Superluminal inertial frames in special relativity

LETTERE AL NUOVO CIMENTO
VOL. 8, N. 2
8 S e t t e m b r c 1973
Superluminal Inertial Frames in Special Relativity.
E. R~C~MI a n d R. )/[IGNANI
I s t i t ~ t o d i . F i s i c a T e o r i e a 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 E i s i e a N u c l e a t e - S e z i o n e di Catania
Centro S i c i l i a n o d i .Fisica N u c l e a t e e 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 e e v u t o il 25 Giugno 1973)
S o m e r e c e n t p a p e r s (1) s h o w e d us t h a t s o m e t i m e s t h e p h i l o s o p h y e x p r e s s e d b y u s
m our series of p a p e r s (~,3) a b o u t t h e classical t h e o r y of t a c h y o n s (i.e. a b o u t special relat i v i t y g e n e r a l i z e d t o 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 ) h a s n o t b e e n well u n d e r s t o o d .
F o r i n s t a n c e , i n ref. (1) it is a s s e r t e d t h a t w e postulated t h e v a l i d i t y of t h e u s u a l
E i n s t e i n i a n v e l o c i t y c o m p o s i t i o n law for S u p e r l u m i n a l velocities, i n s t e a d of d e r i v i n g
it f r o m first p r i n c i p l e s (as w e did).
Since a c t u a l l y our p h i l o s o p h y h a s n o t b e e n clearly f o r w a r d e d i n ref. (~'a), w e w a n t
h e r e t o o u t l i n e i t explicitly.
1) T h e v e r y ( E i n s t e i n i a n ) p r i n c i p l e of r e l a t i v i t y refers t o i n e r t i a l f r a m e s w i t h
c o n s t a n t r e l a t i v e v e l o c i t y u, w i t h o u t a n y a p r i o r i r e s t r i c t i o n on t h e v a l u e of u ( u ~ c ) .
W e w i s h t o e x p r e s s t h e r e l a t i v i t y p r i n c i p l e ( R P ) i n t h e f o r m : (( P h y s i c a l laws of mec h a n i c s a n d e l e c t r o m a g n e t i s m are r e q u i r e d t o b e covariant w h e n p a s s i n g f r o m a n i n e r t i a l
f r a m e ]1 t o a n o t h e r f r a m e ]~ m o v i n g w i t h c o n s t a n t r e l a t i v e v e l o c i t y u , w h e r e
- - o o < u < -t- co ~>.
2) W e a s s u m e t h e R P a n d t h e following p o s t u l a t e : s p a c e - t i m e is h o m o g e n e o u s
a n d s p a c e is i s o t r o p i c .
3) F r o m t h e p r e v i o u s a s s u m p t i o n s 2), t h e e x i s t e n c e of a n i n v a r i a n t s p e e d ~ ]ollows (4); a n d e x p e r i e n c e s h o w s t h a t s u c h a v e l o c i t y is t h e s p e e d of l i g h t : ~ ~ e.
(~) G . A . tt~V~NUJ~ and N. N~SlVAY~M: Left. Nuovo Cimento, 6, 245 (1973).
(a) E. REOAMIand R. MIGNAI~I: in 1)reparation; R. MmNANI and E. RE0~MI: 1VUOVOCtmento, 14A,
169 (1973); E. RECAMI: iI~ Eneicloloegia E S T , Annuario 1973 (Milano, 1973), p. 85; E. RECAMI an4
R. MION~NI: Lett. NUOVOCimento, 4, 144 (1972); R. )Im~ANI, E. REO~I and U. LOMBXRDO:Left. Nuovo
Cimento, 4, 624 (1972).
(~) R. MIONANIand E. REC)~I: Lett. Nuovo Cimento, 7, 388 (1973); E. RECAMI and R. I~ImNANI:preprint PP/368 (Catania, March 1973); V. S. 0LKHOVSKu and E. R E C k : Lett. Nuovo Cimento, 1, 165
(1971); Visnlk Kivskogo Universiteta, Seria Fisiki, 11, 5S (1970); M. BALDO,G. FONTE and E. RECAMI:
Lett. Nuovo Cimento, 4, 241 (1970); E. REC~II: Accad. Naz. Lincei, Rend. Sci., 49, 77 (1970); M. BALDO
and E. •ECAMI: Left. Nuovo Cimento, 2, 643 (1969); V.S. OLKHOVSKYand E. REC~I: NUOVOCimento,
63 A, 814 (1969); E. REC)AV~: Gtornale di Fisica, 1@, 195 (1969).
(4) See, e.g., V. GORINI an4 A. ZECOs Journ. Math. Phys., 11, 2226 (1970); V. BERZI and V. GOmNI:
Jonrn. Math. Phys., 1@, 1518 (1969).
110
SUPERLUMINAL INERTIAL FRAMES IN SPECIAL RELATIVITY
111
4) F r o m p o i n t s 2) and 3), it follows a :( d u a l i t y principle )> (DP) (2,~) : (( 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 n o t
h a v e any absolute m e a n i n g , b u t on]y a r e l a t i v e one. L i g h t speed i n v a r i a n c c allows
an c x a u s t i v e p a r t i t i o n (6) of all i n e r t i a l (u/>, e) frames in two sets {s}, {S}, which arc
e x p e c t e d to be such t h a t a (subluminM) 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}
respectively into themselves, and a S u p e r l 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} i n t o {S} and vice versa >). A b i u n i v o c a l correspondence m a y be set b e t w e e n
f r a m e s s(u) and S ( U ) , w i t h u l t U , where u ~ - - U = e2/u, such a m a p p i n g being an inv e r s i o n (i.e. a p a r t i c u l a r c o n f o r m a l mapping).
5) F r o m RP, light speed i n v a r i a n c e and D P it follows (2.5.~) t h a t t r a n s f o r m a t i o n s
b e t w e e n two frames /1,/2 m u s t be linear and such t h a t , for every t e t r a v e c t o r {fourposition, f o u r - m o m e n t u m , four-velocity, ...),
(1)
c~"t2 - x 2 = ~ (c2t ' 2 - x '2)
( u ~ c) .
6) I n eq. (1), t h e sign p l u s holds for u < c , and m i n u s for u > e . I n fact, w h e n
going f r o m a f r a m e s to a f r a m e S, t h e t y p e of t h e f o n r - m o m e n t u m v e c t o r associated
w i t h t h e s a m e observed object changes f r o m t i m c l i k e to spacelike, or vice versa, as
follows f r o m p o i n t 4).
7) I t is easy to v e r i f y (2) t h a t , in t h e simple case of S u p e r l u m i n a l , eollinear, relat i v e m o t i o n along t h e x-axis (*), eq. (1) breaks into the two r e q u i r e m e n t s
c2t~2 + (ix') 2 = (ict)~ + x ~ ,
(2a)
(u 2 > e 2) .
(iy,)2 + (iz,)2 = y2 + z2 ,
(2b)
B y t h e way, f o r m u l a e (2), (3) of rcf. (1), i.e. t h e sign c o n v e n t i o n s a), b) of ref. (1), seem
to violate our conditions (2), a n d t h e r e f o r e - - a c c o r d i n g to u s - - a r e n o t acceptable. As
a consequence, we still t h i n k t h a t , for a boost, t h e t r a n s f o r m e d , transversal v e l o c i t y
c o m p o n e n t s m u s t f o r m a l l y c o n t a i n an i m a g i n a r y u n i t as a factor, in t h e sense of ref. (2.s).
8) T r a n s f o r m a t i o n s satisfying eq. (1), in t h e case of eollinear (subluminal or Superluminal) m o t i o n w i t h v e l o c i t y u along t h e x-axis, can be shown (2) to be t h e (( generalized L o r e n t z t r a n s f o r m a t i o n s )) (GLT):
(3)
y'=
(0
See
(6)
(7)
(*)
(6)
(~)y
,
z'= (~)z,
L. PARKE~: Phys. Rev., 188, 2287 (1969); A. F. A~TIPPA: Nuovo Cimento, 10A, 389 (1972).
also ref. (~).
See also A. AGODI: Lezioni di ]isica teorica, C~tania, 1972. unpublished.
See, e.g., W. RINDLEtr ,Special Relativity (Edinburgh, 1966).
We always neglect space-tinle translations.
R. MIGNA~I a~d E. I~ECA~rI: NuOVO Cimento, 14A, 169 (1973).
112
E. I % E C A M I
and
~. :~IIG.N&NI
holding for fl~=--(u[c)~l. I n writing eqs. (3), we set
(-~14<9< 88
{ fl~-tgg,
(4)
n -= v ( 9 ) ~ ( c o s 9 / l c o s 9 1 ) " 63 ,
and
(5)
Equations (3), in particular, allow us to deduce the extended velocity composition law (2).
9) F o r m (3)--which improves our eqs. (1 his) of rcf. (S)--results to parametrize
G L T ' s in a <~continuous ~>fashion, where our p a r a m e t e r 9 runs (with continuity) from 0
to 2~t rad. I n particular, form (3) allows a straightforward (continuous) geometrical
i a t e r p r e t a t i o n of generMized Lorcntz transformations, for f l ~ 1 (see Fig. 5, 8 o~ ref. (s)):
(6)
t =
x'
t'cosg+--sin9
,
( 0 < 9 < 2~r),
where
(7)
H ~ [(1 + tg 29)/11
--
tg 2 91] 89.
The previous statement will be expanded in a forthcoming paper. This letter should
be essentially considered as an a n s w e r to ref. (1). F o r the developments of our t h e o r y
we refer to ref. (8.9,2,3)~
,g~gc
The authors t h a n k Prof. A. AGODI and Dr. R. BALDI~I for their interest in this work.
(o) E. RECA.~: I tachioni, in A n n u a r i o di Scienza e Tecnica 1973, Eneiclopedia E S T
(Milano, 1973).