LETTERE AL NUOVO CIMENTO VOL. 24, N. 7 17 Febbra io 1979

On the Viability of the Bimetric Theory of Gravitation.

G. CALL]~GAP,,I and M. SAIJRI~

Is t i tu to di P'igiea dell ' U n i v e r s i t h - F e r r a r a

(rieevuto il 15 Dicembre 1978)

The fundamen ta l a ssumpt ion of Rosen 's b ime t r i e t heo ry of g rav i t a t ion (1) is t he existence in eve ry po in t of space- t ime of a R iemann ian tensor g/~ and a da t e space- t ime metr ic tensor ~ , . I n a local ly Loren tz ian sys tem of t he met r ic giJ E ins te in ' s equi- valence principle holds.

L]~]~ e ta l . (=) h a v e inves t iga ted i ts pos t -Newton ian l imi t and found tha t , wi th a suitable coundary condit ions, i ts P P N pa ramete r s are t he same as in genera l re la t iv i ty .

As shown by WILL (3) t h e P S R 1913 + 16 sys tem can give i m p o r t a n t clues as to the v iabi l i ty of Rosen 's theory .

In order to give a comple te dynamica l descr ipt ion of th is b ina ry sys tem the ~ol- lowing quant i t ies are needed: t he masses 3/1 and M 2 of t he two stars , t he s emima jo r axis a 1 and the inc l ina t ion i.

Among these four quant i t i es two expe r imen ta l re la t ions are known (4):

(1) a 1 sin i = 7.0043- 101~ c m ,

(M2 sing) a (2) (M, + M,)= - - 0 " 1 3 1 2 ~ r ~ "

If one ~ssumes t h a t the companion can be t r ea t ed as a poin t mass, genera l r e l a t i v i t y gives a th i rd re la t ion i .e . the per ias t ron shif t (5)

(M1 + M2'~ ~ {3) e b = 2 . 1 1 \ ) f O / d e g - y -1.

Equat ions (3), (2) and the observed va lue (b = 4.22 degree /years shows t h a t t he to t a l mass M 1 + M 2 is 2 .83M o and t h a t t he inc l ina t ion i mus t be be tween 21 ~ and 90 ~

(1) IV. ROSEN: Jo~rn. Gen. Rel. and Gray., 4, 435 (1973). (~) D. LEE, C. CAVES, W . T. NI a n d C. ]Vl. WILL: .Astro~hys. Jo~rn., 206, 555 (1976). (8) C. IVL WILL a n d D. 1~. EARDLEY: ~lslrophys. Journ., 212, L91 (1977). (4) J . H . TAYLOR, let. A. ItULSE a n d L. A. FOWL]~R: Astrophys. Journ., 206, L53 (1976). (~) T, DAMOUR a n d R . IRUFFI~I: Compl. Re~d., 279, 971 (19740.

237

238 G. ChLLEGAI~I and ~. SAu~I]~

The fundamen ta l difference be tween b ime t r i c t heo ry and genera l r e l a t iv i ty is t ha t t he former , in cont ras t wi th t h e la t te r , p redic ts the emission of dipole g rav i t a t iona l rad ia t ion by b inary sys tems wi th neu t ron stars .

Besides i t p redic ts for such systems, subs tan t ia l correct ion to s tel lar masses as deduced f rom orbit data. These differences be tween general r e l a t iv i ty and b ime t i i c theory are due essent ia l ly to t he fact t h a t in the l a t t e r ine r t i a l masses are not equal to g rav i t a t iona l masses for s t rongly g rav i t a t i ng masses. On the ground of these dif- ferences WiLL (a) could show t h a t the theore t ica l predic t ions of the b ime t r i c theory are in ag reement wi th observat ions on P S R 1 9 1 3 + 16 only if th is sys tem is fo rmed :

a) by two neu t ron stars of masses

M_~ ~ _ M 2 ~ 3 .S M o ,

b) by a neut ron s tar wi th mass M ~ 0 . 4 M o and a rap id ly ro ta t ing whi te dwarf or an he l ium star wi th mass M 2 ~ 2.4 M o.

In th is le t te r we show t h a t even these two possible cases lead to values for i t ha t seem to be s t rongly in cont ras t wi th t he expe r imen ta l evidence.

Fo r case a) we deduce f rom eq. (2) i . e . t he mass func t ion t h a t

in case b) we have

i ~ 3 2 ~ ,

i--~ 22 ~ .

Both these resul ts t u rn out to be too smal l to be consis tent wi th expe r imen ta l evidence. I n fac t t he sys tem P S R 1913-~ 16 exhib i t s g rea t va r i a t i on in t h e radia l veloci t ies ( ~ 1 0 -3 C)(6) and this is in ag reement wi th i~-~90 ~ ra the r t han wi th i~-~(22--32) ~ as p red ic ted by Rosen theory .

W e not ice t h a t genera l r e l a t iv i ty allows, on the contrary , i as h igh as 90 ~ This fact casts therefore v e r y s t rong doubts as to t he agreement be tween b ime t r i e

theory and exper iment . A direct ev idence of t he v iab i l i ty of b ime t r i c t heo ry can come wi th in two or three

years when i t will be possible to measure the combined effects of t he Doppler shif t wi th t h e g r av i t a t i ona l shift. Such effects are descr ibed by the p a r a m e t e r

where

e is t he eccent r ic i ty of the orbit ,

Pb is t he b ina ry per iod,

c is t he speed of l ight

which, once measured, will g ive the four th re la t ion to be added to (1), (2) and (3). ~ h e n i t wil l be possible to de te rmine comple te ly the orbit pa ramete r s as sug-

ges ted by (4).

(s) R. A. HULSE and J. H. TAYLOR: Astrophys . Journ . , 195, L51 (1975).

ON THE VIABILITY OF THE BIM]~TI:~IC TI-IEORY OF GRAVITATION 2 3 9

The situation o~ eases a) and b) depends critically on the omission of dipole gravi- tational radiation as predicted by Rosen's theory by Ros]~I~ (7), if matter in the uni- vers is not sufficient to absorb gravitational radiation completely, the theory by WnEELE~ and FEYNMAN (s) predicts that a system like PSR 1913 + 16 cannot radiate gravitational energy, as in such case one predicts the existence of advanced gravita- tional radiation which supplies the system with the energy radiated as retarded gu'avi- t~tion waves (stationary gravitation waves).

The possibility of experimental verification of this assumption has been sug- gested by FoI~INI et al. (o).

In case that the suggested experiments confirm the assumption, the criticism of Will (3) and the present letter will be no longer valid.

In this case other interpretations of gravitational theories as well as of fundamental physics will be possible.

(7) N. ROSEN: Left. N u o v o Cimento, 19, 249 (1977). (s) J . A. WHEELER a n d R . P . F E Y ~ t A X : Roy. lIIocL Phys . , 17, 157 (1945). (9) P. FORTINI, •. FULIGNI a n d C. GUALDI; Left. Nuovo Cimenlo, 23, 345 (1978).