problem in superluminal Lorentz transformation - UniBG

LETTERE AL NUOVO CIMENTO

VOL. 29, ~. 8

25 0 t t o b r e 1980

Second Contribution on Solving the r Imaginary Quantities ~ Problem in Superluminal Lorentz Transformations (*). P . CALDIROLA

Istituto di F i s i c a dell'Universit5 - V i a Celoria 16, 20133 Milano, I t a l i a

G. D. MACC~,~ON~. and E. R]~cA~I Istituto di F i s i c a Teorica dell'Universitd - Uorso Italia 57, 95129 Catania, I t a l i a Istituto Nazionale di _~isica Nucleate - Sezione di Uatania, Italia Uentro Siciliano di Fisica Nuoleare e Struttura della Materia . Catania, Italia

(ricevuto il 30 Luglio 1980)

(1) we proposed our geometro-physical interpretation of imaginary quantities entering the Superluminal Lorentz transformations (2)

In a previous paper

the

(u=--e2/Y < c) ct

-

-

ux/e

Vi-- (ulc) 2 x--ut

(1)

t'= •

x

-

-

Ut

:~ V ~ ' __

t - - UX/C 2

~= V~2_ --' i

(

U2

)

fl=-- Ule; 7 > 1

y'= • ~iy , z' = • ~ i z ,

where the operator ~ : 1/fl-*fl (formally necessary) can be skipped for our present purposes. These Superluminal Lorentz transformations (SLT) were derived in the four(*) W o r k p a r t i a l l y s u p p o r t e d b y C N R ( c o n t r a c t no. 79/02741.02). (1) E. REaAMI a n d G. D. MAOOARRON]~: Left. Nuovo Cimento, 28, 151 (1980). (3) E. RECAMI a n d R. MIGNANI: Riv. Nuovo Cimento, 4, 209, 398 (1974); R. MIGNANI a n d E . R~.CAMI: NUOVO Cimento A, 14, 169 (1973); 16, 208 (1973), a n d references t h e r e i n ; E. REOAMI ( E d i t o r ) : Taehyons, ,~ionopoles and Related Topics ( & m s t e r d a m , 1978); E. RECAMI: Relativity and beyond, c h a p t . 16, in Albert Eir~slein 1879-1979: Relativity, Quanta and Cosmology in the Development o/ the Scienti/ic Thought o/ A. Einstein, e d i t e d b y F. DE FINIS, Vol. 2 (New Y o r k , N. Y., 1979), p. 537; a p p e a r e d also i n I t a l i a n l a n g u a g e as C h a p t . 18 i n Centenario di Einstein: Astro/isica e cosmologia, gravitazionc, quanti c relativitd, e d i t e d b y M. FANTALEO (Firenze, 1979), p. 1021; H. C. CORBEN: Nuovo Cimento A , 29, 415 (1975). See also P. CALDIROLA a n d E. RECAMI: i n l t al i an Studies in the Philosophy o] Science, e d i t e d b y M. DALLA CHIARA (Boston, Mass., 1980); G. D. MACCARRONE a n d E. RECAMI: Nuovo Cimenlo A, 57, 85 (1980).

241

2~2

P. CALDIROLA~ G. D. MACCARRONE a n d E. RECAMI

d i m e n s i o n a l case (2,3) f r o m t h e (~p r i n c i p l e of e x t e n d e d r e l a t i v i t y ~, i.e. f r o m t h e cons i d e r a t i o n ( p r e v i o u s l y a p p l i e d t o t h e t w o - d i m e n s i o n a l case (4)) t h a t t h e y h a v e t o i n v e r t t h e sign of t h e q u a d r a t i c f o r m s (a). O u r i n t e r p r e t a t i o n of t h o s e i m a g i n a r y q u a n t i t i e s w a s b a s e d on i) t h e o b s e r v a t i o n t h a t p o i n t l i k e o b j e c t s find n o r o o m i n r e l a t i v i s t i c theories, e x c e p t as l i m i t i n g cases of e x t e n d e d p a r t i c l e s (5,6); ii) t h e f a c t t h a t u n d e r a S L T a n y s p h e r i c a l (or ellipsoidal) object transforms into the region bound by a two-sheeted hyperboloid and a doubl6 cone, i n t h e sense t h a t a n y s p h e r i c a l (or ellipsoidal) s u r f a c e is t r a n s f o r m e d i n t o a (two s h e e t e d ) h y p e r b o l o i d . I n t h e p r e s e n t p a p e r we w a n t 1) t o d e e p e n t h e m e a n i n g of p o i n t ii) a b o v e , a n d g e n e r a l i z e o u r p r e v i o u s r e s u l t s (~); 2) to a d d a n o t e , h a v i n g b o t h a p h y s i c a l a n d a h i s t o r i c a l i n t e r e s t , o n a v e r y e a r l y i n t r o d u c t i o n of t h e (~p r i n c i p l e of e x t e n d e d relat i v i t y ~) (2-4).

B e f o r e g o i n g on, let us e m p h a s i z e t h a t o u r r e s u l t s i n ref. ( ~ ) - - e v e n if o n l y t h e case of a S u p e r l u m i n a l b o o s t a l o n g t h e t a c h y o n m o t i o n line w a s t h e r e explicitly m e n t i o n e d - h a v e a q u i t e g e n e r a l v a l i d i t y (since o u r eqs. (6), (8), (9), (11) i n ref. (') w e r e w r i t t e n i n L o r e n t z - i n v a r i a n t form). N a m e l y , t h e e q u a t i o n of t h e (~w o r l d - t u b e )~ of a s u b l u m i n a l p a r t i c l e P B (which, e.g., is s p h e r i c a l i n i t s o w n r e s t f r a m e ) , e n d o w e d w i t h t h e v e l o c i t y v a l o n g a g e n e r i c direction, r e a d s (/~2~-V2/C2< 1)

(2)

( x - - tv~) ~ + (y - - tv~) 2

0< -

-

1 -- f12

1 -- fl~

(xvJc-- yvJc) ~ 1 -- f12

< r~

(v ~ < e 2) ,

w h e r e for s i m p l i c i t y ( a n d w i t h o u t a n y loss of g e n e r a l i t y ) we s k i p p e d t h e t h i r d s p a c e axis. A n d , as e x p e c t e d , eq. (2) i n L o r e n t z - i n v a r i a n t f o r m is still w r i t t e n as ( X ~ -~ (ct, x, y, z))

(3)

0