Volume 35A, n u m b e r 6 P H Y S I C S L E T T E R S 12 Ju ly 1!)71

f hO

o.8h~

kr

1.6

_,'2- 'E

1.2

t,..,-,, 0.8 I00

Fig.2. Frequency f((L) and real wavenumber kr(A ) of the ionization waves against the current I.

obtained for each mode of co l l i s iona l drift waves where the re a re the mode compet i t ions between different modes [ 1,3]. Suppression by other

modes of the ionizat ion waves does not occur in our exper iments , because only one mode of the wave is s e l f - exc i t ed in the p lasma. No theory is avai lable for nonl inear behav io r s of the ion iza- tion wave, so that phys ica l explanation will be fur ther problem.

The author would like to thank P r o f e s s o r Y. Hatta for his encouragement .

References [1j H.W. Hendel. T.K. Chu and P.A. Politzer, Phys.

Fluids. 11 (1968) 2426. [2] K. C. Rogers and F. F. Chen. Phys. Fluids 13 (1970)

513. [3] R. E. R o , ~ e r g and A. Wong, P h y s . F l u i d s 13 (1970)

661. [4] T. H. Dupree, Phys. Fluids 11 (1968) 2680. [5] M. Sato, Beitr. Plasma Phys. 9 (1969) 371. [6] M. Sato, Phys. Rev. Lett. 24 (1970) 998.

I N T E R A C T I O N O F A H A R M O N I C O S C I L L A T O R W I T H G R A V I T A T I O N A L W A V E S *

E. FREHLAND Abteilung Theoretische Physik, Physikalisehes Institut der Universitttt, Freiburg im Breisgau, German3,

R ece i ved 10 May 1971

For a simple model of a harmonic oscillator the energy-momentum-stress tensor is determined and the energetic interaction with a plane gravitational wave of the same frequency is calculated.

The pos i t ive r e su l t s of J o s e p h W e b e r ' s g r a v - i tat ional wave expe r imen t s [1] could poss ibly open a comple te ly new f ie ld for a s t ronomica l ob- se rva t ions . But, apar t f r o m the di f f icul t ies con- cern ing the t heo re t i ca l desc r ip t ion of such ex- t r e m e events , which alone come into quest ion as ' radia t ion s o u r c e s ' , t he re does not yet exis t a r igo rous theory for the in te rac t ion of so l id - s t a t e bodies (detectors) with grav i ta t iona l waves , ne i ther within a ' c l a s s i c a l ' g e n e r a l - r e l a t i v i s t i c theory of e las t i c i ty nor on quan tum-mechan ica l foundations. Weber h imse l f has made the fo l low- ing ansatz for an approx imate ' c l a s s i c a l ' t r e a t - ment of this p rob lem [2]: Under the influence of

* Work supported by the Deutsche Forschungsgemein- schaft.

432

a dr iving fo rce an e las t i c body begins to osc i l la te . The dr iving fo rce with 'damping ' and ' r e s to r ing f o r c e ' is the Riemann t ensor R~7 5 of the g r a v i - ta t ional field.

But the only thing we know is that R~75 de t e r - mines the 'geodesic dev ia t ions ' of in f in i tes imal neighbouring, 'freely fa l l ing ' p a r t i c l e s , which in a ce r t a in sense can be looked upon as the r e l a - t ive acce l e ra t ions . In this sense R~75 is ana lo- gous to the f i r s t de r iva t i ve s of a c l a s s i c a l f ie ld s t rength and the equation of geodes ic deviat ion c o r r e s p o n d s to the f i r s t t e r m of some T a y l o r ' s s e r i e s .

Under the assumpt ion that this approximat ion is val id , it s t i l l r e m a i n s an open quest ion wheth- e r the in te rp re ta t ion of 'geodesic devia t ion ' as the dr iving fo rce in the case of (elast ic) inter-

Volume 35A, number 6 P H Y S I C S

acting mat t e r can be just i f ied. It should be shown whether and under which condit ions f r o m the r igo rous equat ions of motion for the ene rgy - m o m e n t u m - s t r e s s t ensor T ~ u of the de tec to r

#u T , v = 0 (1)

(u, ~t = 1, ... 4;,,v covar ian t differentiat ion) W e b e r ' s heu r i s t i c equations of motion rea l ly fo l - low as a good approximat ion on the bas is of a g e n e r a l - r e l a t i v i s t i c theory of e las t ic i ty .

As a f i r s t step for invest igat ing this p rob lem we d i scuss the energe t ic in te rac t ion of a s imple harmonic o sc i l l a to r (two equal m a s s e s rn of length (b - a) connected by a spr ing of length 2a, e igenf requency co, vanishing damping) with a plane grav i ta t iona l wave of the same frequency. The o s c i l l a t o r s f r ee osc i l l a t ion ( d x dt = + Acoswt ) is to be sl ightly d is turbed by the wave. The f ield of the o sc i l l a to r i t se l f is neglected.

F i r s t we ca lcu la te the energy absorpt ion of this s imple de tec to r model using the eq. (1). In the ' r e s t f r a m e ' the energy change is given by the d i f fe rence between the two volume in tegra l s over T 44 before and af te r the overf lowing of the wave, and t h e r e f o r e [3] by

l 0 f ( ~ T 4 4 ) d V = s f ( ~ g p u ) T"u dV (2)

where g/xu is a s s u m e d to be the me t r i c of the plane wave t r ave l l ing in the z - d i r e c t i o n [4]. With the wel l -known ansatz T Ix u = PoC2 uPu u .e#.z u (Po: poper density of ma t t e r , u tz: 4 -ve loc i ty of ma t t e r , cr gu : s t r e s s tensor) we get in the lowest approximat ion in co and without taking into ac - count g rav i ta t iona l in te rac t ions the following nonvanishing components of TtXU :

Ixl a: TXX=-PoAco2(b - a ) sincol (3)

a < Ixl --< b :TXX= PoA2w2cos2cot

-Aco 2 (b - t x i) sinco t

T x4 =T4X=±Oo A w c . cos co/,

T44 = Po ( c2 + A2 c°s2 wt)

LETTERS 12 July 1971

With gxx = a c o s w t + fl sin ¢0t the energy change h E ( T ) pe r one per iod T then becomes :

T

t=0 a+b

Accord ing to the phase d i f fe rence between wave- and de t ec to r -o sc i l l a t i on , de te rmined by c~, AE is pos i t ive , negat ive or even zero .

We compare this r e su l t with W e b e r ' s theory, where we have to solve the e l emen ta ry d i f fe ren- t ia l equation

d2x i ~2 (5) + w 2 x _c 2 x dl 2 = R4x 4 .x = ~ c~12Yxx • x =

= - a w 2 c o s w / - ~w 2 sin wl

The energy change can be ca lcu la ted f rom the change of the ampli tude. Indeed the r e su l t is the same as above in eq. (4). Also one can show a g r e e m e n t for o ther d i rec t ions of incidence of the wave.

This resu l t s e e m s to indicate that W e b e r ' s ansatz could yield a good approximat ion under the condit ions, that a) the de tec tor is r emo te f r o m the radia t ion source and b) the d imens ions d of the de tec tor a r e sma l l compared with the wavelength (i.e. w d / c << 1).

Fu r the r de ta i l s and the extension to the actual s i tuation in W e b e r ' s expe r imen t s , that the de- t ec to r , or ig inal ly being at r e s t , is exci ted by the in te rac t ion with the wave, will follow.

RcfeYence8 [1] J.Weber. Phys. Rev. Lett. 22 (1969) 1320: Phys.

Rev. Lett. 24 (1970) 276. [2] J. Weber, General relativity and gravitational

waves (New YGrk, 1961) p. 124ff. [3] A. Einstein, Sitz. Ber. Pr. Ak. Wiss. (1918) p. 154. [4] Landau-Lifsehitz, Klassisehe Feldtheorie (Berlin,

1967).

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