An introductory view about superluminal frames and tachyons

LETTERE AL NUOVO CIMENTO VOL. 8, N. 2 8 Se t t embrc 1973

Superluminal Inertial Frames in Special Relativity.

E. R~C~MI and R. )/[IGNANI

I s t i t~ to d i .Fis ica Teor iea dell' U n i v e r s i t h - Catania

I s t i t u to Naz iona le di E i s i e a Nuc lea t e - Sezione di Catania

Centro S i c i l i a n o d i .Fisica Nuc lea te e S t ru t tura della M a t e r i a - Catania

( r ieevuto il 25 Giugno 1973)

Some recen t p a p e r s (1) showed us t h a t some t imes t h e ph i losophy expressed b y us m our series of pape r s (~,3) abou t t he classical t heo ry of t a ch y o n s (i.e. a b o u t special rela- t i v i t y genera l i zed to Super lumina l r e fe rence f rames) has n o t been well unde r s tood . F o r in s t ance , in ref. (1) i t is a s se r t ed t h a t we postulated t he va l id i ty of t he usua l E ins t e in i an ve loc i ty compos i t ion law for Super lumina l velocit ies, i n s t e ad of deriv ing

i t f rom first p r inc ip les (as we did). Since ac tua l ly our ph i lo sophy has no t been clear ly f o r w a r d e d in ref. (~'a), we w a n t

he re to out l ine i t explicit ly.

1) The very (Einste inian) pr inc ip le of r e l a t i v i ty refers to ine r t i a l f r ames w i t h c o n s t a n t re la t ive veloci ty u, w i t h o u t any a pr ior i r e s t r i c t i on on the va lue of u (u~c) .

W e wish to express t h e r e l a t iv i ty p r inc ip le (RP) in t he fo rm: (( Phys ica l laws of mechanics and e l ec t romagne t i sm are r equ i r ed to be covariant when pass ing f rom an ine r t i a l f r a m e ]1 to ano the r f r ame ]~ mov ing w i t h co n s t an t re la t ive veloci ty u , whe re - - o o .

2) W e assume the R P and t h e fol lowing pos tu la t e : space - t ime is homogeneous

and space is i so t ropic .

3) F r o m the prev ious a s sumpt ions 2), t h e ex is tence of an i n v a r i a n t speed ~ ]ol-

lows (4); a n d exper ience shows t h a t such a ve loc i ty is t he speed of l ight : ~ ~ e.

(~) G.A. tt~V~NUJ~ and N. N~SlVAY~M: Left. Nuovo Cimento, 6, 245 (1973). (a) E. REOAMI and R. MIGNAI~I: in 1)reparation; R. MmNANI and E. RE0~MI: 1VUOVO Ctmento, 14A, 169 (1973); E. RECAMI: iI~ Eneicloloegia EST, Annuario 1973 (Milano, 1973), p. 85; E. RECAMI an4 R. MION~NI: Lett. NUOVO Cimento, 4, 144 (1972); R. )Im~ANI, E. REO~I and U. LOMBXRDO: Left. Nuovo Cimento, 4, 624 (1972). (~) R. MIONANI and 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. OLKHOVSKY and E. REC~I: NUOVO Cimento, 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 po in t s 2) and 3), i t fol lows a :( dua l i ty pr inc ip le )> (DP) (2,~) : (( The t e r m s b radyon (B), t a chyon (T), sub lumina l f r ame (s), Supe r lumina l f r ame (S) do no t have any absolute meaning , bu t on]y a r e l a t ive one. L igh t speed inva r i ancc allows an cxaus t ive pa r t i t i on (6) of all ine r t i a l (u/>, e) f rames in two sets {s}, {S}, which arc expec ted to be such t h a t a (subluminM) Loren tz t r ans fo rma t ion (LT) maps {s}, {S} respect ive ly in to themselves , and a Super lumina l (( Loren tz t r ans fo rma t ion >) (SLT) maps {s} in to {S} and vice versa >). A b iun ivoca l correspondence m a y be set be tween f rames s(u) and S ( U ) , wi th ul tU, where u ~ - - U = e2/u, such a mapp ing being an inve rs ion (i.e. a pa r t i cu l a r conformal mapping) .

5) F r o m RP, l ight speed invar iance and D P i t follows (2.5.~) t ha t t r ans fo rmat ions be tween two f rames /1,/2 mus t be l inear and such tha t , for every t e t r a v e c t o r {four- posi t ion, f o u r - m o m e n t u m , four-veloci ty , ...),

(1) c~"t 2 - x 2 = ~ (c2t ' 2 - x '2) ( u ~ c) .

6) In eq. (1), the sign p l u s holds for u < c , and m i n u s for u > e . In fact , when going f rom a f rame s to a f rame S, the type of t he f o n r - m o m e n t u m vec to r associa ted wi th the same observed object changes f rom t imcl ike to spacelike, or vice versa, as

follows f rom po in t 4).

7) I t is easy to ve r i fy (2) tha t , in the s imple case of Super lumina l , eoll inear, rela- t i ve mot ion along the x-axis (*), eq. (1) breaks in to the two r e q u i r e m e n t s

(2a)

(2b)

c2t~2 + ( ix ' ) 2 = (ict)~ + x ~ ,

(iy,)2 + (iz,)2 = y2 + z2 ,

(u 2 > e 2) .

By the way, fo rmulae (2), (3) of rcf. (1), i .e. the sign convent ions a), b) of ref. (1), seem to violate our condi t ions (2), and t he r e fo re - - acco rd ing to u s - - a r e no t acceptable. As a consequence, we sti l l t h ink tha t , for a boost, the t ransformed, transversal veloc i ty components mus t fo rma l ly con ta in an i m a g i n a r y uni t as a factor , in the sense of ref. (2.s).

8) Trans format ions sat isfying eq. (1), in the case of eol l inear (subluminal or Super- luminal) mot ion wi th ve loc i ty u along the x-axis, can be shown (2) to be the (( genera- l ized Loren tz t r ans format ions )) (GLT):

(3)

y ' = ( ~ ) y , z ' = ( ~ ) z ,

(0 L. PARKE~: Phys. Rev., 188, 2287 (1969); A. F. A~TIPPA: Nuovo Cimento, 10A, 389 (1972). See also ref. (~). (6) See also A. AGODI: Lezioni di ]isica teorica, C~tania, 1972. unpublished. (7) See, e.g., W. RINDLEtr ,Special Relativity (Edinburgh, 1966). (*) We always neglect space-tinle translations. (6) R. MIGNA~I a~d E. I~ECA~rI: NuOVO Cimento, 14A, 169 (1973).

112 E. I%ECAMI and ~. :~IIG.N&NI

holding for fl~=--(u[c)~l. In wri t ing eqs. (3), we set

(4)

and

(5)

{ f l ~ - t g g ,

n -= v (9 ) ~ ( c o s 9 / l c o s 9 1 ) " 63 ,

( - ~ 1 4 < 9 < � 8 8

Equat ions (3), in part icular , allow us to deduce the extended velocity composition law (2).

9) Form (3)--which improves our eqs. (1 his) of rcf. (S)--results to parametr ize GLT's in a <~ continuous ~> fashion, where our parameter 9 runs (with continuity) from 0 to 2~t rad. In par t icular , form (3) allows a s t ra ightforward (continuous) geometrical i a te rpre ta t ion of generMized Lorcntz t ransformations, for f l ~ 1 (see Fig. 5, 8 o~ ref. (s)):

(6) x' (0< 9 < 2~r), t = t ' c o s g + - - s i n 9 ,

where

(7) H ~ [(1 + tg 2 9)/11 -- tg 2 91] �89 .

The previous s ta tement will be expanded in a forthcoming paper. This le t ter should be essential ly considered as an answer to ref. (1). For the developments of our theory we refer to ref. (8.9,2,3)~

,g~gc

The authors thank Prof. A. AGODI and Dr. R. BALDI~I for their interest in this work.

(o) E. RECA.~: I tachioni, in Annuar io di Scienza e Tecnica 1973, Eneiclopedia E S T (Milano, 1973).

LETTERE AL NUOVO CIMEI~TO VOL. 4, ~. 13 29 Lugl io 1972

S u p e r l n m l n a l F r a m e s and t h e Group o f Genera l i zed L o r e n t z

T r a n s f o r m a t i o n s in F o u r D i m e n s i o n s .

R. MIG~ANX a n d E. RECAMI

Is t i tu to di Fis ica Teorica dell 'Universith - Catania Centro Sici l iano di Eis ica Nucleare e di Struttura della Materia . Catania Is t i tu to Nazionale di .Fisica 5Vucleare . Sezione di Catania

TJ. LOMBARDO

Is t i tu to di .Fisica Teoriva dell' Universit~ - Catania

( r ioevu to il 3 Lug l io 1972)

1. -- I n a r e c e n t l e t t e r (t), wh i l e a n s w e r i n g to a p r e v i o u s c o m m e n t b y I~AMACHAN- DRAN et al. (2), w e i n t r o d u c e d a new g roup G of L o r e n t z t r a n s f o r m a t i o n s (LT) in four d i m e n s i o n s ( * ) gene ra l i zed for b o t h s u b l u m i n a l ( f i g 1) a n d s u p e r l u m i n a l ( f l > 1) veloci t ies .

B u t in p a p e r I - - i n o rde r to l i m i t i t s l e n g t h - - w e could do n o t h i n g b u t scarce ly men- t i o n i n g our g roup G of gene ra l i zed L o r e n t z t r a n s f o r m a t i o n s (GLT). Ana logous ly , some o t h e r de ta i l s were exp lo i t ed n o t enough .

T h e a im of t h i s f u r t h e r l e t t e r is to cas t more l i g h t on t he new group G, and to c l a r i fy a few o t h e r p o i n t s of p a p e r I .

Such p r o b l e m s - - a s wel l as m a n y r e l a t e d o t h e r o n e s - - w i l l be e x t e n s i v e l y dea l t w i t h in a f o r t h c o m i n g paper , to be p u b l i s h e d e lsewhere .

L e t us cal l S t h e re fe rence f r ames t r a v e l l i n g fa s t e r t h a n l i g h t w i t h r e spec t to t he usua l c lass of i ne r t i a l f r a m e s s. T h e ph i lo soph ica l i n v e s t i g a t i o n d e v e l o p e d in p a p e r I s h o w e d t h a t - - i f s t a n d a r d space - t ime m e a s u r e m e n t s m u s t be p e r f o r m a b l e b y S - - t h e n a (~ s y m m e t r y ,) b e t w e e n f r a m e s s a n d S m u s t hold . I n t he sense t h a t pa r t i c l e s b e h a v i n g as t a c h y o n s w i t h r e spec t to obse rve r s s wil l b e h a v e as b r a d y o n s w i t h r e s p e c t to ob- s e r v e r s S, a n d vice versa (principle o] duality). Actua l ly , t h e words b r a d y o n (B), t a e h y o n (T), f r a m e s, f r a m e S h a v e on ly a relative m e a n i n g (1). T h e ve loc i ty of l igh t c p r e s e r v e s of course i t s c h a r a c t e r of i n v a r i a n t q u a n t i t y for b o t h s a n d S f r a m e s (4).

(1) E. RECAI~[I and R. MIG~A~I: Left. Nuovo Cimento, 4, 144 (1972). This paper will be referred to in the following as * paper I *. (3) G. RA~kCHA~DRA~, S. G. TAGARE and A. S. KOLAS~: Lett. Nuovo Cimento, 4, 161 (1972). (*) Our extended LT's happened indeed to generalize Parker's (a) ones from the bidimensional to the four-dimensional case. (*) Z. PARKER: Phys. Rev., 188, 2287 (1969). (~) See Paper I, and references therein.

624

S U P ] ~ R L U M I N A L F R A M E S AND T H E G R O U P ETC. 625

W e shall call ~ iner t ia l ~) all the (physical) f r ames w i t h re la t ive speeds bo th u < e and u > e. D u e to t he ~( pr inc ip le of dua l i ty ,, f r ames S are supposed to have at t he i r disposal exac t ly t he same phys ica l objec ts as f rames s have .

I n paper I , i t has been a rgued t h a t a ~ pr inc ip le of r e l a t i v i t y ~> mus t hold for t he whole class {I} of (~ iner t ia l ,) f rames, since the phys ica l laws (when genera l ized also for *achyons) are to be covar ian t (~) for G L T ' s of t he whole class {I}. Actua l ly , if b o t h s and S observe the same object (as r equ i red in r e l a t iv i ty ) , b radyon ic laws will t r ans fo rm into tachyouic laws under a super lumina l LT , and vice versa. Therefore , the t o t a l i t y of relativistic phys ica l laws (wr i t ten in the form va l i d for bo th B ' s and T's) wil l be ~ G- covar ian t )) (1).

I n these senses, we m a y say t h a t all our iner t ia l f rames are equivalent.

2. - The condi t ion for hav ing the (( pr inc ip le of dua l i ty ~ satisfied is the fol lowing (~) : W h e n pass ing ]rom s to S , spacelike intervals m u s t t rans]orm into t imel ike in tervals , and vice versa.

W h e n in pape r I we spoke (not too correct ly) about (( change of me t r i c )) or (~ me t r i c inversion ,, we m e a n t no th ing b u t th is fact . Such an invers ion opera tes indeed on the sym- m e t r y wi th respect to the l ight-cone. 1Vfathematically, the G L T ' s mus t be such t h a t

(1) c 2 t ' 2 - - x ' 2 = -r ( e 2 t ~ - - x ~) for u ~ e .

The l inear t ransformat ions , connect ing iner t ia l f rames and sa t is fying eq. (1), are, roughly speaking, i) the usual, o r thochronous (homogeneous) Lo ren t z t r ans fo rma t ions A< (and the ones - - A < ~ ( P T ) - A < ) for u < c , ii) t he genera l ized Loren tz t ransformat ions ~ iA> ~ 4- i l .A> for u > c (1), where (*)

(2) A< =--A(I~I< 1 ) , A> --= A(] f l [> 1) .

F o r example , in paper I we h a v e shown t h a t - - i n t he s imple case of col l inear mot ion a long the x-axis condi t ion (1) is satisfied by

(3)

x - - u t t - - nx /e 2 x" - - - - t" - - - -

VI1- ~1' ~/11-8'1'

= , / = ,

V II-- /J~l [ l l - - f l ~ l

[ ~ ~ 1 ] ,

for r e l a t ive speed both u < c and u > c. The GLT ' s , eq. (3), are precisely of the forms A < and i A > for f12 < 1 and f12 > 1, respec t ive ly .

I n general , le t us consider a un iverse free of charges and represen t t he A ' s by 4 x 4 matr ices . Since mat r ices A> are fo rmal ly iden t ica l w i t h usual L T ' s , b u t cor responding to values ]fl[ > 1, i t is i m m e d i a t e to see t h a t

(4)

Thence

(Sa)

A<Z(fl) -~ A < ( - - f l ) , [ iA>(f l ) ] -x= - - i A > ( . f l ) ~ - - i A > X ( f l ) .

[ iA > @ ] . [ - - i A > ~(p)] = I ,

(*) Let us explicitly recall (1) that the matrices A> are complex.

626 R. MIGNANI, E. RECAMI and v . LOMBARDO

but

(5b) [iA>(fl)]. [iA>X(fl)] = -- I ~ P T ,

so that our generalized (1) group G will contain the total-inversion operator as an element. Precisely, by considering successive applications of GLT's of the types A< and iA>, i t is easy to realize tha t the group G consists of four subsets:

(6)

where

SU~< ~ SO~< =-- SO~+(1, 3; R; ]ill <> I), SUt> -~ SU~(I,3; C; I~I > I),

_-- = i A > ; 4 e

and so on. All the elements L of G are rotations in the four-dimensional space-time, ~.e. the transformations L arc unimodular (with det L = -b 1).

The structure of G will be clarified in a forthcoming article. Here let us simply mention that a correspondence exists between subluminal LT's from a frame So to a frame s, moving with velocity u (0 c). Such a biunivocal correspondence between frames s and S is the particular con]ormal mapping (inversion)

( 7 ) u ~ e 3 / u .

In the case ot a charged universe, interesting observations may be made about the C P T covarianee.

3. - Afterwards, it is worth-while to clarify the following. When generalizing (1) physical laws for tachyons (fl > 1), one should pay at tent ion that a priori ~/f13_ 1 = = • i - v / 1 - 33. Always (*) we consistently choose the sign minus, in order, e.g., to get positive values of the relativistic mass (see eq. {4') of paper I). I t is understood that ~/1--33 represents, for f l> 1, the upper-half-plane solution.

Lastly, let us notice that relativistic laws may be easily generalized for taehyons. In fact, from our discussion about the <~ equivalence, of all the inertial frames, it is immediate to get the following taehyonization principle: ~ The relativistic laws (of mechanics and electrodynamics) for tachyons follow by applying the GLT's to the corresponding laws for bradyons ,)(**).

* * *

The authors are grateful to Prof. A. AGo])I and Dr. M. ]3ALDO for many useful discussions.

(*) A misprint occurredin t h e s e c o n d eq.(7)of paperI,whioh ought to reaxl v'u,z = +vy,z%/~--fll/(1 -uvz/c') for u > c . (**) A f t e r t he comple t ion of p a p e r I , we b e c a m e a w a r e of t h e exis tence of paper s (~,o), which approached ou r p rob l em too. Crit icizing ref . (5) is the subs tan t i a l con ten t of ref . (6). (6) J . G. GILSON: Mathem. Gazette, 52, 162 (1968). (*) S. NARANAN: Left. Nuovo Cimento, 3, 623 (1972).

IL NUOV0 CIMENTO VOL. 14A, N. 1 i Marzo 1973

Generalized Lorentz Transformations in Four Dimensions and Superluminal Objects.

1~. MIG~ANI a n d E. RECAMI

Is t i tu to d i E i s ica Teorica dell'U~Hve~'sitd - Catania Ce~tro Sic i l iano di l~isica Nucleare e di S trut /ura della Mater ia - Catania

Is t i tu to Zrazionale d~ Fgsica Nucleare . Sezione di Cata~Ha

(ricevuto il 16 Agosto 1972)

S u m m a r y . - - A new group G of Lorcntz transformations (LT) in four dimensions, gcnerMized also for Superluminal frames, is introduced and particularly studied in its physical implications. With the help of a (~ principle of duality ))--implied by G--between sublumin~l and Super- luminal frames, the meanings of (, inertial frame ~, (~ equivalence ~, <~ principle of relativity ~>, (( covariance ~) may be correspondingly extended. A biunivocal correspondence exists between bradyonic and tachyonic velocities, which we find to be a particular conformal mapping (inversion). Since the group G consists of generic rotations in space-time, it includes, e.g., also the total-inversion operation (PT) . Moreover (for a non (~ eharg,',>- free universe), it is shown ~hat our gelmralized sp('ci~d relativity requires covariance under CPT. A <~ tachyonization principle )) is formulated, on the basis of which relativistic physical laws (of mechanics and electrodynamics, ~ least) can be easily extended to tachyons. Many simph' applications are performed of the generalized LT's (wdocity composition law, comparison of the length and time units. Doppler effect, refraction index .... ), eith('r useful to clarify our problem or interesting in astro- physics.

1 . - I n t r o d u c t i o n .

I n ~ r ecen t p~per of olu's (~), ~ gene ra l i za t ion of the Loren tz t r a n s f o r ma -

t ions (LT) h~s b e e n pe r fo rmed i ~ f o u r d i m e n s i o n s , for reference f rames S

(1) E. RnCAMI and R. MI~NANI: Lett. Nuovo Cimento, 4, 144 (1972). This paper will be referred to ill the following as (( paper I ~). See also V. S, OLKHOVSKY and E. R~:CAMI : Lett. ~ u o v o Cimento, l , 165 (1971).

169

170 R. MIONANI and E. R~CAMI

t r a v e l l i n g f a s t e r t h a n l igh t w i th r e s p e c t to t h e u sua l class of i n e r t i a l f r ames s (*).

T h e p h i l o s o p h i c a l a n a l y s i s d e v e l o p e d in p a p e r I showed t h a t , if S u p e r l u m i n a l

obse rve r s S a re to be p h y s i c a l (in p a r t i c u l a r , if s t a n d a r d s p a c e - t i m e m e a s u r e -

m e n t s m a y a c t u a l l y be p e r f o r m e d b y S), t h e n a (( s y m m e t r y ,) or (( d u a l i t y ,~

b e t w e e n f r a m e s s a n d S m u s t ho ld , in t h e sense t h a t p a r t i c l e s b e h a v i n g as

t a c h y o n s (s) w i t h r e s p e c t to s u b l u m i n a l obse rve r s s wil l b e h a v e as b r a d y o n s (~.s)

(*) After the completion of paper I , we became aware of the existence of papers (3.4), which approached our problem too. Criticizing ref. (2) is the content of ref. (3). Paper (4), b y ANTIt'PA, puts forward a theory yielding the same results as Parker ' s (7), both works being confined to bidimensional space-time. The original features of ref. (4), i.e. introducing unidireet ionali ty in t ime for bradyons and in space for tachyons, seems extraneous to our own philosophy. (2) J. Cr. CvlLSON: Mathem. Gazette, 52, 162 (1968). See previous footnote. (3) S. ~TARANAN: Lett. 1Vuovo Cimento, 3, 623 (1972). See previous footnote. (4) A . F . ANTIPPA: A one-dimensional causal theory o/tachyons, prepr int UQTR-TH-3, Universi ty of Quebec (Trois Rivieres, 1971). See previous footnote. In a preceding paper (5), ANTIPPA and EVERETT went back to Feinberg 's philosophy (~), taking advantage of the fact tha t Lorentz-covariancc violation implicit in i t does not come out in a bidimensional space-tirae. (5) A. F. A~TIPPA and A. E. EVERETT: Phys. Bey. D, 4, 2198 (1971). See ref. (a). (e) CA. FEINBERG: Phys. Rev., 159, 1089 (1967). See ref. (4). (7) L. PARKER: Phys. Rev., 188, 2287 (1969). (s) See paper I and re/erences therein. For a good bibl iography about tachyons see also the references quoted in J. S. DANBURG, G. R. KALBFLEISCtt, S. R. BORENSTEIN, R. C. STRAND, V. VANDERBURG, J. W. CHAPMAN and J. LYS : Phys. Rev. D, 4, 53 (1971), and in J. S. DAN]3URG and G. R. KALBFL]~ISn: prepr int B:NL-16394 (Upton, 1971).

Problems connected with causality and taohyons have been essentially dealt with in the following papers : E. C. G. SUDARSHAN: The theory o/particles traveling /aster than light, I, in Symposia ou Theoretical Physics and Mathematics, Vol. 10 (New York, 1970) ; ,.1. A. PARMENTOLA and D. D. H. YEE: Phys. Rev. D, 4, 1912 (1971); P. L. CSONKA: Nuel. Phys., 21B, 436 (1970); R. G. ROOT ~nd 5. S. TREFIL: Lett. JYuovo Cimeuto, 3, 412 (1970); L. S. SCHUL~AN: Am. Journ. Phys., 39, 481 (1971); R. G. NEWTON: Science, 167, 1569 (1970); E. RECAMI: Lett. Nuovo Cime~to, 4, 73 (1970), and references therein.

The most impor tan t papers a t tempt ing a ]ield theory for tachyons seem to be: ref. (e) ; J . DHAR and E. C. G. SUDARSHAN: Phys. Rev., 174, 1808 (1968); G. ECKER: An n . o/ Phys., 58, 303 (1970); B. SCrmOER: Phys. Eev. D, 3, 1764 (1971); L. STRmT and ft. R. KLAUDER: Tachyou quantization, prepr in t (Syracuse University, 1972).

Among the more recent papers concerning taehyons and tha t have not ye t been quoted in our previous paper , let us mention: R. PERRIN: Phys. Rev., 140, B 199 (1965); D. KORFF and Z. :FRIED: ~UOVO Cimer~to, 52 A, 173 (1967); C. FRONSDAL : Phys. Rev., 171, 1811 (1968); E. C. G. SUDARSHAN and N. MUKUNDA: Phys. Rev. D, 1, 571 (1968); C. FRONSl)AL: Phys. Eev., 182, 1564 (1968); E. C. G. SUDARS~A~: Eleme¢~tary particle theory, in Proceedings o/the V I I I ~obel Symposium (1968), p. 335; ]:)roe. Indian Acad. Sei., 69, 133 (1969); Phys. l~ev. D, 1, 2428 (1969); Y. AHARONOV, A. KOMAR and L. SUSSKI~rD: Phys. Bey., 182, 1400 (1969); B. M. BILANIUK and C. G. SUDARSltAN: 2~alure, 223, 386 (1969); N. MUKUNDA: Ta ta prepr int (12/4/1969); A. M. GLE~SON and C. G. SUDARSRAN: Phys. Rev. D, 1, 474 (1970); ~ . CAMENZIND: Gen. Relat. Gravit., 1, 41 (1970); E. ~ARX: Int. Journ. Theor. Phys., 3, 299 (1970); R. G. NEWTON:

G E N E R A L I Z E D L O R E N T Z T R A N S F O R M A T I O N S IN F O U R D I M E N S I O N S ETC. 171

wi~h r e spec t to S u p e r l u m i n a l obse rve r s S, a n d viceversa. A c t u a l l y , t h e words

b r a d y o n s (B), t a c h y o n s (T), f r a m e s s a n d f r a m e s S h a v e no ~bso lu te m e a n i n g ,

b u t on ly a r e l a t i v e one (~) ((( principle o] duality ~).

Besides , in p a p e r I i t was shown t h a t , if .~ f r a m e S is S u p e r l u m i n a l as seen

f rom a f r a m e s, i t wil l s t i l l be S u p e r l u m i n a l when seen b y a n y o t h e r (( sub-

l u m i n a l )~ f r a m e , a n d so on. T h e v e l o c i t y of l igh t c p r e se rves of course i ts ctmr-

uc te r of i n v a r i a n t q u a n t i t y for bo th s nnd S f r a m e s (18).

W e sh~ll ca l l ~ inertial ,~ all t h e (phys ica l ) f r a m e s s a n d 8, w i th r e l a t i v e

ve loc i t i es u ~ c.

B u t , a cco rd ing to p a p e r I , we sha l l for t h e m o m e n t r e g a r d t h e re fe rence

f r a m e s m o v i n g w i th t h e i n v a r i a n t v e l o c i t y c as ( the on ly ones) <>,

b e c a u s e of t h e i r singular c h a r a c t e r i s t i c s (~.s).

I n p a p e r I i t has been a r g u e d t h a t a << principle >> of relativity does a c t u a l l y

ho ld for our whore class {I} of <( i n e r t i a l >> f r ames , s ince t h e p h y s i c a l laws (when

g e n e r a l i z e d also for t a c h y o n s ) a re to be c o v a r i a n t for t h e whole class (I} (*).

W e shal l come b a c k to th i s po in t . ] t is w o r t h - w h i l e t o a d d t h a t - - d u e to

t h e <( p r inc ip l e of d u a l i t y >>--frames +~ a re a s s u m e d to h a v e a t t h e i r d i sposa l

e x a c t l y t h e s ame p h y s i c a l ob j ec t s as f r ames s have .

( )wing to t h e h+st two fac ts , we can ge t t h e +< eqmivalence o/reJerence/rames ~>

e x t e n d e d to a l l our <( i n e r t i a l f r a m e s )>. I n t h e fo l lowing we sha l l show how

t h e c o m p a r i s o n m a y be p e r f o r m e d b e t w e e n m e a s u r e m e n t s t a k e n in sy s t e ms

s a n d ~q.

T h e m a i n poin t of our g e n e r a l i z a t i o n is t h e fo l lowing : w h e n passing f rom

a s u b l u m i n a l f r a m e s to a S u p e r l u m i n a l one N (or vice versa, s ince t h o s e t e r m s

Scie~ce, 167, 1569 (1970); A. PERES: Phys. Lett., 31 A, 361 (1970); S. ]. BEN-ABRAHAM: Phys. Re~,. Left., 24, 1245 (1970); F. A. E. PIRANI: Phys. Rev. D, 1, 3224 (1970); A. M. GLEESON, M. G. GUNDZIK, E. C. G. SUDA]~SHAN alia A. I)AGNAMENTA: Particles a~d Nuclei, 1. I (1970); J. STRAND: Fortseh. Phys., 18, 237 (1970); M. GLt~CK: Nuovo Cime~to, 1 A, 467 (1971); L. S. SCHUL~.~N: Nuovo Cime~lo, 2 B, 38 (1971); K. KAMOI and S. KAMEFUCHI: Progr. Theor. Pl*ys.. 45, 1646 (1971); P. V. RAMANA ~[URTHY: Lett. Nuovo Cimet~to, 1, 908 (1971); l t . K. WIMMEL: Lett. Nuovo Cime~to. 2, 363 (1971); B. A. HUBERMAN: Pllys. Lett.. 36B, 573 (1971); R. G. CAWLEY: prepr int USN-2 (Silver Spring, Md., 1971); 1t. C. CORBEN: CSU preprint (Cleveland, 0. , 1971); J . E. MURPHY: LSUNO preprint (Sake,front. New Orleans, 1971). [*) As we know, particles tha t look like B's (T's) to frames s will look like T's (B's) to frames S. and t, ice versa. Therefore- -and we shall come back to this point in the fol- l o w i n g - i f both s and S observe the same object as required in relat ivi ty, bradyonic laws will t ransform into tachyonic laws under a Superluminal LT, and vice versa. On the contrary, each observer (cith~r s or S) will use the same law to desaribe di]]ere~t objects: namely, objects having vclocity u < c (or u > c ) relative to him himself: 4.e. to describe ((bradyons ~) or (~ tachyons ~). In this second sense, the whole of rela- t ivistic physical laws--when writtett in a form holding for both B's and T ' s - -has to be G-covariant. An example may be easily worked out. e.g., in relation to eq. (7) of the text. More in gem~ral, one may be convinced of the last assertion by remembering the+ possibility of writ ing laws in tensorial form.

172 ~. MIGNANI and ~, RECAMI

have no absolute meaning), we have to admit tha t space (time)-like intervals with respect to s must be t ime (space)-like with respect to S (~). Such an ~ invers ion ~ is required in order to realize the above-mentioned symmet ry between s and S frames (~.s). Actually, t ha t (( inversion ~) operates the symmet ry

with respect to the light-cone (both in the configuration and te t ra impulse

spaces). ~amely , in paper I it has been shown th a t LT 's between two inertial f rames

must be such tha t (~.~)

(1) c2t '2 - - x '~ = :]: (c~t ~ - x 2) for u < c ,

respectively. The linear t ransformations connecting inertial frames and satisfying eq. (1)

a re - - rough ly speak ing- - the usual proper, orthochronous (homogeneous) Lo- rentz t ransformat ions A< for u < c (*), and ] .orentz t ransformations iA>

=~ i l A > for u > c (*)(~).

For instance, in paper I we have shown t h a t - - i n the simple case of col-

linear mot ion along the x-axis - -condi t ion (1) is satisfied (**) by

(1 his)

X - - Ul t - - u x / e e

Z ! • y ' = ± y ~ , , = ± z

Let us rewrite relation (1) for f l 2>1 (i.e. for t ransi t ion from a s to a ~) as

c~t,2 ÷ (ix,)~ + (iy,)~ ~_ (iz,): = (ict)2 + x ~ + y2 + z ~ ,

and expl ic i t ly not ice the following. Since we considered for s impl ic i ty the

case of coll inear m o t i o n a long t he x-axis, our eqs. (1 h i s ) m u s t b e - - a s t h e y

a r e - - s u c h t h a t

(1 ')

and tha t

(1")

c~t'~ + (ix')2 = (ict)2 + x 2

( iy ') ~ = y".. (iz')~ = z ~ .

(*) And the ones t i mes the operator - - 1 = ~ P T . See Sect. 2. (**) See, R. MIGNANI, E. REeAMI and U. LOMBARDO: Let t . N u o v o C imen to , 4, 624 (1972). We shall see in the following (see Fig. 4 and 2) that in our eqs. (1 his) the sign mi~tus

has to be taken only for ((~ transfinite ~)) transformations, bypassing the point Po~ according to the observer.

GENERALIZED LORENTZ TRANSFORMATIONS IN FOUR DIMENSIONS ETC. 1 7 ~

Of course, bo th s and S will observe only real quant i t ies! The imaginary units mere ly record the relat ive sign of the var ious couples of the corresponding-

component squares. One of the aims of the present work is extensively i l lus t ra t ing the new

generalized LT gn'oup G, since it was scarcely ment ioned in paper ]. Another

a im is clarifying some points of paper I .

I n our terminology G-covar ian t means covar iant under the whole group G.

By the way, let us r em em ber t h a t only measu remen t results are supposed

to be expressed by real quanti t ies, bu t Um generalized Lorentz t ransformat ions

(GLT) m a y well be represented b y matr ices built up with imaginary quantities too (1).

2. - The group G of the generalized Lorentz transformations in four dimensions.

I n order to fix our ideas, let us first consider a universe free of (~ charges )> (*)

and let us represent the A's b y 4 × 4 matrices. I f we put

A< A(tfll < ~), A> =-- A(lfll> 1),

where A(lfl] > 1) are ]ormal ly identical with usual proper, or thochronous LT ' s bu t correspond to values [fll > 1, it is possible to see t h a t

A-<l(fl) - - ~ A<(- - f l ) , [iA>(fl)] -1 = - . i A > ( - - f l ) ~ - - i A ; l ( f l ) .

Thence

but

[~A> (fl)]. [-- ~A;I(~)] = 1,

and the generalized group (1) G of pape r I has in par t icular to contain also the

total- inversion opera tor

P T -- - -1 •

Precisely, b y considering successive applicat ions of GLT ' s of type A< and

i A > , i t is easy to realize t h a t the group G of the GLT ' s consists of four subsets:

(2) (7 ~ (~,_) u ( , ,~ ) u ( , ,~ ) u ( ~ G ) ,

(*) In this work, tile word ~ charge ~ is used ill its widest sense.

174 a. MIGNaX~ and ~. RnCAM~

where

£/' , = {A<} , .(2'3 ~ ( - - A < } ,

f f~ ~ (iA>} , £ f , - - {-- iA>} .

I n fact , if A< is a generic LT for ]ill< 1, i.e. a subluminal LT, (LT), then the generical GLT for Ifl] > 1, i.e. the generic Superluminal LT , (STL), will(I) be iA>. Moreover, if JSeG, then a l s o - - Z ~ ( t ' T ) I ; e G . I n the fol-

lowing, we shall indicate s imply b y L the generic element (GLT) of G.

B y the way

d e t Z = ~ - 1 , V L e G .

I t is easy to recognize t h a t a correspondence exists between subluminal

L T ' s f rom a f rame So (*) to a f r ame s moving with veloci ty u (0 c), in the sense t h a t

(3) iA~>(Y) = K . A < ( c ~ / Y ) ,

/i: being a ma t r ix independent of the veloci ty magni tudes ~, U. For simplici ty we consider eq. (3) only for collinear motion.

The ma t r i x K represents a (~ transcendental SLT ~. I n fact , for U - > ÷ c%

eq. (3) becomes

(3 his) ~5.~ ~-- iA>(=J= co) = K .

This accords with the observat ion tha t , if a t achyon moves with velocity v ~ v= relat ive to us, it will appea r with divergent veloci ty to the observer s"

having (collinear) veloci ty ~ ~ u= - - c2/v relat ive to us (see Fig. 3). For in-

stance, in the simple case of collinear motion, u ~ u=, U ~ U~,

Z ~ K =

0 - - 1 0

- - 1 0 0

0 0 i

0 0 0

0 ~

0

, °

0

i

(*) For simplicity, in the following we shall consider ourselves as (( the observer s o )~.

G E N E R A L I Z E D L O R E N T Z T R A N S F O R M A T I O N S I N F O U R D I M E N S I O N S E T C , 175

A t last , in the col l inear case, t he m a t r i x K opera tes the exchanges

,~'(u) -+-t(c2/u) ,

t ( u ) - + - - ~'(e'-/u)

a n d y(u), z(u)-+iy(c2/u), iz(c2/u), t h a t accords wi th eqs. (1')-(1").

I n the eol l inear case, for in f in i te speed, as well as for zero speed, the n o t i o n

of d i rec t ion becomes mean ing less (7)(*).

I t is a l r eady rea l izable t h a t the tools of p s e u d o - E u c l i d e a n geome t ry are

no t t he bes t ones for dea l ing wi th our p rob l ems ; we shal l therefore bor row a

b i t f rom p ro jec t ive geomet ry .

F r o m w h a t precedes, i t is a p p a r e n t t h a t our (< d u a l i t y pr inc ip le ~> is char-

ac ter ized b y the fac t t h a t - - w i t h reference to a f r a me so - - the re is a b iun ivocM

cor respondence b e t w e e n observers wi th ve loc i ty u a n d those wi th ve loc i ty

U -- c+:/u. Precisely, t he << s y m m e t r y >> (see Fig. 1) b e t w e e n 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 f rames is a p a r t i c u l a r conformal mapping (<< invers ion ~>):

(4) u " ~, c * / u .

P

x ,\ , / <,

s0

Fig. 1. - Representation of the conformal mapping (inversion) u +-+c2/u in the sim- plified case of collinear velocities u------u~? 0 relative to a frame s 0. Since lulT~ c and we have to deal also with the transcendental frame, we project from a pole P the axis u onto the circle having re-- ± c as diametral points. The chords AB (where A, B refer respectively to a subluminal velocity u and a Superluminal velocity U = c2lu, i.e. to corresponding velocitics in the conformal mapping) are normal to the axis u.

I n such a m a p p i n g , the veloci t ies u = U -- c are the united ones (as requi red) ,

a n d velocit ies zero, in/ inite cor respond to each o ther :

(4 bis) u = e + - + U = c , u = O+->U = c~.

(*) We might also define the LT to a << luminal frame >>. But, as already said, we cannot consider the luminal frames s t / , as physical, even if mathematical use of ~infinite- momentum frames ,> spreads out. In any case, from such frames the space-time should appear as a bidimensional space free of photons (projection of the instantaneous 3-dimensional space onto a plane normal to the ray direction).

176 R. MIGNANI and ~. ~CAM~

The l a t t e r tells us the k n o w n fact t h a t t he d ive rgen t ve loc i ty p lays for t a c h y o n s

t he same role as p l ayed b y the null ve loc i ty for b r adyons . The m a p p i n g (4)

was realized also b y previous au thor s (H.7.~) in different ways.

I n order to i l lustrate the phys ica l m e an ing of the four subsets in eq. (2),

let us for s impl ic i ty confine ourselves to the case of f rames wi th coll inear ve-

locities. W e can represent such velocit ies u ~ u~ ~ 0 (relative to a f r ame So)

a long an axis; see Fig. 2. B u t , since lul ~ c and we have t o m e e t also t he diver-

.~(-1)

#o. iA>( U)~ 7 ~ 2 2,

~- iA~. (U) t ~ , ' ~ ~ > (U =- c / u)

A<(-u)~

'R S o

x(+ l)

Fig. 2. - Illustrating the physical meaning of the four sets (see eq. (2)) that form our generalized Lorentz transformation groups G in four dimensions. For simplicity, we consider only frames with collinear velocity u - - % relative to a frame So. Let us notice tha t A<(--u) = A<~(u), A>(--U) = A>I(U). The Figure must be read in counter- elockwisc sense. When we bypass the (,branching point ~ P~o, we start considering eqs. (1 bis) with the opposite sign. See the text.

gen t veloci ty , it is conven ien t t o p ro jec t t he axis u f r o m the pole P ~ P~ on to

the circle h a v i n g u = ~= c as d iamet ra l points (see Fig. 1, where chords A B e _ u). The charac ter i s t ic fea ture of Fig. 2 is t h a t our circle y m u s t be considered

as a (~ double circle 'b wi th a (( b r anch ing poin t ~) at P(u -- c)o) (such a po in t re-

flects the exis tence of t he double sign in eqs. (1 bis).) I n fact , a long y we

pass wi th con t inu i t y f rom f rames s ~ (e.g. with a right-handed spat ia l f rame)

t o to t a l ly inve r t ed f rames sL=~ P T s R (with a le]t-handed spat ial f r ame and a

reversed t ime axis). This could have been expected , since the to t a l inversion

P T is no~hing b u t a p~r t icular ~ r o t a t i o n ~ in ( four-dimensional) space- t ime,

and such a r o t a t i o n m a y be ach ieved when we do no t res t r ic t our a t t e n t i o n

a n y more to sub lumina l re la t ive velocit ies (*).

(9) K .M. MARIWALLA: Am. Journ, Phys., 37, 1281 (1969). For a brief criticism of this work, sec the first footnote in paper I. (*) See also, e.g., 5. J. SAKURAI: lnvariance Pri~tciples and Eleme~ttary Particles (Princeton, 1964), p. 137; V. B. BI~tCl~TESK¥. E. M. LIFSIIITZ and L. P. ])ITAEVSKY: Relativistic Qua~tum Theory (London, 1971), p. 34.

GENERALIZED LORENTZ TRANSFORMATIONS IN FOUR DIMENSIONS ETC. 177

Let us be clearer. I f A< is the L T f rom So to f r ame st - - 1 m o v i n g wi th

ve loc i ty u, t h e n t he (conformal ly cor respondent ) t r a n s f o r m a t i o n iA> will con-

nec t So to the f r ame ~S'~ 2 m o v i n g wi th ve loc i ty U--c2 /u with respect to So

and wi th infinite ve loc i ty (*) wi th respec t to s~. Now, we m a y go back f rom

S~ to the same So b y app ly ing the inverse t r a n s f o r m a t i o n (A>) -~ - - i A > 1, so

t h a t ( i A > ) . ( - - i A > ~) 1. B u t we m i g h t also decide to come back f rom $2 to so

(( bypass ing )) t he t r anscenden ta l f r ame (relative to so), i.e. b y app ly ing the t rans-

f o rma t ion 5 iA>~; however , in this c~se we would go back to the f r ame So

with all its axes rew'rsed, [(PT)so], since ( iA>) . ( iA> t) - - - - ~ _ PT . More gen-

erally, if ]5(So-->/) is t he G L T connec t ing f rames .% and ], we have

L(so -~ 1) -~ A<(u) , L(so -~2) -- iA>(c~-/u).

L(.~'o -~ 3) ~ - - A > ( - - c'/u) = - - iA~*(e2/u),

L(.~,, ~ 4 ) ~ - - A < ( - - u ) = - -AT~(~) ,

L(so ~- 5) ~ - - Sl<(u), L(so --> 6 ) - - - - i A > ( c ~ / ~ ) ,

L(.% --> 7) ~ iA~(c~/~e), L(So ~ 8) ~ A ~ ( u ) ,

L(so ~ 9 - - 1 ) -- A<(~) .

The re la t ive ve loc i ty be tween s~ ~ 1 and $2 -~ 2, or be tween s8 ~ 8 and $7 - - 7,

is infinite; whilst t he re la t ive ve loc i ty be tween ~-qa ~ 3 and S~ = 2, or be tween ss~=8 and s ~ - - I is

2u 2 U

1 + (uleV- 1 ÷ (UIeF'"

E v e r y t ime we cross the u-axis , we have the exchange spacelikee-~timelike, or fo rmal ly a mul t ip l ica t ion b y i i (in t he sense of Fig. 2). I n o ther words, a t

u - + c we get t r ans i t ion f rom { :~ A<} to { :~ iA>}, and a t u ---- - - c t r ans i t ion

f rom {:J-iA>} to {~:A<}. Besides, when wc cross the pole P ~ , we get t h e PT- inve r s ion , or fo rmal ly a mul t ip l i ca t ion by - -1 .

Before going on, we w a n t t o re -emphas ize t h a t t he pr inciple of r e l a t i v i t y - - i n

its wider s ense - - a sks t h a t phys ica l l~ws (~t least in their fo rm val id fo~ bo th B ' s

and T's) bc covar i an t u n d e r tell t h e d e m e n t s of G. Therefore , it m a y seem

t h a t covar iance :flso um[cr t he P T - s y m m e t r y is required.

Le t us inspect the p rob lem more closely. Le t us now consider a ~o~

(~ charge ~)-free universe. Fo r going" (from so, in t he counter -c lockwise sense:

see Fig. 2) to a ~( PT-ed ~) f rame, we m u s t bypass the (, b r anch ing po in t ~> P ~

i.e. consider ing GLT 's , eqs. (] his), with negative sign. Therefore , if we consider

(*) The c o r r e s p o n d i n g f r ames s, S such t h a t ~_ l_ u, h a v e inf in i te re la t ive veloci ty . The SLT f rom s to S will be ill t h i s case f ~ = L ~ (see Fig. 2).

12 l l N u o w ) C i m e M . A .

178 ~. MIGNANI and ~. ~CAM~

a particle, also i ts t r a n s f o r m e d ene rgy will be nega t ive . This f u r t he r f ac t ha s

to be t a k e n in to accoun t : t o g e t h e r w i th P T , i t a l lows t h e t r a n s i t i o n (s) f r o m t h a t pa r t i c le to i ts antiparticle (*). I n o the r words, e x t e n d e d r e l a t i v i t y tel ls

us t h a t e l e m e n t a r y - p a r t i c l e laws are expec t ed not to change on ly u n d e r t he

c o m b i n e d ope ra t ion : i) pa r i t y , ii) t i m e reversa l , iii) p a r t i c l e - an t i pa r t i c l e

exchange . W e shall come b a c k to th is ques t ion l a te r on.

3. - Extended velocity composit ion law.

A t this po in t we w a n t t o spend some m o r e words a b o u t t h e genera l i za t ion

of t h e ve loc i ty compos i t i on l aw for speeds exceed ing the speed of l ight . Fo l lowing p a p e r I , wi th re fe rence to t w o col l inear s y s t e m s m o v i n g wi th r e l a t ive ve loc i t y

u ~- u~, l u l > c, we h a v e

(5) v~ = c ~ v , , - u , fl = ~ e 1 C~--~tVll ~ V± ~ t~2--UVH

L e t us confine ourselves only to v - - v , ; eqs. (5) hold for ~<>c (see Fig. 3).

I t is a l r e a d y well k n o w n (**) t h a t a t a c h y o n T m a y a p p e a r as a (( par t ic le ~ (e.g.

in t h e ini t ia l s t a t e of a ce r t a in reac t ion) to a n obse rve r s a n d as an (( an t i pa r -

t icle ~) in t he oppos i t e s t a t e (i.e. ou tgo ing f r o m t h a t i n t e r ac t ion region) t o

o the r obse rve r s s ' . I n fac t , t he f irst of eq. (5) yields, for [~t]< c a n d v ~ > c,

(6)

for u = c ~ / v ::>v' = ~ ,

for - - c sign (v') = sign (v~),

for c > u > c~/v~ ::> sign (v') = - - sign (v~).

! !

T h e s a m e wou ld of course h a p p e n for v and v . N o w we w a n t to under l ine a n o t h e r pecu l i a r i ty of the first of eq. (5). L e t us

consider our own res t f r a m e So a n d two col l inear S u p e r l u m i n a l f r ames S, S '

h a v i n g (posi t ive) veloci t ies U ~ U~ > c and V ~ V~ > c, respec t ive ly . I n t h e

case t h a t S ' is f a s t e r t h a n S (with r e spec t to us), t h e f r a m e S will not

(*) If we introduce in relativity the notion of (~ particle ~, then relativity itself leads to the concept of (~ antiparticle ~: in such a sense, the concept o] antiparticle is a purely relativistic one. Remember the (~ switching principle ~--whieh assumes that (~ physical signals are t~'ansported only by positive-energy objects ~--and related problems: see, e.g., O. M. P. BILANIUK, V. K. D]~SI-IPANDE and E. C. G. SUDA~SnAN: Am. Journ. Phys., 30, 718 (1962), and ref. (1% (lo) See M. BALDO, G. FO~T]~ and E. RECA~I: Zett. Nuovo Cimento, 4, 241 (1970), and references therein. See also, e.g., E. C. G. SUI)ARSHA~: Ark. Phys., 39 (40), 585 (1969); in Symposia on Theoretical Physics and Mathematics, Vol. 10 (New York, 1970), p. 129. (**) See, e.g., ref. (lo).

GEN:EI{ALIZED LOR]~NTZ TRANSFORMATIONS IN FOUR DIM:ENSIO~'S ETC, 1 7 ~

tV'il

. . l j - ~ - --4- . . . . . . . .

~ c21v

-D ~ 1 °

I \ . !

:!\

i

Fig. 3 . - Generalized velocity composition law, in the ease of eollinear motions: # !

u ~-u~, v ~v~, v ~v~. The curve represents the behaviour of v as a function of u. The velocity v is taken as a fixed positive parameter. Lines a) refer to v -=~ < c; lines b) to v ~ V = c ~ / ~ > c . See eqs. (5) of the text.

observe the f r ame S' m o v i n g wi th pos i t ive velocity~ as suggested b y u sua l

i n t u i t i o n ! On the con t ra ry , i n fact~ if we call v ' ~ v ' t he S ' ve loc i ty as seen

b y S, we have

v ' ~ 0 when I '~ U,

v ' > 0 when V < U.

This fac t has to be b o r n e in m i n d in order to u n d e r s t a n d Fig. 2 be t te r . Le t

us ana lyse i ts s implif ied vers ion, Fig. 4~ in which on ly {~ r i g h t - h a n d e d >> f rames

A~ 5~ A.

I/1.17/~ ~A.(U=c2/U)

Fig. 4. - The same as in Fig. 2 when confining ourselves to only , right-handed ~ frames, i.e. to eqs. (1 bis) with only the positive sign. With self-evident symbols: u12 = co and L12 = K, us1 = 2 u / [ l + (u/e) 2] and 2581 = A~. Of course A< ~- Lot. Point u = oo cannot be gone through, otherwise we would have a total inversion of the frame.

1 8 0 R. MIC-I~ANI a n d E. ~ECAMI

appear. Le t our rest f rame be s0, and our t ranscendent f rame S~. Then, if A<

is the LT connecting So and the f rame sl ~-1, t ravell ing with veloci ty u > 0

(relative to so), the same LT = A< connects S~ and the f rame $2 ~ 2, t ravel l ing

with veloci ty U = c2/u wi th respect to so. This is required by the group prop-

ert ies of G, as we have exploi ted t h e m (*). Bu t this means t h a t S~ mus t

judge $2 to move with the posit ive veloci ty u, even if according to us f r ame

$2 is slower t han S~ ( remember t ha t so, sl, $2, S~ are all collinear f rames!) .

B y the way, L(8 --> 1) ----A<.2

t

/ / /

t o

I / / /

~ Ix X 0

Fig. 5. - Where it is seen that, when S' travels faster than S relative to s o, i.e. to us (all the frames are collinear), then S' appears to S moving with negative velocity (even if it moves with positive velocity with respect to us). See the vector corresponding to S'. Analogously, frames s', s", S" will be seen by S as moving forwards (even if they would be expected to be seen travelling backwards by 8, according to the usual intuition).

The above-ment ioned fact m a y be seen also on the basis of Fig. 5, where

S ' is a (SuperluminM) f rame travel l ing faster t han S relat ive to So, i.e. to us (all the f rames are still collinear). B y inspection of the components of a ge-

neric space- t ime displacement vector associated with the mot ion of S ' (relative to So), one realizes t h a t S ' appears to S as moving with negat ive veloci ty (even

if it moves with posi t ive veloci ty with respect to us). Analogously, f rames s',

s", S" will be seen b y S as moving forwards (even if t hey would be expected

to be seen b y S travell ing backwards , according to the usual intuition).

4. - . T a c h y o n i z a t i o n . pr inc ip le .

Le t us assume t h a t we know, besides the class ~ of usual physical laws

(e.g. of mechanics and eleetrodynamics) for bradyons , also the class ~ of the

(*) However, things may be probably done in a more smooth, continuous fashion by using a bit more the elementary concepts of projective geometry: namely, by suitably inverting the signs in eqs. (1 bis) at u = i c . besides at u = co. This will be discussed on another occasion.

GENERALIZED LORENTZ TRANSFORMATIONS IN FOUI~ DIMENSIOlgS ETC. 181

physical laws for tachyons . When we pass f rom ~ sublnminnl f r ame s to a superluminal one S, class ~ will have of course (~7) to t r ans fo rm into class and vice versa. I n this sense the to ta l i ty of physical laws ( d u ~ ) will be covar iant under the whole group G, or s imply <~ G-covariant ~).

Moreover, a pre l iminary inspection suggests t h a t physical laws (of special relat ivi ty) m a y be wri t ten in a fo rm valid for bo th bradyons and tachyons:

a fo rm obviously coinciding with the usual one in the b radyonic case. As an

example let us r e m e m b e r (~) tile law

'1t~ o (7) m - - - -

which reads

(fl~ X 1) ,

mo imo "~ = g ~ - / ~ > r /;- < 1 , ~nd ,~ = : , / y _ _ ~ for /~ > ~ .

Let us explicit ly re-emphasize t ha t the i appear ing for f12> 1 comes f rom

the SLT's , and does not mean at all t h a t ~achyons have an imaginary prop-

er mass (as we know, a t aehyon behaves as a b radyon with reference to

its rest f rame, and therefore all its proper quant i t ies are real). The same hap-

pens for proper t ime and proper length; for example , since dr = tiT' "V 11 _f12[, when passing f rom the t achyon rest sys tem to our f rame, one has

d~'o (l% dTo i d% (s) d~:'-- g l ~ - - f i ' l Vfi- ' - - 1 -- - - i g i - - f l 2 -- V l - b '~ @ > 1),

where of course d% d~o are bo th real.

I t is worth-while to notice t ha t , when generalizing(1) physical laws for t achyons (/?2> 1), one shouht pay a t t en t ion t h a t a pr ior i ~ / f l 2 - - 1 = =k i V ' l - - f l 2,

since (=k i) 2 = - 1. Always we consis tent ly choose (*) the sigrt mi~u., , in order,

e.g., to get posi t ive values of m in eq. (7). I t is unders tood t h a t ÷ ~/1--f12

represents the upper half -plane solution for fl~->l. See, e.g., our eq. (5). I n the first pa rag raph of this Section we expressed in which sense all the

(~ inertial frames~) (with relat ive velocities [u[~c) are equivalent . F r o m such

considerations, it is immedia te to get the <~ pr inc ip l e o] tachyoniza t ion ~, which

extends Pa rke r ' s principle(;) to the four-dimensional space-t ime: ((The re-

(*) See, e.g., eq. (4') in paper I. But a misprint entered eqs. (7) of paper I, regarding the SLT for velocity; these equations should read

vy,~= 1 - - u v d c ~- = 1--u~'~lc 2 = 1 - - u v d c 2 - 1 - - u v , lc 2

182 ~. MIGNANI and ~. RECAMI

lativistie laws (of mechanics and electrodynamics) for tachyons follow by applying a SLT (e.g. the t ranscendent one) to the corresponding laws for bradyons ,>.

In Sect. 3 we have seen that , since PT is a chronotopical rotat ion, rela- t ivistic physical laws are expected to be covariant under the CPT-symmetry. Since the pure PT operat ion onto space-time brings from a subluminal f rame to another subluminal frame, the previous s ta tement refers to usual relativistic laws (even if not wri t ten in the G-covariant form). Due to its importance, let us derive it in a second way (for a non(( charge ~>-free universe).

As pointed out in ref. (lo), let us consider a t achyon and a succession of subluminal frames (for simplicity, all moving collinearly with the taehyon). Le t us call s~ the frame in which the tachyon appears with divergent velocity. I f a f rame moving slower than s~ sees the t achyon travell ing in a certain direction, then a f rame moving faster t han s~ will actual ly see the

(( t achyon )) as an an t i tachyon travell ing in the opposite direction: this is

precisely what is already expressed by our relations (6). W h a t we want to underl ine here is that , when the tachyon appears with reversed velocity, it will also show the opposite charge (lo). Therefore, bypassing the , infinite- veloci ty f rame ~> s~ (in the above sense) implies the operat ion of charge conjugation (C). With reference to Fig. 4, when we consider our f rame so as seen by a succession of Superluminal frames, and we (( cross )) the t ranscendental f rame S~, we have tha t all our charges appear reversed. In other words, for ((transfinite trans/ormations~) we get a C-symmetry. Therefore, when we operate a rota t ion (in the four-dimensional space-time) with the aim to reach the to ta l ly inverted frame P T so (see Fig. 2), really we reach the frame CPT so.

Briefly, the right way for doing P T is doing CPT. The CPT-covariance is required by our mere (( extended re la t iv i ty principle ~> (when we do not confine ourselves to subluminal relat ive velocities).

Analogous considerations are to be kept in mind when investigating the physical law transformations fol transit ion between two generalized inertial

frames.

5. - Comparison of time and space intervals.

With the s tandard procedure, from the GLT's for the four-vector components one gets the GLT's for the measure units of space (standard-rod length) and

t ime (standard-clock time interval). I f Axo and Ato are the proper ir~tervals, the magnitudes of the observed

ones are

(9) ±x_-- IAxl = At_- I±tl = (1 1

C:I-ENERALIZED LORENTZ TI~ANSFORMATIONS IN F O U R DIMENSIONS ETC. 183

For simplicity, we considered two collinear frames. Of course, eqs. (9) do not depend on the sign of ft. F igure 6, which depicts eqs. (9), shows t h a t for fl-~> 1 we can have both <~ Lorentz )> contract ion and di latat ion of space or

I ~ / \

/ \\

J f

I

IAd,IA~I I /bl j \

\ / /

/ /

A Xo

~/2- ~' fl

Fig. 6. - Dependence of the magnitudes of space length and time interval vs. relative velocity tic, in the case of eollinear motion, for both fl~ 1. See eqs. (9) of the text. The geometrical reason for the fact that At '= At, Ax'= Ax also for f l= v 2 is given in Fig. 8.

t ime intervals. The fact t h a t A x - Axo, At = 5to not only for fl = 0 but also for the value f l - - ~ / 2 will be geometr ical ly in te rpre ted in Fig. 8.

Le t us now consider the SLT ' s for space or t ime intervals by ex tending the

usual geometrical in terpre ta t ion of the L T ' s (see Fig. 7). F r o m an algebraic

point of view, following ref. (7,9), we m a y observe tha t for fl~-~ 1

,r'/fl ~- ct' t t ' /~ + x ' /c

There fo re , i f we p u t t g~o 1//? ' ,n, l K : ~/(fl~ + - l ~ i f l = ~ l i , we .ge ~,

(lo) x -= t ( ( x ' s i n ~o -~ ct'(,os~o) , t = K eos~0~-t 's in~o .

Equa t ions (10), by the s tandard reasoning (11), allow us to in terpret these SLT ' s

as in Fig. 8.

For the extension of the geometr ical in terpreta t ion, it is enough to remem-

ber (besides tile s t andard definitions) tha t : i) the Loren tz - t rans formed

space uni t Ax m a y be also derived f rom tile t ime a (relative to us) t aken by

the moving s tandard rod to pass <~ before our eyes ~>: A , r - - I + aflcl: ii) the Lo-

(11) See, e.g., P. CALDIROLA: Istituzio~d di fisica teorica (Milano. 1966).

184 x. MIGNANI and E. RECAM1

' t

.... u ' . ,

l u t . ,

/ / l

a l(0,-Ax//~c)

,, .f ¢s 7, J / . G "

. - ' 4 ' /

"IIZ "~ X~

i ~ g cp =fl)

Fig. 7. - The geometrical interpretation of the usual LT's ( t i< 1), for space or time intervals, in the ease of eollinear frames and /7> 0. One has length contraction and time dilatation. Besides, making recourse to the standard definitions, one must notice that : i) A z = - - @ e =--I@cl, where l a l = - - a is the time (relative to us) taken by the moving standard rod to pass <~before our eyes >>; ii) A t = b/tic-= Ib/flc[, where Ibl----b is the space travelled (in our frame) by a (~ lamp }> which is switched on for a uni tary time (this time being measured in the co-moving frame). Notice that OA--- -Ax= = - -a t ic~ I@cl< Axe, OB=-- At = b/tic=_ Iblticl > Ate, OU~ = OU~ (At, Ax > 0); 0 </7 < 1.

r e n t z - t r a n s f o r m e d t ime u n i t At m a y be also der ived f rom the space b t r ave l l ed

(in our f rame) b y a (, l a m p >> which is swi tched on for a u n i t a r y t ime (this

t ime be ing m e a s u r e d in the co -moving f rame) : At = [bite I. Moreover, the ex-

o o > t i > 1

At = Iblticl >Ato for

< Ato ]or

Ax = la/ticl < A~o /or

> Axe /or

A(-Ax,O)

1 < / 7 < { 2 I t

t i > V~; I , < ti < V~ ~ _~'.

o

x"(/~=~#) 4

X" /' /// ,/~.]' ft,,(n=~-)

2sD -

b x

Fig. 8. - The geometricM interpretation of our SLT's (for fl > I). The fact tha t one has a change in the (spacelike or timelike) nature of intervals when passing from frames s to frames S reflects in the exchanged use of hyperbolas, when considering space and time intervals respectively. Notice that one has still At = Ib/ticl, and A x = ]aft@ bu t now--according to Fig. 6--we get both Lorentz contractions and dilatations. In par- tieular (for /7> 1), we have At=Ate and Ax=Axo when t = ~/2. Moreover, for 1 < f l < ~/2, we have A t> Ate and A x < Axe, whilst, for /7> V'2, we have At< Ate and Ax > Ax o.

G(EN~I~ALIZ~D LOI~JdNTZ T~ANSFOR1V£ATIONS IN F O U R DIMENSIO:NS ETC. 185

changed use (for fl~> 1) of Che hyperbola, when considering space a l ld t ime intervals respectively, reflects the change in the (spacelike or timelike) na ture of intervals t ha t we have when passing from frames s to frames S (see Fig. 8). Consistently with Fig. 6, in Fig. 8 we have both Lorentz contractions and dilatations as fl varies. In particular, from the equations of our hyperbolas it

is immediate to see (for f l > l ) t h a t At=Ato, Ax--Axo when, and only

when, fl = ~/2. An analogous procedure can be used for in terpret ing the other cases. For

instance, the case - - c ~ < f l < - - 1 results to be symmetr ic to the case in Fig. 8.

xt-1) I i i

- - I I < # - oo

X v X Iv ~ A. 0<13<1

// i:,

I - . \ , ,

.~,lj ~ ~<#<_1~'~ ,1>

xt-1)

Fig. 9. - I l lust rat ion of the same ~onsiderations depicted in Fig. 2, through a general look on the geometrical interpretation of eqs. (1 bis). See also Fig. 7 and 8.

Moreover, if we proceed beyond the case + 1 < f l < c~, we get again the whole previous succession of cases, but--according to Fig. 2 - - i t will now regard total ly inver ted frames. See Fig. 9.

6. - T h e g e n e r a l i z e d D o p p l e r e f fec t a n d c o s m o l o g y .

Let us spend some words on the generalization of the Doppler-effect formula

for Superluminal sources, because of its possible astrophysical interest.

F rom the SLT's for a t ime interval, one gets tha t

%.v')1-#:I ( ] ] ) ~ . . . . .

i ÷ f l { ' o s ~

186 R. MIGNANI and ~. ~CAMI

where u ~ tic is the relat ive veloci ty and ~ ~ ul , the vector ! being directed

f rom the observer to the source. I n the par t icular case of relat ive mot ion str ict ly

along the x-axis, since

sign (u)-sign (cos a) = I approach

t + ~ recession,

we obta in the behaviour represented in Fig. 10, where the dashed lines refer to approach and the solid ones to recession. The two solid curves (recession)

-v n

I /

- /

1 / /

~ +

P

/ /

! I

/

Fig. 10. - Doppler-effect extension: observed frequency vs. relative velocity for motion along the x-axis. The sign minus refers to approach (dashed line) and plus to recession (solid line). The interpretation of the negative values appearing for the Superluminal approach is given in Fig. 11.

are one the conformal correspondent of the other, as expected, in the sense t h a t the same frequency will be observed bo th for u = v and for u----c2/v.

The same is t rue for the two dashed curves (approach), except for the sign.

The in te rpre ta t ion of the negat ive sign appear ing for Superluminal approach

is easy in the spirit of what precedes (1); see Fig. 11.

I n fact , ~ (subluminal) observer will receive the radio-emission of a Super-

luminal source in the reversed chronological order. I t is immedia te to th ink

the following. I f a macroscopic phenomenon produces a known radio-emission

obeying a cer tain chronological law, and one happens to detect a reversed radio-emission, we could believe a superluminal source has been observed.

Ex t r apo la t ing our <( dual i ty principle )~ even for gravi tat ion, we would have

a way to deduce also the behaviour of tuchyons in a gravi ta t ional field.

CTI~NI~RALIZED LORI~NTZ TI~ANSFORMATIONS IN FOU]~ DIM:ENSIONS ETC. 187

As r e g a r d s t h e (~erenkov ells)ct of t a c h y o n s in h o m o g e n e o u s m a t t e r (of

course , we sha l l n o t h a v e such a n effect in v a c u u m (~-~)), t h e y wil l r a d i a t e

(~erenkov l i gh t e v e r y t i m e t h e y t r a v e l f a s t e r t h a n l i gh t in t h a t m e d i u m .

This is a l w a y s ver i f i ed for t a c h y o n s in a ~ b r a d y o n i c m e d i u m )>. On t h e con-

.t

//"

\ \ 5 ",

//- / . / n c~ l

z / \\ ~

Fig. 1 1 . - The radio emission of a Superluminal source, approaching the observer along the x-axis, will be received in reversed chronological order. This is the meaning of the negative frequencies entering Fig. 10. The line S is the Superluminal world line.

t r a r y , t ~ c h y o n ' s b e h a v i o u r in a (, t a c h y o n i c m e d i u m >> wil l be s y m m e t r i c to

t h e b e h a v i o u r of ~ b r a d y o n in u s u a l m a t t e r ( r e m e m b e r t h e d u a l i t y p r inc ip le ) .

Such c o n s i d e r a t i o n s m a y be s u b s t a n t i a t e d b y b e a r i n g in m i n d h o w t h e m e d i u m

\ I

-c/No

N

C U

Fig. 12. - Generalization of the formula for the refraction i~dex £V of a medium vs.

the relat ive velocity u. The medium is supposed to move eollinearly with respect to the observer. 5to is the proper refraction index.

(12) Showing tha~ a tacbyo~ D~. vacuum does ~ol emit ~ere~kov radiatio~ is a t r ivial application of our ta( 'hyouization pT'ineipb'. See also C. C. CItANG: prcprint CPT-117 (Austin, Tcx., 1971).

18~ R. ~IGNANI and E. R~CAMI

<( r e f r a c t i o n i n d e x ~) N va r i e s w h e n t h e v e l o c i t y u ( r e l a t ive to t h e observer )

var ies , in spec ia l r e l a t i v i t y . I n t h e s imple case of co l l inear mo t ion , if N0 is t h e

proper r e f r a c t i o n index , one has , for b o t h ]u[<>c,

~V(u) ~ i V = -N°c + u c ÷ Nou

a n d v ' - - - -c /N wil l be t h e v e l o c i t y of l i gh t in t h e m e d i u m (wi th r e s p e c t to t h e

obse rver ) . See F ig . 12.

L a s t l y , we w a n t to m e n t i o n t h a t <~ complex t r a n s f o r m a t i o n s >> (:a) h a v e

b e e n s h o w n to p r o v i d e a n a t u r a l c o n n e c t i o n b e t w e e n e lec t r i c a n d m a g n e t i c

cha rge (:~), as wel l as b e t w e e n b r a d y o n i c a n d t a c h y o n i e sources . A n a l o g i e s

b e t w e e n magnetic monopoles a n d t a c h y o n s h a d a l r e a d y b e e n n o t i c e d in ref. (7).

T h e a u t h o r s a r e g r a t e f u l to P rofs . Y. A ~ o ~ o v , M. BALDO, P. CALDII~OLA, ~ . DALLAP01~TA~ G. DANTOI~I~ V. :DE SAB:BATA~ F. DUI]~IIO~ J. PLEBANSKI, G.

S c ~ : F F ~ E ~ a n d to Drs . G. FONTE~ P. FOI~TI~I, U. LOMBAI~DO~ A. PAL~ERI, H .

TILGNEE for m a n y s t i m u l a t i n g d iscuss ions . P a r t i c u l a r l y , t h e y d e e p l y t h a n k

Prof . A. AGODI for v e r y m a n y usefu l c o m m e n t s a n d for his k i n d i n t e r e s t

t h r o u g h o u t th i s work .

(:a) D. W~aNOART]~N : preprint ~BI-HE-71-3 (N. Bohr Inst i tute , Copenhagen, Dec. 1971).

• R I A S S U N T 0

Un nuovo gruppo G di trasformazioni di Lorentz (LT) in quattro dimensioni, genera- lizzato anche per sistemi di r iferimento Superluminali, ~ introdotto e studiato par t i - colarmente nelle sue implieazioni fisiche. Con l 'a iuto di un <~principio di dualit~ ~> - - implicato da G - - t ra sistemi subluminali e Superluminali, 5 possibile estendere il significato di << riferimento inerziale >>, <~ equivalenza >>, <~ principio di relativit~ >>, <~ eova- r ianza ,>. Tra velocit~ bradioniche e tachioniche esiste una corrispondenza biunivoea, ehe risulta essere una part icolare corrispondenza eonforme (inversione). Poich6 il gruppo G eonsiste di rotazioni generiche nello spazio-tempo, esso include per esempio anche l 'operazione di inversione totale (PT). Inoltre (per un universo con << cariche >>), si mostra ehe ia nostra relat ivi t~ r is t re t ta generalizzata riehiede la eovarianza per CPT. Si formula un <~ prineipio di tachionizzazione >>, in base al quale le leggi fisiehe re- lativistiehe (quelle almeno della meecanica e dell 'elet trodinamica) possono essere faeilmente estese al easo dei tachioni. Si applieano le LT generalizzate ad alcuni semplici casi (legge di composizione delle velocit£, confronto di unit~ di tempo e di hnghezza , effetto Doppler, indiee di rifrazione . . . . ) util i per chiarire il nostro problema o di interesse in astrofisiea.

G]~N]~RALIZED L O R E N T Z T R A N S F O R M A T I O N S IN F O U R D I M E N S I O N S ETC. 189

O606ulennbm npeofpa3oBaHU~ ~lopeHTna B qeTb~pex H3Mepemmx u cBepXCBeTHmHecn

O~lbeKTbl.

Pe3mMe (*). - - BBO~HTC~I HOBa,q r p y n n a (; n p e o 6 p a 3 o B a n n ~ t 5IopeHTLta B ~IeTblpex I43Me-

peHHflx, o6061J.teriHa~ TaK~Ke ~,q~ CBepxcBeY~tttHXCn CHCTeM OTCqeYa. ldcc.~e~yvoTcn qbH3~J-

neckt ie npHMCHeHH~I HOBO~I rpyrlrlbi G. I/IcllOJlb3yfl << rlpl~IHilHl-I ,zlyaJlbHOCTH)) Me~K,~y

cy6cBeTfllLtHMI,ICfl a cBepXCBeTfltitHMHC~ CrlCTeMaMH OTCqeTa, MO~eT 6bIYb pactur tpeH

qbH314qeCKH~l CMblCYl rIOH~ITI,I~i << nHepHHaJabHO~I CVlCTeMbI oTcqeTa )), ~< 3KBIaBa27eHTHOCTH )),

<< rlpnHHnlaa OTHOC!4TeYlbHOCTI4 )~" /4 (( KOBap!~aHTHOCTH );,. CylJ2eCTByeT COOTBeTCTBHe Me~,~y

cKopoc ' rnMH 6 p a ~ n o u o B vt TaXnOHOB, KOTOpOe nony , aaexc~ KaK p e 3 y n s T a T ~OHKpewrIoro

KOHqbOpMHOYO OTO6pa~eHVlS ( n r m e p c m t ) . TaK Ka~ r p y n n a ¢; co~ep~rI - r BpameHrln B

IIpOCTpaHCTBe 14 BpeMeHH, TO OHa BKJILoqaeT, HanpHMep, TaK~e o n e p a u m o HOJIHO~[

n n B e p c n n (PT). B c n y n a e 3 a p ~ o ~ o ~ n H a e p c r m H a m a o 6 0 6 m e H r m s cnet~Ham, n a n weopn~

OTHOCHTeYlbHOCTI4 Tpe6yeT KoBapHaHTHOCTVI OTHOCHTeYlbHO ~P .T . dPopMyalapyeTcn ~<npvIH-

I/l/In TaXHOHH3atlHH )), Ha OCHOBe KOTOpOFO peJIflTHBHCTCKrle qbI, I3t'lqeCKHe 3aKOHbI (1IO

r p a ~ H e f i Mepe, MexaurI~rl ~t 3nenTpo,~vlHaMrI~rI) MOryT 6f,tTb a e r K o O606nleHbI a n n wa-

X~JOnOB. PaCCMOTpeHo MHOFO rlpoCTblX rlp~aMeneHnfi o6o6meHm,~x npeo6pa3oBaHH~t

. l IopeH'rua (3aKOH CJ~O~KetiHn cKopocre f i , cpaBneHrle e~rtnvitt ~nVlHSt VI BpeMeHl4, 3qbqbeKT

~ o n n n e p a , ioaqbqbHu~er~T r IpenoM neHnn n T.~.), none3H~,~x ni~6o a n n n p o s c n e H n ~ a a m e ~

r~po6aeMbL J~n6o HHrepecHb[X B acTpoqbH3n~e.

(*) fIepeeeOeno peOatcltueh.

IL NUOV0 CIMENTO VoL 16A, N. 1 1 Luglio 1973

Generalized Lorentz Transformations in Four Dimensions and Superluminal Objects.

t~. MIGNANI and E. RECA~II

Ist~tuto di Fisica Teorica dell'Universit5 - Catania Istituto iYazionale di Fisica Nucleare- Sezione di Catania Centvo Sic4liano di Fisica Nucleate e di St'~'uttura della Materia - Catania

(Nuovo Cimento, 14A, 169 (1973))

Erra ta Corrige

p. 170, line 10 from the bot tom paper papers

p. 174 cq. (3bis) L~ ~ i A > ( + c ~ ) = K

p. 175 lines 5 and 6 eliminate

p. 175, line 2 from bottom free of photons eliminate

p. 176, line 2 of Fig. 2 groups G group G

p. 186, line 7 from bottom of a Superluminal of art approaching Superluminal

p. 187, line 3 faster slower

p. 187, line 4 always never

Moreover, at p. 187, line 6, add the following sentence: (( In fact, from the duality principle and Fig. 12, one may deduce that a) bradyons will emit Cereukov radiation only in bradyonie media (when their speed is larger than the light speed in those bradyonic media), b) tachyons will emit ~erenkov radiation only in taehyonic media, when their speed is slower than the light speed in those tachyonic media (cf. Fig. 12) ,>.

(~ by Societ~ Italiana di Fisica Proprict~'~ letteraria rlservata

Direttore rcsponsabili: GIULIANO TORALDO DI FRANCIA

Stampato in Bologna dalla Tipografia Compositori col tipi della Tipografia Monograf Questo fascicolo ~ stato licei~ziato dai torchi il 2-VII-1973

208

IL NUOVO C I ~ E N T O VOL. 73 B, N. 1 11 Gennaio 1983

Formal and Physical Properties of the Generalized (Subluminal and Superluminal) Lorentz Transformations (')

G. I) . MACCA.B, RONE, M. PAV~I(~ ('*) a n d E . RECAMI

Is t i tu to di F i s i ca Teorica, Universi t~ Statale di Catania - Catania, I t a l i a Is t i tu to Naz ionale di F i s ica Nucleate - Sezione di Catania, Catania, I ta l ia Centre S ic i l iano di l~'isica Nucleare e S tru t tura della Mater ia - Catania, I ta l ia

(riccvuto il 23 Giugno 1982)

Summary . - - We investigate the mathematical and physical properties of the generalized l, orentz t ransformations (both subluminal and Super- luminal). The form here adopted for the Supcrluminal Lorentz transformations is the one-- recent ly introduced by us - -which satisfies the requested group-theoretical properties. We clarify the role of the interpretat ion procedure of the imaginary quanti t ies also froin the formal point of view, for both the ~ lon~ tud ina l ,~ and the <~ transverse ~ co-ordinates. Careful a t tent ion is devoted to define four-momentum and three.veloci ty for tachyons. At last, the shape of a t achyon- -ob ta ined by applying to an ordinary particle a generic Superluminal Lorentz t ransformation (without ro ta t ions) - - i s studied. As a simplifying tool, we make recourse also to the ~ light-cone co-ordinates ~ and to ~ di lat ion-invariant ,~ co-ordinates.

1 . - I n t r o d u c t i o n .

R e c e n t l y t h e gene ra l i zed L o r e n t z t r a n s f o r m a t i o n s (~) [GLT] ( b o t h sublu-

m i n a l [LTJ a n d S u p e r l u m i n a l [SLT]) h a v e been r e w r i t t e n in a f o rm sa t i s fy ing

(*) Work par t ia l ly supported by M.P.I. and C.N.R. (**) On leave from (permanent address} J. Stefan Inst i tute , E. Kardelj University of Ljubl jana, L jubl jana , Yugoslavia. (*) E. RECAMI and R. MIC,.~-ANI: Riv . N'UOVO Cimento, 4, 209, 398 (1974) and references therein; R. MIGNANI and E. RECA.~I: NUOVO Cimento .4, 14, 169 (1973); 16, 208 (1973); Nuovo Cimento B , 21, 210 (1974); Lett . Nuovo Ci~nento, 9, 357, 367 (1974); E. RECAMI

91

92 G. I). MA.CCARRONE, ?,~, PAVSI~ and E. RECAIMI

the r eques ted g roup- theore t i ca l proper t ies (~), such a new fo rm reduc ing to

the p rev ious one by MI~NA~-I and ICEOhMI in the ps r t i cu la r case of coll inear

boosts . 1N~smely, in rcf. (~,-") i t has been shown t h a t

S L T ( U ) - - • ,,

(1) ~ : : i l , c'z-/V) ~

so t h a t t he g roup G of all subluminal and Super lumina l Loren tz t r ans fo rma-

t ions resul ts to be

(2) z ( l ) - ~r :=- { + ] , - ~, + 4 - / } ,

where .Lf+ ~ represents the o rd ina ry (proper, o r thoehronous) Loren tz group. The g roup G has, in par t icu lar , the propert ies (,3)

(3a) d e t G = + 1 , VG~ G ,

(3b) G e G =>-- G e G , VGe G ,

(3c) (I ~ G :~ iG E G, VG c- G ,

~md of course

• G .

and R. MIG.~'ANI: l~ett. Nuovo Cimento, 8, 110 (1973); 4, 144 (1972); R. MIGNANI and E. RECAMI: Lett. Nuovo Cime~to, 7, 388 (1973); R. 3IIG~A.~I, E. RECAMI and U. LOM- BARDO: Lett. Z~uovo Cimento, 4, 624 (1972); E. RECAMI: in Topics in Theoretical and Experimental Gravitation Physics, edited by V. DE SABBATA and J. Wr~BER (New York, N. Y., 1977), p. 305; P. CALDI~tOLA and E. RECA~II: in Italian Studies in the Philosophy o] Science, edited by M. I)ALLA CIIIARA (Boston, Mass., 1980), p. 249. See also H. C. CORBEN: Nuavo Cimento A, 29, 415 (1975); Int. J. Theor..Phys., 15, 703 (1975); L. PARKER: Phys. Rev., 188, 2287 (1969); M. I)AV~I~: The Exte~ded Special Theory o] Relativity, University of Ljubljana preprint, January 1, 1971 (unpublished). (2) G. D. ~ACCARRONE and E. RECAMI: The Introduction o] Superluminal Lorentz Trans]ormations: A Revi~itation ~, University of Catania preprint PP/693 (to be sub- mitted for publication); G. D. MACC~RRON~; and E. RECA~I: Lett. Nuovo Cimento, 34, 251 (1982). (3) E. I~ECA~I and W. A. RODRI(~U]~S: Found. Phys., 12, 709 (1982), and references therein; E. RECA~I: Found. Phys., 8, 329 (1978); E. RECAMI: Chapt. 16 in A..Einstei~ 1879-1979: Igelativity, Quanta and Cosmology in the Development of the Scienti/ic Thought o] A. Einstein, Vol. 2, edited by F. Dn Finis (New York, N.Y. , 1979), p. 537. See also O. i~L BILANIUK, V. K. DESHI'AI~'DE and E. C. G. SUDA~S~A~': Am. J. Phys., ]O, 718 (1962).

FORMAL AND P H Y S I C A L P R O P E R T I E S ETC. ~ 3

F r o m eqs. (1) i t follows t h a t c j is the (~ t r a n sc enden t ~ SLT (i.e. i t corresponds

to U ~ c~). Wi th in the fo rmal iza t ion a d o p t e d in ref. (re,o), the four-pos i t ion x~ is sup-

posed to be a vec to r even wi th respec t to G (i.e. to be a G fottr-vector) , so

t h a t the quadra t i c fo rm dx , dx~, is a scalar under LTs and a pseudoscalar under SLTs (in par t icu lar , unde r the t r anscenden t t r a n s f o r m a t i o n c~). ] n

o ther words, the G L T s are unimoduh~r and special and such t h a t the LTs are o r t h o g o n a b whils t the SLTs are an t io r thogona l :

(4a) GTG ~ -[- 1 (subluminal case: u 2 ~ c 2) ;

(4b) GTG ---- - - 1 (Super luminal case: u 2 ~ c ~) .

Le t us not ice t h a t G is n o n c o m p a c t , nonconnee ted and wi th discontinuit ies

on the l ight -cone; moreover , its cen t ra l e lements are

(5) r --- ( § 1 , - 1, § i l , - - i l ) .

The new groul) G of the GLTs is the following extens ion of the g roup ~+t:

(2') A = ~ ( ~ : , CPT, o~),

since the opera t ion - - 1 has been shown (1-3,4) to be equ iva len t to the ordi- n a r y C P T (~-4):

- 1 - - P T =- C P T .

Notice t h a t our ((new); GI,Ts, eqs. (1), do agree wi th our ref. (~-4), bu t sl ightly disagree wi th the fo rm a d o p t e d in ref. (5).

I n lhe pa r t i cu la r case of boosts a long x, the SLTs t ake the simple form

(4) .~[. l'Av~1~ and E. R~CA~I: Left. Nuovo Cimento, 34, 357 (1982); E. RF, CAM~ and G. ZIIxo: Nuovo Cimento A, 33, 205 (1976); R. MIGxA~,-I and E. RECAMI: Int. J. Theor. Phys., 12, 299 (1975); Nuovo Cimento A, 24, 438 (1974); Lett. Nuavo Cime~to, 1], 421 (1974). (5) ,'v[. PAvgI~ and E. RECAMI: 5~ett. Nuovo Cimento, 19, 273 (1977); E. RECAMI: in Tachyons, Monopoles, and Related Topics, edited by E. RECAMI (Amsterdam, 1978), p. 3; P. C~LDIROLA, G. D. MACC,tRRO~E and E. R~CAMI: Lett. Nuovo Cimento, 29, 241 (1980); E. RECAMI and G. D. 3[ACCARRO~E: Lett. NUOVO Cimento, 28, 151 (1980); E. RECA~I: Chapt. 18 in Centenario d i Einstein: A stro]isica e Cosmologia, Gravitazione, Quanti e Rel~tivith neUo svih~ppo del pensiero scienti]ieo di A. Einstein, edited by ~[. PANTALEO (Firenze, 1979), p. 1021.

94 G.D. MACCARRONE, M. PAV~I~ and E. R]~CAMI

(in na tura l units: c ~ 1)

(6)

t - - u x _ , x ~ Ut

.r -=i"

x - - u t t - - U x

--= z ' ~- _l. i z ;

( Superluminal ease~

U ~ 1' u 2 < 1 / --Uu /

the prob lem of interpret ing eqs. (6) has been exploi ted in ref. (2,e). I n ref. (2) we also justified why we call eqs. (6) ~, t ransformat ions ~>. Here, let us only recall t h a t in the 2-dimensional case the in te rpre ta t ion is s t ra ightforward,

since the t ranscendent operat ion ~ = ~ 2 -~ goes into through the similari ty t ransformat ion

(7)

where T is a uni tary t ransformat ion.

2. - The GLTs by discrete scale transformations.

W h a t precedes can be rewri t ten in ~ more compac t form b y following the phi losophy outlined in ref. (v), i .e. by making recourse to the language of the discrete {real or imaginary) scale t ransformat ions (v):

I ds'~ = ~ ds~ ' (s) [ e'---= =t=1.

For instance, the GLTs can be of course rewri t ten as follows:

l where ~ is the discrete group of the dil~tions D : x ~- r with ~o = ~ ] , • i.

(e) A. O. BARUT, G. D. MACCAttRO.~ and E. R~CAMI: Nuovo Cimento A (in press); E. RzCA~r and G. D. ~ACCARRONE: Lett. Nuovo Cimento, 28, 151 (1980); P. CALDmOLA, G. D. MACCARRO~. and E. R~CAMI: Lett. Nuovo Cimento, 29, 241 (1980). See also G. D. ~ACCARRON]~ and E. RZCA~I: Nuovo Cimento .4, 57, 85 (1980). (7) M. PAv~x~ and E. R~CAMI: Zett . Nuovo Cimento, 19, 273 (1977); M. PAynim: in Tachyons, mo?wpoles, and Related Topics, edited by E. R~CAMI (Amsterdam, 1978), p. 105; Le~t. Nuovo Cimento, 30, 111 (1981); J . Phys . A , ]4, 3217 (1981).

FORMAL ANI) PHYSICAL PROPERTIES ETC. 9 ~

More formally, let us introduce the new, scale-invariant (or di lat ion-invariant) co-ordinates

(9) ~ ~. rx~ (K = 4- 1, 4- i) ,

where K is the intrinsic scale factor of the considered object (7,B). t ?

Notice tha t , under a dilation D, it is U~ = U, with ~, ~e K'X~,, while KI ~= O--1K.

The i m p o r t a n t characterist ic of the present formal ism is tha t , under all the GLTs of the group G, the quadrat ic form da 2 - - d u , d~a is invariant ('):

(10) da '" = da" , VG e G .

_Notice, moreover , that , under a generic proper or thochronous Lorentz t ransformat ion L e (~f+t, i t holds U '~ := L~U"; K ' = K.

I t follows t h a t - - w h e n going back f rom the co-ordinates U5 r to the ordinary co-ordinates x , - - t h e generic GLT = G can be expressed as (x'--Gx)

(1]) { G--r'-~LK'LE.Lfl ( K , ~ = 4 - i , 4 - i ) r , = ~-*K "

As mus t be, the subluminal Lorentz t ransformat ions are the LT = + L and the Supcr luminal ones the SLT---- 4 - iL . I n other words (7), b radyons (antibradyons) correspond to K- - - -+ ] (K = - ]), whilst tachyons and anti- tachyons correspond to r = 4- i.

3. - Generalized (subluminal and Superluminal) boosts B in the c~ l ight-cone co-ordinates ~).

I t is a l ready known (') t ha t the ordinary subluminal boosts along x can be rewrit ten in a more symmetr ic , compact form in te rms of the co-ordinates

(12) ~ ~=t- -x , ~ - t + x , y, z,

where the first two co-ordinates refer to two new axes which are obtained f rom

(s) See, e.g., also A. 0. BA~UT and R. B. IIAL'GE~': Ann. Phys. (hr.:F.), 71, 519 (1972); H. A. KASTRVP: Ann. Phys. (Leipzig), 7, 388 (1962). (*) Its sign included. (9) J . B . KOC~I:T and D. E. SOeER: .Phys. Rev. D, 1, 2901 (1970); J. D. BJORKE.~', J. B. KOGITT and D. E. SORER: Phys. Rev. D, 3, 1382 (1971); P. J. STEINHARD: Ann. Phys. (N.iY.), 128, 425 (1980); M. PAV~I~: private communication.

96 G . D . bfACCAHRONF~, hi. PAV~I~ a n d E. RECAM[

t h e axes t, x by m e a n s of a E u c h d e a n , an t i c l o c kw i se 45 ~ ro t a t i on . (See fig. 1.)

W e sh~ll cal l ~, $ <( l i gh t - cone co -o rd ina t e s }~ (even if t h e y are some t imes , r a t h e r

i n c o r r e c t l y , n a m e d <( i n f i n i t e - m o m e n t u m - f r a m e c o -o rd ina t e s }>). h~amely, a

p r o p e r o r t h o c h r o n o u s b o o s t a long x, w i th r e l a t i v e ( sub lumina l ) speed u, can

be w r i t t e n ~s

(13)

c( -- ~-i ( J 3 ' ) ~- ~-1

~' ~ - 1 ~ , y ' = y , z' z ~ ' . = ~ , = = .

= i t , '/~2 < 1

( O < a < + ~ ) ,

( - - l ~ ~ t - - x and r ~ t + x correspond to two axes ~, r which are obtained from the axes t, x through an Euclidean, anti-clockwise 45 ~ rotat ion.

I t is i n t e r e s t i n g to no t i ce t h a t in t he p r e s e n t f o r m a l i s m the L o r e n t z b o o s t s

a long x c o r r e s p o n d j u s t to a d i l a t i o n of t he c o - o r d i n a t e s ~, ~ (by t h e f ac to r s ct

a n d :r r e spec t ive ly ) . I n p a r t i c u l a r , the i d e n t i t y t r a n s f o r m a t i o n (u = 0)

c o r r e s p o n d s to ~---- - ] -1 , t h e boos t s a long t h e p o s i t i v e x - d i r e c t i o n c o r r e s p o n d

to 1 < :r < - 4 - c~ and t h e boos t s a long t h e n e g a t i v e x -d i r ec t i on c o r r e s p o n d

to 0 < a < 1. F o r ct -7 ~- cx) we h a v e u - 7 1 - a n d for :r -+ 0 + we h a v e u -~ - - (1-).

I t is a p p a r e n t t h a t

(14 ) cr = e R ,

where R is t h e (( r a p i d i t y , .

B y us ing this f o rma l i sm , i t is i m m e d i a t e to r ecogn ize t h a t t h e p r o p e r anti- ehronous (= n o n o r t h o e h r o n o u s ) s u b l u m i n a l L o r e n t z boos t s a long x wi l l cor-

r e s p o n d to t he n e g a t i v e (real) a va lue s : - - 0o < a < 0; t o g e t h e r wi th y ' ---- - - y,

z ' = - - z. Of course , t h e r e l a t i v e speed u will r u n a g a i n w i t h i n t h e s a m e r a n g e :

- - 1 < u < q - 1 , even in c o r r e s p o n d e n c e wi th t h e new (nega t ive) va lues of a.

FORMAL AND P H Y S I C A L P R O P E R T I E S ETC. 97

Similarly, the Superluminal boosts Mong x will co r r e spond- -now- - to the imaginary o: values, together with y ' = -4-iy, z ' = -4-iz.

More precisely, eqs. (13) can be extended to express in synthet ic form all generalized (both subluminal and Superluminal) Lorentz boosts along x, by means of the discrete scale parameter ~, as follows (*):

(15) { ~ ' = o@, $ ' = ~a-~$, y ' = Qy, z ' = ~z,

= • -4-i ( 0 < a < § oo, u~=u:~l),

where it should be noticed tha t a is any real, positive number. Such eqs. (15) represent the generalized boosts (2) be]ore their re in terpre ta t ion; t ha t is to say, they are equivalent in the Superluminal case to eqs. (6). Equat ions (13') must be generalized as follows:

(15') $ ( u 2 = u , ~ l , O < a < + c o ) ,

where u represents here the (relative) speed both of sublnminal and of Super- luminal boosts. Equat ions (15') reduces of course to eq. (13') in the subluminaL boost case. I n the Superlumiual-boost cases, however, eqs. (15') can be derived only after the reinterpretation of the first couple of eqs. (15) (i.e. after the interpretat ion of the meaning of Q ~- -t- i in the first couple of eqs. (15)). F o r such a delicate question, see ref. (2); soon we shall touch again this point. Here, let us anticipate tha t the re interpreta t ion procedure (**) of the first couple of equations (15)--as given in ref. (2)--is equivalent to rewriting them as follows:

(15 bis) { ~ ' = ozd~, $'=- Q~-~', y'--=- Qy, z ' = ~z,

a ~ • o --= + 1 , • a~(O, ~-oo) (U2=U ~ ~ 1),

wherefrom eqs. (15') can be straightforwardly derived. See the following

(eqs. (22)). In conclusion, if B represents a generic boost along x, then all generalized

(subluminal and Superluminal) boosts can take the form (15) with (s

(*) In this section, for convenience, we shall represent by u the boost relative speeds both in the subluminal and in the Superluminal cases. (~ The reinterpretation procedure we are dealing with in this paper has nothing to do, of course, with the Stiickelberg-Feynman-Sudarshan (~ Reinterpretation Principle (RIP) also known as ~ Switching Principle ~.

7 - I I N ~ v o C i m e n t o B

98

~-~+~u ~.~f+~, cr O < a < . 6 o o )

(16a)

(16b)

(16c)

G. D. MACCARRONE, M. P&V~I~ a n d E. lCECAMI

Be.~~162 0 < ~ < . 6 o o -r Q = . 6 1 1

J Be~f+~: --oo<~t<O r ~------I

Bei.~+: a--i~;--oo<~< .6co .> Q----q-i

(u s ---- u s < 1) ;

2 (u ~ = % > 1) .

In particular, it is immediate to check tha t in the case of Superluminal boosts (e = :k i) from eq. (15') i t actually follows t h a t

a + a -I (17) U - - a _ a _ l > l ( Q : :J: i , O < a < . 6 c o ) .

Of course, all generalized x-boosts (cqs. (15)) preserve the quadratic form, except for its sign:

(18) 2 '$ ' - - y,S _ z , S : e 2 ( ~ _ y 2 _ z s) (Qs = -t- 1 ) .

Let us briefly come back to the problem of deriving eqs. (15') in the case of Superluminal boosts, by observing tha t the change of the quadratic-form sign can be obtained either by wr i t ing down eqs. (15) and (18) with Q = -4-i, so as we did before, or by writing (instead of eqs. (15), and only for the case of Superluminal boosts)

(19) ~' =- a S , $' = - - a - 1 5 , y ' -~ - - i y , z' ---- - - i z ~0 < a < . 6 c o ] '

where the real a a (0, + oo); or ra ther - - in more complete form (1)--

z' (19') S ' = ~ , ~ . . . . ~-1~, y'=-i~y, =-i~z (u s -=u~>l , - o o < ~ < . 6 o o )

now with real ~ e (-- oo, -6 co). Equations (19') are the transcription of eqs. (6) in terms of the co-ordinates given by eqs. (12). I t follows that , in particular,

t '= �89189 x'= �89

so tha t for the relative boost speed one obtains

(Ix ~'-r ~ ~ + ~-1 (20) u = ~ ~-_ ~_~, u~ > 1,

FORMAL AND PHYSICAL PROPERTI]~S :ETC. ~

where eq. (20) should be compared with eq. (13'). Notice explicitly tha t the procedure expressed by these equ'~tions (19)-(20) does correspond to our reinterpretation of the first couple of eqs. (15) given in ref. (5) (i.e. eqs. (19') coincide with eqs. (15 bis) for the Superluminal case).

To formalize the whole ma t t e r (i.e. the previous re interpreta t ion problem), let us take a dva n t age - - a t this po in t - -o f the (discrete) scale t ransformation language in t roduced in sect. 2. Tha t is to say, by subst i tut ing the dilation- invariant co-ordinates 7], ~ Kx, for x , (and thus by (< generalizing ~) definitions (:12)), let us eventually define the following scale-iavariant (<light-cone co-ordinates ~:

( 2 1 ) ~ ~ 7]0 __ 711 , y __-- 7]6 _~ 711 , 7]~, 713.

In terms of co-ordinates (21), t ransformations (15) can be writ ten as

{ (p! = ~(p, ~)I __ o~--l~p, 7]12 = 7]2 7]t3 = 713, (22) (lul ~1, U = u ) ,

K ' = ~-IK, a ~ Q a , ~)= -{:1, -4-i, a ~ ( 0 , + o o )

where, as usual, ~ = -4- :1 yields the subluminal and ~ = -_]: i the Superhuninal x-boosts. :Now all generalized boosts (eqs. (22)) preserve the quadrat ic form, its sign included:

(23) ~ , ~ , (7],~)~ (7],~)~ = ~ _ (7]~)~_ (7]~)2.

:It is impor tan t to emphasize tha t eqs. (22) in the Superluminal case yield just eqs. (19r), tha t is to say they automatically include the reinterpretat ion of the first couple of eqs. (15) or (16), as given in ref. (5). In particular, in the Superluminal-boost eases, eqs. (22) have the advantage over eqs. (15) of yielding the correct Superluminal relative speed without any need of re interpretat ion; actually, from eqs. (22) one derives exact ly eq. (15'), for bo th subluminal and Superluminal boosts (without any explicit need of reinterpretat ion).

The more difficult problem of the generic velocity composition law will be considered in sect. 6.

We want here to observe tha t our co-ordinates ~, ~ (or ~, ~) are so deiined that u is subluminal whenever in eqs. (22) the quantities a and a -1 have the same sign: sign (:r sign (a); and u is SuperhLminal whenever ~ and a -1 possess opposite sig-ns: sign (a-l) = _ sign (a).

:In what follows we shall touch the question of interpret ing the second couple of eqs. (15), or (6), or (22), following ref. (e). The problem of geometrico- physically interpret ing in the Superluminal case the second couple of eqs. (6)~ (15), (15bis), (19'), (22) has been exploited in ref. (6), bu t only for the case of Superluminal boosts along a space axis (let us call it x): cf. fig. 2, and ref. (6). Below, we shall extend those results.

100 o .D . ~tACC~RONE, ~t. PAV~I~ and E. ~ECA~X

' ' ' ' �9 X

y ! J J /

a) b) c) a)

Fig. 2. - I, et us consider a particle which is intrinsically spherical, i.e. that is a sphere in its rest frame (a)). Under a subluminal x-boost it appears---of course---as ellipsoidal (b)). Under a Superluminal x-boost it will appear as in d). c) refers to the limiting case when the boost relative speed u --+ c. (It is understood that these figures refer to the solid objects got by rotating them around their axes of abscissas.) Cf. also ref. (e) and the text.

A n o t h e r p rob lem we shall deal wi th is general iz ing eqs. (15'), (17) and

(20) for the case when the Super lumina l ve loc i ty is composed wi th a nonze ro init ial ve loci ty .

W e axe going to consider also some appl ica t ions of the previous formal ism.

4. - A simple application.

A first example to show the power of the present fo rmal i sm is f inding ou t how (9) a 4-dimensional (space-t ime) sphere

(24) t 9 § x ~ § y~ § z 2 = Am,

t h a t is to say

(24')

deforms under a Loren tz t r ans fo rmat ion . Le t us first consider a sub lumina l

boos t (eqs. (13)). Since the tirst two co-ordina tes resul t to be mere ly scaled

by the f ac to r a E (0, cx~), we immed ia t e ly get t h a t eq. (24') in terms of the new (primed) co-ordinates rewri tes

(25a) �89 § �89 tz+2~,9 _~. y,9 -4- z '9 = A S ( sub lumina l case, (z e (0, + cx:))) ,

which in the new f rame is a 4-dimensional ellipsoid.

I n the case of a Super lumina l boos t (eqs. (19'), (15bis)), eq. (24') can be

F O R M t L AND PHYSICAL PROPERTIES ETC. 101

rewr i t t en- - in terms of the new, pr imed co-ordinates--as

(25b) � 8 9 1 8 9 ~ (Superluminal case, a e ( 0 , + ~ ) ) ,

which in the new frame is a 4-dimensional hyperboloid. Notice explicitly tha t this example (i.e. t ransforming under GLTs a

4-dimensional set o/ events) has nothing to do with what one performs usually (in fact, ordinarily, one considers a world-tube and then cuts i t with different 3-dimensional hyperplanes).

5. - O n t h e p h y s i c a l i n t e r p r e t a t i o n o f S L T s .

We would like to extend the whole in terpreta t ion procedure (~,e) (of the whole set of four equations const i tut ing a SLT) to the case of Superluminal I~orentz t ransformations without rotations, i.e. of a Superluminal boost Z(U) along a generic motion line 1. Le t us first realize such an aim in terms of the ordinary co-ordinates x, . A Superluminal Lorentz t ransformat ion (without rotations) Z(U), according to eqs. (1), as a 4 • 4 matr ix will write (u [[ U, u -- 1/U)

[ 7 u ~ ' n 8 (26) L(U; x " ) = i L ( u ; x " ) : ~- i (__

-- - - u T n ~ ~ : - - ( 7 - - 1 ) n ' n J ' (u~ < 1 ~ -1]\ Y ' >

(26') ~ ------ (1 - - u2) - j ,

where Z(u) is the dual (subluminal) boost along the same (generic) direction 1. The quant i ty n is the unit vector characterizing the boost mot ion line l: % n r --~--1--~--In21. The unit vector n points in the (conventionally) positive direction along l. Notice tha t u, U m a y be both positive and negative.

Let us observe tha t eq. (26) expresses L(U) in its (( original ~ form, not yet reinterpreted. Of course, Z(U, x,) can be considered as obtained from the corresponding Superluminal boost .5o(x, U) ~ B(x) along x through suitable rota-

tions (Lo(x, U) ---- iLo(x, u)):

(27) Z( U; x~') ~-- R - I B ( x ) R , I~ ~--

(27') A ~ (1 § n~) -1,

0

n= n~ n z

--n~ 1 -- An~ --An~n~

- - n z - - A n ~ n~ 1 - - A n ~ /

where B(x) is given by eqs. (6). I t is impor tan t to underline tha t in ref. (2,6) we have been able to re interpret

the SLTs, eqs. (1), only in the case of Superluminal boosts along an axis

102 G.D. MACCARRONE, M. PAV~I~ and ~. R~.CAMI

(so as assumed in sect. 3). ~ o w , to re interpret also the Superluminal t ransformat ion L ( U ; x,) in eq. (26), let us compare L ( U ) with /~(U):

(2s) ~(U; x.) ~-/~-~9(x)R,

where B(x) is now the (partially) reinterpreted version (~,~) of eqs. (6) for the Super luminal case (~,%s):

(6bis)

x - - ut t - - Ux t ' = • -= :

X I ~ ~- t - - u x x - - U t

y ' = • iy ,

z' = -k i z .

Superluminal c'tse

u ~ < l , U * > I , U = l / u ]

We adop t eqs. (6bis) even if only the first couple of t hem appears as actual ly (~ re in terpre ted ~) in real terms, since in ref. (~) we already showed how to in- t e rp re t the imaginary units appear ing in the last couple of eqs. (6bis) (at least in some re levant eases); we shall take account of t h a t in the following.

i n connection with the (partial ly reinterpreted) equat ions (6 his), let us recall (~,6)-- incidental ly-- that the generalized Lorentz boosts, both subluminal and Superluminal , can be wri t ten down in a compac t form and in t e rms of a

cont inuous pa ramete r 0 c [0, 2~] as follows:

with

I x ' = f 2 7 o ( X - - t t g O ) , t' = ~27o( t - -x tgO)

y' = - - s ~y , z' = - - s ~z

u , ~ l

0 < 0 :< 2~]

u ~ t g O , cos 0 ^ _L

~'o = -t- (11 - - t g * 0 I)-~,

V l - - t g 2 0

I1 - - tg*O I '

0-d 0 ~ 2~r;

such a fo rm (1) of the G L T s shows explicitly how the various (positive or negative) signs in f ront of x ' and t' and the various (real or imaginary, posi t ive or negative) (( signs ~) of y ' and z' do succeed one another (l.e) as functions of u, or ra ther of 0. (~otice t ha t in this last section u ~ ~.1.)

F r o m eqs. (28) and (6bis) we get for the Super luminal t ransformat ion f~

(29a) L ( U ; xa) == _~ ( - u y --~'na 1 7n" i6: -i- ( i + u y ) n ~ n s ]

( u ~ < 1 ; r, s = 1, 2, 3) ,

FORMAL AND PHYSICAL PROPERTIES ETC. 1 0 S

where 7 is defined in eq. (26') with [u] < J. Equat ion (29a) can, however~ be writ ten also as follows:

(29b) i ( U ; x . ) = • ( U ~ - - l / u , U z > I ) ,

where now : ~ z ( U 2 - - 1 ) - i with U - - ] / u ~ u 2 < l ~ U 2 > 1 . . N o t i c e explicit ly

that, even i f the SLTs in their original mathematical form are always purely imaginary~ the SLTs in their (((partially) reinterpreted ~ form appear to con-

tain on the contrary complex quantit ies: Bu t this is not a problem~ because

the origin of those (( complex quantit ies ~) is evident and we know---of course- -

how to interpret them. We have just to compare the matrices (29a) or (29b) with the matr ix in

eq. (26)~ including in it its imaginary coefficient, in order to get an interpretation of eq. (26) analogous to the one forwarded in ref. (x,2.6) for the Super- luminal boosts along x. Iqamely, the reinterpretat ion will proceed--as usua l - - in two steps: the first step consists (cf. also sect. 3) in reinterpret ing the space co-ordinate along the motion line 1 and the t ime co-ordinate~ the second step consists in interpret ing (e) the imaginaries entering the transverse space co-

ordinates. For instance~ let us compare eq. (26) with eq. (29a)~ apar t f rom their

double signs:

t' ~ iTt ~ iTun . x ' 9

(26) x " = - - iuTn't ~- i(~:x'-- i(~,- 1 )n~n , x ~ ;

t' - - : - - u T t - - ~n~x ~

(29a) x"---- 7n~t ~ iS :x ~ -4- (u~ ~- i )n~n,x" .

_l~irst Step. To reinterpret (in terms of real quantities only) the t ime coordinate and the space co-ordinate along the motion line~ one has to adopt

the following recipe (notice tha t r H ~ r . n ~ - n,x~

:You can eliminate the imaginary uni t in all addenda containing ~ as a

multiplier~ provided that you substitute t ]or r~ ~ - n ,x" and - - n , x " for t.

Let us emphasize (following ref. (~)) t ha t - -when dealing with a chain of GLTs- - such a re interpretat ion rule has to be applied~ if necessary~ only at

the end of the chain (*).

Second Step. In the second ones of eqs. (26) and (29a)~ if we pu t r ---- x ---- -=~(x~y,z) and r ' ~ x ' = (x'~y'~z'), we can write r - - - - r , ~ - r . , where r D-~

(') Let us recall also that, after the reinterpretation, the SLTs loose their group- theoretical properties (2).

1()4 G . D . MACCARRONE, M. PAV~I~ and V.. RECAMI

- - ( r=)n and r • ( r . n ) n . Then eq. (29a), e.g., can be wri t ten as

(29v) ! !

r'---- r, ~- r• : ?(t-- u r , ) n ~- Jr•

After having applied the ~ first-step recipe ~, we are left only with the following relat ion:

e : ir• (30) r•

to be reinterpreted yet, i.e. only with the imaginary terms (not containing as a multiplier):

(30') i ( r • ~ ~-- i((~: ~ n ' n , ) x ' ,

which enter only x" . Of course, r• is a space vector lying on the plane orthogonal to the boost motion line and, therefore, corresponds to two fur ther co- ordinates only.

Since those terms (eq. (30')) refer to the space co-ordinates orthogonal to the boost direction, their imaginary (( sign ~> has to be interpreted so as we did in ref. (6) for the transverse co-ordinates y', z' in the case of Superlnminal x-boosts. (Cf. fig. 2).

This means that , if the considered SLT is applied to a body Ps initially a t rest (B ---- bradyonic = slower than light; for simplicity, let it be spherical in its rest frame), we shall finally obtain a body Pv (T = tachyonic) moving along the boost motion line l with Superluminal speed V---- U, such a body Pv - -ho we ve r - - be ing no longer spherical or ellipsoidal in shape: the tachyon PT will appear, on the contrary, as occupying the spatial region confined between a two-sheeted hyperboloid and a double cone, both having as symmet ry axis the boost motion line l. See fig. 3 and ref. (6). More precisely, let us consider the vector r~ in eq. (30), once eliminated its imaginary (~ sign ,> (i.e. the vector r• since r• lies on the plane orthogonal to l, it can be described by the two co-ordinates r• r• such tha t

(30") I/ ' ---- i Y , Z ' : i Z ,

and the co-ordinates ]Y'] ~ ~ ' / i ~ Y and [Z' I --~ Z ' / i ~ Z express the fundamenta l sizes of the ~, fundamenta l rectangles ~> (e) which individuate the double- cone shape, i.e. the fundamenta l asymptotes of the two-sheeted hyperboloid (see also the following). In other words, quantit ies Y ' / i and Z ' / i (together with the quant i ty AX------V1V=: cf. fig. 3) allow us to determine the shape of the tachyon. Figure 3 refers to the simple case in which P~ is intrinsically spherical: more in general, the axis of the tachyon shape will not coincide with 1 (but will depend on the tachyon speed V ~ U). The double-cone semi-angle ~ is given (6) in our present case by the relation t g ~ -~ (V = - 1) -j.

FORMAL AND PHYSICAL PROPERTIES ETC. 1 0 ~

l

Fig. 3. - If we s tar t again from a spherical particle Ps as in fig. 2a, then- -a f t e r a generic SLT without rotations, i.e. under a Superluminal boost along a generic motion line l - -we get what is represented in this figure. In this case, the tachyon PT occupies the spatial region confined between a two-sheeted hypcrboloid and a double cone, both having as symmetry axis the boost motion line I. Such a structure (the , tachyon shape ,~) travels of course along l with the speed V = U of the Supcrluminal /-boost. h~otice that , if P~ is not intrinsically spherical (but, e.g., ellipsoidal, in its rest frame), then the tachyon shape axis will not coincide with l and its position will depend on the speed V of the tachyon itself. For the cases in which the space extension of the tachyon is finite, see ref. (6).

To c la r i fy t h e a b o v e (second) s t ep of our r e i n t e r p r e t a t i o n , i t is neces sa ry

to a d d some c o m m e n t s : i) we do not a i m to c o n s i d e r - - a n d r e i n t e r p r e t - - t h e

G L T s when t h e y a rc a p p l i e d to a vacuum point: in fac t , t h e m a i n t e a c h i n g

of specia l r e l a t i v i t y is t h a t each o b s e r v e r has a r i g h t to cons ide r t h e v a c u u m

(i.e. t he space , or t h e e t h e r if y o u l ike) as at rest w i t h r e s p e c t to h imse l f (1o);

ii) we do a p p l y - - a n d r e i n t e r p r e t - - - t h e G L T s (in p a r t i c u l a r , t h e SLTs) on ly

to t r a n s f o r m t h e s p a c e - t i m e reg ions a s s o c i a t e d w i th physical objects, where we

a s sume t h e e x i s t i n g ob jec t s to be e s sen t i a l l y extended (as r e q u i r e d b y t h e re la-

t i v i s t i c t heo r i e s (4,e,1~)), so to cons ide r t h e p o i n t l i k e s i t u a t i o n on ly as a l i m i t i n g

(lo) See, e.g., D. I. BLOKHINTSEV: Space and Time in the Microw~Id (Dordreeht, 1973), p. 1. See also ref. (6). (11) See, e.g., J. ~ . JAvci~: .Foundations o/Quantum Mechanics (London, 1968); A. J. K~,LNAV: in Tachyons, Monopoles, and Related Topics, edi ted by E. RECA~I (Am- sterdam, 1978), p. 53, and references therein; Phys. liev. D, 7, 1707 (1973); J. C. G ~ - LXRDO, A. J. K~,LNAY, B. A. STEC and B. P. TOLEDO: NUOVO Cimento A, 48, 1008 (1967); A. J. K~.LNAY and B. P. TOLEDO: Nuavo Cimento A, 48, 997 (1967); V. S. OLK~OVSKY and E. RECA~I: Lett. Nuovo Cimen$o, 4, 1165 (1970); E. RECA,'~I: in Quarks and the Nucleus (Progress in Particle and Nuclear Physics), Vol. 8, edi ted b y D. WILKINSON (Oxford, 1982), p. 401; P. C~LDIROLA, M. PAV~I5 and E. Rv.CAMI: .NUOVO Cimento B, 48, 205 (1978); Phys. I~ett. A, 66, 9 (1978); P. CALDIROLA and E. RECAMI: Lett. ~Vuovo Cimen$o, 24, 565 (1979); P. CALDIROLA: Riv. .~uovo Cimento, 2, 13 (1979).

106 o . D . ~ACC~RRO~V., M. PAV~I6 and E. Rv.CA~X

ease; iii) when considering an extended- type physical object, we adopt the simplifying convenction of referring the frame axes to its symmet ry centre.

We have finally to pass f rom co-ordinates x~ to co-ordinates of the type (12). We deiined eqs. (12) for the ease of x-boosts. In the case of boosts along the generic motion line l, let us generalize definitions (12) as follows:

(3x) { p~-~-t-r, , ~l-~-t+r, ,

where r , - r . n and rz ~ r - - ( r . n ) n , l~otice tha t ~ ~ (~2,~a) is a spacelike vector characterizable by two components only, so as r_..

In terms of these new co-ordinates, eqs. (26) can be rewri t ten as (a e (0, ~ or))

(32) e=+i;

eqs. (32) represent the SLTs (without rotations) in the original, nonreinter- prcted form. To reinterpret the first couple of eqs. (32) it is enough to recall definitions (31) and apply the rule in the previous recipe, i.e.

omit i; and t ~- r u �9

The consequences in eqs. (32) are

(33) ~,o = • ( _ a~O), ~,1 = ~ a - ~ , ~,2,8 = -5 i~ 2,3 ,

where $,o, ~,x are now real bu t nevertheless correspond to Superluminal relat ive motion, due to the change of sign in ~,o (see sect. 3).

Such a re interpretat ion can be easily formalized (i.e. ~ automat ized )>) by making recourse to dilat ion-invariant light-cone co-ordinates, and proceeding in analogy to eqs. (21), (22).

As to the imaginary ~( signs ~> of ~,,,s, the in terpreta t ion procedure is just the same as for y', z' above.

6. - T h e v e l o c i t y c o m p o s i t i o n p r o b l e m .

In sect. 3 we left open the velocity composition problem. Firs t of all, let us observe tha t dx~/ds does not represent a G four-vector

(since dx, is a G four-vector, but ds ~ is a pseudoscalar under SLTs); therefore, the four-velocity and the four-momentum, in order to be G four-vectors,

FORMAL AND PIIYSICAL PROPERTIES :ETC. 107

are to be defined as

dx~ (34) .~

d~o - - - - ~ p ~ ~ - m 0 t t ~

for bo th t achyons and bradyons , where dr0 is the (G-invariant) proper- t ime element (~,2).

Now, f rom our SLTs in their original (not ye t re interpreted) form (26)~ it

is immedia te to obtain that , if . ~ - - - - ~ :l, then

so t ha t we expect a SLT to t ransform bradyons B into tachyons T~ and vice versa.

However , since the velocity (in par t icular the 3-velocity) does refer to the already re in terpre ted situation, i t is be t te r to s tar t f rom the reinterpreted

form (29) of the SLTs. ( In any case~ when s tar t ing f rom eqs. (26) one has soon after to app ly the re in terpre ta t ion rule contained iu the (, first-step rec-

ipe ,>, sect. 5.) Let us, for instance~ s tar t f rom SLTs (without rotat ions) in their form (29b)~

and apply a SLT along a generic mot ion line / with Superluminal speed U - - 1 / u ( U - ' ~ ] , .~2~1) to the case of a b radyon PB with (initial) four-velocity ~ and (initial) velocity v. Fo r the purpose of generali ty, it is essential

tha t v and U are not parallel. We get ( r , s > ] , 2 , 3 ) ( * )

(35) . '~ -~ ~- ~(U~ o ~- . .n~)n~ ~- i(u~.]- . ~ n , n ~) == -- ~(tt,,-- U.~ ~ - i ~

(u -:llu, U ~ > l ; u ~ < ] ) ,

r where .~ ---- - - . ' n . ; . • ~ .~-{- -sns~r - - ~ - - "iiT~ r, where n is stiff the uni t vector along l, and :~ ~ (U 2 - ~)-t so as in eq. (29b). Le t us observe tha t .,0

is real and t h a t the second one of eqs. (35) rewrites

I

! " u ~ - - - . " n . ---- - - ~ ( ' u - - U ' ~ = " ~ U ' ~

(36) ~r ~ - 1 ' ( U 2 > 1) , -'l ~ i . •

! ! ! !

w h e r e . , is real too and only " l is purely imaginary. Notice t h a t . , , ~, (-• -• are the longitudinal (transverse) components of the space pa r t of the object

four-velocity with respect to the boost mot ion line 1.

(*) One should pay attention to not confuse the boost speeds u. U with the 4-velocity components .~ of the considered object.

108 o . D . ~ACO~RRON~., M. PAV~X~ and ~.. REOAMI

At this point~ let us define the 3-velocity V' for tachyons in terms of their 4-velocity ~'u as follows (1,~):

V '~ 1

F r o m eqs. (36), (37) there follows

(38)

U - - v~ 1 - - uv, Vi ~- Uv, - - 1 ~ v I - u

(U~ > 1 , u~< l , u ~ l / U ) ,

v', = i % V U ~ - I iv• V/l ~.-u ~

Uv~ - -1 v K - - u

where once more I[ and J_ mean parallel and orthogonal, respectively~ to the boost motion line I. I t should be noticed tha t

, 1 , ~7• (38') v , = ~-~, v~_ = i ~ - ,

where ~ is the t ransform of v under the dual (subluminal) Lorentz transformation L(u) with u -~l /U, ul[U.

Again, V I is real and V'~ is purely imaginary. However, V '2 is always positive, so tha t IV'[ is real; actually, from eqs. (38') there follows

v ' , = v ' / + v ; ' = v ' , - I V:,.l> ~ .

More in general, from eqs. (38) one derives for the magnitudes the <~ Terletsky relation ,>

(39) ] - - V '2 : (1 - - v2)(1 - - U 2) ( 1 - U . v ) "

(v 2 < 1 , U ~, V ' 2 > 1 ) ,

which- - inc identa l ly- -has been shown elsewhere (1.2) to have general val idi ty and to be G-covariant (i.e. to hold for any values, sublnminal or SuperluminM, of v, U, V').

I t is worthwhile recalling explicitly tha t eqs. (35), (36) and (38), since they have been derived from the (partially) reinterpreted form of the SLTs, do not possess any longer (2) their group-theoretical properties. For instance, eqs. (38) vannot be applied when transforming (under a SLT) a speed initially Super- luminal (~).

We do not pass hero to the light-cone co-ordinates, since nothing would essentially change.

FORMAT. AND PHYSICAL PROPERTIES :ETC. 109

Equat ion (39) shows that , under a Superlnminal Lorentz t ransformation ( U 2 > ]), a bradyonic speed v goes into a tachyonic speed V'. But we have still to discuss the presence of imaginary units in the components of the taehyon 3-velocity t ransverse to the SLT motion line (cf. the second one of eqs. (38)). To such an aim, we have to recall what said in sect. 5 for the transverse co- ordinates, in connection with the (, second step ~) of the re interpreta t ion procedure: see eq. (30") and the comments following it (we are going to work under the same conditions i)-iii)).

Under those circumstances and conditions, we can interpret V~ and V~ in analogy with :Y', Z ~, or ra ther in analogy with p~, p~.

Namely let us consider, in its centre-of-mass frame, an initial, spherical object with centre at 0 whose external surface expands, however, in t ime for t>~ 0; tha t is to say, let us consider in the initial f rame the following (~ sym- metrically exploding spherical bomb ~):

(4o) O<x'- § y~ § z*<(R § vt), (t>0),

where the initial (t -- 0) radius R of the (( bomb ~) and the speed v of the , spherical explosion ~> are fixed, constant quantities. Let us now pass to a second observer, moving, e.g., along the x-axis with Superluminal relat ive speed - -U. The first limiting equali ty in eq. (40) gives r ise--as we already know- - to a double cone with the x-axis as its symmet ry axis and moving with speed V----U along the axis x ~ x'. The second inequali ty in eq. (40), when expressing it in terms of the Superluminal frame (primed) co-ordinates, trans-

forms into

(41) (1 - - v ~ V 2 ) x '2 - - ( V 2 - - 1)(y,2 ~_ z,~) _ 2t' V(1 - - v 2) x ' - - 2 ~ v V ~ x '

~ ( V 2 - - 1 ) R ~ - - 2 t ' i R v V / ~ V 2 - - i - - ( V 2 - - v 2 ) t '~ ( x ' > t ' / V ) .

The same result rhay be obtained, more elegantly, by expressing eq. (40) ---or, rather, the equation of the (( bomb ~) w o r l d - c o n e - - i n Lorentz- invariant form (for the subluminal observers):

f ] (x~,-[- bl , ) (x" ~- b a ) ~ - -

(40') / [(x. 4-

b")'~J~-:] ( 1 - v2)-l(x~, + b , ) (x~ ~- b~) , ~tt U/t

and then passing to the Superluminal observers just recalling tha t the SLTs invert the quadrat ic-form sign. Equat ions (40), (40') refer, actually, to a t runcated , world-cone ~). In cq. (40'), the quant i ty x~ ~ (t, x, y, z) is the generic event vector inside the world-cone, the vector ~, is the four-velocity of the (( bomb ,) centre-of-mass and ba - - - -~ 'R / v .

110 G . D . MkCCARRONE, M. PAV~I~ a n d ~., RECAMI

W h e n in eq. (41)

(4:[') v V ~ I ,

t h e e q u a l i t y s ign in eq. (4]) co r r e sponds to a t w o - s h e e t e d h y p e r b o l o i d , whose

p o s i t i o n r e l a t i v e l y to t h e d o u b l e - c o n e - - h o w e v e r - - n o w changes w i t h t ime .

(See fig. 4.) The d i s tunce b e t w e e n t h e t w o h y p e r b o l o i d ver t ices , for e x a m p l e ,

r e a d s

(42) V. . - -V~ = 2(1 - - v ~ V~)-~[t 'v( V ~ - - 1 ) § R V V ~ - - - 1 ] .

y!

% Fig. 4. - Let us s tar t from a spherically symmetric object Pn whose radius, however, for t ~ 0 changes with t ime: r ~ R -t- vt. In its rest frame, I)B remains always spherically symmetric. Under a Superluminal x-boost we get a tachyon PT with a complicated shape and t ime evolution. This figure refers to the case in which v V < c 2, the quant i ty V being the tachyon speed (i.e. the relative speed of the Supcrluminal boost). I t actually depicts the simple case in which v<< c2/V. In all cases, however, the initial (bradyonic) (~ exploding bomb ~) Pn transforms into a final (tachyonic) (~ bomb )) PT which (~ explodes ~ in two jets tha t remain confined within the double cone. Notice tha t the l imitat ion x'>~ t ' /V should be added to these pictures. The (~ arrows )) in this figure indicate velocities tha t are slower than light with respect to 0 ' ; tha t is to say, the vertex O' of the double cone travels (of course) with the Superluminal speed V, but the hyperboloid sheets move with subluminal speed with respect to 0' .

W h e n in eq. (41), on t h e c o n t r a r y , v V > ] , t h e g e o m e t r i c a l s i t u a t i o n is

m o r e c o m p l i c a t e d . B u t , in a n y case, t h e , e x p l o d i n g b o m b ~) is seen b y t h e S u p e r l u m i n a l ob-

se rvers to , exp lode )) a l w a y s r e m a i n i n g con t ined w i t h i n t h e d o u b l e - c o n e (e,12).

Th is m e a n s t h e fo l lowing : i) as seen b y t h e s u b l u m i n a l obse rvers , t h e

( b r a d y o n i c ) b o m b exp lodes in a l l space d i rec t ions , s end ing i ts (~ c o n s t i t u e n t s ~),

e.g., also a long t h e y a n d z axes w i t h speeds v, , v , r e s p e c t i v e l y ; ii) as seen b y

F O R M A L A N D P H Y S I C A L P R O P E R T I E S ]~TC. l l l

t h e S u p e r l u m i n a I obse rve r s , h o w e v e r , t h e ( t aehyon ic ) b o m b looks to explode

in two ((jets ~> which remain con]ined within the double cone, in such a w a y

t h a t no c o n s t i t u e n t s of i t m o v e a l o n g t h e y ' or z' a x i s : in o t h e r words , t he

speeds V' V ' of t h e t a c h y o n i c - b o m b c o n s t i t u e n t s <~ m o v i n g >> a long t h e y ' , z ' Ir~ z

axes , r e s p e c t i v e l y , w o u l d r e su l t to b e i m a g i n a r y (e,~).

F o r different a s p e c t s of our r e i n t e r p r e t a t i o n , see sect . 8 in t h e f irst of ref . (~).

T h a n k s a r e duo to A. O. BARU% H . C. CORBErr, J . KOWALCZY~rSa-I,

P . -O. L~W-D1N, l:~. ~IGNANI~ ]~. 3[ILEWSKI, D. 1"~. PETRESCU, W . A. RODRIGUES,

E . C. G. SU~A~SH_~,', D. WILKINSON for s t i m u l a t i n g d i scuss ions a n d t o L.

R. ]3ALDINI, S. MARLETTA, M. SEDITA for k i n d co l l abo ra t i on .

(x~) See, e.g., H. C. CORBEN." Zett. Nuovo Cimento, 11, 533 (1974); Nuovo Cimento A, 29, 415 (1975); YA. P. TERr.~TSKY: in Tachyons, Marwpoles, and Related Topics, edited by E. R~,CA~I (Amsterdam, 1978), p. 47; V. A. GLADKIKH: •izika (Is. Tomsk Univ.), No. 6, p. 69 (1978); No. 6, p. 130 (1978); No. 12, p. 52 (1978); C. C. CHIA~G: private communication. See also N. FLZURV, J. LEITE-LoPES and G. 0BERr.~.CHNER: Acta Phys. Austriaca, 38, 113 (1973); J. R. GOTT I I I : Nuovo Cimento B, 22, 49 (1974).

�9 R I A S S U N T 0

Si esaminano le propriet~ matematiehe e fisiehe delle trasformazioni generalizzate di Lorentz (sia subluminali , sia Superluminali). La forma qui ado t ta ta per le trasformazioni Superluminali di Lorentz ~ que l l a - -da noi recentemente in t rodo t t a - -che sod- disfa le richieste propriet~ gruppali. Quale mezzo di sempliticazione, si fa use anche delle (c coordinate sul cone d i l u c e ~> e di coordinate (~ invariant i per dilatazioni *). Si ehiarisce il ruolo della procedura di reinterpretazione anche dal punto di vista formale, e cib per le coordinate tanto (( longitudinali ~) quanto (~ trasversali >>. Si dedica opportuna attenzione a definire quadrimomento e trivelocits per i tachioni. Infine, si s tudia la ]orma dei tachioni ot tenut i applicando a una particclla ordinaria una generica trasfor- mazione Superluminale di Lorentz (senza rotazioni).

Pe3ioMe He IIOrly~eHo.

I L N U O V 0 CIMF.I~TO VoL. 9 2 B , 1~. 2 11 Apri le 1986

Formal and Physical Properties of the Generalized (Sublomlnal and Superlumlnal) Lorentz T r a n s f o r m a t i o n s .

D. G. ~ACOA~RO~, M. PAV~I6 an d E. RECAp1

I s~ tu to di Pis ioa Yeorica dell' Unlversi th Statale d i Catwrda - Gata/rda, I t a l i a Is t i t~ to 2~azionale di ~ i s i v a _~,~oleare - Sezione vii Catanla, I t a l i a Centre S iz i l iano di f f i s iea ~ue leare e S t ru t tura della Mater ia - Catania, Itabia

(_~Yuovo Cimento B, 73, 91 (1983))

At page 93, eqs. (4a) a n d (4b) should read

(4a) G ~ G ~ ~- ~ ( sub lumina l case: u 2 ~ c~),

(db) ( ~ ~G --~ - - 77 (Super luminal case: u s > o 3) ,

respectively. I n fact, /~he Germs (~ or~hogonal ~ a n d (( an t io r thogonaI ~ are, of course, to be un4er -

s tood as mean ing pseudo-orthogonal a n d pseudoantiorthogo~al, respectively, in Min- kowski space-t ime 07 is the Minkowski metric).

The whole paper r emains unaffected b y the present correction.

239

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