field according to the Maxwell–Ampère relations in (3.32) and Eq. (3.34). In fact, one arrives at the

Arctan(Ey /Ex ) field, E˜ = (E˜ x (z, t), E˜ y(z, t), E˜ z), (3.37) 1.0004 0 0 0 ˜ z ˜ x ( , ), ˜ y( , ) where E0 is a constant and E0 z t E0 z t are given by 1.0002 √ − f+ −g ˜ x = ˜ x − ∂ 1√ , E0 C0 exp z (3.38) 1.0000 −g 1 − f+ √ + − 0.9998 ˜ y ˜ y 1 f+ g E = C exp − ∂z √ , (3.39) 0 0 −g 1 + f+ 0.9996 t where C˜ x and C˜ y are constants of integration. These func- 0.02 0.04 0.06 0.08 0.10 0 0 tions clearly depend on time and therefore the generalized Fig. 2 The dynamical polarization of the electric fluctuations gener- Maxwell displacement current cannot be zero leading to a ated by the GW is represented here for a fixed position in the (x, y) plane. non-vanishing magnetic field. When considering a generic On the vertical axis we have arctan(Ey/Ex ) normalized to the respec- tive value in the absence of GW and in the horizontal axis we have time electric field with three components as in (3.36), one cannot −5 −5 ∂ ˜ k = in seconds. The GW parameters are a = 3.6 × 10 , b = 2.6 × 10 , assume that t E0 0 neither a zero magnetic field. w/2π = 31.48 Hz, α = 0.26π.WehaveusedEqs.(3.30)and(3.31), Therefore in the general case one needs to consider the ˜ ( , ) = ˜ ( , ) = where for simplicity we assumed C1 z t C2 z t 1 and electric influence of the electric field on the magnetic field, through field magnitudes E = E = 10−3V/m 0x 0y the generalized Maxwell displacement current. An excep- tion to this is the special case first considered in Sect. 3.2.1, respectively, where where the electric field is aligned with the direction of the propagation of the GW. √ √ ˜ j jj 00 ¯ iijjk ii jj k E ≡−g g −gEj , B ≡ g g −gB , (3.33) 3.3 Magnetic field oscillations induced by GWs and the generalized Maxwell displacement current density is The passage of the GW can induce a non-vanishing time j i = ε ∂ ˜ i , D 0 t E (3.34) varying magnetic field, even for an initially static electric 123 Eur. Phys. J. C (2017) 77:237 Page 9 of 14 237

field. In general the full system of the Maxwell equations Making the Ansatz can be explored numerically to compute the resulting electric ∂ ˜ z =−∂ ˜ z, and magnetic oscillations. These magnetic fluctuations could x E0 y E0 (3.46) be measured in principle using SQUIDS (super conducting the Faraday equations then imply that ∂ Bx = ∂ B y,from quantum interference devices) that are extremely sensitive to t t which one derives an equation for I˜(x, y, z, t) with the gen- small magnetic field changes. eral solution To get a glimpse of the gravitationally induced magnetic ⎧ field fluctuations, we can consider for simplicity only the (+) ⎨ 2 1 − f+ GW mode and take the generalized Maxwell–Ampère law I˜ = C − exp ⎩ ( − )( − ) in the form of Eq. (2.17). We will be considering an electric x y 1 f+ ⎡ ⎛ ⎞ ⎛ ⎞⎤ ⎫ field aligned with the z axis given by the following solution − + ⎬ to the Gauss law: × ⎣ ∂ ⎝ 1 f+ ⎠ + ∂ ⎝ 1 f+ ⎠⎦ , ⎛ ⎞ x t y t dt⎭ 1 − f 2 1 − f 2 E˜ z(x, y, t) + + = ⎝ , , 0 ⎠ ,∂˜ k = . E 0 0 k E 0 (3.40) (3.47) 2 1 − f+ where C is an integration constant. In order to illustrate the We can also consider an electric current I along the z axis general effect of the GW on the magnetic field, consider such that, in principle, by symmetry we expect a magnetic without great loss of generality that I˜ = I is a constant. = ( x , y, ) field in the xy plane, B B B 0 . Then the Maxwell– Then the magnetic field in the background of the harmonic Ampère equations (2.17)are GW considered in (3.16) has the following fluctuations: √ ¯ ˜ ∇×B = μ0( −gj + ε0∂t E0), (3.41) μ Iy Bx =− 0 [1 + a cos (kz − wt)] , (3.48) π( 2 + 2) where B¯ ≡ (B¯ zzyyx, B¯ zzxxy, 0), while the Faraday law yields 2 x y μ Ix the equations B y = 0 [1 − a cos (kz − wt)] , (3.49) 2π(x2 + y2) ∂ ˜ z ∂ ˜ z x y E0 y x E0 ∂t B =− ,∂t B = . (3.42) respectively. 2 2 1 − f+ 1 − f+ We can easily see that, for any point (x, y) fixed on any magnetic field line, the x and y components of the magnetic Then we can perform an integration over an “amperian” field will oscillate in time out of phase, such that when one is closed line coincident to a magnetic field line (in perfect at its maximum value, the other is at the minimum, and vice analogy with the method taken in usual electromagnetism) versa. The overall result is that the magnetic field lines will to integrate the Maxwell–Ampère law, assuming axial sym- oscillate with the passage of the GW, following the deforma- metry, around the charge current distribution and electric flux tions of the spacetime geometry, perfectly mimicking the (+) (Maxwell displacement) current. mode deformations. Figure 3 illustrates this phenomenon and We obtain the following solution to Eq. (3.41): was obtained using the expressions in Eqs. (3.48) and (3.49). μ ˜ The strength of the effect as a function of time is independent ¯ 0 Itot B = cos φ ey − sin φ ex , (3.43) from the current I and it depends on the position (x, y) as π x2 + y2 2 well as on the GW parameters (see Fig. 4). It can easily be √ ˜ shown that the strength of the fluctuations are much stronger where Itot(x, y, z, t) = −gI + ID(x, y, t) and ID = j z j z = in specific regions of the (x, y) plane. Ddxdy. I is the (constant) electric current and D ε ∂ ˜ z 0 t E0 is the Maxwell displacement current density. We then get the magnetic field components 3.4 Charge density fluctuations induced by GWs ˜ 1 + f+ μ0 Itot(x, y, z, t) In the previous analysis we considered the behavior of elec- Bx =− y 2 2 (3.44) 2 2π(x + y ) tric and magnetic fields in vacuum regions and did not take 1 − f+ into account the effect of the propagating GWs on charge and distributions. The effect of spacetime geometry can be under- stood from the charge conservation equation in curved space- ˜ 1 − f+ μ I (x, y, z, t) ∇ μ = B y = 0 tot x , time ( μ j 0). As a result of this equation even in the 2 2 (3.45) 2 2π(x + y ) 1 − f+ absence of (intrinsic) currents, a non-static spacetime will induce a time variability√ in the charge density according to respectively. ∂t ρ =−∂t (log( −g)ρ, so we can write 123 237 Page 10 of 14 Eur. Phys. J. C (2017) 77:237

3 3

2 2

1 1

y y B 0 B 0

–1 –1

–2 –2

–3 –3

–3 –2 –1 0 1 2 3 –3 –2 –1 0 1 2 3 B x B x

Fig. 3 These vector plots illustrate the changes of the magnetic ﬁeld lines on the (x, y) plane which follow the GW (+) mode. The two patterns are separated in time by τ/2, where τ is the period. The GW parameters are a = 0.312, w/2π = 26.80 Hz. We have used Eqs. (3.48)and(3.49), with I = 4.6A

llδB/B As an example, for the harmonic GW modes considered pre- 2.×10–12 viously, we obtain

2 2 1.5×10–12 ρ(z, t) = ρ0 1 − b cos (kz − wt + α) − 1 − a2 cos2(kz − wt) 2 . (3.52) 1.×10–12 Consequently, one naturally predicts charge density fluc- 5.×10–13 tuations and, therefore, currents due to the passage of GWs. Such density oscillations propagate along the z direction fol- t lowing the GW penetrating a conducting material medium. 0.02 0.04 0.06 0.08 0.10 This is analogous to Alfvén waves in plasmas [29], which Fig. 4 Here we see the strength of the magnetic fluctuation induced are density waves induced by magnetic disturbances which by the GW as a function of time (in seconds). In the vertical axis propagate along the magnetic field lines. In this case, astro- | − |/ , we have the non-dimensional quantity BGW B0 B0 where physical sources of GWs such as gamma ray bursts or generic B = (Bx , B y) is the magnetic field in the presence of the passing GW coalescing binaries that happen to be surrounded by plasmas GW,obtained from Eqs. (3.48)and(3.49), and B0 is the control magnetic field in the case without GW. The GW parameters are a = 2.0 × 10−6, in accretion disks or in stellar atmospheres, might gener- w/2π = 41.38 Hz ate similar mass density waves and charge density waves induced by the GW propagation. A more realistic treatment % would require the equations of Magneto-Hydrodynamics in g ρ(t) = ρ 0 , the background of a GW (see [13]). An interesting study 0 ( ) (3.50) g t would be to consider the backreaction of the relativistic plasma and electromagnetic fields on the GW properties such where ρ is the initial charge density before the passage of the 0 as the frequency, amplitude and polarizations, so that the wave and g is the determinant of the initially unperturbed 0 traveling wave after detection could, in principle, contain background metric. For the simpler case of GWs traveling information as regards the physical properties of the medium along the z direction, seen as disturbances of Minkowski through which it propagated. spacetime, we have the simple result The above expression can also indicate another window − 1 for GW detection. Conductors in perfect electrostatic equilib- 2 2 2 ρ(t) = ρ0 1 − f× − f+ . (3.51) rium or superconducting materials at very low temperatures 123 Eur. Phys. J. C (2017) 77:237 Page 11 of 14 237 might reveal very dim electric oscillations with well-defined ciently high frequency GWs. Simulations are required to see characteristics, induced by GWs. the feasibility or not of such methods.

3.5 GW effects on electromagnetic radiation 4 Discussion and final remarks The vacuum equations for the 4-potential in the presence of a background GW can be derived from Eq. (2.23). In terms GW astronomy is an emerging field of science with the of the electric and magnetic components of the 4-potential, potential to revolutionize astrophysics and cosmology. The we have construction of GW observatories can also effectively boost major technological developments. Given the extremely low μ f× (∂t f×) + f+ (∂t f+) ∂μ∂ φ + ∂ φ GW amplitudes reaching the solar system, incredibly huge 2 2 t f× + f+ − 1 laser interferometers have been built and others are under (∂ ) + (∂ ) − f× z f× f+ z f+ ∂ φ = development in order to reach the required sensitivities. In 2 2 z 0 (3.53) f× + f+ − 1 fact, the biggest of all, LISA is expected to be achieved in space possibly through an ESA–NASA collaboration. ESA’s and LISA-Pathfinder science mission officially started on March (∂ ) + (∂ ) 2016 and during the following six months it conducted many μ k f× t f× f+ t f+ k ∂μ∂ + ∂ A 2 2 t A experiments to pave the way for future space GW observa- f× + f+ − 1 tories, such as LISA. These huge observatories represent an f× (∂ f×) + f+ (∂ f+) − z z ∂ Ak = 0, (3.54) amazing technological effort. A natural question is the fol- 2 + 2 − z f× f+ 1 lowing: Can we amplify the GW signal? respectively. In the absence of GWs we recover the usual One fundamental prediction of the coupling between grav- wave equations. ity and electromagnetism is the generation of electromag- The resulting expressions simplify significantly if one netic waves due to gravitational radiation. Therefore, in prin- considers only one of the two possible GW modes. For exam- ciple under the appropriate resonant conditions, the electro- ple, for an electromagnetic wave traveling in the z direction magnetic signal thus produced can be amplified allowing us and the harmonic GW in Eq. (3.15) with no (×) mode, we to measure GWs, not through the motion of test masses but get the following wave equation for the electric potential: rather by transferring the GW signal directly into electromagnetic information. This fact might represent an impor- w 2 (w − ) (w − ) 2 2 a sin t kz cos t kz tant change in perspective for future ground and space GW ∂ φ − ∂ φ − ∂t φ tt zz a2 cos2(wt − kz) − 1 detectors. ka2 sin(wt − kz) cos(wt − kz) The fact that GWs can generate electromagnetic waves is − ∂zφ = 0, (3.55) of course not evident if one restricts the analysis to the propa- a2 cos2(wt − kz) − 1 gation of light rays (in the geometrical optics limit) in curved which can be studied by applying Fourier transformation spacetime. On the other hand, the full Einstein–Maxwell sys- methods. tem of equations have to take into account the curved space- In order to study in depth the physical (measurable) effects time within Maxwell’s equations and also the contribution of of the passage of the GW on electromagnetic wave dynam- the electromagnetic stress-energy tensor to the gravitational ics, one needs to solve these equations and then compute field. The first aspect of this coupling was considered in this the gauge invariant electric and magnetic fields. We see in work, and it is sufficient to show that GWs can be sources of the above wave equations, the presence of terms propor- electromagnetic waves. The full gravity–electromagnetism tional to the first derivatives which are completely absent coupling also shows the reverse phenomenon. in the electromagnetic wave equations in flat spacetime (in In this work, we obtained electric and magnetic field oscil- cartesian coordinates). These terms are always induced by lations fully induced by a GW traveling along the z axis. For gravitational fields, but in this case the gravitational field is simplicity we assumed harmonic GWs. We considered the dynamical which represents a much richer electromagnetic Gauss law for the cases of an electric field along the z axis, wave signal with the signature of the GW (see also [18]). along the x axis and in the (x, y) plane. In the first case, Such signals in the radio regime might possibly be detectable the solutions in Eqs. (3.10) and (3.17) allowed one to make through methods of long baseline interferometry, in order to an estimation of the energy flux of the resulting radiation. amplify it. Nevertheless, we can see from the expressions It is important to emphasize the fact that the electric fluc- above that the extra terms on the electromagnetic wave equa- tuation thus produced corresponds to a longitudinal wave. tions, induced by GWs are proportional to the frequency. This means that a non-zero longitudinal mode in electro- Such gravitational effects might become important for suffi- magnetic radiation can in general be induced by gravitational 123 237 Page 12 of 14 Eur. Phys. J. C (2017) 77:237 radiation. One should search for these GW signatures in the principle be measured with SQUIDS (superconducting quan- electromagnetic counterpart of GW sources. The solution we tum interference devices), which are sensitive to extremely obtained shows the dependence on the amplitudes of the two small magnetic field changes [30–33]. SQUIDS have amaz- GW modes, a and b, as well as on the frequency w and phase ing applications from biophysics (in particular to biomag- difference α. An important aspect of hypothetical electric- netism) and medical sciences but also to theoretical physics: GW detectors is the fact that in general although the signal studies of majorana fermions [34], dark matter [35], gravity is very weak for any GW reaching the solar system, under wave resonant bar detectors [36], cosmological fluctuations appropriate resonant conditions it can be amplified. In fact, [37,38]. this can be used to improve the signal to noise ratio since The calculations in this work point to electromagnetic a system analogous to optical resonators can act as a filter effects induced by GWs such that privileging the signal with a specific (resonant) frequency. The changing electric field in Eq. (3.10) inside a capacitor, δE δB − ∼ h, ∼ h, h ∼ 10 21, (4.1) for instance, would also generate alternate currents in any E B conductor placed between the capacitor’s charged plates. In particular, a diode placed in the appropriate orientation would where h is the amplitude (strain) of the GW reaching the solar allow a current signal in a single direction intermittently, system. SQUIDS have an incredible sensitivity [30–40] being following the rhythm of the GW fluctuations. In the (x, y) able to measure magnetic fields of the order of 10−15T or plane the electric field can be generated by two independent even 10−18T for measurements performed over a sufficient capacitors in perpendicular configuration. The approximate period of time (the SQUIDS used in the GP-B experiment had solutions obtained show electric field oscillations generated this sensitivity). Using these values for the SQUIDS sensi- by the GW which propagate along the z axis with non-linear tivity, in order to be able to measure the tiny GW effects on polarization. We can expect a spatial polarization pattern in magnetic fields we would require magnetic fields of the order our detector which changes with time. This contrasts with the of B ∼ 106T or in the best case B ∼ 103T . Presently, the other cases where the resulting wave was linearly polarized. highest magnetic fields produced in the laboratory have val- This effect is shown in Figs. 1 and 2. ues of B ∼ 45T (continuous) and B ∼ 100T –103T (pulsed). In all cases, the resulting electromagnetic signal has the therefore, although SQUIDS are extremely sensitive, there is signature of the GW that produces it, depending on a, b, w a real limitation to perform these measurements coming from and α. In any of these examples, time varying electric fields the huge magnetic fields required. Nevertheless, the science are generated, which can contribute to the magnetic field of SQUIDS and ultra-sensitive magnetometers is very active via the Maxwell–Ampère law. In particular, they appear in and evolving [39,40] and it is natural to expect improvements the generalized Maxwell displacement current vector density, in terms of sensitivities and noise reduction and modeling. Eq. (2.18), induced by the GW. This in turn can generate a For B ∼ 10T laboratory magnetic fields we would require time varying magnetic field even in the absence of electric extremely higher sensitivities (δB ∼ 10−20), which are not in currents. Accordingly, GWs also induce magnetic field oscil- the reach of present magnetometers. Besides these consider- lations. We made an estimation of such an effect considering ations, intrinsic and extrinsic noise should be extremely well the case of a diagonal metric by setting the (×) GW mode to modeled and if possible reduced by advanced cryogenics and zero. We assumed a certain electric current I along the z axis filtering processes. and the electric field in Eq. (3.40) along the same direction. We may consider the use of electromagnetic cavity res- The magnetic field thus generated lies on the (x, y) plane and onators to amplify the electromagnetic waves induced by the it is easy to see that, for any point (x, y) fixed on the magnetic GWs. For magnetometers with δB ∼ 10−18T sensitivities field lines, the x and y components of the magnetic field will and 10T reference magnetic fields, it means that the ampli- oscillate in time out of phase, such that when one is at its fication of the signal would have to be about 2 orders of maximum value, the other is at the minimum, and vice versa. magnitude. Even if this cannot be achieved by present day The overall result is that the magnetic field lines will oscillate electromagnetic resonators it might be in the near future. with the passage of the GW, following the deformations of An important advantage of these cavities is that in practice spacetime geometry. Figure 3 illustrate this phenomenon and they work as filters being able to amplify a signal centered was obtained using the expressions in Eqs. (3.48) and (3.49). around a specific frequency which corresponds to the funda- In Fig. 4 we see the strength of the effect as a function of mental frequency of the resonator. For cylindrical resonators time. The signal to be measured is independent from the cur- with size L, the wavelength of the fundamental frequency is rent I and it depends on the position (x, y) as well as on λ ∼ 2L, meaning that different resonators of different sizes the GW parameters. It can be easily shown that the strength would be sensitive to the different parts of the GW spectrum. of the fluctuations are much stronger in specific regions of By effectively filtering and amplifying the signal around a the (x, y) plane. Such small magnetic field changes could in certain frequency far from the noise peak, it is in principle 123 Eur. Phys. J. C (2017) 77:237 Page 13 of 14 237 possible to substantially increase the signal to noise ratio, coupling between light and gravity. More generally, all elec- which is essential for a good measurement/detection. tromagnetic waves can experience gravitational effects on Let us consider the case where we use electric fields the amplitudes, frequencies and polarizations. Besides, as instead of magnetic fields in our electromagnetic detectors, shown in [22], electric and magnetic wave dynamics can be for example the electric field inside a charged plane capacitor. coupled due to the non-stationary geometries, as is the case By measuring the Voltage signal instead of electric field, we of GWs. Important studies have been made regarding the have the advantage of being able with the present technology electromagnetic counterpart of GW sources (see for exam- to, on one hand, easily produce 103V or higher static fields ple [41,42]), but there is much to explore in the landscape of and on the other hand, to reach sensitivities of δV ∼ 10−15V . (multi-messenger) gravitational and electromagnetic astron- This means that the signal should be amplified 2 to 3 orders omy. of magnitude. The combination of electromagnetic cavity In general, one expects that GWs induce very rich elec- resonators and electronic amplifiers (for the voltage signal) tromagnetic wave dynamics. These effects might become could make this a real possibility for GW detectors. more significant for very high frequency GWs as one can Moving now from human made laboratories on earth or in see from Eq. (3.55). Moreover, the terms proportional to the space to natural astrophysical observatories, we call the atten- first derivatives of the 4-potential have space and time vary- tion to the fact that the highest magnetic field values (indi- ing coefficients. For the harmonic GWs considered in this rectly) measured so far are those of neutron stars with values work, these coefficients oscillate between positive and nega- around 106T − 1011T . Radio and X-ray astronomy is able tive numbers, a fact that might imply a very distinctive wave to indirectly measure these astrophysical magnetic fields by modulation pattern of the resulting electromagnetic wave. considering the properties of cyclotron radiation. A stochas- This hypothesis and its implications require further investi- tic GW background signal due to innumerable sources in the gation as it might provide very rich GW information codified galaxy and beyond is expected to leave a measurable imprint in the electromagnetic spectra of different astrophysical and on the magnetic field of normal pulsars and magnetars. In even cosmological sources. fact, this method could be used in a complementary way to that of PTA (pulsar timing arrays) to measure a stochastic Acknowledgements FC acknowledges financial support of the Fun- GW signal. The huge magnetic fields in the surroundings of dação para a Ciência (FCT) through the grant PD/BD/128017/2016 and Programa de Doutoramento FCT, PhD::SPACE Doctoral Network pulsars makes them natural laboratories to study the effects for Space Sciences (PD/00040/2012). FSNL acknowledges financial of GWs on electromagnetic fields. The use of arrays of pul- support of an Investigador FCT Research contract, with reference sars could be advantageous in order to distinguish the GW IF/00859/2012, funded by FCT/MCTES (Portugal). This article is based signal from intrinsic fluctuations of the magnetic field and upon work from COST Action CA15117, supported by COST (Euro- pean Cooperation in Science and Technology). to better deal with extrinsic noise. Pulsars are extremely pre- cise clocks and if they behave as very stable dynamos, then Open Access This article is distributed under the terms of the Creative it might be possible to generalize the methods and years of Commons Attribution 4.0 International License (http://creativecomm ons.org/licenses/by/4.0/), which permits unrestricted use, distribution, improvement in PTA by measuring the interaction of GWs and reproduction in any medium, provided you give appropriate credit with magnetic fiels. 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