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General Relativity and Gravitation , Vol. 22, No. 1, 1990 Interaction of Gravitational Waves with a Superconducting Cylindrical Antenna Huei Peng 1 and Bo Peng 2 Received April 4, 1989 In this paper we investigate the effects of gravitational waves (GW) on a superconducting cylindrical antenna (S-antenna). We suggest that the electric fields induced by GW of dimensionless amplitude h --- 10 -24 in the interior of existing cylindrical antenna might be detectable. 1. INTRODUCTION The effects of gravitational waves (GW) on a superconducting antenna have been proposed recently [1]. In a superconducting antenna, negative and positive charges exhibit independent vibrational responses to GW, be- cause there is no resistance offered to superelectrons while the ions can still be treated as damped harmonic oscillators. These independent vibrations establish a net current with a resultant magnetic vector potential A(r, t). While the potential A is changing, an electric field E(r,Q is induced in the interior and reacts on ions and superelectrons, hence changing the be- havior of the superconducting antenna. This effect is based on a model in which ions and superelectrons are treated as idealised harmonic oscillators and free particles, respectively. But there are several differences between an idealised oscillator and a normal antenna (see for example Ref. 2). A normal cylindrical antenna 1 Physics Department, The University of Alabama in Huntsville, Hunstville AL35899, USA, and Institute of Applied Mathematics, Academica Sinica, Beijing, P. It. ChTma 2 Department of Mathematics, University of Arizona, Tucson AZ85721, USA 45 0001-7701/90/0100-0045506.00/0 @ 1990 Plenum Publishing Corporation

Interaction of gravitational waves with a superconducting cylindrical antenna

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Page 1: Interaction of gravitational waves with a superconducting cylindrical antenna

General Re la t i v i t y and Grav i ta t ion , Vol. 22, No. 1, 1990

Interaction of Gravitational Waves with a Superconducting Cylindrical Antenna

H u e i P e n g 1 a n d B o P e n g 2

Received Apr i l 4, 1989

In this paper we investigate the effects of gravitational waves (GW) on a superconducting cylindrical an tenna (S-antenna). We suggest tha t the electric fields induced by GW of dimensionless amplitude h --- 10 -24 in the interior of existing cylindrical an tenna might be detectable.

1. I N T R O D U C T I O N

The effects of gravitat ional waves (GW) on a superconducting antenna have been proposed recently [1]. In a superconducting antenna, negative and positive charges exhibit independent vibrational responses to GW, be- cause there is no resistance offered to superelectrons while the ions can still be t reated as damped harmonic oscillators. These independent vibrations establish a net current with a resultant magnetic vector potential A(r, t). While the potential A is changing, an electric field E ( r , Q is induced in the interior and reacts on ions and superelectrons, hence changing the be- havior of the superconducting antenna. This effect is based on a model in which ions and superelectrons are t reated as idealised harmonic oscillators and free particles, respectively.

But there are several differences between an idealised oscillator and a normal antenna (see for example Ref. 2). A normal cylindrical antenna

1 Physics Department , The University of Alabama in Huntsville, Hunstville AL35899, USA, and Inst i tute of Applied Mathematics, Academica Sinica, Beijing, P. It. ChTma

2 Depar tment of Mathematics, University of Arizona, Tucson AZ85721, USA

45

0001-7701/90/0100-0045506.00/0 @ 1990 Plenum Publishing Corporation

Page 2: Interaction of gravitational waves with a superconducting cylindrical antenna

46 Peng

has a set of normal modes. The fundamental longitudinal mode is anal- ogous to the mode of the idealised oscillator. The mass, frequency and length are not independent for practical antennas. It is reasonable to ex- pect that there are similar differences between idealised oscillators and a superconducting antenna.

In this paper we derive the generalized London equations which de- scribe the electromagnetic properties of a superconducting cylindrical an- tenna. By use of those equations, the quantitative estimate of the induced E field is provided. The E field might be detectable. Then the relations between the parameters of an idealised oscillator and a superconducting cylindrical antenna and restriction on the size of the antenna are derived.

We let the speed of light c = 1.

2. I N T E R A C T I O N O F G W W I T H A N I D E A L I S E D O S C I L L A - T O R

We briefly review the effect of GW on a superconducting cylindrical antenna [1].

We assume that: (i) Ions are still represented by damped simple har- monic oscillators; (ii) GW will penetrate the antenna; (iii) Small vibrations of ions will not destroy the superconductivity of the antenna; (iv) Since the gravitational fields of the Earth are independent of time, the effects of the Earth 's gravitational fields are negligible. (v) We ignore the effect of normal electrons.

Consider an oscillator which consists of two points carrying equal masses m~ and electric charges e on the ends of a spring of equilibrium length 21, natural frequency wi, and a damping time ~-~. The oscillator is oriented along the x-axis. This model represents ions in the superconduct- ing antenna under the influence of GW. Now imagine another model made of two points which carry equal masses m e and charges - e , situated at xe and - x e respectively, and vibrate along the x-axis without external forces acting on them, except the incoming GW. This model, which we will call the free oscillator, represents superelectrons.

The incoming GW of frequency w cause the ions and superetectrons to vibrate differently along the axis of the antenna, and as a result induces a net electric current which creates a time-dependent magnetic vector poten- tial A, and thus an electric field E which will react on the superelectrons and ions. Taking into account the effect of the x-component of the induced electric field on ions and superelectrons, the equation of motion of ions

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A S u p e r c o n d u c t i n g C y l i n d r i c a l A n t e n n a 47

is [1], d2xi dxi e -,tt--e + ~ + ~ ' ~ = aow + --E,m, (1)

and the equation of motion of superelectrons is

d 2 x e e : a Q w - - - E , (2) dt 2 me

where aaw is the gravitational driving acceleration that results from pro- jecting the tidal gravitational force due to GW onto the antenna, mi and me are masses of ion and electron, respectively, and we have assumed that the mass of a Cooper pair is approximately equal to 2me [3].

Throughout this paper we will assume all t ime-dependent quantities vary with time as

u(r , t ) = u(r) e-i~t. (3)

The solutions of eqs. (1)-(2) are respectively

e E ) , xi(t) = - - G(~) a~w + -~i

z ~ ( t ) - ~ 2 - '

(4)

(5)

where a(~) = (~2 _ ~ + i~/Ti)-l, (6)

is the harmonic oscillator response function. The net current induced by incoming GW is a combination of the

electron-current and ion-current,

To find the explicit expression of xi and xe, we need to derive the induced E field.

Since GW are very weak, instead of using the general relativistic Maxwell equations, one may substitute the net current into the Maxwell equations to derive the generalized London equations. By solving those equations, the expression for the induced E field is obtained

E(O ~- _m~ aow(~? - i~/',-i) (1 + d).~2 _ ~ + i~/~-i' (8)

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48 Peng

where d = m e / m i , (9)

and we have ignored small terms. The Eint is a narrow-band resonant func- tion centered at w i / y / ~ + d) with a full width at half maximum of wi /Qi , where Qi = wivi. One might expect that a more significant bandwidth restriction comes from the readout system.

The effects of the induced E field may be seen by substituting the E into eqs. (4)-(5) to find the displacements of the ion and superelectron. The displacement of the ion is

(1 + d)aGw z i ( t ) ~-- (1 + d)oa 2 - ~2 + ioalri' (10)

where the small terms have been ignored. The induced E field will change both the displacement of the ion's vibration and the resonant frequency.

3. I N T E R A C T I O N OF G W W I T H A S U P E R C O N D U C T I N G C Y L I N D E R

For simplicity, we consider longitudinal vibrations propagating along the axis of a superconducting cylinder whose length, 2L, is much greater than its radius.

Taking into account the reaction of an induced E field on ions and superelectrons in the cylinder, the equations of motion for ions and super- electrons are, respectively,

O~zic 1 0 ~ e Ot ~ - p cgz + aow + --E,mi (11)

O2Zec e Ot ~ = a a w - - - E , (12)

me

where &r/cgz is the force per unit volume along the z-axis, due to the sur- rounding material, acting on an ion, p ~ nmi is approximately the density, n is the concentration of ions, subscript c denotes the displacements of the ion and superelectron in the cylinder, respectively, ~ is the z-component of the stress tensor.

Eq. (12) has the same form as eq. (2), since there is no resistance on the superelectrons in the cylinder. The expressions of displacements of su- perelectrons in both models of the free oscillator and the superconducting cylinder have the same form,

zec -- -(aGw - e E / m e ) / w ~. (13)

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A S u p e r c o n d u c t i n g Cylindrical Antenna 49

The bulk modulus of elasticity, e, is slightly different in the supercon- ducting and normal states [4], but the effects are extremely small. Thus the bulk modulus e and speeds of sound (v~ = ~ ) can be considered to be the same in both states. Therefore, Hooke's law still holds for a superconducting cylinder

axi~ 50__ Oxir (14) ~ = e - ~ - - z + Ot Oz '

where 5 represents all the effects of the dissipative processes. Substituting eq. (14) into eq. (11), we obtain

02xic e 02xi~ 5 8 82xi~ e__E. (15) 8t 2 -- p 8X 2 Jr fl Ot 0 2 ~ + aGw + m i

We can rewrite eq. (15) by use of eq. (3) as

w2xi ~ + e 02xi~ . 5 82xic e___E POx 2 Z W p ~ + aGw + mi = 0. (16)

The solution of eq. (16) is

Xic --~ Xl e - i ~ x "3 t- X2E ic~x - - (aGw -~- (17)

where

w2p ~ (18)

The two constant xl and x2 can be determined by imposing two boundary conditions. Because of the quadrupole character of GW, the center of mass of the cylindrical antenna does not move, i.e.

xi<(O, t) = O. (19)

At the ends of the antenna, the stress vanishes

& = o. (20)

The first boundary condition, eq. (19), gives

xl = -x2 . (21)

Since we have not yet derived the explicit expression of the induced E field yet, we cannot use the second boundary condition until the generalized London equations are derived and the induced E field is found.

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50 P e n g

4. G E N E R A L I Z E D L O N D O N E Q U A T I O N S F O R A S U P E R - C O N D U C T I N G C Y L I N D R I C A L A N T E N N A

Following the method of [1], the net current is

J=ne\ Ot Ot ] =2nex2wsin~x+--(l+d)E,mew (22)

and the generalized London equations for the superconducting cylindrical antenna are

OJ ne 2 Ot -- i2nex2w2 sinc~x + - - ( 1 + d)E, (23)

m e

n e 2 V • J - - - (1 + d)B, (24)

m e

V2J=( p~ (1 +d)-w2)j + 2nex2w 3sinolx (25) \ me

V2E=( #~ (l+d)-w2)E-i2#onex2w2sinax (26) \ m e

V2B=( #~ (1 +d)-wU)B. (27) \ me

This is a complete set of equations which can be used to describe the elec- tromagnetic properties of a superconducting cylinder under the influence of GW. It can be shown that the Meissner effect is still valid for a super- conducting cylinder. Here we are interested in the induced E field in the interior which is, from eq. (26),

i2wUmex~ sin ax (28) Eint "~ e(1 + d)

Substituting the Ein t and eq. (21) into eq. (17), one obtains

i2x2 sin ax aGw (29) xie -- 1 + d w 2 "

For simplicity and without loss of generality, we consider polarized GW propagating along the z-axis. Then we have the acceleration due to the GW

1 0.) 2xh, (30) a~w = - 2

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A S u p e r c o n d u c t i n g C y l i n d r i c a l A n t e n n a 51

where h is the dimensionless ampl i tude of the incoming GW. Now one can apply the second b o u n d a r y condit ion, eq. (20), to find

x2. Subs t i tu t ing eqs. (29)-(30) into eq. (20), we find

h x2 - - i4c~ cos a L " (31)

Then inser t ing x= into the expressions of zir and Eint , we obta in

h ( sin a x x~c= ~ x d ~ s ~ L ] '

w2 h me s i n a z Ein t = -- _ _ 2 e ceconsceL

W h e n w = wm reaches its m a x i m u m value

I E i n t m a x l ~ - -

(32)

(33)

= (m + 1/2)rrv, L -~ such tha t cos(wL/v , ) = O, Eint

2w2h me LQc rr 2 e (m + 1/2) 2, (34)

where Q~ = ~(~6) -1, (35)

is the quali ty factor of the superconduc t ing cylinder, m is an integer num- ber. T h e different values of m correspond to different resonant frequencies of the superconduc t ing cylinder.

T h e superconduc t ing cylinder becomes a "generator" under the in- fluence of GW, in the sense tha t there is a voltage drop, V, be tween the center of mass and the end faces of the cylinder.

I f it is technical ly feasible to measure V --- 10 -22 volt by SQUIDs [5], then the sensi t ivi ty of a superconduc t ing cylindrical an tenna with m = 0, Qc -- l0 s, L _~ 3, is given as

2 7r 2 v ' ~ e V (h)all directions -- Q ~ mew~L2 ~- 10 -24, (36)

where h is dimensionless ampl i tude of unpolar ised G W incoming in an a rb i t r a ry direction, (h) implies an average over all directions.

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52 P e n g

5. E Q U I V A L E N C E B E T W E E N A N I D E A L I Z E D O S C I L L A T O R A N D A S U P E R C O N D U C T I N G C Y L I N D R I C A L A N T E N N A

We now demonstrate that a superconducting cylinder is equivalent to a number of idealised oscillators, provided certain conditions which are slightly different from that of a normal cylindrical antenna are satisfied.

This can be done by calculating the square modulus of the transfer functions for a superconducting cylinder, from eq. (32)

IKcl 2 Ix s in~x 2 16/2 w~) (37) = d ~ ~-- ~r4 4 w 2 ( w o _ w ) ~ + w 4 / Q ~ '

and for an idealised oscillator, from eqs. (10)-(30),

IK~l 2 _~

where we have used

/~(1 + d)2w 4 [ ( l + d ) w 2 w~] 2 2 2 2, - + ~ wi I Q i

(3s)

~_ wv7111 + i(2Qc)-1], (39)

wo = 7r%L -1. (40)

The expressions, eqs. (37) and (38), are identical, provided

Qo = Qi, w = ~Vo, (41)

wi = ~ + d), 4L

l - 7 r 2 ~ , (42)

namely, a superconducting cylindrical antenna with length 2L is equivalent to an oscillator with length 21, provided 1 satisfies eq. (42) and natural frequency wi is given by eq. (41). The d terms in eqs. (41)-(42) represent the effects of the induced E field.

R E F E R E N C E S

1. Peng H. (1989). Gen. Rel. Gray., 22, 33 2. Blair, D. (1983). "Resonant Bar Detectors for Gravitational Waves." In Gravita-

tional Radiation ed. N. Deruelle and T. Piran, (North-Holland Publ. Company). 3. Tare, J. e ta] . (1989). Phys. Rev. Lett., 62, 845. 4. Lynton, E. (1969). Superconductivity (Methuen & Co. Ltd.). 5. Jain, A., Lukens, J. and Tsai, J. (1987). Phys. Rev. Left., 58, 1165.