A&A 396, 10451051 (2002)
DOI: 10.1051/00046361:20021427
Acceleration and cyclotron radiation induced by
gravitational waves
D. Papadopoulos
Department of Physics, Aristoteleion University of
Thessaloniki, 54006 Thessaloniki, Greece
Received 16 July 2002 / Accepted 5 September 2002
Abstract
The equations that determine the response of a
charged particle moving in a magnetic field to an incident
gravitational wave (GW) are derived in the linearized approximation
to general relativity. We briefly discuss several astrophysical
applications of the derived formulae, taking into account the
resonance between the wave and the particle's motion that occurs
at
,
whenever the GW is parallel to the
constant magnetic field. In the case where the GW is perpendicular
to the constant magnetic field, magnetic resonances appear at
and
.
Such a resonant
mechanism may be useful in building models of GWdriven cyclotron
emitters.
Key words: relativity  gravitational waves
Papadopoulos & Esposito (1981) discussed the perturbations of
the Larmor orbits in the presence of a gravitational wave (GW) and
estimated the resulting magnetic bremsstrahlung. They have shown
that it is possible to identify the presence of GWs in macroscopic
systems by detecting the shifts in the spectrum of the
electromagnetic radiation (cyclotron) given off by charged
particles as they interact with a GW.
Recently, the motion of a relativistic charged particle in a
constant magnetic field perturbed by a GW incident along the
direction of the magnetic field has been examined (van Holten et al. 1999 and references therein). In the same work, a
generalized energy conservation law to compute the variations of
the kinetic energy of the particle during the passage of the GW
has been derived and explicit computations of the orbit of the
charged particle due to the GW have been obtained.
In this paper, we discuss the interaction of GW with a charged
gyrating particle in the presence of a constant magnetic field
across the zaxis in the framework of linearized theory of
gravity.
When the GW propagates parallel to the magnetic field with a
frequency
,
the coupling of a gyrating particle with the GW becomes very strong at the resonance that occurs between the
GW and the Larmor orbits. The resonance is at twice the Larmor
frequency (
)
e.g.
(Macedo
& Nelson 1990).
In the case that the GW propagates perpendicular to the magnetic
field, the interaction again becomes extremely efficient at the
resonances,
and
.
In both cases we verify that close to gyro resonances, the
obtained spectrum of the produced cyclotron radiation becomes
comparable to the spectrum of the initially gyrating particle,
especially in the vicinity of a source producing the GW.
Our results suggest that a) the linear theory breaks down at the
resonances and the interaction of the GW with gyrating particles
becomes very efficient and, b) the linear theory supports the
discrepancy on the estimations for the cyclotron damping
radiation recently discussed by M. Servin (Servin et al.
2001) and Kleidis (Kleidis et al. 1996; Kleidis et al.
1995), since in Kleidis work, the problem is examined in the
nonlinear theory where magnetic resonaces occur and some of them
overlap.
The paper organized as follows. In Sect. 2, we derive the
equations of motion in the linearized theory. In Sect. 2 we
discuss the interaction of the GW and the magnetic field when the GW is parallel to the magnetic field. In Sect. 4 we discuss the
same problem assuming that the GW is perpendicular to the
magnetic field. The obtained results are discussed in Sect. 5
In the linearized approximation to general relativity the metric
tensor is decomposed in the fashion



(1) 
where the elements h_{ij} are small compared to unity. By
imposing the condition



(2) 
we reduce the vacuum field equations to homogeneous wave
equations for all components of h_{i}^{j}. The gravitational
field is then described by a symmetric, traceless, divergenceless
tensor with two independent space components. Thus, the square of
the line element is



(3) 
where Greek indices take values 1, 2, 3 and Latin 0, 1, 2, 3.
The components of the covariant fourvelocity, consistent with
the linearized theory, are



(4) 



(5) 
where u^{0},
are the components of the fourvelocity
and the same quantities with the subscript M distinguish the
specialrelativistic Minkowski values.
The equations of motion of a test particle moving in the presence
of an electromagnetic field in the spacetime defined by Eq. (1)
are given by



(6) 
where the righthand side is the inhomogeneous driving term
determined by the electromagnetic field in the spacetime defined
by the Eq. (1).
For the metric (1), the nonzero Christoffel symbols are







(7) 
From Eqs. (6), (4), (5) and (7), the equations of motions take the
form:



(8) 
and
where



(10) 
Finally, the equations of motion (8) and (9), in the Newtonian
and linearized limit, reduce to the equations:



(11) 
To make further progress with the derived equations of motion (11), we consider the gravitational wave which is characterized
by the wave vector



(12) 
and one of two possible states of
polarization given by



(13) 
where h_{0} is the amplitude of the gravitational wave and
is the angular frequency of the GW.
The vectors
and
have space
components only and satisfy the conditions



(14) 
Conditions (14) imply



(15) 
Under the above consideration, we proceed to the following two
cases: a) the GW is parallel to the magnetic field and b) the GW
is perpendicular to the magnetic field, which will be analyzed in
the following two sections.
We choose the electromagnetic field to be



(16) 
where
.
We choose the gravitational wave to propagate parallel to the
magnetic field e.g., in Eq. (13) we obtain
and thus
.
Equation (11)
with the aid of Eq. (16) yields:



(17) 



(18) 



(19) 
where
.
To solve the system of Eqs. (17)(19), we decompose the
components of the 3velocity as follows:



(20) 
where the subscript zero means zero
order in the sense that h_{0}=0, while the subscript one means
first order in the sense that .
The perturbed equations of motion are derived from Eqs. (17)(20). Thus, after some straightforward calculations (see Appendix
B), we find the solution (Macedo & Nelson 1990):



(21) 



(22) 



(23) 
where
and
.
It is evident that in Eqs. (21)(23) if h_{0}=0, we obtain the
components of the space velocity of the initial gyrating charged
particle. If ,
the Eqs. (21)(23) reveal that the
gyrating charged particle diverts from its initial plane orbit
and moves into a helical trajectory. Now the vector
does not move
in a circle, but on the surface of a cone with its axis along the
.
From Eqs. (21)(23) we conclude that the particle is
accelerated at the resonace
.
The existence of
the resonace at
in Eqs. (21)(23) is due to the GW. Because of the resonance, the gyrating particle gains kinetic
energy. Thus, if E_{0} and E_{1} are the kinetic energy of the
particle before and after the interaction with the GW,
respectively, the energy gained by the particle in a period, let's
say T, is given by the average of the ration
e.g.
Obviously, as
approaches ,
taking values
between
and ,
the factor
,
becomes negative,
making the term multiplied by h_{0} approach plus infinity with
positive values. This suggests extra emission of cyclotron
radiation which will change the spectra distribution of the
radiation of the initial gyrating charged particle and transfer
energy from the GW to the particle.
Integrating the Eqs. (21)(23) we obtain the parametric equations of
motion of the charged gyrating particle interacting with a GW in
the presence of a constant magnetic field across the zaxis.
These are



(25) 



(26) 
and



(27) 
where x_{(01)}, y_{(01)} and z_{(01)} are constants of
integration.
The intensity of the radiation per solid angle per unit interval
of frequency produced from a charge test particle moving in the
presence of a magnetic field which interacts with the GW maybe
obtained from the relation (Landau 1975):



(28) 
where
is the frequency of the outgoing radiation,
,
is a
vector that joins the charged particle with the observer,
is the velocity of the charge particle.
We carry out the integral of Eq. (28) neglecting terms of the
order
.
Thus, we find:



(29) 
where
,
,
J_{j} is the Bessel function of first kind and J_{l}^{'} its
first ordinary derivative in terms of its argument. For
simplicity we call
L = 
(30) 
and
T = 
(31) 
It is evident that, as
approaches
,
the
factor
making the term T tend to minus infinity. But, if
approaches
taking values between
and ,
the factor
becomes negative, making the term T approach plus infinity with
positive values. Also, from Eq. (31), we see that the divergence
of the term T is faster as we approach the source producing GW.
Nevertheless, in the linearized theory of gravitation, we have to
approach the resonant in such a way that the term T remains
always below the term L, otherwise the linear theory breaks
down.
We shall now consider the case where the GW propagates
perpendicular to the unperturbed magnetic field
Thus, in a reference frame where
has the zdirection (as for the previous case), but the
GW propagates along the xdirection, the two nonvanishing
components of the GW are given by Eqs. (13) setting
and



(32) 
Subsequently, from Eqs. (11), (20) and (32) we obtain the following
equations of motion:



(33) 



(34) 
and



(35) 
where,
and may be chosen equal zero,



(36) 



(37) 
and



(38) 
Obviously, if in Eqs. (33)(35) h_{0}=0, then we obtain the space
velocities of the initial gyrating charged particle.
If
,
then we have gyrating motion again, but magnetic
resonances appear at
,
and
.
The existence of the above resonances in Eqs. (33)(35) is due to
the GW. Because of those resonances, the gyrating particle gains
kinetic energy. Thus, as in Sect. 3, we verify that the
energy gained by the particle in a period, let say T, is given
averaging the ratio
e.g.



(39) 
Obviously, if
approaches
from the right or
from the left, the factor
becomes
positive, making the term multiplied by h_{0} approach plus
infinity with positive values, suggesting changes to the spectra
distribution of the radiation of the initial gyrating charged
particle.
Integrating Eqs. (33)(35) we derive the parametric equations of
motion which are:
x(t) = 
(40) 
y(t) = 
(41) 
z(t) = 
(42) 
where
and may be chosen equal to zero; the
expression for X_{h} and Y_{h} are given explicitly in the
Appendix A.
In the Eqs. (40) and (41) a drift term, with resonances at
,
is present. This drift term
can generate electric currents. Those currents are sources of
secondary electromagnetic waves. For further details see Macedo
& Nelson (1990).
Following the same procedure as in Sect. 3, Eq. (28) reduce to
As in Sect. 3, we call



(44) 
and



(45) 
Now the term T_{2} has two magnetic resonances. It is evident
that, as
approaches
or
and
,
the term T_{2} tends to plus
infinity. Also, from Eq. (45), we see that the divergence of the
term T_{2} becomes faster as we approach the source producing the
GW as we are dealing with ultra relativistic particles where the
ratio
takes higher values. Again, as in
in the Sect. 3, in the linearized theory of gravitation we
have to approach to the resonances carefully, in the sense that
the term T_{2} should not exceed the term L, otherwise the
theory breaks down.
In this article we pose the following problem. If h_{0}=0, we
reproduce a well known formula for the spectrum distribution of a
gyrating charged particle in both cases e.g. when the GW is
parallel and perpendicular to the magnetic field. In this case
the angular distribution of the gyro radiation is highly
anisotropic. The radiation is concentrated mainly in the plane of
the orbit.
If
,
we are dealing with the interaction of a GW with
a gyrating charged particle. We have distinguished two cases:
(a) The GW propagates parallel to the constant magnetic field:
Because of the GW, the gyrating charged particle is diverted from
its initial plane orbit and starts to move across a helical
trajectory. Also, because of the GW, the spectrum of the produced
cyclotron radiation is isolated by a factor proportional to h_{0},
namely T, in which a resonance at
appears. If
approaches
taking values in the interval
,
the term T approaches plus infinity,
indicating that the existence of the resonant interaction between
the charged particle and the GW can lead to a strong emission of
cyclotron radiation, even in the linear theory of gravity. The
suggested mechanism of cyclotron radiation could be useful to
astrophysicists, especially they make simultaneous observations
of the cyclotron radiation and the GW so obtained. Therefore,
knowing that astrophysicists are looking to detect GWs at
frequencies between
Hz (Cutler Thorne 2002),
magnetic resonance may occur whenever the magnetic fields are
between
(
Hz),
(
Hz) and in both cases the T term becomes positive
and comparable to the L term. But, if we want to use
electromagnetic radiation for indirect detection of GW, then
other frequencies are also important, e.g. lower frequencies of
Hz coming from binary neutron stars before
coalescence. The corresponding magnetic fields are
(
Hz),
(
Hz) or high frequencies
Hz, due to
supernovae explosions, normal modes of pulsating neutron stars or
starsize black holes. In the later case the corresponding
magnetic fields are
G.
Outside the interval I_{1} the particle seems to lose energy due
to destructive interference of the two oscillators e.g. the
sinusoidal gravitational wave and the gyrating particle. In this
case, the actual loss described by the negative term T is of
the order of h_{0}.
(b) The GW propagates in the xdirection e.g. perpendicular to the
constant magnetic field.
In this case, we verify that 1) the gyrating particle remains on
the initial plane of orbit described by Eqs. (33)(35) and for a
certain
,
let's say
,
at
and
the
corresponding term T_{2} diverges. 2) If
approaches
from the left or
from the right, whereas
,
the term T_{2}becomes positive, and approaches plus infinity, unless
.
For the frequencies mentioned
above, two magnetic resonaces may occur for the same values of
at the same range of the magnetic field. But if
approaches one or other resonance, taking values in
interval I_{1}, where
,
the term T_{2} becomes negative (unless
), indicating that the particle loses energy due to
the same reason mentioned in paragraph (a). Another interesting
feature is that the term T_{2} is proportional to the ratio
,
which means that high relativist
particles support the term T_{2} to become more significant.
Nevertheless, in both cases, the strength of the cyclotron
radiation described by the ratios
and
remain constant for large values of l(high
frequencies). This may seen in Fig. 1 where we plot the
mentioned above ratios for
,
obtaining the
amplitude of the GW to be
h_{0}=10^{21} and its frequency
.

Figure 1:
The strengths of cyclotron radiation in the cases where
H is parallel to the k(T/L1) and H is perpendicular to the
k 
Open with DEXTER 
However, we have to point out that, in both cases, the terms
multiplied by h_{0} should not exceed the term L, otherwise the
linearized theory of gravity breaks down.
In the nonlinear theory, the problem could be more interesting.
The problem somehow has been examined from a dynamical point of
view, considering the equations of motion from a Hamiltonian, (Varvoglis & Papadopoulos 1992; Kleidis et al. 1993; Kleidis et al. 1995 and Kleidis et al. 1996), and
integrating them numerically. In this case, the interaction of
the GW with a gyrating charged particle exhibits resonances and in
several cases chaotic behavior. We intend to discuss the problem
in nonlinear theory in a forthcoming paper.
Acknowledgements
The author would like to thank Loukas
Vlahos, Kostas Kokkotas L. Witten and Nik Stergioulas for their
comments, criticism and beneficial discussions.
In the Eqs. (33)(35) theX_{h} and Y_{h} are:
and
We start with Eqs. (17)(19)



(48) 



(49) 



(50) 
We write Eqs. (48), (49) as follows:



(51) 
In order to solve the Eq. (51), we decompose the components of the
3velocity as follows:



(52) 
where the subscript zero means zero
order in the sense that h_{0}=0, while the subscript one means
first order in the sense that .
We substitute Eqs. (52) into Eq. (51) and the perturbed equation now
reads:



(53) 



(54) 
This yields



(55) 
where
and
.
From Eq. (53), we have
= 
(56) 
Notice that on the right hand side, the factor
gives
Now from Eqs. (56) and (57) we have

= 

(58) 
We treat Eq. (58) as an ordinary first order differential equation
with the initial conditions: if t=0 then
.
Thus, we have
Upon consideration of the initial conditions we have



(60) 
Eventually, from the Eqs. (59) and (60) we
obtain

= 

(61) 
or



(62) 



(63) 
Furthermore, following the same method, from Eq. (19) we find



(64) 
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Copyright ESO 2002