LETTERS AL NUOVO CIMENTO VOL. 23, N. 9 28 Ot tobre 1978

On the Detectability of Advanced Gravitational Radiation.

P. FORTINI

Is t i tu ta di A s t r o n o m i a dell' Universi th - Bologna

Is t i tu to di .Fisica dell' Universi th - Bologna

F. FULIGNI

.Laboratorio T E S R E del C . N . R . - Bologna

C. GUALDI

Is t i tu to di F i s i c a dell' Universi th - ~'errara

(r icevuto il 2 Giugno 1978)

The absorber theory of WH~ELV, R and FEYNMAN (1) for e lec t romagnet ic radia t ion has been ex tended by RosEN (3) to g rav i t a t iona l waves. As i t is wel l known Maxwell equations, be ing symmet r ica l under t i m e reversal , admi t bo th r e t a rded and advanced solutions which, f rom a ma thema t i ca l po in t of view, are equa l ly acceptable. This seems to conflict w i th t he fact t ha t r e ta rded po ten t ia l s only are in agreement wi th experi- ment .

To account for th is asymmet ry , WHEELER and FEYNMAN (1) assume tha t the radia t ion field is always bui l t by an advanced and a re ta rded par t and expla in t he nonobservabi l i ty of advanced solut ions wi th the fact t h a t m a t t e r in the universe can complete ly absorb e lec t romagnet ic radiat ion. The source (e.g. an accelerated po in t charge) generates a symmet r ic po ten t ia l , the re ta rded pa r t of which is absorbed by surrounding ma t t e r (absorber) which in its t u rn produces a symmet r i c field, the advanced park of which arr ives at the source at the same ins t an t of emission of the or iginal field and com- bines wi th i t to g ive the full r e ta rded observed potent ia l . The correct radia t ion reac- t ion on the e m i t t i n g charge is also ob ta ined in this way.

All this is however possible only if enough m a t t e r is present to absorb the radia- t ion completely.

As the l inear ized Einste in equat ions show the same t ime symmet ry , so admi t t ing bo th advanced and re ta rded solutions, R o s ] ~ (2) has inves t iga ted whether m a t t e r in the universe is enough to produce the same effects on g rav i t a t iona l waves, as for the e lec t romagnet ic radia t ion . The result is tha t , because of the ex t reme weakness of g rav i ta t iona l in terac t ion , the effects of advanced potent ia ls are always present. The

(1) J. A. WHEELER and R. P. FEYNMA-~:: Rev. =~lod. Phys., 17, 157 (1945). (2) N. ROSEN: Lett. Nuovo Cimento, 19, 249 (1977).

345

346 P. F O R T I N I , F. FULIGNI and c. GUALDI

conclusion is then that , ex tend ing the W h e e l e r - F e y n m a n suggestion to the grav i ta - t ional field, g rav i ta t iona l waves can exist only as s t and ing waves.

I n th is paper we inves t iga te whe ther exper imenta l effects exis t which allow, at least in pr inciple , to d iscr iminate be tween the two emission mechanisms, i .e . the usual (retarded) and the symmet r ic (Rosen's) one.

To this purpose let us consider a d is tant source of bo th e lec t romagnet ic and gravi - t a t iona l r ad ia t ion as, e.g . , an a symmet r i ca l pulsar or a b ina ry system. As we shall see below, a g rav i t a t iona l and an e lec t romagnet ic wave t r ave l l ing in the same di rec t ion do not in teract . Therefore if the convent iona l theory of emission ( i .e . only r e t a rded potent ia l s are considered) holds for g rav i ta t iona l waves, there will be no resul t ing effect. If, on the o ther hand, Rosen suggest ion applies, there wil l be an in teract ion be tween the advanced par t of the s tand ing grav i ta t iona l w a v e and the e lec t romagnet ic w a v e which propagates in the opposi te direct ion.

Because of the large dis tance of the source, the rad ia t ion emi t t ed can be t r ea t ed in t he p lane-wave approximat ion . We shall therefore consider e lec t romagnet ic and g rav i t a t iona l plane waves p ropaga t ing along y-axis and t r ea t the problem in the l inear approximat ion .

The met r ic tensor of the g rav i t a t iona l wave can then be wri t ten as

(0) (1) gi~ = gil~ + hi1~ ,

where ~Ik = (-- l , l , 1, 1) is t he Minkowski tensor and hiz= are quant i t ies of the first order. The nonvanish ing components of h~k are

(2) hll = - - h3a, h13 = h31.

In th is approx imat ion the e lec t romagnet ic tensor Fik can be split , like the met r ic (0) (1)

tensor, in a fiat space par t /~k and in a first-order pe r tu rbed par t F ~ :

(o) (1~

(3) F i k : F i k q - F i k �9

By the use of Maxwell equat ions in vacuum, it is easi ly verif ied tha t the pe r tu rbed par t of the e lect romagnet ic field is g iven by the fol lowing equat ions (3):

{1) (0) (0) (4) F k = h S ~ F h ~ F i,k i, sr-~ ir,s

(lJ (1) (1)

(5) Fik,m -~ ]~mi,k ~- Fkm,i = 0 .

The nonvan i sh ing components of the field in a plane e lec t romagnet ic wave propaga t ing (o) (o) (o) (o)

in the pos i t ive y direct ion are F10, F12 and F3o, F32 sat isfying

(0) (0) (0) (0) (6) Fto = - - FI, . , /~'30 = - - Fa2"

(3) F . I . COOPERSTOCK: An?t . O] Phys., 47, 173 (1968) .

ON T H E D E T E C T A B I L I T Y OF A D V A N C E D G R A V I T A T I O N A L R A D I A T I O N 347

I t fol lows i m m e d i a t e l y tha t , if the g rav i t a t iona l wave also propaga tes along the posi- t ive y direct ion, t he r igh t -hand side of eq. (4) vanishes ident ical ly , thus leaving the e lec t romagnet ic field unper turbed, as i t was ant icipated.

According to Rosen suggestion, to the above considered par t of the grav i ta t iona l field, corresponding to re ta rded solution, one must add the advanced one, which, in this case, is a p lane w a v e of the same a m p l i t u d e but propagat ing along t h e nega t ive y-axis.

Wi thou t loss of general i ty , we assume circular polar izat ion for the grav i ta t iona l wave and l inear polar iza t ion for the unpe r tu rbed e lec t romagnet ic wave, so tha t

(7)

(S)

and

(9)

(lO)

hll = - - h ~ = A[cos ( k , y - - % t ) + cos (kgy + wgt)] ,

h13 = h31 ~ B[sin ( k g y - - % t ) + sin (k~y + %t)]

(o) (0) --~'10 : E~ = E cos (ky - - cot) ,

~o~ (o) F ~ = B~ = - - E cos (ky - - ~ot) .

Insert ing these expressions for the fields in eq. (4) and taking into account condit ions (5), one obtains for the per turbed field a set of D 'A lember t nonhomogeneous equations. A p lane-wave solut ion of these can be obta ined ei ther by direct subs t i tu t ion or by in tegra t ing over the contr ibut ions due to planes perpendicular to the direct ion of propagat ion (4).

Af te r a s imple and tedious algebra, one f inally gets for the nonvanish ing corn- (0)

ponents of Fi~:

(1) (1) A E Fie = E~ -- ~ [k~ sin (}:gy § r sin (ky - - ~ot) + k cos (k~y + ~%t) cos ( k y - - wt)],

(1) (1) A E F12 = B~ = ~ [kg sin (k~y § oJ~t) sin (ky - - cot) - - k cos (]%y § w~t) cos (ky - - ~ot)] ,

/F3o = E~ ~ - [ - - k, cos (k~y + % t) sin (ky - - cot) + k sin (kgy + % t) cos (ky - - wt)] ,

i~'23 = B~ = ~ - [kg cos (key § wgt) sin (ky - - ~ot) -t- k sin (k~y § wgt) cos (ky - - (ot)] .

W h a t one genera l ly measures is the in tens i ty of the radiat ion. To evalua te the effect of the in te rac t ion one can then de te rmine the energy densi ty in a local or thonormal frame, by forming there the 00-component of the ene rgy -momen tum tensor out of (0) (1) Fi~ + F~k. This gives to first order in h~k , af ter averaging over one period of the e lec t rmnagnet ie wave , which is expected to be much shorter t han t h a t of the gravi ta- t ional one,

W = ~'[1 + A cos (k~y + o~ t ) ] ,

where IV is the to t a l energy dcnsi ty and the unper tu rbed one.

(~) R . P . F E , ' , ~ ) I A ~ : Lectures ore Physics, Vol . l , C h a p . 30 ( R e a d i n g , 51ass . , 1969) .

34S P. F O R T I N I , F . FULIG!WI and C. G U A L D I

This result shows that the electromagnetic energy density is modulated by the gravitat ional wave with an amplitude A (the gravitational component with amplitude B gives in this case only a second-order effect}. This modulation propagates toward the source, as one could expect as the interacting part of the gravitational wave travels just in tha t direction.

This phenomenon could in principle be observed by forming coincidences between two distant detectors placed along the path of the radiation. Unfortunately the small- ness of the effect places it below the actual experimental possibilities. Remembering in fact that , due to the presence of the advanced wave, a radiating system does not lose energy via gravitational emission (with a consequence of a rather long life), we can consider one of the most favourable cases, i . e . a very. close binary system made up of two neutron stars. This hypothetical object can exchange with the waves as much as 10 4e erg/s, thus giving on the earth, for a distance of about 1 kpc, A ~ 10 -18 (5).

0n the other hand, this verifieability in principle of Rosen suggestion seems attrac- tive, as only in the domain of gravitat ional waves one can possibly hope to check the assumption of advanced radiation. In fact, electromagnetic radiation, being com- pletely absorbed by matter, cannot exhibit the (( advanced ~ behaviour, while gravita- t ional radiation (~ should )) exhibit it, as being practically unabsorbed by the mat ter in the universe.

(5) G. CALLEGARI and A. M. NOBILI; M t m . deZla Soc. Astv . ~rtal., to be published.