More about Lorentz transformations and tachyons: answer to the

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by 0LKHOVSKY and one of us (2), regarding Lorentz transformations (LT) and faster- ... composition law in the form reported, e.g., by TE~LETSKY (n):. (1) c~--(v'V .... dencing the ultimate consequences of the assumption of eq. (3). Incidentally ...
L]~TTEI~E AL NUOVO CIME~TO

VOL. 4, N. 4

27 :MaggJo 1972

More About Lorentz Transformations and Tachyons: Answer to the Comments by Ramachandran, Tagare and Kolaskar. E. ~ECAMI I s t i t u t o di ~ i s i c a Teorica dell' U n i v e r s i t h - C a t a n i a I s t i t u t o N a z i o n a l e di F i s i c a N u c l e a t e - S e z i o n e di C a t a n i a Centro S i c i l i a n o di .Fisica N u c l e a t e e di S t r u t t u r a della M a t e r i a - C a t a n i a

R. MIGNANI I s t i t u t o di .Fisica Teorica dell' Universith - Catania Centro S i c i l i a n o di _Fisica N u c l e a t e e di S t r u t t u r a della M a t e r i a - C a t a n i a

( r i c e v u t o il 9 M a r z o 1972)

I n a n o t e (1), RAMACHA~DI~AN et al. h a v e c o m m e n t e d a b o u t a r e c e n t , b r i e f l e t t e r b y 0LKHOVSKY a n d one of u s (2), r e g a r d i n g L o r e n t z t r a n s f o r m a t i o n s (LT) a n d fastert h a n - l i g h t - p a r t i c l e s (3.~). E v e n if t h e i r c o m m e n t s d i d n o t a p p e a r to us to b e v e r y p e r t i n e n t , we are g r a f e f u l to t h e m for t h i s occasion, since i t allows u s to clarify o u r p r e l i m i n a r y l e t t e r (2), w h i c h r e m a i n e d i n d e e d a t t h e surface of t h e p r o b l e m s . I n p a r t i c u l a r , we shall show t h a t o u r p r e v i o u s c o n s i d e r a t i o n s c a n b e f o u n d e d m u c h b e t t e r t h a n in ref. (2). 1. - L e t u s first analise ref. (1). T h e w h o l e c e n t r a l b o d y , f r o m eq. (4) to eq. (14), does n o t a t t a i n a t all to o u r l e t t e r , since i t deals w i t h a c o m p l e x s p a c e - t i m e , following a p h y l o s o p h y w h i c h w a s not ours. N a m e l y , c o n s i d e r i n g (5) c, i.e. also w h e n going from a subluminal s to a superluminal S. W e assumed as well t h e v a l i d i t y of eq. (1) for b o t h t h e frames being superluminal, u n d e r t h e hypothesis t h a t only t h e r e l a t i v e v e l o c i t y u m e a n t in this context. E q u a t i o n (1), in its wider i n t e r p r e t a t i o n , can be a good e x t r a p o l a t i o n of t h e usual one, since it obviously coincides w i t h t h a t equation, in the usual v e l o c i t y domain. Besides, the c o n t i n u i t y condition at the b o u n d a r y are well satisfied s i n c e - - f o r a couple of (~inertial frames )) either s-type or S - t y p e - - w h e n u - + c ~, one has v ' ~ c for a n y value of v. B u t we m u s t first o v e r c o m e the following difficulty, connected w i t h using eq. (1). A c c e p t i n g this equation for u ~ c means assuming some k i n d of generalized L T also for u ~ c. T h e trivial generalization of the LT, although consistent w i t h eq. (1), does n o t work. I n fact, let us consider one (free) B and two inertial frames s and S, r e s p e c t i v e l y sub

(7) T h e w o r d * b r a d y o n s ,), for u s u a l (v < c) particles, has been i n d e p e n d e n t l y i n t r o d u c e d in: R. G. CAWLEY: A n n . of P h y s . , 54, 132 (1969); E. RECAMI: Accad. N a z . L i n c e i (Rendic. Se.) 49, 77 ( R o m a , 1970); M. BALDO, G. •ONTE a n d E. RECAMI: L e f t . N u o v o Cimento, 4, 241 (1970). (B) G. FEINBERO: P h y s . Rev., 159, 1089 (1967). (9) On t h e c o n t r a r y , for a n e x a m p l e of a t r u e discussion of c o m p l e x s p a c e - t i m e , see A. DAS: J o u r n . M a t h . P h y s . , 7, 45, 52, 61 (1966), a n d references therein. For a quite different philosophy, see also, e.g., A. J. KXLNAY, J. A. GALLARDO, B . A . STEC a n d B. P. TOLEDO: N u o v o Cimento, 49 A, 393 (1967); 48 A, 997 (1967); ]~I. BALDO a n d :E. RECAMI: Lett. NUOVO Cimento, 2, 643 (1969); V. S. OLKHOVSKY a n d E. RECAMI: L e t t . N u o v o Cimento, 4, 1165 (1970), a n d references therein. (10) Analogously, for a n e x a m p l e of a t r u e i n t r o d u c t i o n of c o m p l e x rest-masses, see A. IVI. GLEESON a n d E. C. G. SUDARSHAN: P h y s . Rev. D, l , 474 (1970); E. C. G, SUDARSItAN: P h y s i c s of Complex M a s s P a r ticles, R e p o r t CPT-41 (Austin, 1970); U n i t a r y Problems i n I n d e f i n i t e 3letric Theories, I4eport CPT-42 (Austin, 1970); A. M. GLEESON: R e p o r t CTP-77 (Austin, 1971); H . YAMA~OTO: Progr. Theor. P h y s . , 42, 707 (1969); 43, 520 (1970); 44, 272 (1970), a n d references therein. (tl) YA. 1D. TERLETSKY: Paradoxes i n the Theory of R e l a t i v i t y (New York, 1968). See as well rcf. (1,).

146

n. RECAM~ and n. ~ N A ~ I

and superlumin~l, w i t h identical space orientation ~nd m o v i n g for simplicity along t h e x-direction w i t h c o n s t a n t v e l o c i t y u. Because of t h e a s y m m e t r y of t h e usual v e l o c i t y t r a n s f o r m a t i o n formulae for t h e B v e l o c i t y c o m p o n e n t along the f r a m e r e l a t i v e veloc i t y u and in the plane n o r m a l to u, respectively,

(2)

v,, -

1 - - uv~/c 2 '

v~,.~ =

1 - - q~v~/e ~

'

[u ~ u~ > c ] ,

we shall m e e t i m a g i n a r y ~ transversal ~>velocities vv.~' w h e n passing from s to S. L e t us explicitly notice since now t h a t this a s y m m e t r y is due to the a s y m m e t r y b e t w e e n t h e n u m b e r of t h e spatial and t e m p o r a l dimensions in the chronotopieal space (3, 1). I n a fictitious bidimensional chronotope (1, 1), where space-time dimensional s y m m e t r y holds, any such difficulty is not met. Therefore, one has to be careful in avoiding reasonings based on a bidimensional space-time: on the contrary, t h a t misleading procedure has been till now followed b y almost all t h e authors (~~*) who approached the problem. tn particular, t h e t r a n s f o r m t t i o n group proposed b y PARKER (la) (i.e. his (~e x t e n d e d L o r e n t z group in one spatial dimension ~) has been shown to h a v e no s t a n d a r d (real) analogous in t h e pseudo-Euclidean four-dimensional space-time (~s). By the way, also t h e assertions (*) for~,arded by MARIWAZLA (14) h a v e been p o s t u l a t e d or worked out only in t h e case of one-dimensional space. Therefore, we m u s t seek for different generalized L o r e n t z t r a n s f o r m a t i o n s (GLT). A t this purpose we h a v e to bear in m i n d t h a t GonIN1 (10) showed t h a t in four dimensions no real G L T exist, which connect (~) frames s to frames S (i.e. for u > c). As a guide in our research for G L T let us choose the following observation. T h e almost universal starting point for s t u d y i n g T ' s - - s i n c e t h e first p a p e r b y SUDHARSttA~N"and coworkers (~) is generalizing the relativistic e q u a t i o n m = 1~o/7 also for fl~ > 1 u n d e r t h e c o n v e n t i o n t h a t /~o= m0"0(1 _ f i e ) + i m o . O ( f l ~ 1), i.e. assuming for u > c:

(3)

m - - %/~__f12

re~

fl ~ c

'

u > c

.

This i m p o r t a n t generalization has been c o m m o n l y accepted in an uncritical way. Now, let us t e m p o r a r i l y assume t h a t superluminal frames S do exist w i t h s t a n d a r d properties. W e shall show in Sect. 3 t h a t physical frames S m u s t regard (, our ~>T ' s as B's. T h e r e fore, we shall h a v e S-frames in which (~our ~) T's are actually at rest (behaving as t a c h y o n s w i t h regard to us, b u t as B ' s w i t h regard to S, in particular to their rest systems). Of course, t h e proper mass of t a c h y o n s will be real, m s (as well as t h e proper lengths

( ~ ) 1~. T . J O ~ E S : J o u r n . FraukliJ~ l n s t . , 275, 1 (1963). (~*) a ) L . P A ~ R : P]~ys_ B e y . , 188, 2287 (1969); s e e a l s o : b) D. L E I T E R : Left. Nuovo Cime.~to, ] , 395 (1971)_ ( ~ ) X . iV[. ~ A R I W A L L A : A m . ,]ourn. P h y s . , 37, t 2 8 1 (1969). ( ~ ) V. GOR[N[: Comm- ~Iath. P h y s . , 21, 159 (1971). (*) S i n c e i n t h e f i r s t p a r t of r e f . (L~) s u p e r i u m i n a [ f r a m e s w e r e n o t y e t c o n s i d e r e d , i t s e q s . (6) a r e n o t even understandable. I n a n y c a s e , t h e y l o o k t o b e w r o n g , a n d n o t e a s i l y c o m p a t i b l e w i t h t h e cqs. (1) o f t h e s a m e Dapet'. 5 s t h c s e c o n d p a r t of t h a t n o t e is tautological, s i n c e i t a s s u m e s t a e h y o n s h a v i n g a l w a y s v > c w i t h r e s p e c t to b o t h B ' s a n d T ' s , a n d a f t e r w a r d s i t f i n d s o u t of c o u r s e ~ - - i n eq. ( 8 ) - - t h a t t h e r e l a t i v e v e l o c i t y b e t w e e n t w o T ' s is g r e a t e r t h a n c. I n t h e f o l l o w i n g of t h e t e x t , w e s h a l l c r i t i c i z e that hyDothesis. (~s) V. GORINg: D r i v a t e c o m m u n i c a t i o n (1971). 0 : ) U n l e s s w e w i s h t o m a k e r e c o u r s e t o discrete s e t s of c h r o n o t o p i c a l - c o - o r d i n a t e v a l u e s . Cf., e.g., Y . AH~AVAARA: J o u r a . -]Iath. P h y s . , 6, 87 (1965); 7, 197; 201 (1966).

147

MORE ABOUT LORENTZ TRANSFORMATIOI~S AND TACIIYONS ~TC.

a n d t h e p r o p e r t i m e s : see t h e following). L e t us explicitly r e p e a t t h a t t h e p r o p e r m a s s of T ' s is n o t i m a g i n a r y , in t h i s f r a m e w o r k (is). Now, w h e n p a s s i n g f r o m t h e r e s t s y s t e m of a T to a s u b l u m i n a l s y s t e m , w e g e t t h a t its r e l a t i v i s t i c m a s s t r a n s f o r m s a c c o r d i n g to eq. (3) if w e assume, for t h e e n e r g y t h e g e n e r a l t r a n s f o r m a t i o n law

[ me always

,,0

(4)

m = Vil__fl,[

v a l i d b o t h for u < c

(4')

a n d for n > c .

I n fact, for u > e

m . . . . . . . . . ~//il - - f121 ~/f12-- 1

real,

fl ~

o,

u ~

c

,

eq. (4) r e a d s

--iV~fl

2

Vl--fl =

t h u s yielding eq. (3). I n t h i s w a y w e g a v e a m e a n i n g to t h e (real) r e s t m a s s of T's, evid e n c i n g t h e u l t i m a t e c o n s e q u e n c e s of t h e a s s u m p t i o n of eq. (3). I n c i d e n t a l l y , t h e i m a g i n a r y u n i t e n t e r i n g cq. (3) a p p e a r s n o t b e c a u s e of a p r o p e r m a s s s u p p o s e d l y i m a g i n a r y , b u t b e c a u s e of t h e m o d u l u s s y m b o l in t h e GLT, eq. (4). I n o r d e r to show t h a t t h e p r e v i o u s p r o c e d u r e is n o t m e r a l y a f o r m a l trick, let us c o n s i d e r t h e t r a n s ] o r m a t i o n s L b e t w e e n a f r a m e s a n d a f r a m e S m o v i n g w i t h v e l o c i t y u, In > e] along t h e x - d i r e c t i o n (for simplicity) u n d e r t h e c o n d i t i o n s t h a t a) L b e l i n e a r ; b) t h e s p e e d of l i g h t be e in b o t h s a n d S; c) 4 - d i m e n s i o n a l s p a c e - t i m e be h o m o g e n e o u s , a n d 3 - d i m e n s i o n a l space be isotropieal. T h e n , p a r t i a l l y following ref. (~aa), one m a y s h o w t h a t t r a n s f o r m a t i o n s L m u s t c o n s e r v e t h e n o r m of f o u r v e c t o r s , e x c e p t for a sign; i . e . t h e L ' s m u s t b e s u c h t h a t , e . g . ,

(5a)

C2g2--X 2~

~- ( c 2 t ' 2 - - X '2)

for u
c ,

(e2t'2--x

'2)

e,

for i n s t a n c e , w i t h t h e p s e u d o - E u c l i d e a n m e t r i c ( ~ - - - - - - - ) . I n eq. (5b) w e chose t h e sign (( m i n u s ,~, i n s t e a d of t h e sign (~plus ~), b e c a u s e a) w e w a n t ( a m o n g t h e others) to g e t eq. (4), i . e . to save t h e v a l i d i t y of eq. (3); b) t h i s choice is in a c c o r d w i t h our (~p h y l o s o p h y ~>, as w e shall soon see in t h e 4 - m o m e n t u m space. Besides, t h i s choice, in t h e b i d i m e n s i o n a l case, coincides w i t h P a r k e r ' s . I [ w e w a n t to a s s i g n s t a n d a r d m e a n i n g t o / r a m e s S , t h e p r e v i o u s one s e e m s to be t h e only possible choice. I n fact, c o n d i t i o n (5) is satisfied b y t h e usual L T for u < c a n d b y t h e t r a n s f o r m a t i o n s (*) x -- ut x ' = - -

(6)

t -- ux/c 2

t'--

~/I 1 - - f l q '

-

-

,

y'=iy,

z'=iz

[f12>l]

~/I 1 - - fl'~l

for u > c. W e u n d e r l i n e t h a t precisely t h e m o d u l u s [1--f121 e n t e r s eqs. (6).

(*) (6')

By the way, the GLT for both u > c and u < c may rcad, e.g., x'=

x -- ut

VI1 zfl,i

;

t'=

t -- ux/c ~

,

y ' = (--i)Vy,

z'= (--i)Vz,

[u ~ c],

VI1 - f l ' r

where a (, tachyonle index ,> (function of tho only relative vdoeity u) has been formally introduced: z-- 0 for B's and v = ,1, for T's.

148

E. RECAMI

and

R.

MIGNANI

These t r a n s f o r m a t i o n s (6) m a y well be chosen as generalized L o r e n t z t r a n s f o r m a t i o n s for u > c, since t h e y i) form a group t o g e t h e r w i t h t h e usual L T , ii) are u n i m o d u l a r (with d e t e r m i n a n t + 1), iii) are (( p s e u d o u n i t a r y ~), iv) satisfy our f u n d a m e n t a l equation (1) and of course eq. (3). W h y i m a g i n a r y units appear (~s) in y', z' will become clear later on. I n our terminology, p s e u d o u n i t a r y m e a n s satisfying condition (5b); notice t h a t here (( u n i t a r y ~ does n o t h a v e the c o m m o n significance, since t h e chronotope is n o t a H i l b e r t space (it does not even h a v e a positive-definite metric, and its n o r m is n o t a H e r m i t i a n form). I n this f r a m e w o r k usual L T will be also valid between frames S. I f we write condition (Sb) in the f o u r - m o m e n t u m space (5'b)

E 2 - - c2 p 2 = - - ( E '2 - - e 2 p '2) ,

[ u ~ c] ,

we see i m m e d i a t e l y t h a t our p s e u d o u n i t a r y G L T operate (for u ~ c) a s y m m e t r y w i t h respect to t h e ~ light-cone ~ p 2 ~ O, i . e . t r a n s f o r m two-sheeted hyperboloids p2 = m~ ) 0 into single-sheeted hyperboloids p ~ = - - m e ~ ~ 0. This fact accords w i t h our philosophy, as exploited, e . g . , in Sect. 4. As regards velocities, from eqs. (6) the usual L T will follow for u ~ e. I n t h e case u > c, from eqs. (6) we shall get the v e l o c i t y G L T (*) r

Vx -- U

[u > c] ,

1 -- uv~/c ~

(7) ,

iv~.~"

l~2--~'l _ i%~.v/~

1 -- uvJc 2

~-

-

1 -- uv~lc 2

%o.VVA--~ 1 -- uv~lc 2

T r a n s f o r m a t i o n s (7) are consistent w i t h eq. (4) and m a y be verified to satisfy eq. (1). Therefore, our eq. (1) is not only L T c o v a r i a n t (u ~ v), b u t also G L T c o v a r i a n t (u > c); briefly, we shall say t h a t it is ~ G c o v a r i a n t ~). Our Table in ref. (2) has a quite g e n e r a l validity, contrarily to w h a t claimed b y ]~AMACHANDRAN et a l . (~). W h y i m a g i n a r y units f o r m a l l y enter v~,~r (as well as y', z') will be clear soon. I t is ] i r s t interesting to observe t h a t our GLT, eq. (6), in t h e bidimensional case reduce (**) to the t r a n s f o r m a t i o n s (~3) b y P A R K E R , who possibly did n o t realize t h a t eqs. (4)-(5) of ref (~aa) m a y be w r i t t e n in t h e simple form of our first two eqs. (6). L e t us explicitly p o i n t out t h a t our philosophy too coincides w i t h the P a r k e r ' s one (~3). T h e fact t h a t our eqs. (6) generalize t h e P a r k e r t r a n s f o r m a t i o n s to the four-dimensional chronotope is not in contrast w i t h Gorini's t h e o r e m (~6), since we i n t r o d u c e d G L T n o t real b u t c o m p l e x . T h e circumstance t h a t t h e G L T for u ~ c are c o m p l e x - - p r e c i s e l y t h e y are j u s t t h e usual L T , for u ~ c, multiplied b y t h e i m a g i n a r y u n i t / - - r e f l e c t s in the f o r m a l w r i t i n g of eqs. (6) and (7). W h e n going f r o m one f r a m e to another, b o t h either subluminal or superluminal, we h a v e no change in t h e m e t r i c ( i . e . in t h e w a y t h a t t h e two frames (ts) T h i s w e l l - k n o w n f a c t w a s , e . g . , r e p o r t e d b y *vV. P A U L I : R e l a t i v i t i i l s t h e o r i e , i n E,T~c. d e r • l a l h . W i s s e n . , 5, C h a p t . 19 ( L e i p z i g , 1921). (*) F r o m P a r k e r ' s t r a n s f o r m a t i o n s o n e g e t s t h e v e l e c i t y t r a n s f o r m a t i o n t a w i n t w o d i m e n s i o n s : .... i

v~

Ca(v,+c)+b(v,

c),

a ~

~

a-l[u~

c]

--c

w h i c h of c o u r s e c o i n c i d e s w i t h t i l e f i r s t of o u r eqs. (7). (**) I n t h e b i d / m e n s i o n a l e a s e , P a r k e r ' s t r a n s f o r m a t i o n s a r e t h e u n i q u e o n e s s a t i s f y i n g t h e b i d i m e n s i o n a l e q . (5b); t h i s m a y i n d i c a t e t h a t o u r G L T t o o a r e t h e u n i q u e o n e s s a t i s f y i n g e q . (Sb).

MORE ABOUT LORENTZ TRANSFORMATIONS AND TACHYONS ETC.

possess for d e s c r i b i n g t h e M i n k o v s k y s p a c e - t i m e ) : f r a m e s are c o n n e c t e d b y L T , w h i c h s a t i s f y eq. (5a): e 2t~2_x~2=

c 2t 2-x

namely,

14~

two s - - s '

(or S - - S ' )

2 .

I n t h a t ease, if we w o u l d use a E u c l i d e a n m e t r i c ( + + + + ) , w e s h o u l d w r i t e t h e t i m e l i k e c o m p o n e n t s in a real f o r m , a n d t h e spacelike c o m p o n e n t s in a n i m a g i n a r y f o r m ,

and this ]or both ]rames. W h e n going, on t h e c o n t r a r y , f r o m a s to a S f r a m e

(or f r o m S t o s), we h a v e a

change of m e t r i c Qs); for i n s t a n c e , we go f r o m a f r a m e , s, d e s c r i b i n g t h e M i n k o v s k y space-time with the metric (+------), t o a f r a m e , S, d e s c r i b i n g it w i t h t h e m e t r i c ( - - § + ~-). T h i s e v e n i e n c e is r e p r e s e n t e d b y t h e f a c t t h a t o u r G L T (for u > c) s a t i s f y eq. (5b). c 2 t'2 __

x,2

=

--

(c 2 t 2 --

x 2)

.

T h i s s a m e e v e n i e n c e is r e p r e s e n t e d as well b y s a y i n g t h a t t h e 3 spacelike c o - o r d i n a t e s will go i n t o o n l y l, a n d t h a t t h e t i m e l i k c c o - o r d i n a t e will go i n t o 3 ( a n d vice versa): (a, 1)-+ (1, 3) T h e o t h e r , p r e v i o u s c o n s i d e r a t i o n s m a y b e easily m o d i f i e d for t h i s case. p o r t a n t is t h a t i t m u s t b e s y m b o l i c a l l y : (8)

[ u > c], W h a t is im-

G L T -- i - L T ,

a n d t h i s is w h a t we h a v e in eqs. (6) a n d (7). Of course, t h e i m a g i n a r y (or real) u n i t s m e r e l y r e c o r d h o w t h e s e c o n d o b s e r v e r is s u p p o s e d to u s e his o w n o b s e r v a t i o n s for b u i l d i n g u p f o u r - v e c t o r squares. T h e y do n o t m e a n , o b v i o u s l y , t h a t t h e s e c o n d f r a m e will r e a l l y o b s e r v e i m a g i n a r y or re.a] q u a n t i t i e s ! ( T h e o b s e r v e d q u a n t i t i e s , briefly s p e a k ing, will b e g i v e n b y eqs. (6)-(7) except for a n i m a g i n a r y u n i t . ) I n c i d e n t a l l y , l e t u s a d d t h a t t h e c o m p o n e n t s of v e l o c i t y b e h a v e e s s e n t i a l l y as t h e 3 spacelike c o m p o n e n t s of t h e f o u r - d i m e n s i o n a l v e l o c i t y in t h i s r e s p e c t . I n o u r p h i l o s o p h y , w h e n p a s s i n g f r o m a s to a S f r a m e , B ' s go i n t o T ' s a n d vice versa, w h i l s t (( l u x o n s >>(4) t r a n s f o r m i n t o luxons. Since t h i s q u e s t i o n of symmetry is e s s e n t i a l for a t t r i b u t i n g a standard meaning to s u p e r l u m i n a ] reference frames, we w a n t to discuss it f u r t h e r t y . 3. - T h e preliminary p o i n t to b e a n s w e r e d w h e n c o n s i d e r i n g f r a m e s S is: m a y t h e S be (~p h y s i c a l ~>frames, or do t h e y h a v e to b e n e c e s s a r i l y r e g a r d e d as c exist. E.g., e i t h e r w equals e also for u > e [case I] ; or w i n c r e a s e s as u i n c r e a s e s [case I I ] ; or w decreases [case I I I ] ; or w = 0 for n > c, as for u < c (case IV), a n d so on. (L,) Cf., e.g., V. FOeK: The Theory o] Space, T i m e and Gravitation (Oxford, 1964).

150

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R. MIGNANI

T h e k i n c m a t i c a l l y s y m m e t r i c a l case I V is t h e o n l y one c o m p a t i b l e w i t h S to b e a n d w i t h t h e e x i s t e n c e of ( s u p e r l u m i n a l ) r e s t s y s t e m s for t a c h y o n s . O t h e r wise, a s u p e r l u m i n a l o b s e r v e r w o u l d see e v e r y o b j e c t (~ (which, a c c o r d i n g

physical,

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F i g . 1. - R e l a t i v e u w of t w ~ o b j e c t s P , Q, t r a v e l l i n g s i d e b y s i d e a t t h e s a m e v e l o c i t y u w i t h r e s p e c t t o a s u b l u m i n a l f r a n l e s. ~Vc h a v e w ~ 0 f o r 0 < u < c a n d w ~ c f o r u = c. S o m e c a s e s of p o s s i b l e e x t r a p o l a t i o n s f o r a > e a r e s h o w n . O n l y t h e c a s e I V ( k i n e m a t i c a l s y m m e t r y ) is c o m p a t i b l e w i t h s u p e r l u m i n a l r e f e r e n c e f r a m e s S t o b e a priori p h y s i c a l .

to us, is w i t h h i m , s i d e - b y side) m o v i n g off f r o m h i m s e l f ; e v e n his i n s t r u m e n t s a n d his o w n b o d y ! R e f u s i n g a n d t h e v(t), t h e n T will receive (~ ( r e t a r d e d ) I~S ~>u n t i l l a c e r t a i n 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 ; a f t e r w a r d s T will s t a r t r e c e i v i n g t h e (( a d v a n c e d RS )~. c) If v ( T ) < v(t), t h e n p r e v i o u s s e q u e n c e will b e r e v e r s e d . T will s t a r t r e c e i v i n g (~ ( r e t a r d e d ) RS ,) o n l y since a c e r t a i n i n s t a n t .

For example,

3) T will receive, f r o m ~ s e c o n d , , unparallel ~ t a c h y o n t (~( r e t a r d e d ) RS )) e i t h e r n e v e r , or u n t i l l a c e r t a i n i n s t a n t , or since a c e r t a i n i n s t a n t , a c c o r d i n g t o t h e i r p o s i t i o n s a n d velocities (relative to t h e s u b l u m i n a l f r a m e ! ) . I n general, t h e s e q u e n c e of emission will r e s u l t h i g h l y m i x e d u p a t t h e r e c e p t i o n . I t is v e r y i m p o r t a n t to n o t i c e t h a t t h e conclusions a b o v e are d r a w n on t h e b a s i s of t h e s - f r a m e o b s e r v a t i o n s , a n d t h e y are a priori q u i t e i n d e p e n d e n t of t h e t r a n s f o r m a t i o n s b e t w e e n f r a m e s s a n d S. I n paxticlflar, t h e y t h e m s e l v e s a priori f a v o u r n e i t h e r t h e ~ k i n e m a t i c a l a s y m m e t r y ~>, n o r t h e ~ s y m m e t r y )>. F o r i n s t a n c e , WI~MEL (~1) c l a i m e d t h a t two o b j e c t s P , Q, t r a v e l l i n g side b y side w i t h t h e s a m e velocity, along p a r a l l e l p a t h s , m a y decide if t h e y are b o t h B ' s or T ' s , since t h e t w o T ' s could not e x c h a n g e p h o t o n s e a c h o t h e r (cf. case B.2.a)). B u t t h e w h o l e d e s c r i p t i o n is v a l i d o n l y w i t h r e s p e c t to a (particular) s u b l u m i n a l o b s e r v e r s. A priori,

(2~) I~. K, WIIVIX~EL: p r i v a t e c o m m u n i c a t i o n .

152

x. REcA~n a n d 1~. MIGNANI

we do n o t k n o w (if, P , Q are ~ t a c h y o n s ~) for s) w h a t e a c h t a c h y o n will (~see ~), since a priori we w o u l d n o t k n o w w h i c h d e s c r i p t i o n will b e d e v e l o p e d b y a n y s u p e r l u m i n a l o b s e r v e r S (if t h e y p h y s i c a l l y exist). I n P a r k e r ' s a n d o u r o w n p h i l o s o p h y , (( t a e h y o n )) P, e.g., will see (( t a e h y o n ~ Q a t r e s t (i.e. as a b r a d y o n ) ; n m r e o v e r , we ~ b e l i e v e ~) t h a t t h e s u p e r l u m i n a l f r a m e s S will p i c t u r e t h e (~M i n k o v s k y space ~) e s s e n t i a l l y in t h e s a m e m a n n e r as t h e f r a m e s s. T h i s is s u p p o r t e d , a n d r e q u i r e d , b y o u r Sect. 2. T h e r e f o r e , a c c o r d i n g to us, to every observer 0 t h e i m p o s s i b i l i t y of e x c h a n g i n g p h o t o n s will a p p e a r for a n y couple of side-by-side o b j e c t s P , Q t h a t t r a v e l a t a v e l o c i t y u > c with respect to him himsel]. L e t us e x p l i c i t l y n o t i c e t h a t only p h y s i c a l l a w s - - a n d n o t descriptions of p h e n o m e n a ( * ) - - a r e r e q u i r e d b y t h e p r i n c i p l e of r e l a t i v i t y to b e c o v a r i a n t (22). T h e r e f o r e , t h e s a m e P, Q may a p p e a r as e x c h a n g i n g p h o t o n s e a c h o t h e r to a n o b s e r v e r h a v i n g (relative) v e l o c i t y u < c, a n d u n a b l e to e x c h a n g e p h o t o n s e a c h o t h e r to a n obs e r v e r h a v i n g (relative) s p e e d u > c. A t last, in o u r p h i l o s o p h y , t h e a s s e r t i o n s (( two bradyons (for a c e r t a i n o b s e r v e r ! ) can always exchange photons each other )~, a n d ~ two taehyons (for a c e r t a i n o b s e r v e r ! ) can not alxays exchange photo~s each other ~ c o u l d b e physical laws, as well as t h e c o n s t a n c y of l i g h t speed, since t h e y a p p e a r v a l i d to a n y o b s e r v e r (s or S). A t t h i s p o i n t , e v e n t h e w o r d s (~s u b l u m i n a l ~ a n d (~s u p e r l u m i n a l ,~ w o u l d n o t h a v e a n a b s o l u t e m e a n i n g , b u t o n l y a r e l a t i v e one.

T h e a u t h o r s are v e r y i n d e b t e d to Profs. A. AGODI a n d M. BALDO for v e r y useful discussions. T h a n k s are also d u e to Profs. G. FONTE, D. GUTKOWSKI, U. LOMBARDO, S. NOTAt~RIGO, G. SCHIFFREI~ for s o m e i n t e r e s t i n g c h a t s ; a n d to Drs. A PALIaERI, C. SPITALERI for scores of discussions w i t h i n t h e confines of ~(T r a t t o r i a F i r e n z e ~ a n d elsewhere. U n f o r t u n a t l y , t h e c o - a u t h o r of t h e p r e v i o u s l e t t e r , V. S. 0 L K H O V S K Y , could n o t p a r t e c i p a t e in t h e p r e s e n t w o r k b e c a u s e of t h e e n o r m o u s p o s t a l d e l a y s in c o m m u n i cating between Catania and Kiev.

(*) I n general r e l a t i v i t y we h a v e a n analogous situation, e.g., for t h e g r a v i t a t i o n a l collapse. I n t h e p r o p e r f r a m e of t h e collapsing s t a r , t h e S c h w a r t z s e h i l d r a d i u s will bc r o a c h e d in a finite t i m e ; while this t i m e will a p p e a r as infinite to o t h e r observers. T h e m o r e suitable f r a m e w o r k foc dealing w i t h such our p r o b t e m s m a y be n o t t h e special r e l a t i v i t y . See, e.g., L. FANTAPPt~: R e n d i c . L i n t e l , 17, 158 (Reran, 1954). (~2) See, e.g., M. BALDO, G. FONTE a n d E. RECANII: L e f t . N u o v o C i m e n t o , 4, 241 (1970), a n d references therein.

(~) b y Soeiet~ I t ~ l i a n a di Fisica Propriet5~ l e t t e r a r i a r i s e r v a t a Direttore rcsponsabile: G I U L I A N O T O R A L D O DI FRANCIA S t a m p a t o i n Bologna dalla Tipografia Compositori coi tipi della Tipogralia Monograf Questo fasoicolo 6 s t a t e licenziato dai torchi il 22-V-1972