Gravitational waves and electrodynamics: new perspectives
Eur. Phys. J. C
Gravitational waves and electrodynamics: new perspectives
Francisco Cabral 0
Francisco S. N. Lobo 0
0 Instituto de Astrofi?sica e Cie?ncias do Espac?o, Faculdade de Cie?ncias da Universidade de Lisboa , Edifi?cio C8, Campo Grande, 1749016 Lisbon , Portugal
Given the recent direct measurement of gravitational waves (GWs) by the LIGOVIRGO 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 as regards 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.

A century after general relativity (GR) we have celebrated
the first direct measurement of gravitational waves (GWs)
by the LIGO?VIRGO collaboration [1], and ESA?s
LISAPathfinder [2] science mission which officially started on
March, 2016. For excellent reviews on GWs see [3,4]. 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,
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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 [5] as well as those including ?quantum corrections?
from quantum field theory which can predict a GW signature
of the nonclassical physics happening at or near the black
hole?s horizon (see [6,7] for recent claims on the detection of
GW echoes in the latetime 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 two other possible 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 [8]
will most probably provide positive detections in the near
future.
However, it is crucial to keep investigating different routes
toward GW measurements (see [8?12]) 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 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
GWelectromagnetic 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 [13]). 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
electrodynamics, 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 [14]. In particular,
gravitomagnetism is associated to spacetime metrics with timespace
components. Similarly, as we will see, the (?) polarization
of GWs is related to spacespace offdiagonal 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 shows 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 (offdiagonal) 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 [15], 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) [16,17] so, if these
results are robustly confirmed then superconductivity and
superfluidity 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 [13,18]), 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 [19,20]).
It has been shown that gravitational waves have an
important effect on the polarization of light [12]. 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 Sect.
2, we briefly review the foundations of electrodynamics and
spacetime geometry and present the basic equations that will
be used throughout this work. In Sect. 3, we explore the
coupling between electromagnetic fields and gravitational
waves. In Sect. 4, we discuss our results and conclude.
2 Electrodynamics: general formalism
In this section, for selfconsistency and selfcompleteness,
we briefly present the general formalism that will be applied
throughout the analysis. We refer the reader to [21?27] for
detailed descriptions of the deep relation between
electromagnetic fields and spacetime geometry. We will be using a
(+ ? ??) signature.
Recall that in the premetric formalism of
electrodynamics, the charge conservation provides the inhomogeneous
field equations while the magnetic flux conservation is given
in the homogeneous equations [23?27]. The field equations
are then given by
dF = 0, dG = J.
Note that these are general, coordinate free, covariant
equations and there is no need for an affine or metric structure
of spacetime. J is the charge current density 3form; G is
the 2form representing the electromagnetic excitation, F is
the Faraday 2form, so that F = dA, where A is the
electromagnetic potential 1form; the operator d stands for exterior
derivative.
The constitutive relations (usually assumed to be linear,
local, homogeneous and isotropic) between G and F,
provide the metric structure via the Hodge star operator ,
which introduce the conformal part of the metric and maps
kforms to (n ?k)forms, with n 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
where we have used in the inhomogeneous equations the
general expression for the divergence of antisymmetric
tenso?rs? gin ?ps?eu.dToheRiheommaongnengeeooumseetrqyu,at?io?ns ?a?re =inde?pe1?ndge?n?t
from the metric (and connection) due to the torsionless
character of Riemann geometry.
We introduce the definitions
1 Ek
F0k = c ?t Ak ? ?k A0 ? c , (2.4)
Fjk = ? j Ak ? ?k A j ? ? i jk Bi ,
where i jk is the totally antisymmetric threedimensional
LeviCivita (pseudo) tensor, Bi is a vector density (a
natural surface integrand) and Ek is a covector (a natural line
integrand). Then the homogeneous equations are the usual
Faraday and magnetic Gauss laws
?t Bi +
i jk ? j Ek = 0, ? j B j = 0,
while the inhomogeneous equations can be separated into the
generalized Gauss? and Maxwell?Amp?re laws. These are,
respectively,
?k E j g0k g j0 ? g jk g00
+ E j ? ji ? Bk kmn? mni = ?0 j i ,
? j ? ?k g0k g j0 ? g jk g00
??g) g0k g j0 ? g jk g00
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
nonvanishing offdiagonal timespace components g0 j , which in
i jk gii g j j ? j Bk
??g)F ?? = ?0 j ? , ?[? F?? ] = 0,
linearized gravity correspond to the gravitomagnetic
potentials. These terms are typical of axially symmetric geometries
(see [22,28]).
For diagonal metrics, the inhomogeneous equations, the
Gauss and Maxwell?Amp?re laws, can be recast into the
following forms:
+ c12 g00gii ?t Ei + i jk ? jii Bk + Ei ? ii
??g) + ?k gkk g00
??g) + ? j (g j j gii ),
The Einstein summation convention is applied in Eq. (2.13)
only for j and k, while the index i is fixed by the right hand
side. Also, no contraction is assumed in Eq. (2.14) nor in the
expression for ? jii .
New electromagnetic effects induced by the spacetime
geometry include an inevitable spatial variability
(nonuniformity) of electric fields whenever we have nonvanishing
geometric functions ? k , electromagnetic oscillations
(therefore waves) induced by gravitational radiation and also
additional electric contributions to Maxwell?s displacement
current in the generalized Maxwell?Amp?re law. This last
example becomes clearer by rewriting Eq. (2.13) in the form
where j i ? ??g j i and
j Di ? ??0??g g00gii ?t Ei + c2 Ei ? ii ,
is the generalized Maxwell displacement current density and
B? ii j jk ? gii g j j ??g Bk .
Again no summation convention is assumed for the index
i in Eqs. (2.17) and (2.18). The functions ? ii 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. (2.12) can be written as
E? j ? ?g j j g00??g E j ,
These are physical, observable effects of spacetime
geometry in electromagnetic fields expressed in terms of the
extended Gauss and Maxwell?Amp?re 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 4potential which in vacuum are convenient to
study electromagnetic wave phenomena. From the definition
of the Faraday tensor and Eq. (2.3), we get
???? A? ? g?? R?? A? ? ?? ?? A?
where R?? is the Ricci tensor. Using the expression for
the (generalized) Laplacian in pseudoRiemann manifolds,
????? = ?1?g ?? ??gg????? , and assuming the
generalized Lorenz gauge (?? A? = 0) in vacuum, we arrive
at
??gg?? ?? A? ? g?? R?? A? = 0.
For a diagonal metric, we get
??gg?? ?? A? ? g?? R?? A? = 0,
with no contraction assumed in ?. In general, and contrary
to electromagnetism in Minkowski spacetime, the equations
for the components of the electromagnetic 4potential are
coupled even in the (generalized) Lorenz gauge. Notice also
that, for Ricciflat 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. (2.23) are significant.
This completes the main axiomatic (foundational)
formalism of electrodynamics in the background of curved
(pseudoRiemann) spacetime.
3 Gravitational waves and electromagnetic fields
As mentioned in the Introduction, GWs have been recently
detected by the LIGO team using laser interferometry [1].
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 toward 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 linearization of gravity is usually applied which allows
one 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 g?? = ??? +h?? ,
with h??  1, so that
where the perturbation corresponds to a wave traveling along
the z axis, i.e.,
ds2 = c2dt 2 ? dz2 ? [1 ? f+(z ? ct )]dx 2
?[1 + f+(z ? ct )]dy2 + 2 f?(z ? ct )dx dy,
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
socalled TT (TransverseTraceless) Lorenz gauge. For this
metric, we get
ds2 = c2dt 2 ? dx 2 ? dy2 ? dz2 + h?? dx ?dx ? ,
??g) =
??g) =
These quantities will be useful further on.
3.1 GW effects in electric and magnetic fields
Consider an electric field in the background of a GW traveling
in the z direction. The general expression for Gauss? law
(2.7), in vacuum, is now given by
[1 ? f+(z, t )]?1?x Ex + [1 + f+(z, t )]?1?y E y
which clearly shows that physical (possibly observable)
effects are induced by the propagation of GWs.
As for the Maxwell?Amp?re law, Eq. (2.8) provides the
following relations in vacuum:
and magnetic fields. Let us start by considering the effects of
GWs in electric fields.
f??1?t E y ? (1 ? f+)?1?t Ex + Ex ? x x + E y ? yx
? (1 ? f+)?1 (1 + f+)?1?y Bz ? ?z B y
= 0,
f??1?t Ex ? (1 + f+)?1?t E y + E y ? yy + Ex ? x y
? (1 + f+)?1 (1 ? f+)?1?x Bz ? ?z Bx
= 0,
? c12 ?t Ez + Ez ? zz ? f??1 ?y B y ? ?x Bx
+ (1 ? f+)?1?x B y ? (1 + f+)?1?y Bx
= 0,
with the nonvanishing geometric coefficients given by
= ?
= ?
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. (2.7), (2.8) and the
homogeneous equations in (2.3) have to be taken as a whole.
As we will see from Eq. (3.5), 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?Amp?re relations in Eqs. (3.6)?(3.8), 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
3.2 Electric field oscillations induced by GWs
We will consider electric fields in the following scenarios.
3.2.1 Electric field aligned with the z axis
An electric field along the z 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. 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
GW, in general the electric field is perturbed by both the (+)
and the (?) 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. (3.5), we have
?z Ez + Ez
??g) = 0,
where ?1?g ?z (??g) is given by the expression in Eq. (3.3).
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
nonuniform, oscillating with the same frequency of the passing
GW. In fact, the general solution is
Ez (x , y, z, t ) = ??g =
where in the most general case, E0 = E0(x , y; t ). To get
the full description of the electric field one has to consider
also both the Maxwell?Amp?re relations in Eqs. (3.4)?(3.8)
and the Faraday law. Nevertheless it is already clear from Eq.
(3.10) that GWs induce propagating electric oscillations.
We will consider the most simple case in which E0 is a
constant (without any dependence on x , y or t ). Indeed, one
can easily verify that the fields E = (0, 0, Ez ), B = (0, 0, 0)
constitute a (trivial) solution of the full Maxwell equations,
namely Eqs. (3.5)?(3.8) together with the homogeneous
equations in (2.3). Notice that for zero magnetic field the
z Maxwell?Amp?re equation (3.8) is
??g),
which is verified by the solution in (3.10) for a constant E0.
This can easily be seen when one considers that
in accordance with the expressions previously shown for ? zz
and Eq. (3.4). 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 c2, then Eq. (3.11)
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 z axis, Bx , B y = 0 and due to the
gravitational factors in Eqs. (3.6)?(3.8) the magnetic field is
dynamical (time dependent). Therefore, this field necessarily
affects the electric field via the Faraday law,
which implies that in general E0 = E0(x , y, t ). Since E0 is
time dependent, in such a case the electric field contributes
to the magnetic field via the (nonnull) generalized Maxwell
displacement current, in accordance with Eq. (3.8), where
now
As a practical application consider the following harmonic
GW perturbation:
f+(z, t ) = a cos (kz ? wt ) ,
In this case, we get the following electric oscillations:
Ez (z, t ) = E?0 1 ? a2 cos2 (kz ? wt )
? b2 cos2 (kz ? wt + ?) ?1/2,
for a2 +b2 ? 1, which is obeyed by the extremely low
amplitude GWs reaching the solar system. Here E?0 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
analyzing 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
nonintuitive 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 k = kz ez .
In order to have an approximate idea on the energy density
uem of the resulting electromagnetic wave we can use the
usual expression (derived from Maxwell electrodynamics in
Minkowski spacetime). We get
uem ? ?0 Ez2(z, t ) = ?0 E?02 1 ? a2 cos2 (kz ? wt )
? b2 cos2 (kz ? wt + ?) ?1,
and the energy per unit area and unit time through any
surface (with a normal making an angle ? with the z axis) is
approximately expressed by
S cos ? = ?0c E?02 1 ? a2 cos2 (kz ? wt )
? b2 cos2 (kz ? wt + ?) ?1 cos ?,
where S is the Poynting vector, and S ? uem c.
If E?0 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 ) ? E?0
1 ? a2 cos2 (kz ? wt )
and in terms of the energy density, we get
1 ? a2 cos2 (kz ? wt )
with ue0m = ?0 E?02.
Substituting in these two expressions the typical
amplitudes for GWs due to binaries (10?25?10?21), 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
w. Then, under appropriate (resonant) geometric conditions,
the signal can be amplified. This might have very important
practical applications for future GW detectors.
3.2.2 Electric field in the x y plane
Suppose we have an electric field in the x direction. The
electric field could initially be uniform and confined within
a plane capacitor. Under these conditions, the Gauss law in
[1 ? f+(z, t )]?1 ? Ex ? ( f?)?1(z, t )
? x
= 0.
A similar expression is obtained if the electric field is
aligned with the y axis. Assuming a separation of variables
Ex (x , y, z, t ) = F1(x , z, t )F2(y, z, t ), where z and t are seen
as external parameters, substituting in the above equation and
dividing it by Ex , we obtain
therefore, one arrives at the following expressions:
Since we can always add a constant to the solution obtained
from F1(x , z, t )F2(y, z, t ), we can write
where in general C? (z, t ) = C? f+(z, t ), f?(z, t ) 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 Ex = E0x .
Therefore
f+ = f? = 0 ? C? (z, t ) = 0.
A natural Anszatz is
C? (z, t ) = ? f+?1 + ? f??2 + ? f+?3 f ?4 ,
?
where ?, ?, ? and ?i (i = 1, 2, 3, 4) 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. (3.25) is given by the following
expression:
E0x C? (z, t ) exp
? (1 ? a cos (kz ? wt )) x
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 nonzero 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 C? (z, t ).
Let us consider now the case where an electric field E1 =
(Ex , 0) is generated by a plane capacitor oriented along the x
axis and a second electric field E2 = (0, E y ) is generated by
another similar capacitor oriented along the y axis. Under this
condition, the resulting electric field in the vacuum between
the charged plates, E = E1 + E2 = (Ex , E y ), obeys the
equation
(1 ? f+)?1?x Ex ? ( f?)?1?y Ex + (1 + f+)?1?y E y
A possible solution to this equation is given by
where for f+ = f? = 0 we get C? 1(z, t ) = C? 2(z, t ) = 0.
The resulting electric oscillations propagate along the z
axis as an electromagnetic wave with nonlinear
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 x axis is then arctan E y /Ex
i.e., for E0x = E0y
?
arctan ?
?
1 + C? 1(z, t )e?[(1? f+)x+ f? y] ?
Even if we had C? 1 = C? 2, we still necessarily get a
nonlinear, 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 E0x = E0y the strength of the effect is given by
?/2 ? (x , y, z, t )/(?/2). 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?
Amp?re relations can provide constraints on the functions
C? 1 and C? 2.
3.2.3 Electric field in the background of a GW with zero
(?) mode
If we consider solely the (+) GW mode, the spacetime metric
(3.2) becomes diagonal and the Gauss and Maxwell?Amp?re
equations simplify to the following expressions in vacuum:
?k E? k = 0,
Fig. 1 This vector plot illustrates the spatial (nonlinear) polarization
pattern on the (x , y) plane for the electric field at a given instant. This
pattern is exclusively induced by the GW. The GW parameters are a =
0.036, b = 0.766, w/2? = 89.81 Hz, ? = 0.11? . We have used
Eqs. (3.30) and (3.31), where for simplicity we assumed C? 1(z, t ) =
C?2(z, t ) = 1 and electric field magnitudes E0x = E0y = 10?3V /m
Arctan(Ey /Ex )
Fig. 2 The dynamical polarization of the electric fluctuations
generated by the GW is represented here for a fixed position in the (x , y) plane.
On the vertical axis we have arctan(Ey /Ex ) normalized to the
respective value in the absence of GW and in the horizontal axis we have time
in seconds. The GW parameters are a = 3.6 ? 10?5, b = 2.6 ? 10?5,
w/2? = 31.48 Hz, ? = 0.26? . We have used Eqs. (3.30) and (3.31),
where for simplicity we assumed C?1(z, t ) = C?2(z, t ) = 1 and electric
field magnitudes E0x = E0y = 10?3V /m
E? j ? ?g j j g00??g E j ,
B? ii j jk ? gii g j j ??g Bk , (3.33)
and the generalized Maxwell displacement current density is
in accordance with Eqs. (2.17)?(2.21). 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? = (E? 0x (y, z, t ), E? 0y (x , z, t ), E? 0z (x , y, t )),
?
E = ?
1 ? f
Furthermore, if ?t E? 0k = 0 (k = 1, 2, 3), the
generalized Maxwell displacement current density j Di is zero,
therefore effectively the electric field does not contribute to the
Maxwell?Amp?re equations. Consequently, in the absence
of electric currents, such an electric field solution seems to
be compatible with the condition B = 0. Let us assume
that this is the case. Regarding the remaining Maxwell
equations, the magnetic Gauss law ?i Bi = 0 is trivially obeyed
but what about Faraday?s law? In this case, one can show that
the condition ?t B = ?curl E = 0, leads to a field E? which
necessarily depends on time which contradicts the
hypothesis of zero magnetic field according to the Maxwell?Amp?re
relations in (3.32) and Eq. (3.34). In fact, one arrives at the
field,
E? = (E? 0x (z, t ), E? 0y (z, t ), E? 0z ),
where E? 0z is a constant and E? 0x (z, t ), E? 0y (z, t ) are given by
E? 0x = C? 0x exp ? ?z 1???fg+ 1???fg+ ,
E? 0y = C? 0y exp ?
?z
1?+?fg+
where C? 0x and C? 0y are constants of integration. These
functions clearly depend on time and therefore the generalized
Maxwell displacement current cannot be zero leading to a
nonvanishing magnetic field. When considering a generic
electric field with three components as in (3.36), one cannot
assume that ?t E? k
0 = 0 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 Sect. 3.2.1,
where the electric field is aligned with the direction of the
propagation of the GW.
3.3 Magnetic field oscillations induced by GWs (3.37) (3.39)
The passage of the GW can induce a nonvanishing 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?Amp?re law
in the form of Eq. (2.17). We will be considering an electric
field aligned with the z axis given by the following solution
to the Gauss law:
?
E = ? 0, 0,
We can also consider an electric current I along the z axis
such that, in principle, by symmetry we expect a magnetic
field in the x y plane, B = (Bx , B y , 0). Then the Maxwell?
Amp?re equations (2.17) are
? ? B? = ?0(??gj + ?0?t E? 0),
?t Bx
= ?
, ?t B y
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?Amp?re law, assuming axial
symmetry, around the charge current distribution and electric flux
(Maxwell displacement) current.
We obtain the following solution to Eq. (3.41):
where I?tot(x , y, z, t ) = ??g I + ID(x , y, t ) and ID =
j Dzdx dy. I is the (constant) electric current and j Dz =
?0?t E?0z is the Maxwell displacement current density. We then
get the magnetic field components
= ?
B y =
1 ? f+
the Faraday equations then imply that ?t Bx = ?t B y , from
which one derives an equation for I?(x , y, z, t ) with the
general solution
1 ? f+
Making the Ansatz
where C is an integration constant. In order to illustrate the
general effect of the GW on the magnetic field, consider
without great loss of generality that I? = I is a constant.
Then the magnetic field in the background of the harmonic
GW considered in (3.16) has the following fluctuations:
respectively.
We can easily see that, for any point (x , y) fixed on any
magnetic field line, the x and y components of the magnetic
field will oscillate in time out of phase, such that when one is
at its maximum value, the other is at the minimum, and vice
versa. The overall result is that the magnetic field lines will
oscillate with the passage of the GW, following the
deformations of the spacetime geometry, perfectly mimicking the (+)
mode deformations. Figure 3 illustrates this phenomenon and
was obtained using the expressions in Eqs. (3.48) and (3.49).
The strength of the effect as a function of time is independent
from the current I and it depends on the position (x , y) as
well as on the GW parameters (see Fig. 4). It can easily be
shown that the strength of the fluctuations are much stronger
in specific regions of the (x , y) plane.
3.4 Charge density fluctuations induced by GWs
In the previous analysis we considered the behavior of
electric and magnetic fields in vacuum regions and did not take
into account the effect of the propagating GWs on charge
distributions. The effect of spacetime geometry can be
understood from the charge conservation equation in curved
spacetime (?? j ? = 0). As a result of this equation even in the
absence of (intrinsic) currents, a nonstatic spacetime will
induce a time variability in the charge density according to
?t ? = ??t (log(??g)?, so we can write
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 ?/2, where ? is the period. The GW parameters are a = 0.312, w/2? = 26.80 Hz. We have used Eqs. (3.48) and (3.49),
with I = 4.6A
Fig. 4 Here we see the strength of the magnetic fluctuation induced
by the GW as a function of time (in seconds). In the vertical axis
we have the nondimensional quantity  BGW ? B0 / B0 , where
BGW = (Bx , B y) is the magnetic field in the presence of the passing
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,
w/2? = 41.38 Hz
where ?0 is the initial charge density before the passage of the
wave and g0 is the determinant of the initially unperturbed
background metric. For the simpler case of GWs traveling
along the z direction, seen as disturbances of Minkowski
spacetime, we have the simple result
1
?(t ) = ?0 1 ? f?2 ? f+2 ? 2 .
As an example, for the harmonic GW modes considered
previously, we obtain
Consequently, one naturally predicts charge density
fluctuations and, therefore, currents due to the passage of GWs.
Such density oscillations propagate along the z direction
following the GW penetrating a conducting material medium.
This is analogous to Alfv?n waves in plasmas [29], which
are density waves induced by magnetic disturbances which
propagate along the magnetic field lines. In this case,
astrophysical sources of GWs such as gamma ray bursts or generic
coalescing binaries that happen to be surrounded by plasmas
in accretion disks or in stellar atmospheres, might
generate similar mass density waves and charge density waves
induced by the GW propagation. A more realistic treatment
would require the equations of MagnetoHydrodynamics in
the background of a GW (see [13]). An interesting study
would be to consider the backreaction of the relativistic
plasma and electromagnetic fields on the GW properties such
as the frequency, amplitude and polarizations, so that the
traveling wave after detection could, in principle, contain
information as regards the physical properties of the medium
through which it propagated.
The above expression can also indicate another window
for GW detection. Conductors in perfect electrostatic
equilibrium or superconducting materials at very low temperatures
might reveal very dim electric oscillations with welldefined
characteristics, induced by GWs.
ciently high frequency GWs. Simulations are required to see
the feasibility or not of such methods.
3.5 GW effects on electromagnetic radiation
The vacuum equations for the 4potential in the presence of
a background GW can be derived from Eq. (2.23). In terms
of the electric and magnetic components of the 4potential,
we have
????? + f? (?t f?) + f+ (?t f+) ?t ?
???? Ak + f? (?t f?) + f+ (?t f+) ?t Ak
respectively. In the absence of GWs we recover the usual
wave equations.
The resulting expressions simplify significantly if one
considers only one of the two possible GW modes. For
example, for an electromagnetic wave traveling in the z direction
and the harmonic GW in Eq. (3.15) with no (?) mode, we
get the following wave equation for the electric potential:
wa2 sin(wt ? kz) cos(wt ? kz)
a2 cos2(wt ? kz) ? 1
ka2 sin(wt ? kz) cos(wt ? kz)
a2 cos2(wt ? kz) ? 1 ?z ? = 0,
which can be studied by applying Fourier transformation
methods.
In order to study in depth the physical (measurable) effects
of the passage of the GW on electromagnetic wave
dynamics, one needs to solve these equations and then compute
the gauge invariant electric and magnetic fields. We see in
the above wave equations, the presence of terms
proportional to the first derivatives which are completely absent
in the electromagnetic wave equations in flat spacetime (in
cartesian coordinates). These terms are always induced by
gravitational fields, but in this case the gravitational field is
dynamical which represents a much richer electromagnetic
wave signal with the signature of the GW (see also [18]).
Such signals in the radio regime might possibly be detectable
through methods of long baseline interferometry, in order to
amplify it. Nevertheless, we can see from the expressions
above that the extra terms on the electromagnetic wave
equations, induced by GWs are proportional to the frequency.
Such gravitational effects might become important for
suffi
4 Discussion and final remarks
GW astronomy is an emerging field of science with the
potential to revolutionize astrophysics and cosmology. The
construction of GW observatories can also effectively boost
major technological developments. Given the extremely low
GW amplitudes reaching the solar system, incredibly huge
laser interferometers have been built and others are under
development in order to reach the required sensitivities. In
fact, the biggest of all, LISA is expected to be achieved in
space possibly through an ESA?NASA collaboration. ESA?s
LISAPathfinder science mission officially started on March
2016 and during the following six months it conducted many
experiments to pave the way for future space GW
observatories, such as LISA. These huge observatories represent an
amazing technological effort. A natural question is the
following: Can we amplify the GW signal?
One fundamental prediction of the coupling between
gravity and electromagnetism is the generation of
electromagnetic waves due to gravitational radiation. Therefore, in
principle under the appropriate resonant conditions, the
electromagnetic signal thus produced can be amplified allowing us
to measure GWs, not through the motion of test masses but
rather by transferring the GW signal directly into
electromagnetic information. This fact might represent an
important change in perspective for future ground and space GW
detectors.
The fact that GWs can generate electromagnetic waves is
of course not evident if one restricts the analysis to the
propagation of light rays (in the geometrical optics limit) in curved
spacetime. On the other hand, the full Einstein?Maxwell
system of equations have to take into account the curved
spacetime within Maxwell?s equations and also the contribution of
the electromagnetic stressenergy tensor to the gravitational
field. The first aspect of this coupling was considered in this
work, and it is sufficient to show that GWs can be sources of
electromagnetic waves. The full gravity?electromagnetism
coupling also shows the reverse phenomenon.
In this work, we obtained electric and magnetic field
oscillations fully induced by a GW traveling along the z axis. For
simplicity we assumed harmonic GWs. We considered the
Gauss law for the cases of an electric field along the z axis,
along the x axis and in the (x , y) plane. In the first case,
the solutions in Eqs. (3.10) and (3.17) allowed one to make
an estimation of the energy flux of the resulting radiation.
It is important to emphasize the fact that the electric
fluctuation thus produced corresponds to a longitudinal wave.
This means that a nonzero longitudinal mode in
electromagnetic radiation can in general be induced by gravitational
radiation. One should search for these GW signatures in the
electromagnetic counterpart of GW sources. The solution we
obtained shows the dependence on the amplitudes of the two
GW modes, a and b, as well as on the frequency w and phase
difference ?. An important aspect of hypothetical
electricGW detectors is the fact that in general although the signal
is very weak for any GW reaching the solar system, under
appropriate resonant conditions it can be amplified. In fact,
this can be used to improve the signal to noise ratio since
a system analogous to optical resonators can act as a filter
privileging the signal with a specific (resonant) frequency.
The changing electric field in Eq. (3.10) inside a capacitor,
for instance, would also generate alternate currents in any
conductor placed between the capacitor?s charged plates. In
particular, a diode placed in the appropriate orientation would
allow a current signal in a single direction intermittently,
following the rhythm of the GW fluctuations. In the (x , y)
plane the electric field can be generated by two independent
capacitors in perpendicular configuration. The approximate
solutions obtained show electric field oscillations generated
by the GW which propagate along the z axis with nonlinear
polarization. We can expect a spatial polarization pattern in
our detector which changes with time. This contrasts with the
other cases where the resulting wave was linearly polarized.
This effect is shown in Figs. 1 and 2.
In all cases, the resulting electromagnetic signal has the
signature of the GW that produces it, depending on a, b, w
and ?. In any of these examples, time varying electric fields
are generated, which can contribute to the magnetic field
via the Maxwell?Amp?re law. In particular, they appear in
the generalized Maxwell displacement current vector density,
Eq. (2.18), induced by the GW. This in turn can generate a
time varying magnetic field even in the absence of electric
currents. Accordingly, GWs also induce magnetic field
oscillations. We made an estimation of such an effect considering
the case of a diagonal metric by setting the (?) GW mode to
zero. We assumed a certain electric current I along the z axis
and the electric field in Eq. (3.40) along the same direction.
The magnetic field thus generated lies on the (x , y) plane and
it is easy to see that, for any point (x , y) fixed on the magnetic
field lines, the x and y components of the magnetic field will
oscillate in time out of phase, such that when one is at its
maximum value, the other is at the minimum, and vice versa.
The overall result is that the magnetic field lines will oscillate
with the passage of the GW, following the deformations of
spacetime geometry. Figure 3 illustrate this phenomenon and
was obtained using the expressions in Eqs. (3.48) and (3.49).
In Fig. 4 we see the strength of the effect as a function of
time. The signal to be measured is independent from the
current I and it depends on the position (x , y) as well as on
the GW parameters. It can be easily shown that the strength
of the fluctuations are much stronger in specific regions of
the (x , y) plane. Such small magnetic field changes could in
principle be measured with SQUIDS (superconducting
quantum interference devices), which are sensitive to extremely
small magnetic field changes [30?33]. SQUIDS have
amazing applications from biophysics (in particular to
biomagnetism) and medical sciences but also to theoretical physics:
studies of majorana fermions [34], dark matter [35], gravity
wave resonant bar detectors [36], cosmological fluctuations
[37,38].
The calculations in this work point to electromagnetic
effects induced by GWs such that
? h, h ? 10?21,
where h is the amplitude (strain) of the GW reaching the solar
system. SQUIDS have an incredible sensitivity [30?40] being
able to measure magnetic fields of the order of 10?15T or
even 10?18T for measurements performed over a sufficient
period of time (the SQUIDS used in the GPB experiment had
this sensitivity). Using these values for the SQUIDS
sensitivity, in order to be able to measure the tiny GW effects on
magnetic fields we would require magnetic fields of the order
of B ? 106T or in the best case B ? 103T . Presently, the
highest magnetic fields produced in the laboratory have
values of B ? 45T (continuous) and B ? 100T ?103T (pulsed).
therefore, although SQUIDS are extremely sensitive, there is
a real limitation to perform these measurements coming from
the huge magnetic fields required. Nevertheless, the science
of SQUIDS and ultrasensitive magnetometers is very active
and evolving [39,40] and it is natural to expect improvements
in terms of sensitivities and noise reduction and modeling.
For B ? 10T laboratory magnetic fields we would require
extremely higher sensitivities (?B ? 10?20), which are not in
the reach of present magnetometers. Besides these
considerations, intrinsic and extrinsic noise should be extremely well
modeled and if possible reduced by advanced cryogenics and
filtering processes.
We may consider the use of electromagnetic cavity
resonators to amplify the electromagnetic waves induced by the
GWs. For magnetometers with ?B ? 10?18T sensitivities
and 10T reference magnetic fields, it means that the
amplification of the signal would have to be about 2 orders of
magnitude. Even if this cannot be achieved by present day
electromagnetic resonators it might be in the near future.
An important advantage of these cavities is that in practice
they work as filters being able to amplify a signal centered
around a specific frequency which corresponds to the
fundamental frequency of the resonator. For cylindrical resonators
with size L, the wavelength of the fundamental frequency is
? ? 2L, meaning that different resonators of different sizes
would be sensitive to the different parts of the GW spectrum.
By effectively filtering and amplifying the signal around a
certain frequency far from the noise peak, it is in principle
possible to substantially increase the signal to noise ratio,
which is essential for a good measurement/detection.
Let us consider the case where we use electric fields
instead of magnetic fields in our electromagnetic detectors,
for example the electric field inside a charged plane capacitor.
By measuring the Voltage signal instead of electric field, we
have the advantage of being able with the present technology
to, on one hand, easily produce 103 V or higher static fields
and on the other hand, to reach sensitivities of ?V ? 10?15 V .
This means that the signal should be amplified 2 to 3 orders
of magnitude. The combination of electromagnetic cavity
resonators and electronic amplifiers (for the voltage signal)
could make this a real possibility for GW detectors.
Moving now from human made laboratories on earth or in
space to natural astrophysical observatories, we call the
attention to the fact that the highest magnetic field values
(indirectly) measured so far are those of neutron stars with values
around 106T ? 1011T . Radio and Xray astronomy is able
to indirectly measure these astrophysical magnetic fields by
considering the properties of cyclotron radiation. A
stochastic GW background signal due to innumerable sources in the
galaxy and beyond is expected to leave a measurable imprint
on the magnetic field of normal pulsars and magnetars. In
fact, this method could be used in a complementary way to
that of PTA (pulsar timing arrays) to measure a stochastic
GW signal. The huge magnetic fields in the surroundings of
pulsars makes them natural laboratories to study the effects
of GWs on electromagnetic fields. The use of arrays of
pulsars could be advantageous in order to distinguish the GW
signal from intrinsic fluctuations of the magnetic field and
to better deal with extrinsic noise. Pulsars are extremely
precise clocks and if they behave as very stable dynamos, then
it might be possible to generalize the methods and years of
improvement in PTA by measuring the interaction of GWs
with magnetic fiels. Is this another window for GW
astronomy through VLBL (very long baseline interferometry)? We
leave this as an open question that deserves more research
from both theorists and observation experts.
We also obtained charge density oscillations induced by
GWs. These can propagate as density waves in the case of
charged fluids, through which a GW is propagating. This
effect deserves to be taken in consideration within more
complete magnetohydrodynamical computations, in order to have
simulations of the effects of GWs in plasmas near the cores
of highly energetic GW sources. These plasma environments
might occur in different astrophysical sources such as gamma
ray bursts and some specific coalescing binaries.
Regarding electromagnetic waves in the presence of
gravity, extra terms appear in the generalized wave equations
which deserves further research to get a full analysis of the
approximate solutions. Indeed, going beyond the
geometrical optics limit, light deflection (null geodesics) and
gravitational redshift are not the only effects arising from the
coupling between light and gravity. More generally, all
electromagnetic waves can experience gravitational effects on
the amplitudes, frequencies and polarizations. Besides, as
shown in [22], electric and magnetic wave dynamics can be
coupled due to the nonstationary geometries, as is the case
of GWs. Important studies have been made regarding the
electromagnetic counterpart of GW sources (see for
example [41, 42]), but there is much to explore in the landscape of
(multimessenger) gravitational and electromagnetic
astronomy.
In general, one expects that GWs induce very rich
electromagnetic wave dynamics. These effects might become
more significant for very high frequency GWs as one can
see from Eq. (3.55). Moreover, the terms proportional to the
first derivatives of the 4potential have space and time
varying coefficients. For the harmonic GWs considered in this
work, these coefficients oscillate between positive and
negative numbers, a fact that might imply a very distinctive wave
modulation pattern of the resulting electromagnetic wave.
This hypothesis and its implications require further
investigation as it might provide very rich GW information codified
in the electromagnetic spectra of different astrophysical and
even cosmological sources.
Acknowledgements FC acknowledges financial support of the
Funda??o para a Ci?ncia (FCT) through the grant PD/BD/128017/2016
and Programa de Doutoramento FCT, PhD::SPACE Doctoral Network
for Space Sciences (PD/00040/2012). FSNL acknowledges financial
support of an Investigador FCT Research contract, with reference
IF/00859/2012, funded by FCT/MCTES (Portugal). This article is based
upon work from COST Action CA15117, supported by COST
(European Cooperation in Science and Technology).
Open Access This article is distributed under the terms of the Creative
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ons.org/licenses/by/4.0/), which permits unrestricted use, distribution,
and reproduction in any medium, provided you give appropriate credit
to the original author(s) and the source, provide a link to the Creative
Commons license, and indicate if changes were made.
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