IL NUOVO CIMENTO

VOL. 106 B, N. 1

Gennaio 1991

NOTE BREVI

Light Phase Shift in the Field of a Gravitational Wave. P. FORTINI(*) and A. ORTOLAN Dipartimento di Fisica dell'Universitd - via Paradiso 12, 44100 Ferrara, Italy I N F N , Sezione di Ferrara - via Paradiso 12, 44100 Ferrara, Italy

(ricevuto l'l Ottobre 1990)

Summary. - - We show that the response of an interferometric device to a monochromatic gravitational wave results from two different contributions: one due to the change of space geometry (mirror motion) and the other one due to the change of time rate (photon red- or blue-shift). An alternative exact derivation of phase shift is given. PACS 04.20 - General relativity. PACS 04.80 - Experimental tests of general relativity and observations of gravitational radiation.

As is well known any reference frame is indifferent to make theoretical calculations (and therefore one is free to choose the most suitable one) but when theoretical predictions are to be compared with experimental results, one is obliged to refer the predictions to the frame of the experimental apparatus. So, for instance, in the case of particle collisions with a fixed target, calculations can be made in the centre-of-mass system (where they are much simpler) but afterwards the results are to be transformed to the laboratory frame. This distinction of reference frames is even more important in general relativity where all measurements a r e devised to exhibit space and time effects. The frame of the laboratory (for Weber antennas, various kinds of interferometers, etc.) is the so-called Fermi normal coordinate system (FNC) [1]; therefore the theoretical predictions concerning the response of an experimental apparatus must be in the end referred to the FNC system connected with it. Here too there exists a particular coordinate system in which calculations can be easily performed, i.e. the TT-gauge coordinate system; however one must be very careful to transform back the TT results to FNC (TT system is not the laboratory system the more so that it is not even realizable by means of material bodies!) [2].

(*) On sabbatic leave at Laboratori Nazionali INFN di Legnaro, Via Romea 12, 35100 Padova, Italy. 101

102

P. FORTINI a n d A. ORTOLAN

The exact transformation formulae between the TT coordinates (primed) and FNC (unprimed) in the linear approximation of general relativity for a monochromatic plane wave propagating along z-axis and ,,plush) polarized (hnIT = - h ~ = A+ sin(kz' - o~t'), all other components of the metric perturbation being zero), as shown in ref. [3], are 9

x 2 _ y2 ( cos (kz - ~t)

t ' = t - A+

2c~

kz

) sin (oJt) ,

cos (oJt)

kz

kz

sin (o)t)) 2

cos (kz - cot)

cos (~ot)

sin(~ot) ),

kz

kz

y ' = y - A+ y z ' = z - A+

cos (~t)

kz

cos (kz - o)t) x ' = x + A+ x.

(1)

~

x2-y____~2(cos(kz-o~t) 2z kz

cos(~t)

sin (~t)).

kz

Equations (1) must be used, whenever one makes calculations in TT-system, in order to make clear the physical contents of theoretical formulae and to get predictions to be compared with experimental data. As an example of this procedure we consider an interferometer of the kind used as a detector of gravitional waves (see, e.g., [4] and references therein). This device consists of two mirrors and a beam splitter (such that the straight lines connecting the beam splitter with the mirrors are mutually orthogonal) suspended as pendula by means of wires; with suitable mirror suspensions the apparatus can be considered as free falling in the field of the incoming gravitational wave. For a wave travelling along the z-axis perpendicular to the plane (x, y) of the interferometer, a FNC system can be attached to the measuring apparatus by choosing the ,,fiducial geodesic, like that of the beam splitter (thought of as the origin of the coordinates) and the orthogonal axes x and y passing through the two mirrors which are at a distance L from the origin. A light beam travelling from the beam splitter to the mirrors experiences a phase shift due to the interaction with the gravitational wave and the motion of the mirrors with respect to the beam splitter. The calculation of this phase shift in the physical FNC frame is quite difficult but it can be performed fairly easily in the TT-frame as shown in ref. [5] by integrating the equation of the null geodesic ds ,2 = 0. It can be seen in fact that mirrors, initially at rest, stay at rest in TT at any subsequent time so that one can write, for a light beam travelling along the x-axis (y-axis) and scattered back by the mirror at the fixed distance x . = L ( y . = L ) at the time t, (see fig. 1), I_~ (2)

=tj-t~

+

|2y,

A+L sin~ sin(~ot~- 7) c ~ A+L

= t~ -

t~

sin v

c

where ~ = o~(t~ - t~)/2 ~- ~oL/c and oJt~ - v = (t~ + t~ )/2 = , t , . It is clear from eq. (2) that the only effect of the gravitational wave is a delay in the photon travel time: this fact is however true in the TT-system in which the total phase shift is A r = 4=(L/~e)(sin V/v)A+ sin (oJt2- V), where )~e is the light wavelength [5].

LIGHT PHASE SHIFT IN THE FIELD OF A GRAVITATIONAL WAVE

103

y! "~ 2

r~

--I

( t ' , ~,':L)

"'

(t' , ~[ =o)

n (t/, ~ x,t =L)

Fig. - The interferometer as viewed from TT-gauge; BS is the beam splitter and M1 and M~.are the two mirrors, t~ is the time at which the photons leave the beam splitter; t~ is their arrival time after the scattering by the mirror at a distance x, L(y,- L) and at a time t,. r

~

t

_

!

To find the phase shift as viewed by the interferomeber (i.e in FNC) one must apply eqs. (1) for the special value z'= 0 and y ' = 0. The TT coordinates of the three events (t{ , 0, 0, 0), (t,,L,O,O) and (t~0,0,0), representing, respectively, the departure of a photon from the beam splitter, its reflection by the mirror and its arrival back again at the beam splitter, are therefore related to the corresponding Fermi normal coordinates by ( t ~ = t i , x l ' = X l = 0 , y~=0, z~=0) (t,-t,+ + A + oJL2 cos(~ot,)/(4c2), x,-'- x, +A+Lsin(~ot,)/2, y,= O, z,= 0) and (t~ = t2, x~ = = x2 = 0, y~ = 0, z'2 = 0). It follows from the above that the extra time delay and the change of space distance due to the gravitational interaction are r

_

[ 2x, A+L sinv tAt~--t2--tl-~---=---~--(1--~ )sin(~t2--~), (3)

l a x ----2L - 2 x , = A+ L sin (~t2 - V).

Repeating the same reasoning for the mirror on the y-axis and remembering t h a t the total phase shift due to the gravitational interaction of photons is he = (2=/~e)[(hx - c Ate) - (hy - c Aty)] (note that Ax differs from c Atx because of the presence of a gravitational field which acts as an optically active medium) we get (4)

Ar = 4~ L

2e

sin ~ A + sin (o~t2 - ~).

This result coincides with the one evaluated in the TT-system in ref. [5] and this is a consequence of the scalar nature of nr Our result makes it clear that the interferometer really measures the combined effects of the displacements of the mirrors and the time delay of the photons.

104

P. FORTINI and A. ORTOLAN

In all physically meaningful situations envisaged for the detection of gravitational waves by interferometric devices (L ~-3kin, o~< 105 rad/s), it turns out that ~ = = ~L/c = L/2 < 1 and therefore the time delay effect in the laboratory frame goes as sin v/v - 1 ~ 72/6 < 1 so it is negligible compared with the space displacement. This is the reason why in the current literature (see, e.g., [6]) people, using the geodesic deviation equation in FNC, attribute the phase shift solely to the mirror motions: Ar = (4~L/~)A+ sin (~ot2). This deduction of the phase shift is approximate because the geodesic deviation equation holds only for interferometric arm length less than a gravitational wavelength while our derivation is exact in FNC. We stress that in the TT-system the phase shift is entirely due to the delay effect, while in the laboratory frame the latter is negligible and the phase shift is mainly due to mirror motions but, no matter how small is the time delay, the quantity which is really measured in this scheme of detection is the combined effect of hx and Ate.

REFERENCES [1] C. W. MISNER,K. S. THORNEand J. A. WHEELER: Gravitation (S. Francisco, 1973), p. 1004. [2] L. BARONI et al.: Proceedings of the Fourth Marcel Grossmann Meeting on General Relativity, edited by R. RUFFINI (Elsevier Science Publ., 1986), p. 641. [3] P. FORTINI and C. GUALDI:Nuovo Cimento B, 71, 37 (1982). [4] A. GIAZOTTO:Phys. Rep., 182, 365 (1989). [5] F. B. ESTABROOK,H. D. WAHLQUIST:Ge~. Rel. Gray., 6, 439 (1975); F. B. ESTABROOK:Gen. Rel. Gray., 17, 719 (1985); J. Y. VXNET:J. Phys. (Paris), 47, 639 (1986); J. Y. VINET, B. MEERS, C. N. MAN and A. BRILLET: Phys. Rev. D, 38, 433 (1988). [6] R. L. FORWARD:Phys. Rev. D, 17, 379 (1978).