IL NUOVO CIMENTO

NOTE BREVI

VOL. 107 B, N. 11 Novembre 1992

Space and Time Measurements in the Field of a Gravitational Wave.

P. FORTINI(1) and A. ORTOLAN (2) (1) Dipartimento di Fisica, Universit& of Ferrara - via Paradiso 12, 44100 Ferrara, Italia (2) INFN, Sezione di Ferrara - via Paradiso 12, 44100 Ferrara, Italia

(ricevuto 1'11 Giugno 1992; approvato il 27 Ottobre 1992)

Summary. -- We give here the separate contributions to the figure pattern of a gravitational-wave interferometer due to time delay and mirror motion in the Fermi frame. Our analysis allows for a test of inertial properties of the reference frame of the interferometer.

PACS 04.20 - General relativity. PACS 04.80 - Experimental tests of general relativity and observations of gravi-

tational radiation.

An interferometer used as a detector of gravitational waves consists of two mirrors and a beam splitter located at the vertices of an isosceles right triangle; with suitable mirror suspensions the apparatus can be considered as free falling in the field of the incoming gravitational wave[l]. A locally inertial system (Fermi Normal Coordinates) can be attached [2] to the measuring apparatus by choosing the ,<fiducial geodesic, as the timelike geodesics of the beam splitter.

Let us consider a plane gravitational wave with wave vector k along z-axis. The x and y axes are taken in the direction of + and x polarizations. In this system the direction n of an arm of the interferometer with length L is n-= (sin~cos~, sin $ sin ~, cos $), where ~ and ~ are the usual polar angles. In what follows we limit our analysis to one arm of the interferometer; the complete figure pattern can be found substituting n with the other arm direction n' and summing both results.

The perturbation to the metric tensor in FNC y,, can be found by means of a gauge transformation from that of the TT-gauge h~

( 1 ) = - -

where e', for a monochromatic plane wave with amplitudes A+ and A• and frequency

1329

1330 P. FORTINt and A. ORTOLAN

o) = I-kl/c, are given by[3]

" o(t, x) = _ 1 ( cos ( k ~ - ~ t ) kz kz

~1 (t, x) = ( cos (kz - o~t) cos (o~t) \ kz kz

(2) cos (kz - o~t) cos (~ot)

~2 (t, x ) = k z kz

_ 1 [ cos (kz-o~t) e3(t, x) = Yz[ kz

(~ot) \ c o s

sin (o)t)) [A+ (x 2 - y2) + 2A• xy] ,

sin (oJt)2) (A+ x + A• y)

sin (oJt)) 2 (A• x - A + y ) ,

(o~t) sin (oJt)) [A + (x 2 _ y 2) + 2A • xy] .

COS

kz /

The effect of the gravitational wave on light rays n ~ = (1, n) is easily found from the equation ds 2= 0

(3) n~ d ~ t - 1 ~ j dx o - 2 Too - ]'oi n~ - ~ ~'ij n n ,

where dz ~ is the perturbation of Minkowsky's coordinates and x ~ is the Minkowsky time. Integrat ing eq. (2) on the path ~'1 of the photon from the beam splitter to the mirror and on ]'2 from the mirror back to the beam splitter (characterized by the unit null 4-vector m ~ = (1, - n ) ) and adding the two contributions one gets

(4) [~0 (t = 2L/c , O) - n . a(t = 2L/c , 0)] - [~0 (t = 0, 0) - n . a(t = 0, 0)] -

S ~%n dx + ?%m dx .

; Y2

In the left-hand side the first four terms give the phase shift while the last gives, as we shall see presently, the space displacement of the mirror so that the right-hand side can be interpreted as a time delay - 2c At. I f we substitute eq. (1) in the above formula we get

(5) - 2czXt = ~ zX~ - O, (sv n ~) dx ~ - O~ (~ m~) dx ~ =

= "~ezx~ -- n" s . ( c t + L, Ln) + m ~ ( c t + L, Ln) = -ff-s - 2n .s(ct + L, Ln) 2 ~ '

where A~ = r~/)~ = I h~ ~n~dx~ + I h~ ~m~dx~ is the phase shift calculated in[4] in TT

gauge and i~ is the photon wavelength. This formula agrees with our interpretation of n ' a ( t = L / c , Ln) as the mirror displacement. We remind in fact that e(t ,x) represents the displacement of any free-falling particle with respect to the fiducial geodesics as can be seen by integrating the geodesic equation in the Fermi frame. F rom eq. (2) we can calculate the figure pattern of the mirror displacement

(6) AL = k - 1 COS (2~) ~sin~ {cos [~ot + ~oL/c(1 - cos ~)] - cos (~ot + o~L/c)}

For kL << 1 this equation gives the usual figure pattern[5] for an interferometer

SPACE AND TIME MEASUREMENTS IN THE FIELD ETC. 1331

showing that this is the leading te rm in the phase shift when the wavelength is g rea te r than L. In the opposite case of short wavelength we can expand (6) in Bessel functions getting

sin (kL ) AL = A + k - 1 sin 2 ~ cos (2~) kL > f(cos ~).,

where f is a series of Legendre polynomials. The -diffractive- dependence on kL of the displacement figure pat tern AL is to be noticed.

The angular dependence of the time delay At is easily calculated from

~ A+ c cos (2~) sin 2 ~/ 1 cos (cot) (7) ~-=Ar co [ l + c o s , $ 2

(cf. [4]) and eq. (6),

cos sin 2

- - - c o s [~t - oiL~c(1 + cos-$)] +

A+ c l 1 cos (cot) (8) 2c At = cos 2~ sin e ~ - - +

co ( 1 + cos ~ 2

1 cos (cot - 2oJL/c)~ 1 - cos J

1 + cos___~ cos [cot - ~L/c(1 + cos ~)] 1 cos

sin 2 ~ cos (cot - 2coL~c) -

cos [(oJt + o~L/c(1 - cos ~)] - cos (o~t + o~L/c) - 2 cos

Expanding in series of powers of V -= kL = ~oL/c, one finds that At is of order 2 [2], while AL is of order 0 in V. The most favourable situation for measuring At is clearly realized when the two mirrors are a distance 2g/2 apart. This condition is fulfilled for a 3 km inter ferometer (today under construction) at a f requency of ~ 5 .10 4 Hz, which is not much bigger than the frequencies expected from gravitational collapses.

The theory here developed allows one in principle to tes t the inertiality of the reference system of the interferometer . In fact, if one performs in the same in ter ferometer measurements of h~ at various distance Li from the beam splitter (i.e. for different location of the end mirrors), one can tes t if relations (6), (7) and (8) are satisfied and therefore separate the different contributions AL and At.

R E F E R E N C E S

[1] R. L. FORWARD: Phys. Rev. D, 17, 379 (1978); A. GIAZOTTO: Phys. Rep., 182, 365 (1989). [2] P. L. FORTINI and A. ORTOLAN: Nuovo Cimento B, 106, 101 (1991). [3] L. BARONI et aL: Proceedings of the Fourth Marcel Grossmann Meeting on General

Relativity, edited by R. RU~FIN[ (Elsevier Science Pub., 1986), p. 641. [4] F. B. ESTABROOK and H. D. WAHLQUIST: Gen. Relativ. Gravit., 6, 439 (1975); F. B.

ESTABROOK: Gen. Relativ. Gravit., 17, 719 (1985); J. Y. VINET: J. Phys. (Paris), 47, 639 (1986); J. Y. VINET, B. MEERS, C. N. MAN and A~ BRILLET: Phys. Rev. D, 38, 433 (1988).

[5] B. F. SCHUTZ and M. TINTO: Mon. Not. R. Astron. Soc., 224, 131 (1987).