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Astrophysics and Tachyons. t~. ~IGi~A~NI and E. RECA~. Istituto eli ~isiea Teorica dell' Universit5 - Catania. Istituto Nazionale di 2'isiea 2~ueleare - Sezione di ...
I L NUOVO CI)/s

VoL. 21 B, N. i

11 lY[~ggio 1974

Astrophysics and Tachyons. t~. ~IGi~A~NI a n d E . R E C A ~ Istituto eli ~ i s i e a Teorica dell' Universit5 - Catania Istituto Nazionale di 2'isiea 2~ueleare - Sezione di Catania Centro Siciliano di Fisica 2r e di Struttura della Materia - Catauia

(ricevuto il 5 Novembre 1973)

Summary. - - F r o m the very special relativity theory, the existence of Superlnminal reference frames and faster-than-light objects can be inferred, when its validity is not a r b i t r a r i l y confined to snblnminal velocities. Recently, in fact, special relativity has been extended to Superluminal inertial frames and to tachyons, thus allowing a self-consistent development of a (~classical t h e o r y ~) of tachyons. In this paper, it is shown t h a t the (~extended t h e o r y of relativity ~) allows in particular extending the Doppler-effect formulae to Superluminal sources. The possibilities of radioeontaet between subluminal and Superluminal objects are considered as well. The behaviour of taehyonie bodies in the gravitational field has been clarified. Taehyons are predicted to suffer a gravitational repulsion (so t h e y might be considered as the (( antigravitational m a t t e r ,) ; however, owing to their dynamics, t h e y happen finally to bend towards the gravitational source, as usual objects do (even if with different curvatures). At last the problem o f gravitational Cerenkov radiation (GCR) is studied. Contrarily to the previous authors, extended relativity predicts tachyons not to emit GCR in v a c u u m or in b r a d y o n i c (subluminal) media.

1. - Introduction. Extended relativity in ten points. t ~ e e e n t l y , s p e c i a l r e l a t i v i t y h a s b e e n e x t e n d e d (1-~) t o S u p e r l u m i n a l i n e r t i a l frames and faster-than-light objects (tachyons). It yielded the framework for (1) E. t~]~cA~r and 1%. ~IG~A~I: R i v . NUOVO Cimento, 4, 209 (1974). See also report INFN/AE-73/7 (Frascati, Nov. 15, 1973). (2) R. MIG~ANI and E. RECA•I: XVuovo Cimento, 1 4 A , 169 (1973); E r r a t u m , 1 6 A , 210

AaTRO]?HYSICS

AND TACHYONS

211

building up a sell-consistent (~classical t h e o r y ~ of t a c h y o n s (L~) (where classical m e a n s relativistic b u t not y e t q u a n t u m mechanical). L e t us here outline t h e logical scheme t h a t guided t h a t special r e l a t i v i t y extension. 1) T h e v e r y (Einsteinian) principle of r e l a t i v i t y refers to inertial frames w i t h c o n s t a n t relative velocity u, w i t h o u t a n y a priori restriction on the value of ~ ( u ~ e ) . W e wish to express t h e r e l a t i v i t y principle (I~P) in t h e f o r m (~physical laws of mechanics a n d e l e c t r o m a g n e t i s m are required to be covariant w h e n passing f r o m a n inertial f r a m e ]~ to a n o t h e r f r a m e /~ m o ~ i n g with cons t a n t relative velocity u, where - - c~ < u < + c~ ~. W e complete this postulate b y adding the r e q u i r e m e n t s t h a t the v a c u u m is c o v a r i a n t when changing inertial frame, a n d t h a t (~physical signals are t r a n s p o r t e d only b y positiveenergy objects ~ (cf. ref. (~)). 2) W e assume i) t h e I~P a n d ii) t h e following p o s t u l a t e : (~space-time is homogeneous a n d space is isotropic ~. 3) F r o m t h e previous a s s u m p t i o n s 2), t h e existence of a n i n v a r i a n t speed ~ follows b y generalizing the procedures in ref. (~); a n d experience shows t h a t such a velocity is t h e speed of light: ~ ~ e. 4) F r o m points 2) a n d 3) there follows a (~d u a l i t y principle ~ (DP) (~.~.6): (~the t e r m s b r a d y o n (*) (B), t a c h y o n (T), s u b l u m i n a l f r a m e (s), S u p e r l u m i n a l f r a m e (S) do not h a v e a n absolute m e a n i n g , b u t only a relative one. L i g h t speed i n v a r i a n c e allows a n e x h a u s t i v e p a r t i t i o n (7) of all inertial (u Xc) f r a m e s in two sets {s}, (S}, which are e x p e c t e d to be such t h a t a (subluminal) L o r e n t z t r a n s f o r m a t i o n (LT) m a p s (s}, (S} separately into themselves, a n d a Superl u m i n a l (~L o r e n t z t r a n s f o r m a t i o n ~ (SLT) m a p s (s} into (~q} a n d vice versa~). A one-to-one correspondence m a y be set b e t w e e n f r a m e s s(u) a n d S(U), w i t h ull U, where u ~ - U ~ c~/u, such a m a p p i n g b e i n g an ~inversion ~ (i.e. a partienlar con]ormal m a p p i n g ) .

208 (1973); E. RF.C~MI: i tachioni, in Annuario Scienza e Teenica 73, Enciclopedia EST-Mondadori (Milano, 1973). (8) ]~. RvC2~MI and R. MIGNANI: Lett. ~q~ovo Cimento, 4, 144 (1972); 8,~ 110 (1973); and references therein. (4) L. P~XK]~R: Phys. l~ev., 188, 2287 (1969). (~) See, e.g., V. Go]~I~I and A Z~ceA: Journ. Math. Phys., l l , 2226 (1970); V. B]~RzI and V. Go~I~I: Journ. Math. Phys., 10, 1518 (1969). These papers are to be generalized. (~) See also F. ANTrm'A: Nuovo Cimento, 10A, 389 (1972). (*) Slower-than-light particle. (7) See, e.g., A. AGODI: Lezioni di ]isica teo~iea, Catania, 1972 (unpublished).

212

R. ~G~AN~ and ~. R~eAI~

5) F r o m RP, light spee4 invariance and D P it follows (1-~.s) t h a t transformations between two frames ]1, ]2 m u s t be linear a n d such t h a t , for every t e t r a v e c t o r (four-position, f o u r - m o m e n t u m , four-velocity, ...)~ (1)

c 2t 2 - x ~ - - • (c at ' 2 - x '2)

(uXe).

The metric ( § will be a d o p t e d t h r o u g h o u t this work. N a t u r a l units ( e = 1) will be used, when convenient. 6) I n eq. (1), the sign p l u s holds for u ~ e , and m i n u s for u ~ e . In fact~ when going from a f r a m e s to a frame S, the type o~ four-vectors associated with the same observed object changes from timelike to spaeelike, or vice versa, as follows from point 4). For instance, in the f o u r - m o m e n t u m space, since for bradyons (2a)

E ~ - - p ~ =~ p~ : m~ > 0

(~ (GET) (*): x ' = :~: ~

x - - ut

tf

,

t - - q~x[c~ = -~ ~ ,

(- o o < ~ _ ~ ~~
fashion, where our p a r a m e t e r ~0 runs (with continuity) from 0 to 2 z r a d . I n particular, form (4)allows a straightforward (contimlous) geometrical interpretation of generalized Lorentz transformations, for fl~ >< 1 (see, e.g., Fig. 5, 8 of ref. (~)). F o r example, in the bidimensional case we have

(6)

x = C((' COS~ ~- ct' s i n a i , ~gr t = C t' cos q~-]- T sin

I

(0b y him. Namely, let us s t u d y (3) the relative position of t h e worldlines of b o t h b r a d y o n s a n d tachyons, a n d of the light-cones associated with one or more events of their history. Notice (3) t h a t - - i n a three-dimensional space-time, for s i m p l i c i t y - - t h e light-cones springing from a b r a d y o n i c worldline are strictly one inside the other, since the locus of the vertices passes inside the cones. This is not trne for the light-cones springing from a tachyonic world-line. I n this second ease, their envelope is constitutec[ b y two

ASTI~OPHYSICS

AND

TACHY0~S

217

planes: t h e v(t), t h e n T will receive (retarded) l~S's n n t i l ~ certain i n s t a n t ; a f t e r w a r d s t h e c o n t a c t will b r e a k . c) I f v ( T ) < v(t), t h e n the previous sequence will be reversed. example, T will s t a r t receiving l~S's only after a certain instant.

For

3) T will receive, f r o m a second, (

1)

brings a b o u t anothe} sigzt change, the equations of m o t i o n for a t a c h y o n in a gravitational field will still r e a d (]0')

d~x'~ r ' ~ d~'0 dx'~ (is, 2 t-~ 0z ds' ds'

0

(ds' spacelike).

Therefore, eqs. (]0) and (]0') are already G-eovariant; t h e y hold for both bradyons and taehyons. T h e y m a y be (still G-covariantly) written

(]3)

a"+r~u u =o

(/~]),

where a and u are four-acceleration a n d four-velocity, respectively. Notice t h a t , whilst dx/ds is a four-vector only with respect to the p r o p e r Lorentz group, on t h e c o n t r a r y u - dx/d% is a G-four-vector. As regards tachyons, let us, e.g., consider a t a e h y o n going towards a gravita-

220

tionM-field source.

R. ) I I G N A N I

and

~.

R:ECA1VII

Since it will experience a repulsive force, its energy (~-3) --

m o Oz

%/f12 __ 1

(fl~ > 1 , mo real}

will decrease ( c o n t r a r y to t h e b r a d y o n i c case); however (see ref. (1.3)) its velocity will increase. Therefore, we spoke a b o u t (~g r a v i t a t i o n a l repulsion ~) since t h e g r a v i t a t i o n a l force applied to t h e t a c h y o n is c e n t r i f u g a l (and not centripetal); however~ as a resul G the tavhyon will accelerate towards (*) the gravitational/ield source (similarly to a bradyon). I n conclusion, u n d e r our h y p o t h e s e s it follows t h a t : a) F r o m the d y n a m i c a l (and energeticM) p o i n t of v i e % t a c h y o n s a p p e a r as g r a v i t a t i o n a l l y repulsed. T h e y might be considered to couple n e g a t i v e l y to t h e g r a v i t a t i o n a l field (the source velocity being inessential!), i.e. to be the (~antigravitational particles ~). b) F r o m t h e k i n e m a t i c a l v i e w p o i n t , however, t a c h y o n s a p p e a r as (~falling down ~ t o w a r d s (*) t h e gravitationM-field source (the source velocity being inessential, as well as for electromagnetic field). L e t us explicitly r e m e m b e r t h a t usual antiparticles do couple positively to t h e g r a v i t a t i o n a l field. There is no g r a v i t a t i o n a l difference b e t w e e n objects a n d t h e i r antiobjects (in accordance with the (~r e i n t e r p r e t a t i o n principle ~): see ref. (1-3,~v) a n d t h e first point in Sect. 1). F o r tachyons, in ~he (( nonre]ativistic limit ~) (speed n e a r c V ~ : cf. eq. (7)) we would find for t h e three-force m a g n i t u d e no longer l~lewton% law F--~ Gmol mo2/r2~ b ~ t -

-

~

-~- G m~176

(fl~ > 1 ; m0~, m0~ real) ,

r 2

since w h e n passing to t a c h y o n s G t r a n s f o r m s into - - G . All t h e foregoing accords with t h e principle of duality. I n fact, given an observer O, all objects t h a t a p p e a r as b r a d y o n s to h i m will also a p p e a r to couple positively to the g r a v i t a t i o n a l field; on the contrary, all objects t h a t a p p e a r as t a c h y o n s to h i m will also a p p e a r to couple negatively to the g r a v i t a t i o n a l field. I n a n y case, bradyons, taehyons and

(*) Note added in proo/s. - See also R. 0. HETTEL and T. M. HELLIWELL: .NUOVO Cimento, 13 B, 82 (1973). In this paper the problem of taehyon trajectories in a Schwartzschild field has been considered within general relativity. Cf. also R. W. FULL~.~ and J. A. WHnEL~R: Phys. Rev., 128, 919 (1962). (1~) 0. M. P. BIT.A~IUK,V. K. D~S~eAND~ and E. C. G. SUDARSgA~: Am. Journ. Phys., 30, 718 (1962).

A S T E O P H Y S I C S AND TACHYONS

221

luxons (*) will all bend towards the gravitational-field sources, even if with different curvatures.

5. - On the so-called ~ gravitational Cerenkov r a d i a t i o n , and tachyons. Case of the vacuum.

All t h a t was previously said holds u n d e r the assumption of negligible gravitational-wave radiation. If we i m p r o v e d eq. (10), for bradyons in a gravitational field, b y allowing gravitational-wave emission, then the tachyonization rule (in the spirit of t h e duality principle) would still p e r m i t us to derive immediately the corresponding law for tachyons (now allowed to emit gravitational waves when i n t e r a c t i n g with t h a t gravitational field). B u t here we w a n t only to deal with t h e problem of t h e so-called ~(gravitational Cerenkov radiation ,> (GC]%). A usual bradyon (e.g. without charges, b u t of course massive) will generally radiate gravitational waves (~s) as follows: A) First of all, when (and if) it happens to enter a sensu late m e d i u m (i.e. a force field), with speed larger t h a n the gravitational-wave speed in t h a t field, then it is expected to radiate a cone of gravitational radiation (somewhat similarly to the sound wave cone emitted b y an ultrasonic airplane in air}. Such a cone should not be con]used with gravitational ~erenkov radiation, which on t h e c o n t r a r y has the following features (~9): i) it is e m i t t e d b y the bodies (or particles) of the ~naterial m e d i u m and not b y t h e travelling object itself, if) it is a function of the velocity (and not of the acceleration) of the travelling object. B) Secondly, when the considered bradyonie object (e.g. a galaxy) B happens for example to enter a cluster of uniformly distributed galaxies, with a speed larger than the gravitational-wave speed inside t h a t ~ material med i u m ~ (the cluster), then B will cause the cluster galaxies to emit gravitational waves such t h a t possibly t h e y will coherently s~m up to form a gravitational Cerenkov cone (19.~o). Igow, we w a n t to investigate i] and when t a c h y o n i c objects can emit cones of gravitational waves, and in p a r t i c u l a r cause emission of gravitational-radiation ~ e r e n k o v cones. (*) ~(Objects ~ travelling at the light speed, like photons and neutrinos. (is) See, e.g., J. WEBER: Gene~'al Relativity and Gravitational Waves (New York, N.Y., 1961); V. DE SABBATA: L'elettroteenica, 57 (9), 1 (Mflano, 1970). {~9) See, e.g, L. D. LA~DA~ and E. M. LIFsnITZ: Eleetrodynamies o] Continuous Media {Oxford, 1960), p. 357. ~20) Cf. J. W. J]~LLEu Ce~'enkov l~adiation and Its Applications (London, 1958).

222

i~. M1GNANI &lld ]~. R ] ~ C A M I

T h e first result is t h a t ]ree tachyons in vacuum will not emit gravitational Cerenkov radiation, or gravitational-wave cones, or any gravitational radiation. This result m a y be i n n n e d i a t e l y o b t a i n e d - - s e e ref. (~.~.9)--by using t h e t a e h y o n i z ~ t i o n rule, i.e. b y a p p l y i n g a SLT to t h e b c h a v i o u r of a /tee b r a d y o n (e.g. at rest) in v a c u u m . As regards t h e g r a v i t a t i o n a l Cerenkov effect, let us consider a n object t h a t a p p e a r s to us (frame so) as a T m o v i n g at c o n s t a n t velocity in v a c u u m . Such a n object will a p p e a r as a B w i t h respect (e.g.) to its rest f r a m e S; according to f r a m e S, therefore, the (bradyonic) differential e n e r g y loss t h r o u g h GCI~ in v a c u u m is obviously zero (~3):

04a)

-d7

= 0

(~' < 1 ) .

I f we t r a n s f o r m such a law b y m e a n s of t h e SLT f r o m S to so (and r e m e m b e r t h e v a c u n m - c o v a r i a n c e postulate), we get t h e differential energy loss t h r o u g h GCI~ for a t a e h y o n in v a c u u m (2.2~):

(14b)

as' }o = o

(fi~ > 1 ) .

Moreover, a n object uni/ormly m o v i n g in v a c u u m does not e m i t any r a d i a tion b o t h in t h e sublumin~l ease and in t h e SuperluminM one, as r e q u i r e d b y (extended) relativity. L e t us r e p e a t t h a t it is necessary not to confuse GCI~ with the radia~ior~ t h a t a n object can indeed e m i t in v a c u u m when it is accelerated. Lastly, let us a g a i n emphasize t h a t GC1B comes o u t f r o m t h e objects of t h e material medium, a n d n o t f r o m t h e (~r a d i a t i n g ~ object itself (~.~0). T h e r e fore, t h e v e r y expression (~GCR in v a c u u m )) would h a v e b e e n meaningless, unless one s i m u l t a n e o u s l y p r o v i d e d a suitable t h e o r y a b o u t a n i m p r o b a b l e vacuum

6. -

s t r u c t u r e a.

Case of material

~( m e d i a - .

Since the previous case A), considered in Sect. 5, is s u b s t a n t i a l l y n o t different f r o m case B) (13.~2), we shall con/ine ourselves to the study o/ Cerenkov radiation cones caused b y t a e h y o n i e objects. The principle of duality, t o g e t h e r w i t h t h e t a c h y o n i z a t i o n rule, allows us to solve also t h e general p r o b l e m of GCI~ f r o m t a e h y o n s in a m a t e r i a l m e d i u m (*~). (~1) Cf. Mso 1~. ]VhGNA~I and E. I~ECA~I: Let& IV~ovo Cimento, 7, 388 (1973); 9, 362 (1974). (23) See, e.g., W. PAULI: Relativit~tstheorie (Leipzig, 1921).

A8TROPHYSICS

AND

223

TA(]HYONS

L e t us first define the proper (~g r a v i t a t i o n a l re]ruction i n d e x ~ No of a m e d i u m M~ with respect to a second m e d i u m M~ as the ratio of the gravitational-wave (GW) speed in M2 over the G W speed in M~. Let us now generalize (Fig. 3) the e~ '

(270//~>1)

where No is still the proper (( gravitational refraction index ~>of the medium, and w the t a c h y o n speed (relative to 0). I n particular, for _7Vo~>1 (i.e, for the v a c u u m and for bradyonic media), one gets

(dw) The (~Cerenkov relation )> giving the cone angle in the bradyonic case (2o) 0sa)

1 cos 0 = ~_~.~

(*) As follows by analogy with the electromagnetic case. Cf. ref. (2o).

(~2 < 1)

AST~Or~VSICS ~ D

m~c~Yoss

225

t r a n s f o r m s u n d e r t h e s a m e S L T i n t o t h e f o l l o w i n g r e l a t i o n , g i v i n g t h e cone a n g l e i n the tachyonic case: (lSb)

cos0 = -~

No

( ~ > 1)

A l l t h e p r e v i o u s p a p e r s (:~'~) d e s l i n g w i t h G C R , i n o u r o p i n i o n , a r e n o t c o r r e c t , s i n c e t h e y ~re b u s e d on a n uncritical e x t e n s i o n of t h e u s u a l p r o c e d u r e for b r a d y o n s t o t h e t ~ c h y o n i c case. They~ e.g.~ c o n c l u d e d a b o u t a G C R from t~chyons in vacuum. We have seen from (extended) relstivity, on the c o n t r a r y , t h a t S u p e r l u m i n a l o b j e c t s do not e m i t G C R i n b r ~ d y o n i c m e d i ~ (e.g. in b r ~ d y o n i c g a l a x y c l u s t e r s ) .

T h e a u t h o r s a r e g r a t e f u l t o P r o f s . A. AGODI, M. BhLDO, P. CALD~O]~A ~ n d V. D]~ S~]~]~Amh, a n d D r . E . MASShlr for u s e f u l d i s c u s s i o n s , ~ n d t o Mr. 1+. A ~ V A , D r . L. 1% B h ~ , ] ) I ~ , ~ n d Mrs. G. G ~ v ~ I ~ ) ~ f o r t h e i r k i n d collaboration.

(~) A. P]~R~s: Phys. Lett., 3 1 A , 361 (1970). (s~) L. S. SCn~:L~A~q: ~uovo Cimento, 2 B, 38 (1971). (ss) A. S. LAPEDES and K. C. JAeoBs: Nature Phys. Soc., 234, 6 (1972). (ss) H. K. WI~MEL: Nature Phys. Sci., 236, 79 (1972). See also R. G o ~ I I I : p r e p r i n t s (unpublished).

9

RIASSUNTO

Dall~ teoria della relativit~ ristretta, quando non se ne restringa arbi~rariamente la vaHditk a velocith sabluminali, ~ possibile dedurre l'esistenz~ di sis~emi di riferimento Superluminali e di oggetti pilk veloei della lnee. Di recente, invero, la relativit~ ristretta ~ sta~a estesa ai sistemi di riferimento inerziali Superluminali e ai tachioni, cosl da consentire Io sviluppo autoconsistente di una (~teoria classica ~) dei taehioni. La (~teoria della rolativits estesa )> consen~e in particolare di es~endere le formnle dell'effetto Doppler a sorgen%i Superluminali. Si prendono in considerazione anche le varie p ossibflit~ di eontatto radio t r a gli oggetti subluminali e Superluminali. Si ehiurisce il eomportamento dei corpi taehionici nel campo gravitazionale. Si predice che i tachioni subiseono una repulsione gravitazion~le (cosi ehe potrebbero essere considerati la ~ maiberia antigravitazionale ~>); t u t t a v i a , a eausa della loro dinamiea, essi finiseono con l'avere traiettorie t h e s'incurvano verso la sorgente del campo gravi~azionale, come gli oggetti soliti (anehe se con eurva%ure diverse). Infine, si studia il problema della rndiazione Cerenkov gruvitaziona]e (GCR). In eontrasto con quan~o affermato dagli autori precedenti, la rela~ivit~ es%esa predice che i tachioni non emettono GCR nel vuoto o nei mezzi bradioniei (sublumin~li). 15 - 1l Nuovo Cimento B.

226

R. ~ I G N A ~ I

~nd

~. ~ECA~

ACTpOdpH3H~a I4 TaXHOH~,I.

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