IL NUOVO CIMENTO VOL. 107 B, N. 11 Novembre 1992

Str ing as a G r a v i t a t i o n a l A n t e n n a ( * ) .

A. NICOLAIDIS and A. TARAMOPOULOS

Theoretical Physics Department, University of Thessaloniki - Thessaloniki 54006, Greece

(ricevuto il 16 Dicembre 1992; approvato il 29 Giugno 1992)

Summary. - - We consider the motion of a string with f'Lxed ends, under the influence of a plane monochromatic gravitational wave. The gravitational interaction is treated perturbatively and by choosing an appropriate orthonormal gauge the equations of motion are simplified to a large extent. A resonant phenomenon is observed when the condition k(E +- Tod) = 2~Ton is satisfied, where k(E) is the energy of the gravitational wave (string) and d is the length of the string projected along the direction of the incident wave. We suggest that the string may be used as a detector for gravitational waves.

PACS 04.80- Experimental tests of general relativity and observations of gravitational radiation.

PACS 11.17 - Theories of strings and other extended objects.

Gravitational waves are continually bathing the Earth. Such gravitational waves, which may be viewed as ripples rolling across the space-time, can be felt through the tidal effects they produce on an extended body. The first detector of the gravitational radiation was proposed and built by Weber[ l~ in the form of a large aluminum cylinder. Other types of mechanical detectors [2] include detectors with the shape of a sphere[3], dumb-bells [4], a hollow square or a tuning fork[5]. Since the simplest extended body is a string (one-dimensional extended body), we examine here the motion of a string with fLxed ends, under the influence of a plane monochromatic wave. Our study indicates that a string detector might be useful in exploring the high-frequency part ((10 ~ + 10 l~ Hz) of the gravitational radiation.

There is an enormous interest in string dynamics, since strings appear in statistical physics [6], hadronic physics [7], cosmology [8] and in the unified theories of all interactions [9]. In all the cases the string appears as a system with infinite degrees of freedom, respecting a large set of symmetries and subject to constraints. As the string propagates in the space-time, it sweeps out a world sheet which is parametrized by the functions X ~ (~, ~). A point along the string is specified by the variable ~(0 ~< ~ ~< ~), while z is a timelike evolution parameter. The interaction of a

(*) The authors of this paper have agreed to not receive the proofs for correction.

1261

1262 A. NICOLAIDIS and h. TARAMOPOULOS

string with an external gravitational field is provided by the curvature of the four-dimensional space-time. By choosing the conformal gauge, the action takes the form of the nonlinear sigma model [9,10]

(1) S - To2 f dvd~[)~')~ - )~"Y~]g~(X~)'

To is the string tension, dot and prime denote differentiation with respect to z and ~, respectively, and g~ is the space-time metric tensor. The variation of the action leads to the equations of motion, a system of four coupled wave equations in two dimensions

(2)

where '~ 1 ~), are the Christoffel symbols

(3) F ~ = l g ~ [ -

r! ! t

s - x ~ + r ~ ( 2 ~ s ~ - x ~ x ~) = 0,

3g~ 3gv~ ag~ ]

~X ~ 3X ~ 3X ~ J "

The equations of motion are supplemented by boundary conditions, which will be specified later.

The gravitational waves that bathe the Earth are all weak in the sense that the metric tensor g~v is nearly that of the flat space-time n~v. Moreover, since the nearest source of significant waves is so far away from us, the gravitational waves impinging on the Earth are plane waves to an extremely good approximation. The metric tensor then takes the following form:

(4) g~ = n~ + Ae,~ exp [iK~,X~'],

where K 2= O. The polarization tensor $~,v is symmetric and traceless and satisfies K .~ $,~ = 0. Let us suppose that the wave propagates in the z-direction so that

(5) K~ = (k, 0, 0, k).

There are two polarizations that correspond to physical gravitational waves. Taking into account eq. (5), we consider in our calculations

(6) $11 = -- $22 = + 1,

all other components vanish. The other polarization is obtained from the above one via a rotation of = /4 around the z-axis. With the above choices, eq. (2) gives

~ ' , (7a) 2 0 - 2 ~ = Akexp [ik(X ~ - X31][(21) 2 - (X1) 2 - (2212 + (X2)21,

p ! r

(7b) s _ .~1 = i A k e x p [ik(X ~ - X 3 ) ] [ s 1 ( s _ s _ X 1 ( X 0 _ X 3 ) ] ,

(7c) 2 2 j~2 = i A k e x p [ik(X o _ x 3 ) ] [ 2 2 ( 2 o _ 2 3 ) _ ~ 2 ( ~ 0 _ ~3)1 '

Pt i t

(7d) J ~ - X 3 = ~Akexp [ik(X ~ -X3)][(X1) e - (X1) 2 - ()~2)2 + (X2)e].

Notice that the variable X ~ X 3 satisfies the D'Alembert (free-wave) equation.

STRING AS A GRAVITATIONAL ANTENNA 1263

The amplitude A is exceedingly small and we look for a perturbative solution in the power series of A:

(8) X~ = X ; + A . Xc ~ + ....

The fwst term obeys the free-wave equation (we suppress the index f )

11

(9) X~ - X ~ = O.

We are interested in a string with f'Lxed ends. One end of the string defines the Origin of the coordinate system, while the other end is placed at the point Ri. Then the boundary conditions take the form

X i (O, ~) = O, X i (re, v) = R i , i = 1, 2, 3, (1O) , ,

X ~ (O, v) = X ~ (~, v) = 0.

The solution of eq. (9) which respects the above boundary conditions is given by the following Fourier expansion:

a o

(11) X~ v) = p~ + i v a n , , o --n-COsnaexp [ - inv],

Ri ~ (12) Xi(a , z) = ~ - + V2 ~ s i n n z e x p [ - inz] .

n ~ O

/~* The reality of X implies a~ = a~_~. It is useful to introduce the variables

(13) ~+ = v + o - , ~_ = v - z .

In terms of $+ (left-moving modes),

Ri i (14) X i = ~ ( G - $ - ) +

1(o R3) (15) X ~ P - ~ G §

~_ (right-moving modes), the X's become

-~-[exp [ - i n G ] - e x p [ - in~_ ]],

i E (a~ - a n) - ~ ~ o n exp [ - i n G ] +

+ ~ p O + _ 7 ~_+ V ~ ~ n

Our calculations are carried out in the conformal gauge, where

!

(16) (X _+ X) 2 = O.

Still there is a residual symmetry left over, since any reparametrization

(17) ~+ ---> ~+ + f ( G ) , ~- --~ ~- + f(~-) ,

exp [ - in~_ ].

where f is a 2~ periodic function, preserves the gauge choice. We can use this freedom

1264 A. NICOLAIDIS and A. TARAMOPOULOS

to eliminate the ~+ oscillations from eq. (15), i .e. we can se t [ l l ]

(18) a ~ = a~ .

Alternatively we impose the conditions

(19) X0 _ ~3 = p0 _ _ _

By using eqs. (16) and (19) we obtain

P

R3 X o _ ~8 = 0.

1

p O R~ a~ + - - ~ ~,

where

(21) L~ E • i i = a m al . i=1,2 m + l = n

m,l # O

The above relations clearly indicate that the independent dynamical variables are the transverse oscillators (in the x, y directions).

The term X~ satisfies the following equation:

(22) 3~+ ~ _

_ i k e x p [ i k ( v + + v _ ) ] [ a + w + a _ v _ + a _ w _ ~ + v + ] , 2

where, due to our choice of gauge,

(28) i v+ ($+) = a_ ~+ ,

v_ (~_) = a+ ~_ + 2h3($_),

R1 hi w• (~• = _+ ~--~• _+ (~•

hi(f)= i ,~,o ai - ~ -~- exp [ - i n ~ ] ,

~• denotes differentiation with respect to ~• Direct integration yields

(24) x : - i 2 V2 exp [ i k ( X ~ - XS)] �9

] �9 - - exp [ - i n $ + ] a..________%_~ exp [ - i n ~ _ ] - k a _ - n k a + - n

c+ r - c_ r ).

We defined above the zero-mode coefficient a~ = R 1/V ~ 7:. The function r and the constants C• are determined from the boundary conditions: X 1 has to vanish at

= 0, =. This requirement is satisfied if we choose for r162 the function obtained from the first term in eq. (24), where we replace ~_ (~+) by $+ (~_). The

STRING AS A GRAVITATIONAL ANTENNA 1265

constants C_+ are given by

(25)

C + 1 - exp [ika_ 27:]

1 - exp [ik(a+ + a_)27:] '

exp [ika_ 27:] - exp [ik(a+ + a_)27:]

1 - exp [ik(a+ + a_)2=]

for Xc 2, while X ~ and X~ ~ are A similar solution holds obtained from the constraints.

Equation (24) displays a resonant behaviour. Since the total energy E of the string is given by

(26) E = r:Top ~ ,

a resonance occurs for the n-th left-moving mode, whenever the condition

(27) k(E - ToR3) = 2rzTon

is satisfied. For the m-th right-moving mode the resonance condition is

(28) k(E + ToR3) = 2rcTom.

From the preceding analysis it is clear that To R is the potential energy of the string and ToRi corresponds to the/-component of the total momentum carried by a string with free ends. Recalling that 2=T0 n is the spacing between the mass squared of the n-th excited level and the mass squared of the ground state of the string[12], eqs. (27), (28) become

(29) (K + P+)2 = M ~ , (K + P_)2 = M ~ ,

P+ (P_) is the total four-momentum carried by the left(right)-moving modes. The interpretation of the above relation is obvious: the graviton hits and excites the string from the ground state to higher vibrational modes. It is also useful to recast the resonance conditions in the following form:

(30) [R 2 + 2r~2L0] 1/2 - R3 = n2,

(31) [R 2 + 2r:2Lo] 1/e + R3 = m2,

where ~ is the wavelength of the gravitational wave. Assuming that both eqs. (30) and (31) are valid, then from a Fourier analysis of the string motion we can extract R~ and )~, i.e. the direction and the energy of the incident wave. For a gravitational wave incident from a direction perpendicular to the string, R3 = 0, m = n, while if it is parallel to the string, R3 = R. An array of strings pointing into different directions should provide us with considerable information about gravitational radiation bathing the Earth.

It is beyond the scope of the present work to examine the technical problems a stringy antenna might pose. With regard to other devices, the proposed detector is a resonant wide-band detector and highly directional. The string will be most sensitive to gravitational waves of wavelength 2 comparable to its length R. A string few kilometers long will be sensitive to the gravitational radiation emitted by a black hole or a supernova (10'~Hz), while a string few meters long will be sensitive to a

1266 n. NICOLAIDIS and ), TARAMOPOULOS

cosmological radiation (10 s Hz)[13]. Therefore, our gravitational antenna will be efficient in detecting the high-frequency part of the gravitational radiation. One can possibly view our device as a ,,violin string~, through which we might ,,listem, to the fluctuations of the space-time and especially the first instants of our Universe.

This work was started while one of us (AN) was visiting the University of Crete and he would like to thank B. Xanthopoulos for the warm hospitality and many useful discussions.

R E F E R E N C E S

[1] J. WEBER: Phys. Rev., 117, 306 (1960). [2] A clear and comprehensive discussion on the detection of gravitational waves is given in C.

MISNER, K. THORNE and J. WHEELER: Gravitation (Freeman, San Francisco, CA, 1973), chapt. 37.

[3] n. FORWARD: Gen. Relativ. Gravit.,, 2, 149 (1971); N. ASRBY and J. DREITLEIN: Phys. Rev. D, 12, 336 (1975).

[4] S. RASBAND et al.: Phys. Rev. Lett., 28, 253 (1972). [5] D. DOUGLASS and J. TYSON: Nature, 229, 34 (1971). [6] A. POLYAKOV: Gauge Fields and Strings (Harwood Academic Publishers, Chur, Switzerland

1987). [7] Y. NhMBU: in Proceedings of the International Conference on Symmetries and Quark

Models, held at Wayne State University (Gordon and Breach, New York, N.Y., 1970); T. GOTO: Progr. Theor. Phys., 46, 1560 (1971); L. SUSSKIND: Nuovo Cimento A, 69, 457 (1970).

[8] A, VILENKIN: Phys. Rep., 121, 263 (1985). [9] M. GREEN, J. SCHWARZ and E. WITTEN: Superstring Theory (Cambridge, University Press,

Cambridge 1987). []0] M. ADEMOLLO et al.: Nuovo Cimento A, 21, 77 (1974). [11] J. ARVIS: Phys. Lett. B, 127, 106 (1983). [12] C. REBBI: Phys. Rep. C, 12, 1 (1974). [13] W. PRESS and K. THORNE: Ann. Rev. Astron. Astrophys., 10, 335 (1972).