gravitation wave interaction with normal & superconducting circuits (1)

7/31/2019 Gravitation Wave Interaction With Normal & Superconducting Circuits (1)

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annals of physics 248, 3459 (1996)

Gravitational Wave Interaction with Normal and

Superconducting Circuits

Pierluigi Fortini* and Enrico Montanari

Department of Physics, University of Ferrara and INFN Sezione di Ferrara, I-44100 Ferrara, Italy

Antonello Ortolan

INFN National Laboratories of Legnaro, I-35020 Legnaro, Padua, Italy

and

Gerhard Scha fer

Max-Planck-Gesellschaft, AG Gravitationstheorie an der Friedrich-Schiller-Universitat,D-07743 Jena, Germany

Received November 12, 1994; revised July 21, 1995

The interaction, in the long-wavelength approximation, of normal and superconducting

electromagnetic circuits with gravitational waves is investigated. We show that such interac-

tion takes place by modifying the physical parameters R, L, C of the electromagnetic devices.

Exploiting this peculiarity of the gravitational field we find that a circuit with two plane and

statically charged condensers set at right angles can be of interest as a detector of periodic

gravitational waves. 1996 Academic Press, Inc.

1. Introduction

Future gravitational-wave astronomy requires gravitational-wave detectors offantastic high sensitivity. Two designs detectors are on the beam of realisation: bar

detectors and laser-beam detectors. Both are promising to become enough sensitivefor the detection and even for the detailed measurement of cosmic gravitationalwaves. Whereas the beam detectors will reach rather high sensitivity the bar detec-

tors will be quite cheap. The sensitivity of the latter should be sufficient to detectgravitational waves from coalescing binaries in the VIRGO cluster and from type-IIsupernovae in our galaxy. Unfortunately, both of them happen not very often

(about one per 30 years). However, there might exist other sources in our galaxy,like slightly deformed rotating neutron stars, in particular young asymmetric

article no. 0050

340003-491696 18.00Copyright 1996 by Academic Press Inc

* E-mail address: fortiniferrara.infn.it.


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pulsars, or even completely unexpected sources which make gravitational-wave sur-

veys in our galaxy, or other galaxies of the Local Group, very desirable. The centersof galaxies are very difficult to investigate with present-day telescopes so that our

knowledge of the centers is very crude. Gravitational-wave detection might revealinteresting objects there.

Purpose of this paper is to work out in details a general formalism for treatingthe interaction of a gravitational wave with an electromagnetic (ohmic and super-

conducting) device in the quasi-stationary approximation. Such a formalism is infact lacking and only partial cases are to be found in the literature (see ref. [1, 3,

11, 16]).As an application of this general theory we introduce and investigate a new

design of detectors, normal and superconducting electromagnetic circuits whichcouple to gravitational waves in non-parametric manner. These devices are circuit-

generalisations of a capacity-device approach invented by Mours and Yvert in 1989.As we shall see in the following our devices could reach sensitivities comparablewith the sensitivities of mechanical detectors for periodic radiation [20, 21].

The paper is organized as follows. In Section 2 we discuss, in quasi-stationaryapproximation, the interaction of electromagnetic fields and currents with weak

external gravitational fields. In particular we discuss under which conditions theconductors, i.e. the ions of the lattices of the conductors, can be treated as freelyfalling in external gravitational wave fields. Section 3 presents the general theory of

normal conducting circuits in the field of external gravitational waves. In this sec-tion the parametric nature of the interaction is shown. Section 4 refines the resultsof Section 3 to superconducting circuits. In Section 5 the influence of dielectric andmagnetic matter is investigated. Finally, in Section 6, we apply our theoreticalresults to concrete, thermal-noise-dominated detectors and derive sensitivity-limits

for the detection of gravitational waves.

2. Electromagnetic Fields and Conductors in Curved Spacetime

The electromagnetic field equations and the equations of motion for charged

matter (ions and electrons) in an external gravitational field g+& (+, &=0, 1, 2, 3)can be derived from the action

S=Sem+Sm+Sint , (2.1)

where Sem=(14?c) d4

x -&g F+& F+&

(g=det (g+&)) is the action for the free elec-tromagnetic field, Sm is the action for the conductors that we will discuss below,

and Sint=1

c2 d4 x-&g j+A+ , with the current density defined by

j+=c\

-&g00

dx+

dx0,

35GRAVITATION AND ELECTRIC CIRCUITS


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is the action for the coupling between the electromagnetic field and the charges and

currents in the conductor. The electromagnetic field-strength tensor F+& is relatedwith the four-vector potential A+ by F+&=A&, +&A+, & and we have the following

relation between F+& and the standard electromagnetic field-strength entities, Ei andBi (i=1, 2, 3),

F+&=\0

E1

E2

E3

&E1

0

&B3

B2

&E2

B3

0

&B1

&E3

&B2B1

0 + . (2.2)By varying the action (2.1) with respect to A+ we get the De Rahm equations.

In the Lorentz gauge, {&A&=0, they read

{&{&A

+&A\R\+= &

4?

cj+. (2.3)

Eq. (2.3) generalizes the inhomogeneous Maxwell equations in the Lorentz gaugeto curved spacetime; R+& is the Ricci tensor defined by

R+&=R:+:& ,

where R:+;& is the Riemann tensor

R:+;&=1:+&

x;&

1:+;

x&+1:';1

'+&&1

:'& 1

'+; , (2.4)

and where 1:+& denote the Christoffel symbols

1:+&=12g:;

\g;&x+

+g;+x&

&g+&x; +

, (2.5)

which enter in the definition of the covariant derivative {+ (notations and conven-tions as in ref. [4] pp. 2234).

Our aim is to study the interaction of a gravitational wave, emitted for instanceby a cosmic source, with electromagnetic circuits. In the interaction region the

metric tensor g+& satisfies the Einstein equations in vacuo, R+&=0. For weak

gravitational fields, to which we shall limit ourselves in this paper, the metric tensorand its determinant can be written as

g+&='+&+h+& ,

(2.6)

-&g=1+h

2,

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where '+&=diag(&1, 1, 1, 1) is the Minkowski metric tensor and where h+& , with

|h+& |


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(see [22]) property of the TT system that if freely falling bodies are at rest before

the arrival of the wave their spatial TT coordinates remain constant in time also inthe oscillating gravitational field. This fact will greatly simplify our treatment as all

the relevant circuit parameter perturbations can be found in TT coordinates byintegration over unperturbed coordinate paths.

To find the free-falling conditions let us consider the normal modes of a solidbody as a collection of non-coupled damped harmonic oscillators. This problem

can be easily solved in the Fermi Normal Coordinates (FNC) frame (see [5]). Thedeformation from the equilibrium shape can be written as a sum

$x=:n, l

Anl(t) nl(x) (2.12)

The l-label refers to the kind of modes: for instance, as far as cylindrical bars areconcerned, it refers to the longitudinal, torsional and flexural modes. The n-labelrefers to the eigenfrequencies of a given mode.

For each specific value of l the fundamental mode shall be denoted by n=1. Inthe following we assume that only one specific l-mode is strongly interacting with

the gravitational wave. For instance, for a spherical body Rnl is different from zeroonly when l=2.

In the presence of a gravitational wave, the coefficients Anl are determined by the

equation (see [4] Eqs. (37.42b

c)):

Anl+1

{nlA4 nl+|

2nlAnl=Rnl, (2.13)

where

Rnl=

1

2

1

M hij |i

nlx

j

\ d

3

x (2.14)

(\ is the mass density of the body and M its total mass). The solution in Fourierspace (!(|)=(2?)&12 +& !(t) exp(&i|t) dt) is given by

Anl(|)= &1

2t ijnlhij(|)

|2

|2nl&|2+i|{nl

, (2.15)

where t ijnl is defined as

t ijnl=1

M|inlx

j\ d3x. (2.16)

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For monochromatic gravitational waves of frequencies |g>>|1l we get

A1l(t)=12 t

ij1lhij(t) (2.17)

that is to say, in its fundamental mode the body behaves as a freely moving system.This is the operating condition for the interferometric devices under construction in

Europe (VIRGO) and in the United States (LIGO).Let us now consider the opposite case where |g|1l,

different from all the other |nl, then from Eq. (2.20) one can see that only thefundamental mode is practically excited. For instance if the first harmonic excited

by the wave is n=3 (see, e.g. [4], Eq. (37.45)) then A3l&1

10 A1l. Therefore thebody is free falling because it follows the incoming gravitational wave like a cloudof dust (see Eq. (2.17)) and so it is practically at rest in TT gauge. One can see

therefore that, by suitably arranging the fundamental mode frequency |1l withrespect to the frequency chosen for observing gravitational radiation, the condition

of free falling can be easily fulfilled. In the rest of this paper we shall restrict our-selves to physical devices where these conditions are satisfied and therefore we

neglect the matter-action term in (2.1).



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3. General Theory of Electromagnetic Circuits

To establish the behaviour of a circuit in the field of a weak gravitational wave

we generalize the lagrangian approach of [15]. Let #a (a=1, 2, ..., N) be a systemof N conducting non-ferromagnetic one dimensional (wires) or extended bodies

(capacitors). The quasi-stationary approximation allows us to neglect surfacecurrents in extended conductors and charged wires. The charge over conductor sur-

face and the current flowing in wires can be treated as not depending on spatialcoordinates. The relations between, on the one side, j+ and the current I and, on

the other side, charge density \ and the charge dQ in an infinitesimal volume ele-ment can be written in any coordinate system as (see for instance [14] 990)

{

I dxi=ji-&g d3x

dQ=\ -3g d3x(3.1)

where 3g is the determinant of the three-dimensional metric 3gij=gij&g0ig0jg00 .The equations for I and Q can be found by substituting in the action (2.10) the (for-

mal) solution of the De Rahm equation (2.3) for gravitational waves. If we can

neglect g+&, : with respect to g+& (i.e. for slowly varying gravitational fields) theng#{

+{+rg:; : ; . In this case we can write

A+(x)=1c |0 K+&[x, x$] j

&(x$) -&g(x$) d4x$, (3.2)

where K+& is the retarded Green function, solution of the equation

gx K+&[x, x$]=&4?g+&(x$)

-&g(x$)$4(x:&x$:) (3.3)

In a synchronous reference frame x#(ct, x), where3

gij=g ij and &g=3

g, we canintegrate Eq. (3.2) over the time coordinate x$0 and get the generalized retardedpotential ([4] p. 500)

A+(x, t)=1

c |V K+&[x, x$] j&(x0, x$) -3g(x0, x$) d3x$. (3.4)

Now we have to introduce the Lagrangian L of a system. From (2.10) and (3.4)

we obtain the total Lagrangian of a system of conductors and wires in curvedspacetime

L=1

2c2 |V-3g(x) d3x |

V

-3g(x0, x$) d3x$

_K+&[x, x$] j+(x) j&(x0, x$) (3.5)

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From the relations (3.1), taking into account that I and Q are only functions of

time in the quasi-stationary-field approximation and that K0i=0 in synchronousreference frames, we get

L=:

N

a, b=1

I#a I#b

2c2 |#a dx

i

|#b dx

$ j

Kij[

x, x$]

+1

2 |#adQ |

#b

dQ$ K00[x, x$]. (3.6)

For quasi-stationary fields charges are distributed over the conductor surfaces A#aand so we can replace in (3.6) the charge integrals with surface integrals. This leadsto

L= :N

a, b=1

[ 12 L#a#b(t) I#a I#b&12 C #a#b(t) Q#a Q#b ], (3.7)

where L#a#b are the generalized coefficients of mutual inductance

L#a#b(t)=1

c2 |#adxi |

#b

dx$ j Kij[x, x$], (3.8)

and C#a#b

Q#a C #a#b(t) Q#b= &|#a

dA#a |#b

dA$#bdQ#adA#a

K00[x, x$]dQ#bdA$#b

, (3.9)

define univocally the coefficients of capacity C#a#b by means of the equationsNb=1 C #

a#

b

C#b#

c

=$ac . From the above formulae one easily recognizes that the

action of gravitational fields in synchronous reference frames (such as TT-gaugeframes) is a change in the geometrical circuit parameters, i.e. the interaction ofgravitational radiation with standard electromagnetic circuits is of parametricnature.

In order to take into account also dissipations of real circuits we go over toOhm's law written in its covariant form ([8] p. 263); for the current in wires onehas

j:&j; u;u:=_F:& u&, (3.10)

where u& is the normalized four velocity of the lattice of the conductor #a , and _ais its conductivity which we assume constant in the range of frequencies we areinterested in. As the lattice has fixed TT coordinates u+=$+0 , the total dissipated

power in our circuits by Joule effect is



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P#|V

-3g d3x j+F+&u&

=:a

1

_a|#a

-3g d3x gijji(xk) j j(xk), (3.11)

which can be rewritten in the form

P= :N

a=1

R#a(t) I2#a

, (3.12)

where Ra(t) are generalized resistances

R#a(t)= gij

-3g |#a1

_#a S#at j#a dx

i; (3.13)

here t i#a is the unit vector (3gijt

i#a

t j#a=1) tangent to the wire #a and dS#a#d3x#a dl,

(dl2=gij dxi dx j) is the cross-section of the wire.

In order to get the EulerLagrange equations from the Lagrangian (3.7) we have

to set up a relation among our charges and currents. From (3.1) and the chargeconversation relation follows I#a=Q4 #a . Hereof the equations of motion result in the

form

:N

b=1

[L#a#b(t) Q#b+L4 #a#b(t) Q4 #b+C #a#b(t) Q#b]=0. (3.14)

Once we have defined the dissipation function 8=P2 we can substitute the r.h.s.of the equation (3.14) with 8Q4 #a and write (see [13] 91.5)

:N

b=1

L#a#b(t) Q#b+L4 #a#b(t) Q4 #b+C #a#b(t) Q#b= &R#a(t) Q4 #a . (3.15)

These equations describe, in TT coordinates, RLC circuits in external gravitationalradiation fields when dissipations are taken into account.

In the weak gravitational field approximation one can easily show (cf. [10]) that

K+&=

'+&+h+&

|x&x$| +

1

2 hkl $k

$l

|x&x$| '+& (3.16)

satisfies Eq. (2.11) through Eq. (3.4), and therefore it gives the right expression of

the Green function K+& in the linear and quasi-static field approximation and for aTT-metric perturbation h+&(t). From Eqs. (3.8), (3.9) and (3.13) it follows that, in

the same approximation, the coefficients L#a#b(t), C #a#b(t) and R#a(t) read

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L#a#b(t)=0L#a#b+l#a#b(t)

C #a#b(t)=0C #a#b+c#a#b(t) (3.17)

R#a(t)=0R#a+r#a(t)

where0

L#a#b ,0

C #a#b , and0

R#a are respectively the usual inductance, capacitance, andresistance coefficients in flat spacetime, defined as

0L#a#b=1

c2 |#a |#b$ij

dx i#a dx$j

#b

|x#a&x$#b |, (3.18)

0R#a=$ ij |#a

t i#a tj#a

_#a

dl

dA#a(3.19)

Q#a 0C #a#b Q#b=|#a |#b

dQ#a

dA#a

dA#a dA$#b|x#a&x$#b |

dQ#b

dA$#b. (3.20)

Here l#a#b(t), c#a#b(t) and r#a(t) are the time dependent perturbations induced by the

gravitational waves on the circuit parameters which can be written as:

l#a#b=hij*ij#a#b

c#a#b=hij/ij#a#b

(3.21)

r#a=hij*ij#a

with

* ij#a#b=1

c2 |#a |#bdx i#a dx$

j#b

|x#a&x$#b |+

1

2c2 |#a |#b$kl $

i$ j |x#a&x$#b | dx

k#a

dx$l#b ,

(3.22)

Q#a/ ij#a#b Q#b=

1

2 |#a |#bdQ#a

dA#a

dQ#b

dA$#b$i $ j |x#a&x$#b | dA#a dA$#b (3.23)

and

*ij#a

=|#a

t i#a tj#a

_#a

dl

dS#a. (3.24)

The coefficients *ij#a#b , /ij#a#b

and * ij#a , defined in (3.22), (3.23) and (3.24), are pure

geometric quantities which play a similar role as the mass-quadrupole tensor Qij=V \(x) (x ix j&(13) $ijxkxk) d3x in the mechanical interaction between a gravita-tional wave and a solid body.

We notice that the equations that describe the interaction between gravitational

waves and electromagnetic circuits are parametric ones. This means that gravity is



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not acting as electromotoric force but is changing the circuit parameters. The effect

of the small perturbation of the parameters of the circuit is a frequency andamplitude modulation (see for instance [18]).

4. The Superconducting Circuit

Below a certain temperature Tt , called transition temperature, some metalsbecome superconducting. The most impressive properties of this state are the sud-

den drop of the electrical resistance and the Meissner effect. F. London in 1935(cfr. [12]) suggested a macroscopic theory of the pure superconducting state which

unified these two experimental facts under the same theoretical scheme. He sup-posed that the density current flowing in the superconductor was a sum of twocurrents: the normal jn and the supercurrent js

j=js+jn . (4.1)

The two kinds of current are related to the electromagnetic field inside the conduc-tor in different ways. The supercurrent satisfies the two London equations

curl(4js)=&

B

c(4.2)

t(4js)=E.

Here 4 is a constant characteristic of the superconductor (its order of magnitudeis >10&31sec2); B and E are the intensities of the magnetic and electric fields. The

normal current is connected with the electric field by Ohm's law

jn=_E (4.3)

where _ is a continuous function of the temperature and has no jump at T=Tt .The total current j satisfies the Maxwell equations and the relation between the twocurrents turns out to be

tjs=

1

_4jn=;jn (4.4)

where ; has the dimension of a frequency (;=1_4r1012 Hz). If the elec-tromagnetic field oscillates at a frequency |em we have

|jn | =|em

;|js | (4.5)

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and if |em


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where r0 is the radius of the wire (r0>>*); the power lost as Joule heat is given

then by

P=|V

1

_j2n dV=RsI

2n (4.11)

where

Inr2?*r0 |jn(t)| (4.12)

and

Rs=|#

dl

_'. (4.13)

' is the effective section in which the current flows. It takes the value

'r4?*r0 . (4.14)

In a wire one has Rsr(r0 *) R, where R is the usual resistance. For the kineticenergy we get

Wkin

=

|V

1

24j2

s

dV=1

2;R

s

I2s

(4.15)

where for the last equality see Eq. (4.4); Is is defined analogously to In .

Therefore we can write the Lagrangian of an RLC superconducting circuit as(I=Is+In , Q=Qs+Qn)

L=1

2 \L+Rs

; + I2s +LIsIn+1

2LI2n&

1

2CQ2s &

1

CQsQn&

1

2CQ 2n . (4.16)

The dissipation function reads

8= 12 Rs I2n (4.17)

and the relation between In and Is is given by (see Eq. (4.4))

I4s=;In . (4.18)

Using the method of Lagrange multipliers (see for instance [13] 92.4) we find

Q+2#sQ4 n+|20 Q=0 (4.19)

where 2#s=Rs L and |20=1LC. The only difference from the equation of anohmic circuit is in the resistance term: in the superconducting case this term is only

due to a part of the current (the normal current).

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Making use of relation (4.18) we find for the total current a differential equation

of the third order in time

I5+;(1+21) I+|20 I4+;|20 I=0 (4.20)

where 21=Rs ;L


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in which '#a is the effective cross-section (4.14) of the ath wire. These equations

allow us to write down the generalization of the kinetic energy

Wkin(t)=1

2;:N

a=1

Rs#a(t) I2s#a

; (4.26)

and finally the Lagrangian of a system of N superconducting wires or bodies can

be written as

L= :N

a, b=1_

1

2L#a#b(t) I#a I#b&

1

2C #a#b(t) Q#a Q#b&+

1

2;:N

a=1

Rs#a(t) I2s#a

. (4.27)

The dissipation function is given by (4.24). To get the EulerLagrange equationsfrom the Lagrangian (4.27) and the dissipation function (4.24) we have to know the

relations between charge and current and between charge and supercurrent. For aset of RLC circuits we have I#a=Q4 #a and because of the general covariant form ofEq. (4.2) (dd{[4(j+s &j

:su:u

+)]=F+& u&), the relation between Qsa and Qna is given

in the curved case also by

Q4 s#a=;Qn#a . (4.28)

Using the method of Lagrange multipliers we get from (4.27) and (4.24)

:N

b=1

[L#a#b(t) Q#b+L4 #a#b(t) Q4 #b+C #a#b(t) Q#b]= &Rs#a(t) Q4 n#a (4.29)

and by means of (4.28) we get the equations for the supercurrent

:N

b=1

[L#a#b(t) Q5s#b+(;L#a#b(t)+L4 #a#b(t)) Qs#b+Rs#a(t) Qs#a

+C #a#b(t) Q4 s#b+;C #a#b(t) Qs#b]=0. (4.30)

5. Conductors in the Presence of Matter

In this section we generalize our theory to the case in which the capacitor is filledby a dielectric and the inductance has a magnetic kernel. Within the usual

approximations the matter can be characterized by a dielectric constant = and amagnetic permeability + which enter into the constitutive equations (see [8]p. 263):

H:; u:==F:; u

:

(5.1)(g:; F#$+g:#F$;+g:$ F;#) u

:=+(g:;H#$+g:# H$;+g:$ H;#) u:

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where u: is the macroscopic 4-velocity of the medium. With these definitions the

Maxwell equations become

F+&, _+F&_, ++F_+, &=0

(5.2)

H

+&

;&=

4?

c j

+

.

Under free-falling conditions u:=$:0 , Eqs. (5.1) reduce to

H0k==F0k Fik=+Hik . (5.3)

In this way the second pair of Maxwell equations becomes

F0&; &=4?

=

j0

c (5.4)

Fk&; &=4?

c+jk.

The action of the free electromagnetic field reads Sem=(14?c) d4x -&g H+&F+& .With these notations Eqs. (3.18) and (3.22) become (see [15] 993033)

0L#a#b=+

c2

|#a

|#b

$ijdx i#a dx$

j#b

|x#a&x$#b |, (5.5)

* ij#a#b=+

c2 |#a |#bdx i#a dx$

j#b

|x#a&x$#b |+

+

2c2 |#a |#b$kl $

i$ j |x#a&x$#b | dx

k#a

dx$ l#b , (5.6)

while (3.20) and (3.23) become

Q#a 0C #a#b Q#b=

1

= |#a |#bdQ#a

dA#a

dA#a dA$#b|x#a&x$#b |

dQ#a

dA$#b. (5.7)

Q#a/ ij#a#b Q#b=

1

2= |#a |#bdQ#a

dA#a

dQ#b

dA$#b$i $ j |x#a&x$#b | dA#a dA$#b . (5.8)

Therefore the presence of matter with scalar dielectric or magnetic properties

changes only the definition of the system parameters; they are increased by = and+ factors. This is important because, in order to obtain great capacitances one can

use suitable dielectrics: standard commercial types of capacitors can reachcapacities of about 10 mF.

6. Application and Discussion

Let us now consider the device of Figure 1, located in the x&y plane with x=x1

and y=x2, and evaluate its response to a periodic gravitational wave using realistic



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Fig. 1. A scheme of the RLC circuit described in the text which can be used as a detector of gravita-

tional waves. A static charge is distributed on the plates of the condensers C1 and C2 .

parameters. This circuit can be described as a set of six conductors: two wireslabelled #1 (the inductance and resistance) and #2 (idealized connection wire) which

connect four plane conductors labelled with #3 , #4 (to form the capacitor C1) and#5 , #6 (the capacitor C2) with charge Q#3= &Q1+Q, Q#4=Q1&Q, Q#5=Q2+Qand Q#6= &Q2&Q; Q1 and Q2 are the constant electrostatic charges on the platesof the two capacitors in the absence of gravitational waves when there is no current

flow which satisfy

Q1+Q2=Q0 ,(6.1)

Q1C1&Q2 C2=0.

The connection between the capacity of a condenser and the capacitance coef-ficients of the conductors i and j reads (see [15])

C&1#C ii&2C ij+Cjj. (6.2)

The Lagrangian of this system can be written as

L=1

2 L(t) Q42&

1

2

(Q&Q1)2

C1(t) &1

2

(Q+Q2)2

C2(t) . (6.3)

Taking into account dissipation, the equation of the circuit becomes

L(t) Q+L4 (t) Q4 +R(t) Q4 +\1

C1(t)+

1

C2(t)+ Q=Q1

C1(t)&

Q2

C2(t). (6.4)

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Let us now consider the case of a weakly coupled gravitational-wave field. Setting

:+, _=(e+, _) ij* ij33L

}+, _= &(e+, _) ij (/ij

11&2/ij

12+/ij

22) C (6.5)

\+, _=(e+, _) ij*

ij3

R,

we can separate the polarization and angular dependences of the interaction (figurepattern). In this way we obtain

L(t)=L[1+=L(t)]=L[1+:+ h+(t)+:_ h_(t)],

1

C(t)=

1

C[1&=C(t)]=

1

C[1&}+ h+(t)&}_ h_(t)], (6.6)

R(t)=R[1+=R(t)]=R[1+\+ h+(t)+\_ h_(t)].

For the sake of simplicity let us put

C1=aC C2=C (6.7)

|20=a+1

a

1

LC.

With this definition of the factor a we obtain

Q1=a

a+1Q0 Q2=

1

a+1Q0 (6.8)

and the equation of the circuit can be written as

(1 +=L(t)) Q+2# \1+=*L(t)

2#+=R(t)+ Q4 +|20 \1&

=C1(t)+a=C2(t)

a+1 + Q

=|20 Q0a(=C1(t)&=C2(t))

(a+1)2. (6.9)

As the time dependent coefficients in the l.h.s. are very small compared with unitytheir unique effect will be an amplitude and frequency modulations of the unpertur-

bed charge motion

Q+2#Q4 +|20 Q=v+ h+(t)+v_ h_(t) (6.10)



19/26

where

v+, _=|20 Q0

a(}1, +, _&}2, +, _)

(a+1)2. (6.11)

In equation (6.10) the interaction between the gravitational wave and the circuit isnon-parametric. The dropped modulations are so small that they cannot be

observed being a second order effect with respect to the solution of (6.10). If weconsider a periodic gravitational wave h+, _(t)=A+, _ cos (|gt+,+, _), then the

solution of (6.10) can be written as

Q(t)=Q+(t)+Q_(t)

(6.12)

Q+, _(t)=

a(}1, +, _&}2, +, _)

(a+1) 2

2A+, _ Q0Q

1+(| 20&|

2g)

2

4#2|2g

|0

|g sin (|g t+,+, _+$)

where Q=|04# is the quality factor of the circuit and

$=arc tan|20&|

2g

2#|g. (6.13)

In this system the effect of the gravitational wave is to cause a current I(t) to flow

in the circuit with

I(t)=a(}1&}2)

(a+1)22A|0 Q0Q

1+(|20&|

2g)

2

4#2| 2g

cos (|gt+,+$). (6.14)

Near resonance (i.e. (|0&|g)2


20/26

For weak gravitational waves we have

(1+=L(t)) Q+=*L(t) Q4 +2#s[1 +=R(t)] Q4 n+|20 \1&=C1(t)+a=C2(t)

a+1 + Q

=|20 Q0 a(=C1(t)&=C2(t))

(a+1) 2. (6.17)

Again the time dependent coefficients in the l.h.s. of equation (6.17) produce only

a very small frequency and amplitude modulation of the solution of the followingequation

Q+2#sQ4 n+|20 Q=v+ h+(t)+v_ h_(t) (6.18)

where

v+, _=020(1+21) Q0

a(}1, + , _&}2, +, _)

(a+1) 2. (6.19)

In equation (6.18) the gravitational-wave electromagnetic-circuit interaction ispurely non-parametric. If we use relation (4.28) in (6.18) we find the equation of the

supercurrent as

Q5s+;(1+21) Q s+020(1+21) Q4 s+;020(1+21) Qs

=;[v+ h+(t)+v_ h_(t)]. (6.20)

If h+, _(t)=A+, _ cos (|gt+,+, _), then the solution of (6.20) can be written as

Qs(t)=Qs, +(t)+Qs, _(t)

(6.21)Qs, +, _(t)=a(}1, +, _&}2, +, _)

(a+1) 22A+, _(1+21) Q0Qeff

T(|g)

00

|g

_sin (|gt+,+, _+$)

where

tan $= (1+21)(0 20&|2g)

2#0 |g \1+020&|

2g

2#0; +, (6.22)

T(|g)=\1+020&|

2g

2#0; + -1+tan2 $ (6.23)



21/26

and

Qeff=00

4#0=

;

4100=

;2

200

L

Rs(6.24)

is the quality factor of the superconducting circuit which is several orders ofmagnitude greater than that of the ohmic circuit. In this way the gravitational wave

gives rise to a supercurrent

Is(t)=a(}1&}2)

(a+1)22A(1+21) 00 Q0Qeff

T(|g)cos (|g t+,+$) (6.25)

in which, for the sake of simplicity, we have considered only one polarization stateand dropped the indices +, _. Near resonance (i.e. (020&|2g)

2


22/26

so that

Q#a/ ii#a#b Q#b=

1

2Q#aC #a#b Q

#b&1

2= |#a |#bdQ#a

dA#a

dQ#b

dA$#b

(x i#a&x$i

#b)2

|x#a&x$#b |3

dA#a dA$#b (6.30)

(no summation over i).

Let us now consider the case of a capacitor made of two circular plates set per-pendicular to the x axis and at a distance d apart. If #a=#b then

/11#a#a=12 C #a#a (6.31)

and

/22#a#a=/33#a#a

= 14 C #a#a . (6.32)

If #a {#b , assuming that d is much smaller than the diameter of the circular plates,by means of (6.28) and the symmetry of the problem, we approximately get

/11#a#b=C #a#b&12 C #a#a

(6.33)/22#a#b=/

33#a#b

= 14 C #a#a

which is valid also for #a=#b as can be seen by comparing (6.33) with (6.316.32).

Similarly one can show that /12#a#b=0.Now let us assume that the wave comes from the z=x3 axis perpendicular to the

place containing our circuit (see Figure 1). In this case we have for the polarizationtensors

(e+)+&=\0

0

0

0

0

1

0

0

0

0

&1

0

0

0

0

0+ (6.34)(e_)+&=

\0

0

0

0

0

0

1

0

0

1

0

0

0

0

0

0+(6.35)

With reference to eq. (6.5) let us calculate }+; one finds

}+= &(e+) ij (/ij11&2/

ij12+/

ij22) C

=[(/2211&/1111)&2(/

2212&/

1112)+(/

2222&/

1122)] C. (6.36)

Taking the x axis parallel to the plates of C1 and the y axis parallel to the platesof C2 we obtain

(}+)1=1, (}_)1=0(6.37)

(}+)2= &1, (}_)2=0



23/26

Substituting (6.37) in (6.15) and (6.26), and taking a=1 one gets

Ires=A+ |0 Q0Q cos (|0 t+,),(6.38)

Ires.s=A+ 00 Q0Qeff cos (|0 t+,).

We now estimate the minimal gravitational amplitude which can be detected by ourdevice. Once the device is isolated from the outside world by means for instance of

mechanical filters, as those used in gravitational bar detectors and Faraday cages,the main noise source is the thermal one, as well for the mechanical part of the

detector as for the electric part, e.g. the back-action noise of the amplifier, atcryogenic temperature and for the best amplifiers, is at most of the order of

magnitude of the thermal noise. In the following we shall derive formulae for thethermal noise in the electric part of the detector. The thermal spectral noise in the

mechanical part (2$ll=h, see equation (2.17)) is given by -16kBTM{l2|4 whereT, M, {, and l denote temperature, effective mass, life-time of the fundamentalmode, and effective length of the detector, kB is the Boltzmann constant and |denotes the angular frequency of the gravitational wave, e.g. see [24]. Assuming the

numbers T=4K, M=104 kg, {=10 sec, l=10 m, and |2?=60 Hz yields about2.1_10&23 for h, measured through a period of 4 months. This number is as small

as the smallest number in the equation (6.45) below, i.e. for the superconductingcircuits in question the mechanical part has to be adjusted to those numbers.

We now make the hypothesis that the only resistance in our circuit is that of ourscheme in Figure 1. The mean square fluctuation of the voltage noise is therefore

given for tobs>>Q|0 by ([25] p. 288)

( V2n) =kBT

Ceff(6.39)

where Ceff is the total capacitance of the capacitors. The mean noise energy is

Ceff( V2n) =kBT (6.40)

The energy dissipated by the ohmic and superconducting current produced by thegravitational wave is

W=|RI2 dt= 12 RI20 tobs(6.41)

Ws=|Rs I2n dt= 12 RsI2n0 tobs

where tobs is the observational time and I0 the amplitude of the current I. Equating(6.40) and (6.41) one finds the minimum detectable current after an observation

time tobs ,

56 FORTINI ET AL.


24/26

I0=2kBT

Rtobs(6.42)

In0=2kBT

Rstobs

At resonance this implies (by means of Eqs. (6.7), (6.15) and Q=|0 L(2R) for theohmic circuit and (4.22), (6.24) and (6.27) for the superconducting one) that

Anoise=1

V0 (a+1)4

a2(}1&}2)2

kBT

QC|0 tobs(6.43)

Anoise s=1

V0 (a+1)4

a2(}1&}2)2

kBT

QeffC|0 tobs

where V0=Q0C. If a=}1= &}2=1 then

Anoise=2

V0 kBT

QC|0 tobs(6.44)

Anoise s=2

V0 kBT

QeffC|0 tobs

Taking V0=105

Volt, T=4 K, C=10&2

F, |0=2?_60 radsec, tobs=4 months=107 sec, Q=103 and Qeff=10

6 one has

Anoise&7.6_10&22, Anoise s&2.4_10

&23 (6.45)

Finally we compute the order of magnitude of the normal and superconductingcurrent, produced by a gravitational wave with amplitudes A+=2_10

&21 and

A+=5_10&23 which correspond respectively to a signal to noise ratio equal to 2;

we find

In&7.5_10&13A

(6.46)Is&2_10

&11A

that are within the possibilities of actual measurements with SQUIDs. If we go

down with the temperature in the ultracryogenic zone (t40 mK) we obtain

Anoise&7.6_10&23 I&7.5_10&14A

(6.47)Anoise s&2.4_10&24 Is&2_10

&12A

Further improvements can come either from different arrangements of the circuit

elements (for instance one can consider a series of many capacitors). As far as thelast question is concerned, today's limitations in the superconducting case are due

to the energy dissipation in the material which acts as a mounting of the circuit.



25/26

For this reason we have set Qeff=106 but there are no theoretical limitations in

thinking of quality factors higher by several orders of magnitude: it is in fact onlya technological problem. We think that a serious research in this field would be

very useful.It is to be expected that when the device is operating near the mechanical

resonance the sensitivity should increase. However this situation deserves a moredetailed investigation.

Recall now that we have calculated the currents in TT-gauge. In actualexperiments we measure currents in the laboratory reference frame (FNC). Because

of the charge conservation we can write

IFNC dy0=ITT dx0 (6.48)

where y+ are the FNC coordinates while x+ are the TT coordinates. As

x0=y0& 14 hij, 0yiy j we have

dx0=\1&k2

4h (2 )ij y

iy j+ dy0&k

2h (1 )ij y

i dy j (6.49)

where the superscript (i) is the ith derivative of h with respect to its argument.Therefore the relation between the current in the two gauges reads

IFNC=ITT

\1&k

2

4h (2 )ij yiyj&k2

h(1)ij yi dy

j

dy0+. (6.50)

In our approximation both first order corrections are negligible and so

IFNC=ITT (6.51)

We point out that our theoretical limits in the magnitude of the gravitational-

wave amplitude detectable from periodic sources like the Crab pulsar are better

than those experimentally obtained by a Japanese research group (A


26/26

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gravitation wave interaction with normal & superconducting circuits (1)

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