Generalized Lorentz transformations in four dimensions and

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 I s t i t u t o d i E i s i c a Teorica dell'U~Hve~'sitd - Catania Ce~tro Siciliano di l~isica Nucleare e di Strut/ura della M a t e r i a - Catania Istituto Zrazionale d~ Fgsica Nucleare . Sezione di Cata~Ha

(ricevuto il 16 Agosto 1972)

Summary. - - 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 Superluminal frames, the meanings of (, inertial frame ~, (~equivalence ~, , (( 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 ( P T ) . Moreover (for a non (~eharg,',>free universe), it is shown ~hat our gelmralized sp('ci~d relativity requires covariance under CPT. A =~ i l A > for u > c (*)(~). F o r instance, in paper I we have shown t h a t - - i n the simple case of collinear motion along the x - a x i s - - c o n d i t i o n (1) is satisfied (**) b y X - - Ul

t -- ux/e e

(1 his) y'= ±y

~,,

Z !

= ±z

•

L e t us rewrite relation (1) for f l 2 > 1 (i.e. for transition from a s to a ~) as c~t,2 ÷ (ix,)~ + (iy,)~ ~_ (iz,): = (ict)2 + x ~ + y2 + z ~ ,

a n d explicitly notice t h e following. Since we considered for simplicity t h e case of collinear m o t i o n along t h e x-axis, our eqs. (1 h i s ) m u s t b e - - a s t h e y are--such that

(1 ')

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

and that

(1")

(iy') ~ = y"..

(iz')~ = z ~ .

(*) And the ones t i m e s the operator - - 1 = ~ P T . See Sect. 2. (**) See, R. MIGNANI, E. REeAMI and U. LOMBARDO:L e t t . N u o v o C i m e n t o , 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.

G E N E R A L I Z E D LORENTZ TRANSFORMATIONS IN FOUR DIMENSIONS ETC.

17~

Of course, b o t h s and S will observe only real quantities! The i m a g i n a r y units m e r e l y record the relative sign of the various couples of the correspondingc o m p o n e n t squares. One of the aims of the present work is extensively illustrating the new generalized L T gn'oup G, since it was scarcely m e n t i o n e d in p a p e r ]. A n o t h e r aim is clarifying some points of p a p e r I. I n our terminology G-covariant means covariant under the whole group G. B y t h e way, let us r e m e m b e r t h a t only m e a s u r e m e n t results are supposed to be expressed b y real quantities, b u t Um generalized L o r e n t z t r a n s f o r m a t i o n s (GLT) m a y well be represented b y matrices built up with i m a g i n a r y 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 )> (*) a n d let us represent the A's b y 4 × 4 matrices. I f we p u t

A
=--A(lfll> 1),

where A(lfl] > 1) are ]ormally identical with usual proper, orthochronous L T ' s b u t correspond to values [fll > 1, it is possible to see t h a t

A- ( - - f l )

~ --iA;l(fl) .

Thence

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

and the generalized group (1) G of p a p e r I has in particular to contain also the total-inversion o p e r a t o r P T -- - - 1 • Precisely, b y considering successive applications of G L T ' s of t y p e A< and i A > , it is easy to realize t h a t t h e group G of the G L T ' 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 c), in the sense t h a t (3)

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

,

/i: being a m a t r i x independent of the velocity m a g n i t u d e s ~, U. F o r simplicity we consider eq. (3) only for collinear motion. The m a 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 t h e o b s e r v a t i o n t h a t , if a t a c h y o n m o v e s with velocity v ~ v= relative to us, it will a p p e a r with divergent velocity to the observer s" h a v i n g (collinear) velocity ~ ~ u= - - c2/v relative to us (see Fig. 3). F o r instance, in the simple case of collinear motion, u ~ u=, U ~ U~, 0

--1

0

0 ~

--1

0

0

0

Z~K=

,°

0

0

i

0

0

0

0

i

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

GENERALIZED

LORENTZ

TRANSFORMATIONS

IN FOUR

DIMENSIONS

ETC,

175

A t last, i n t h e collinear case, t h e m a t r i x K o p e r a t e s t h e e x c h a n g e s

,~'(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 w i t h eqs. (1')-(1").

I n t h e eollinear case, for i n f i n i t e speed, as well as for zero speed, t h e n o t i o n of d i r e c t i o n b e c o m e s m e a n i n g l e s s (7)(*). I t is a l r e a d y r e a l i z a b l e t h a t t h e tools of p s e u d o - E u c l i d e a n g e o m e t r y are n o t t h e b e s t ones for d e a l i n g w i t h our p r o b l e m s ; we shall t h e r e f o r e b o r r o w a bit from projective geometry. F r o m w h a t precedes, it is a p p a r e n t t h a t our (is characterized by the fact t h a t - - w i t h

reference t o a f r a m e s o - - t h e r e is a b i u n i v o c M

correspondence between observers with velocity u and those with velocity

U -- c+:/u. P r e c i s e l y , t h e (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 r a m e s is a p a r t i c u l a r conformal mapping (} t o {~:A r /;- < 1 ,

~nd

imo ,~ = : , / y _ _ ~

for /~ > ~ .

L e t us explicitly re-emphasize t h a t t h e i a p p e a r i n g for f12> 1 comes from the SLT's, and does not m e a n at all t h a t ~achyons h a v e an i m a g i n a r y proper mass (as we know, a t a e h y o n behaves as a b r a d y o n with reference to its rest frame, and therefore all its proper quantities are real). The same happens for proper t i m e a n d proper length; for example, since dr = tiT' "V 11 _f12[, when passing f r o m the t a c h y o n rest s y s t e m to our frame, one has d~'o

(s)

d~:'-- g l ~ --fi'l

(l%

dTo

Vfi-'-- 1 -- - - i g i - - f l 2

i d%

-- V l - b

'~

@>

1),

where of course d% d~o are b o t h real. I t is worth-while to notice t h a t , w h e n generalizing(1) physical laws for t a c h y o n s (/?2> 1), one shouht p a y a t t e n t i o n t h a t a p r i o r i ~ / f l 2 - - 1 = =k i V ' l - - f l 2, since (=k i) 2 = - 1. Always we consistently choose (*) t h e sigrt mi~u.,, in order, e.g., to get positive values of m in eq. (7). I t is u n d e r s t o o d t h a t ÷ ~/1--f12 r e p r e s e n t s t h e u p p e r half-plane solution for fl~->l. See, e.g., our eq. (5). I n the first p a r a g r a p h of this Section we expressed in which sense all the (~ inertial frames~) (with relative velocities [u[~c) are equivalent. F r o m such considerations, it is i m m e d i a t e to get t h e Ato < Ato

Ax = la/ticl < A~o > Axe

for 1 < / 7 < { 2

It

]or

I~ _~'.

t i > V~; /or , < ti < V~

X"

/'4 /// ,/~.]' ft,,(n=~-)

2sD -

/or

A(-Ax,O)

o

b

x

Fig. 8. - The geometricM interpretation of our SLT's (for fl > I). The fact t h a 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@ b u t now--according to Fig. 6--we get both Lorentz contractions and dilatations. I n partieular (for /7> 1), we have A t = A t e and A x = A x o 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 A t < Ate and Ax > Axo.

G ( E N ~ I ~ A L I Z ~ D LOI~JdNTZ T~ANSFOR1V£ATIONS I N F O U R D I M E N S I O : N S E T C .

185

changed use (for fl~> 1) of Che hyperbola, when considering space a l l d time intervals respectively, reflects the change in the (spacelike or timelike) n a t u r e of intervals t h a 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. I n 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 interpreting the other cases. F o r instance, the case - - c ~ < f l < - - 1 results to be symmetric to the case in Fig. 8.

xt-1) I i i

I

- -

Xv

I, >, . Tra velocit~ bradioniche e tachioniche esiste una corrispondenza biunivoea, ehe risulta essere una particolare 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 ), si mostra ehe ia nostra relativit~ ristretta generalizzata riehiede la eovarianza per CPT. Si formula un >, in base al quale le leggi fisiehe relativistiehe (quelle almeno della meecanica e dell'elettrodinamica) 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 h n g h e z z a , effetto Doppler, indiee di rifrazione . . . . ) utili 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

O606ulennbm npeofpa3oBaHU~

ETC.

~lopeHTna B qeTb~pex H3Mepemmx u

189

cBepXCBeTHmHecn

O~lbeKTbl.

P e 3 m M e (*). - - 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 I43MepeHHflx, o6061J.teriHa~ TaK~Ke ~,q~ CBepxcBeY~tttHXCn CHCTeM OTCqeYa. ldcc.~e~yvoTcn qbH3~Jnecktie npHMCHeHH~I HOBO~I rpyrlrlbi G. I/IcllOJlb3yfl