the university ~lms-p~y ~s ~ professor of physics from the Italian Education. Ministry (MPI), thus helping him to survive ,~nd, therefore, indirectly ~llow-.
VOL. 4, ~. 3
I{IVISTA. DEL NUOV0 CIMENTO
Luglio-Settembre 1974
Classical Theory of Tachyons (Special Relativity Extended to Superlnmlnal Frames and Objects). E . RECAMI
Istituto di tJisica Teorica dell' Universith - Catania Istituto Nazionale di ~isica JOC, acleare . Sezione di Catania Centro Siciliano di ffisica N#cleare e di Struttura della Materia - Catania :R. MIGNANI
Istituto di .Fisica dell' Universit~ . R o m a (Rivista det .37uovo Cimento, 4, 209 (1974))
Owing to an editorial error, the Fig. 25 appeared in an incorrect version. I t must be substituted b y the following one:
/
(Ho)z,y
hJ,y ////
, ~ ' y , z H/~p ,
I J/ c). l l'4. Case of interactions among bradyons. 12. Causality and taehyons. 13. Digression. 14. Taehyon mechanics. 14"1. Rest mass and proper quantities. 14"2. Generalized relativistic Newton's law. 14"3. An application. 14"4. (~Virtual particles ~>and taehyons. 15. Tachyons in the gravitational field. 15" 1. I n t r o duetion. 15'2. Tachyons and gravitational field in special relativity. 15"3. On the so-called >and taehyons. 16. About electromagnetic Cerenkov radiation (ECR) from taehyons. 16"1. Taehyons do not emit ECR in vacuum. 16"2. The general problem. 17. Doppler effect for Superluminal astrophysical sources. 18. Generalization of Maxwell equations. 18" 1. AutoduM electromagnetic tensor. 18"2. ~[agnctic monopoles and taehyons. 19. Note on taehyon q u a n t u m field theory. 20. Miscellaneous remarks. 21. Brief discussion of the experiments. 22. Conclusions.
Foreword.
P u b l i s h e d m a t e r i a l c o n c e r n i n g f a s t e r - ~ h a n - l i g h t o b j e c t s ( t a c h y o n s ) is r a t h e r w i d e - s p r e a d a n d a l r e a d y c o n s p i c u o u s , as well as contradictory~ so t h a t a r e v i e w article seems i n order.
I n so doing~ we shall confine ourselves to t h e classical
t h e o r y of S u p e r l u m i n a l objects, w h e r e classical m e a n s >.
CLASSICAL THEORY
OF TACNYONS
211
For linearity purposes, we are essentially exposing the theory elements t h a t we regard as correct, with large reference to our own contributions. Therefore, m a n y papers are only briefly discussed. Some other works, t h a t appear ~o us as inessential or incorrect, are not even mentioned, for compactness' sake. The mMn point we shM1 call attention to is generalization of speciM-relativity t h e o r y to Superluminal (inertial) reference frames and tachyons, since according to us this constitutes the only sound basis allowing a self-consistent development of the classical theory of faster-than-light objects. Another point we shall investigate and clarify refers to the so-called (( causal paradoxes ,~; in fact they, even if easily solw~ble, gave rise to some confusion in the literature. Particular attention shall be paid as well to rend explicit the symmetries required b y (( extended relativity >): for example, the (( crossing relations )> and the ~/5~ theorem will be derived even for usual particle scattering (provided their interaction is rel~tivistically covariant). Moreover, extended relativity will be shown to clarify other points in standard physics, such as the connection between m:~tter and antimatter. In extending mechanics and electromagnetism to tachyons, an interesting point will arise, in connection to generalized (( Maxwell equations ~>, that suggests a close link between the so-called magnetic monopoles and tachyons. The experiments so far performed looking for tachyons are criticized. In :~ !irst reading, Subsect. 8"4 (iii) and Sect. 13 m a y be skipped. Most of the present work is originM, either in the substance or in the form.
2. -
Historical
remarks.
E v e n in pre-rela~ivistie times the possibility of faster-than-light particles ( F T L P ) fascinated physicists' minds. Among the first scientists occupied with this topic, let us mention Ttio~IsoN [1]~ HE_tV[~IDE [2], DE: COUDnES [3], and in particular SO)[3IEm'ELD [4, 5]. Together wi~h relativity theory [6], however, the conviction spread unfort u n a t e l y t h a t light speed in v a c u u m was the upper limit of any velocity, the early century physicists being misled b y the evidence tha~ usual particles (bradyons (*)) cannot overcome t h a t velocity. They behaved like Sudarshan's imaginary demographer studying the popnlalion patterns of the Indian subcontinent [10]: of special r e l a t i v i t y m a y a c t u a l l y be deduced, as the ((reciprocity principle>>[20], the linearity of t r a n s f o r m a t i o n s [20] b e t w e e n two frames J~, ]2~{I}, a n d t h e existence of an i n v a r i a n t square speed. N a m e l y , f r o m the a b o v e - w r i t t e n postulates it f o l l o w s - - b y e x t e n d i n g t h e procedure in ref. [ 2 0 ] - t h a t a real q u a n t i t y exists, h a v i n g the dimensions of a v e l o c i t y squared and showing inv~riant value for all f r a m e s ] ~ { I}: (1)
c~- - i n v a r i a n t q u a n t i t y . 3"2. Duality principle (**). - At this point, experience is k n o w n to h a v e
u n d o u b t e d l y yielded t h a t the i n v a r i a n t speed c is real, a n d a c t u a l l y is the speed of light: (2)
c = speed of l i g h t .
I n t h e cases in which c m a y be a s s u m e d to be (practically) infinite, we would of course get in t h e following t h e Galilean physics. L e t us n o w choose a p a r t i c u l a r inertial f r a m e so. T h e light speed c - - b e c a u s e of its i n v a r i a n t - q u a n t i t y c h a r a c t e r - - M l o w s an e x h a u s t i v e p a r t i t i o n (**) of f r a m e s J e{1} in t w o subclasses {s}, {S} of f r a m e s h a v i n g speeds u < c a n d U > c relative to so, respectively. F o r simplicity, in t h e following we shall consider ourselves as (( t h e o b s e r v e r so >). F r a m e s s c {s} will be called subluminal, a n d f r a m e s s ~ {S} Superluminal. The relative speed of two f r a m e s s~, s~ (or S~, S~) will a l w a y s be smaller t h a n c; a n d t h e relative speed of t w o f r a m e s s, S will a l w a y s be larger them c. The i m p o r t a n t point is t h a t t h e above, e x h a u s t i v e p a r t i t i o n is i n v a r i a n t when So is m a d e t o v a r y inside {s} (or inside {S}); on t h e c o n t r a r y , w h e n we I)ass f r o m So E {s} to a f r a m e So ~ {S}, tile subclasses {s}, {S} are i n t e r c h a n g e d one with t h e o t h e r [19, 22, 23] (***). A t t h e p r e s e n t time, we neglect l u m i n a l fraHleS (u U - c) as (( u n p h y s i c a l ~>, even if m a t h e m a t i c a l use of ((infinite-monlentum f r a m e s >~ has spread o u t r e c e n t l y (***). (*) As early as 1911, it has been shown [21] (in the framework of standard relativity) that the assumption of the existence of an invariant velocity is not necessary in order to derive the ~,Lorentz transformations ,). (**) See also Sec~. 10. (***) The most original feature of paper [22]--i.e. introducing unidircctionality in time for bradyons and ill space for tachyons--docs not nleet our agreement, since irreversibility seems essentially to be a statistica7 output. The correct part of that article, confined to a bidimensional space-time, inainly coincides with the results in ref. [18]. (***) In any case, from such frames the space-time should appear as a bidimensional space, projection of a suitable 3-dimensional hypersurfaee onto a plane normal to the luminal ray direction. On such a plane, both objects and photons appear immobile. Of course, these frames are useful, e.g., for investigating the (at rest) (( partons )>, otherwise relativistic subpartieles.
214
~.
R:ECAMI a n d
1~. M 1 G N A N I
We can immediately deduce a ((duality principle)) [18, 23-25], which m a y be briefly put in the form (, the terms B, T, s, S do not have an absolute meaning, b u t only a relative one ~>. Disregarding the fact t h a t the (~duality principle )) follows from the previous (extended)~[30]. F r o m the r e q u i r e m e n t t h a t SuperluminM frames be physical [] 9], it follows of course t h a t objects m u s t exist, which are at rest relative to S and tachyons relative to frames s. F r o m the further fact t h a t luxons t show the same velocity to a n y observer s or S, it can be deduced t h a t a b r a d y o n relative to an S, B(S), will be a t a c h y o n relative to a n y s, T(s), and vice versa: (6)
B(S) = T(s),
T(S) ~-- B ( s ) ,
[(S) = t~(s).
This accords [18, 23-25] with the duality principle, that we are going to complete b y adding t h a t [23] >. I n conclusion, when frames s, S observe the same event, (~timelike >>vectors t r a n s f o r m into (( spacelike >> vectors and vice versa, in going f r o m s to S, or f r o m S to s. On the contrary, it is well known t h a t usual Lorentz transformations, f r o m s~ to s2, or from S~ to $2, preserve the four-vector type. One is therefore allowed to say t h a t subluminM LT's, i.e. usual Lorentz t r a n s f o r m a tions (LT), are expected to be such t h e t
(5a)
c2t'~+ ( i x ' V = ÷ [c~t~+ (ix) ~]
( ~ < 1),
while (~Superluminal Lorentz trans]ormations ~> (SLT), f r o m s to S or from S to s, are expected to be such t h a t
(5b)
c~t':+ (ix')~ = -
[c2t~-÷(ix) ~]
(fi~> 1).
3"5. Generalized relativistic trans]ormations. - The case (5a), i.e. the case of usual L T ' s , is v e r y well known. L e t us investigate case (5b), i.e. the case of SLT's. This p r o b l e m has been independently approached, for the bidimensionM ease, in ref. [18, 22] and, for the four-dimensioi~M case, in ref. [19]. I n the bidimensionM space-time case, P A ~ E R [18] showed t h a t the linear transformations P satisfying eq. (5b), for /~2~ 1, are unique; in ref. [19] it
(*) See also Sect. 9, 10.
216
~. RtlCAMI an4 R. MIGNANI
has been observed t h a t those P m a y be simply written [19] (7)
x'=
x--ut ± -@ ~- . ~
t'
~
t--ux/c~ = ~ -,~,//~2 1
{ \
u~ /~ ~ - ~ >
~
1]
"
B y the way, let us m e n t i o n t h a t PARKEI~ chose only the sign m i n u s [18]. A t this point, the problem of the generalized L T ' s is t h a t of finding new (~generalized Lorentz transformations ~ (GLT), t h a t reduce to the usual ones for f12 < 1, and to P a r k e r ' s in the case of D 2 > 1 and two dimensions. This p r o b l e m has been solved in ref. [19, 23]. I n the particular case of a boost (collinear motion) along the x-axis, the G L T ' s read [25] (for - - o o < f i < co)
x,=
±
x - - ut
t' =
t -- ux/c 2
(-~ 1),
one can i m m e d i a t e l y see t h a t S L T ' s operate (*) a p-~= 0, or b e t t e r t r a n s f o r m (**) two-sheeted hyperboloids p~ = m~ > 0 into single-sheeted hyi)erboloids p2 = m~ < 0. Of course in the tetraimpulse space, B ' s are characterized b y tim(dike f o u r - m o m e n t a , ~'s b y lightlike fourm o m e n t a and T ' s b y spacelike f o u r - m o m e n t a . (*) Note added i n proo/s. - T h i s (hdicatc p o i n t will be b e t t e r c o n s i d e r e d in :R. MIG~ANI a n d E. RECA~rI: Lett. N u o v o Cimento, 9, 357 (1974). See also G. C. WICK: p r e p r i n t C E R N - 7 3 - 3 ( G e n e v a , 1973); R. :PENItOSE a n d M. A. H. MAc CALLVM: P h y s . Rev. 0 C,
245 (1973). (**) At the finite. ~t least. 15 - Rivisla del Nuovo Cimento.
218
~. RECAMI and R. MIGNANI
3"7. Conservation laws. - F r o m P o s t u l a t e 1 the usual conservation laws (of energy, m o m e n t u m and angular m o m e n t u m ) follow for an isolated s y s t e m even in generalized r e l a t i v i t y theory, i.e. even w h e n Superluminal velocities are allowed. F o r instance, with the help of t h e results of Sect. 6, the derivations contained in the second of ref. [29] m a y be i m m e d i a t e l y generalized. Therefore, the G L T ' s (of the group G of the n e x t Section) will preserve the validity, inside a n y isolated system, of energy, m o m e n t u m and angularm o m e n t u m conservation laws, in accord with the generalized definitions of Subsect. 7"2. No different f u r t h e r results m a y be derived f r o m Postulate 1 and consideration of space-time translations. This it the reason w h y we are explicitly investigating only the group of (~space-time rotations ~.
4. - Group G of generalized Lorentz transformations (GLT). 4"1. T h e g r o u p elements. - The G L T ' s f o r m a new [19, 25] group G of t r a n s f o r m a t i o n s in four-dimensionM space-time. This group is the group of all (( rotations )) in Minkowsky space. I n our terminology, (~ G-covariant ~) will m e a n eovariant u n d e r the whole group G. L e t us represent the t r a n s f o r m a t i o n s L b y 4744 matrices, and consider first, for simplicity, a universe free of (~charges ~)(*). I f we call usual proper, orthochronous (homogeneous) L T ' s (12a)
A< ~_ A(fl z < 1 ) ,
t h e n it m a y be realized [23] (see eqs. (8)) t h a t S L T ' s have the form SLT :
-4- iA> ,
where (125)
A> =_A @ > 1)
are matrices (**) f o r m a l l y identical to the A c ) , in the sense t h a t
-- iA>(U, O, (p) = / ~ + ( 0 , q)).A 0),
/~+ being a matrix independent of the magnitudes of velocities u, U, and dependent only on their common direction (that we indicated by the two angles O, ~). I n the case of col]inear motion, one has [23]
(20 his)
-- iA>(U) = K+'A O),
iA> are boosts, K+ ~-/~+(0, 0), and ~+(0, ~) = R(O, ~o).K+.R-I(O, ~) ,
the matrix R(O, q~) being the space rotation which brings u, U parallel to the x-axis. The matrices K+, K+ represent (( transcendent SLT's ~> (we m a y call them also (~transluminal LT's ~) for the reasons we shall see later). In fact, for /7-->-}- c~, eqs. (20), (20 bis) become L ( + c~, 0, ~) ~ L + ~ ~ l i m [-- iA>(U, O, ~ ) ] = K+(O, ~),
(20')
L ( + ~ ) ~ L+~ =_-- iA>(+ c(U=c2/u)
6~ 2
\ /,,-,: / -,) -~ ~ + iA> ~ , in such a w a y t h a t (-- i A > ) ( + i A ~ ~) = 1. B u t we m i g h t
a)
b)
Fig. 3. - a) and b) correspond to the ease considered in Fig. 2 b). They must be re~d only in the counterclockwise sense, a) and b) symbolicMly, formMly depict transformations --L-l(fl) and + L-l(fl), respectively, with f l > 1 (see the texti eq. (13b); and eq. (31b) below). Notice that transformations L(O, A), L(A, B), L(A, O) correspond to transformations Z(So-+ S~), L(so-+ SB), L(so-~ So), respectively, of Fig. 6 b).
also decide to go b a c k f r o m $2 to so (~b y - p a s s i n g ~) t h e transcendent ]rame Poe (relative to So: see F i g 4 b)) ; in this case, however, we would reach the f r a m e so with all its axes reversed, (Drf)so, since (-- i A > ) - ( - - iA> 1) ~ -- 1. W e shall see m o r e clearly (Sect. 7) t h a t an ideal frame, t h a t u n d e r w e n t such a (( trip ~), w o u l d c o m e b a c k with spatial axes i n v e r t e d (space parity) and with particles transformed into antiparticles. More generally, if LIs~)-->/) is t h e L T c o n n e c t i n g f
X(-1)
l
!b" I I
1
X(+l)
O,(so)
a)
b)
Fig. 4. - a) snd b)correspond to the cssc considered in Fig. 2 b), but a.re now to be read only in the clockwise sense, a) symbolicully shows the transformation L ( - - 8) = -- L-l(fl) ~ -- iA> (-- 8) ~ -- iA>t(fi) . b) symbolically shows the transformation - - L ( - - f l ) = L-~(fl) ~ + iA>(--fl) ~ + iA>~(fl). Always - - 8 < - - 1 , i.e. 8 > 1.
226 frames
:E, R E C A M I
and
R.
MIGNANI
~ n d ], w e h ~ v e
so
L (So --> 1)=+A ( c ~ / u ) , L( so --> 3 ) ~ - i A > ( - - c ~ / u ) = + i A ~ ( c 2 / u ) ,
(28)
L(so
4)~-A
the
relative
-A
e2
V2 ~
C 2 ==> V t2 ~
C2
V2:>
e 2=:>v'2 c, respectively. L e t us call L(so--~ S) and L(So--~ S') the G L T ' s connecting so to S and S', respectively. In the case t h a t S' is faster t h a n S with respect to us, f r a m e S will not observe S' m o v i n g with positive velocity! This m a y seem c o n t r a r y to c o m m o n sense. B u t transformations L(so--~S, S') h a v e the f o r m - - i A > . I n s p e c t i o n of this fact, a n d Lorentz comparison of space and t i m e distances between couples of events, lead to the conclusion t h a t : W i t h respect to So, the f r a m e S appears as possessing parallel t-axis and r e v e r s e d - p a r i t y space axes (an antiparallel x-axis, and 270°-rotated y, z axes). Such analysis is enough to satisfy intuition, which m a y in a n y ease get help also f r o m the G L T geometrical interpretation p u t forth in the n e x t Section. Therefore, n a m i n g v ' - : v: the S' velocity as seen b y S, we h a v e [23] [ v' v / 2 ; ~ for 1 < fl< ~/2 f > oV2 ra
T I , C-
-d
x"(fl=~)
x" / ~ 4/ ~ ~ ~ /_/t / " /~ ~( P t,'(p=~) -- ~ ?, ~-, ~-!
~ /
/
-,,t,,
#
~JB
A(-Ax,O)
0
b
x
Fig. 10. - The geometrical interpretation of the 8LT'B (for fl> 1). Notice the exchanged r51c of hyperbolas, in considering space and time intervals, respectively, due to the change in the (spacelike or timelike) type of intervals that one meets when passing from frames s to frames S. Here one has still At = [b/flcI, and Ax = [aflcI, but now--according to Fig. 8--we get both Lorentz contractions and dilatations [Ax, At> 0]. In particular (for f l > l ) , we have At=At o and Ax=Ax o when fl= ~/2. Moreover, for 1 < # < ~/2, we have A t > A t o and Ax ~/2, we have At A x o.
and by putting
fl -- tg F ,
C ~ -}- [ f l 2
1.
We get [23] x---- (~(x' sinv2 + ct' cos Y~i'
(40) ,
where now ~f runs ]rom 0 to 45 °. Of course, also eqs. (40) can be interpreted as
in Fig. 7, 10. Let us go back Co GLT's, eqs. (38) and (38'). The (~geometrical >~analysis of space and time unit transformations becomes easier if we remember (besides the standard definitions [30]) t h a t [41] i) the Lorentz-transformed space unit Ax m a y be also derived from the time a (relative to us) taken by the moving standard rod to (~pass before our eyes >>: Ax = laficl; ii) the Lorentztransformed time unit At m a y be also derived from the space b travelled (in our frame) by a lighting (( lamp >>which is switched on for a unitary time (this time being measured in the co-moving frame): A t = [b/flc[. F r o m Fig. 10 we can deduce what already got from Fig. 8; in particular, it is apparent t h a t At ---- At0, Ax = Ax0 (for f12> 1) when fl = V'2. Equations (8 his) correspond to eqs. (8) t a k e n alternatively with the sign plus or m i n u s - - b e c a u s e of the effect of coefficient ~--as depicted ia Fig. 11 and 6. The (substantial) interpretation of the effect of the sign entering
235
CLASSICAL THEORY OF TACHYONS
x(-1)
x(+i)
\0.< I
@2
< ,\/-' 0
(fl~ < 1 ) ,
from the duality principle it follows t h a t , for a tachyon, .i.e. after a SLT, we shall have [42] (42b)
E ' 2 - - p '2 : - p ' : = - - m ~
1, mo real).
Besides, it is k n o w n t h a t in the luxonic case
(42c)
E ~ - p~ ~p ~ = m~ = 0
(#~= 1).
236
E. RECA~X and R. ~IGNANI Therefore, one has [15]
(43a)
p~ = m o2> O
for b r a d y o n s (case I, or timelike),
(43b)
p2_~ 0
for luxons (case I I , or lightlike),
(43c)
p~=--
m2< 0
for t a e h y o n s (case I I I , or spacelike).
I n f o u r - m o m e n t u m space (see Fig. 12), eqs. (42) represent r e s p e c t i v e l y i) for b r a d y o n s , a t w o - s h e e t e d h y p e r b o l o i d of r o t a t i o n a r o u n d t h e E - a x i s ;
(C
f
J,
Py
c~) I
b)
Fig. 12. - :Representation of the hypersurfaces E ~ - - p 2 - p ~ , for a) bradyons, with p~ ~ m ~ > 0 (timelike case); b) luxons, with p2 ~mo2 = 0 (lightlike case); c) tachyons, with p2 ~ _ m o ~ < 0 (spaeelike case), where m0 is always real. In a), the points A ' and A" represent the particle kinematical states obtained by applying the operations C P T and CT, respectively, to the kinematical state A. In the case when we confine ourselves to subluminal frames and to usual LT's, then it happens that the (( matter ~>or (~antimatter )> character is invariant for B's, but relative to the observer for T's. When eliminating the previous restriction, we may pass from particles to their antiparticles (through GLT's) even in the case of bradyons.
ii) for luxons, a double indefinite cone, h a v i n g E as axis; iii) for t a c h y o n s , a single-sheeted r o t a t i o n h y p e r b o l o i d . I n all cases, we h a v e iv[ = [p/E I. F o r o b v i o u s reasons, in Fig. 12 o n l y t h e t h r e e - d i m e n s i o n a l (*), in a c c o r d a n c e w i t h t h e d u a l i t y principle: in particular, t r a n s c e n d e n t t r a n s f o r m a t i o n s K ± , K ( 0 , ? ) o p e r a t e a symmetry w i t h respect to t h e light-cone p~---~ 0: (44)
E,2 _ p,~. ___ _ (E: _ p2)
(fl.~ > 1).
7"2. F o u r - m o m e n t u m space. - T h e usual relativistic f o r m u l a e for t h e e n e r g y a n d t r i m o m e n t u m of a free t a c h y o n t r a v e l l i n g parallel to t h e x-axis w i t h (')
At least, this happens at the ]inite.
237
CLASSICAL THEORY OF TACI:IYON8
velocity v ~ v~ :- tic, when fl'-'> 1, will read [15, 43, 44[ m o C2
,
m0/~c
(f12 > 1, mo real) .
I n Fig. 13, the relativistic energy behaviour is represented, for a free particle (bradyon or tachyon), vs. its velocity v. Figure 14 shows the behav-
c
Fig. 13.
0
c
v
c
v
Fig. 14.
Fig. 13. - Magnitude of relativistic total energy vs. velocity v for a free particle, either bradyonic (Iv] < c) or tachyonic (Ivl > c). For simplicity, we may refer to a velocity directed along the x-axis of the reference frame. Fig. 14. - Three-momentum magnitude vs. velocity, for both bradyons and taehyons.
iour of the t h r e e - m o m e n t u m m a g n i t u d e vs. velocity for b o t h bradyons and ~achyons. I t m a y be noted t h a t a) the speed c preserves of course its character of l i m i t kinematieal p a r a m e t e r of the four-dimensional universe (even if we know t h a t such a limit has two ~ sides ~); b) tachyons will slow down when energy increases and accelerate when energy decreases. I n particular, divergent energies are needed to slow down the tachyonic veloci~,y t o w a r d s the (lower) limit c. On the contrary, when t a c h y o n ' s velocity tends to infinity, its energy tends to zero ; this prevents violation of the c o m m o n postulate t h a t (( energy m:~y be t r a n s m i t t e d only at ]inite sI)eed ~, since a t a c h y o n shows zero energy to the same observers relative to w h o m it presents divergent velocity. Notice t h a t a b r a d y o n m a y h a v e zero m o m e n t u m (and minimal energy moC"-) and :~ t a c h y o n m a y h a v e zero energy (and minimal m o m e n t u m moC); however (Fig. 12), b r a d y o n s B eannot exist at zero energy, as well as t a c h y o n s T cannot exist at zero m o m e n t u m (with respect to the observers relative to whom t h e y a p p e a r :m tachyons!). I t is i m m e d i a t e to see *hat infinite veloeity belongs to t a e h y o n s corresponding to the interseetion of the h y p e r b o l o i d in Fig. 12 c) with the plane ]A = 0. B y the way, since t r a n s c e n d e n t taehyons do t r a n s p o r t m o m e n t u m , t h e y allow getting the rigid-body b e h a v i o u r even in special r e l s t i v i t y [ 4 3 ] ; as a consequence, in e l e m e n t a r y - p a r t i c l e physics (see the following), tachyons m i g h t a priori result to be useful for int, erpreting diffractive seattering or the so-called p o m e r o n - e x e h a n g e reactions [45-49].
~
:E. RECAMI a n d
R. MIGNANI
Before going on, let us de]ine the negative-energy points of hyperboloids in :Fig. 12 a)-c) as representing the possible kinematical states o] the (( antiparticle )~ (of the particle represented b y the corresponding positive-energy points). W e shall see the reason for such a definition; in particular, it will be shown to coincide with the usual definition in the bradyonic case. B y the way, let us r e m e m b e r ~hat (in the usual language) the operation of (( changing particles into antiparticles ~>, and vice versa, is the ~(C/~T operation ~ (see the following). 7"3. The (~reinterpretation principle ~: the third postulate. - L e t us lastly r e m e m b e r t h a t the kinematical state of a generical ]ree particle (with param e t e r m0) will be represented b y a point on one of the hypersurfaces in Fig. 12. And the kinematical states of t h a t free particle with respect to all the subluminal inertial frames will be r e p r e s e n t e d b y all the points of the same surface sheet. I n fact, usual (subluminal) L T ' s do not effect transitions f r o m a sheet to another. A simple look at Fig. ]2 c), which shows a connected hypersurface, imposes the following considerations. For tachyons, (subluminal) L T ' s will exist t h a t operate with continuity transitions f r o m u p p e r - s e m i - s p a c e points to lower-semi-space points. I n other words, it m a y seem t h a t a tachyon, regularly a p p e a r i n g to observer 0 as h a v i n g positive energy (see point A of the u p p e r semi-hyperboloid), will a p p e a r to other observers 0 ' as bearing negative energy (see, e.g., point A ' of the lower hyperboloid). However, if a LT, eqs. (8), is such as to invert the energy sign, the same LT, eqs. (8), will invert the sign of any other tetraYector fourth component, associated with the same observed object; in particular, t h a t L T will invert also the sign of t i m e [15, 46, 47]. This fact is visually shown in Fig. 15. 1Namely, if a t a c h y o n moves, e.g., along the x-axis with positive velocity U with respect to us, the a b o v e - m e n t i o n e d sign inversions h a p p e n [38, 50] for all boosts cor-
Xr
0
X
Fig. 1 5 . - World-line OT of a tachyon. With respect to observcrs 0'-----(t', x') the tachyon appears to move backwards in time (relative to the time arrow, uniquely determined by the (~thermodynamical~) behaviour of usual macrosystems). In fact, projection of OT on the axis t' is directed towards the negative semi-axis. However, observcrs O' are the same ones to which the taehyon will show (~negative energy ~>. Those two paradoxical occurrences compensate each other, easily allowing an orthodox physical interaction. See thc following.
CLASSICAL THEORY OF TAC~YONS
2~
r e s p o n d i n g to positive velocities u ~ c~-/U (along the x-axis, a n d with reference
to us). I n conclusion, if a t a c h y o n is e x p e c t e d to show negative e n e r g y r e l a t i v e to a c e r t a i n observer, it is also e x p e c t e d to a p p e a r to t h e s a m e o b s e r v e r as m o v i n g backwards in time (with r e s p e c t to t h e t i m e a r r o w u n i v o c a l l y determ i n e d b y usual m a c r o s y s t e m s ' b e h a v i o u r ) . I t is v e r y e a s y to convince ourselves t h a t those t w o p a r a d o x i c a l occurrences allow a quite orthodox r e i n t e r p r e t a t i o n , w h e n t h e y are (as t h e y a c t u a l l y are) s i m u l t a n e o u s . T h a t ~ reinterpretation (or " s w i t c h i n g " ) principle ~), t h a t we shall call ~ l~I1) >), has been first p u t forth b y SUDAI~SHAN a n d co-workers [] 5, 541, in t h e spirit of p r e v i o u s i n t e r p r e t a t i o n s b y DIlCAC [51], ST{'TCKELBERG [52 I a n d FEY]NMAN [53]. ~Vamely, let us suppose (see Fig. 16 a)) t h a t a particle P, with n e g a t i v e e n e r g y (and, e.g., ~ charge ~>-- e), a n d t r a v e l l i n g b a c k w a r d s in t i m e , is e m i t t e d
[t0;fi~>0 ,Vv] (+~);v>0
z
ph = [ , 4 j
(t,x)
x -=
x,
x2
=
-
(x')
~?(h)
X
r-~(-q);E>0;~;pn e g a t i v e e n e r g y a n d (( charge ~ -- e, i.e. loses e n e r g y a n d (( charge ,) -~ e. Such a p h y s i c a l p h e n o m e n o n will of course a p p e a r to be n o t h i n g b u t t h e e x c h a n g e ]rom B to A of a ( s t a n d a r d ) particle Q, w i t h positi'ce e n e r g y (and (~charge ~) -[- e) a n d t r a v e l l i n g f o r w a r d in time. W e h a v e t h e r e f o r e seen t h a t Q has t h e o p p o s i t e ((charge ~) of P ; this
240
E. R E C A M I a n 4 R. M I G N A N I
means t h a t operates, among others, a charge conjugation C. inspection of ) -
A closer
F~F,
where b y E and :~ we mean the operations of energy reversal and m o m e n t u m reversal, respectively [38, 9, 54-56]. Notice that, in our terminology, ~ m e a n s conjugation of all charges (e.g. also of magnetic charge, if it exists). I n the present case of tachyons, we call Q the antiparticle of P : Q -- P
(v~ > e~),
thus rending precise (in the tachyonic case) what was defined in Subsect. 7"2. In other words, let us consider a t a c h y o n T traveUing, e.g., along the x-axis, and a continuous series of subluminul frames s, moving collinearly with o u r frame so. Let us call s o the (critical) frame in which T becomes transcendent. As we go f r o m so to s~, the t a c h y o n T appears with increasing velocities. As we by-pass s~, with a L T t h a t we shall call/~, the new frames should observe a t a e h y o n T (still with positive >[15], however, we shall change the previous s t a t e m e n t into the following one: (~As we by-pass the critical frame s~, the new frames will judge the observed particle as the antitachyon (*) T (now with negative " c h a r g e " ) , travelling in the opposite direction ~> (cf. also eqs. (34)). In fact, let us, e.g., consider the (subluminal) L T ~ ~ making transition from A to A' of Fig. ]2 c) (by the way, ~ i s t h e x-axis boost with positive relative velocity u -- u~ 2Vj ( 1 -~ V i~e ~), if V is the velocity of the considered t a c h y o n in the kinematical state A). I t is easy to see (cf. Table II) t h a t L acts kinematically on the observed tachyonic object T as the product E T a , where is the velocity-reversal operation. When we apply (~]~IP ~, we get eventually t h a t the second frame observes--as regards the t a c h y o n a n a l y s e d - - t h e same effect as produced (in the first frame) b y a mere C T operation [57],
(0~p)(k2+) -> C2, applied only to the observed object T. Such a CT-action, as shown b y Fig. 16b), does prove our previous s t a t e m e n t in quotation marks. The emerging fact that, given a particle 1), the concept of P is a purely relativistic one will be soon revisited. The previous analysis suggests us to introduce into the special-relativity t h e o r y a third postulate (besides ~he two in Sect. 3), i.e. the (~r e i n t e r p r e t a t i o n principle ~
(*) Cf. also the second of ref. [55].
241
CLASSICAL TtI]BORY OF TACHYONS
(RIP), in the form [39, 50, 23, 57]: (~Physical signals are actually transported only by positive-energy objects (i.e. b y the objects t h a t appear to us as carrying positive energy and going forward in time) ,~. The meaning of such a principle within information t h e o r y is straightforward. The (( R I P ~ can be inserted harmoniously (*) into special relativity: it is indeed necessary to the self-consistency of generalized special relativity. For example, the generalized velocity composition law, in particular the above-mentioned eqs. (34) of Sect. 5, hohl for the reinterpreted objects. The same happens, e.g., for the electric charge q; t h a t is to say, a LT making a transition between a frame/1, (( preceding ~ the critical frame s~, and a frame ]~, (~following ~ s~ (i.e. such as to (( overcome ~ the critical velocity), will automatically yield the final electric charge shown b y the reinterpreted objects. In fact, t h a t LT, being such as to invert the fourth-components' sign, will inveI~ also the sign of charge density (, and therefore of the particle (total) (.h~rge q: (47)
0 -~- ~ ,
q - > - q,
where we have defined
(47,)
q +feldVI.
This accords to the fact t h a t the above-considered LT transforms tachyons into antitachyons, and vice versa. We shall come back to the R I P in Subsect. 8"5. 7"4. Antimatter and matter. - The R I P is appropriate (*) in tile bradyonic case as well [46, 47, 8]. Namely, a particle P in the kinematical state corresponding to a point of the lower hyperboloid (Fig. 12 a)) will be shown to appear as the antiparticle P of P in the usual sense. The fact is quite interesting that, once the notion of particle is introduced (as is usually done in special relativity), merely from special relativity itself the concept of antiparticle follows [23, 46, 47, 8]. Precisely, since 1905, on the basis o] the double sign entering the relation (48)
E = ± v~p 2 ÷ m~,
the existence, /or any particle, o] its antiparticle could have been expected, provided the third postulate (RIP) had been used. Moreover, let us emphasize that---when we limit ourselves to subluminal
(*) It is indeed necessary, even in usuM special relativity, to avoid transmission of information into the past! Deprived from this (~third postulate,), standard relativity would allow sending signals into t,he. past (~ts observed, e.g., by L. ]~ANTAPPI~).
242
E. RECAMI an4
R. M I G N A N I
]rames--the clean separation between matter and antimatter is confined only to bradyons, owing to the fact t h a t the hyperboloid in Fig. 12 a) consists (*) of two disconnected sheets [9, 46]. On the contrary, in the case of tachyons, the character m a t t e r / a n t i m a t t e r is no longer absolute, but relative to the (subluminal) observer [9, 46]. However, if we consider also Superluminal/tames, since the product of two suitable SLT's m a y yield a G L T of the t y p e -- A< ~ (/ST)A density, t r a n s f o r m e d b y the G L T considered. I~ is immediate to conclude t h a t - - s i n c e - - 1 :: ~/5~/, both for b r a d y o n s and t a c h y o n s - - u n d e r the G L T (, strong reflection ~), particles (B or T) in the initial st:~te of an interaction process will be t|'ansformed into antiparticles (B or T) in thc final state of the same iuteraction process, and vice versa [9, 38, 54-581. F o r instance, the two reaetions
ad-b-->c÷d,
(5~)
~ ÷ ~ - ~ +~
are the two difi'erent descriptions of the same phenomenon us seen b y the two different inertial frames ,% and --,% (CPr[')so, respectively. B y the way, from the foregoing it follows t h a t usual symbols like /5 ~nd
244 have too
~. ~v.CAM~ and R. MIGNANI
restricted
a m e a n i n g [57]; one o u g h t o n t h e c o n t r a r y t o i n t r o d u c e s y m b o l s
m e a n i n g t h e sign i n v e r s i o n p r o d u c e d b y a G L T in
(e.g.
components
all
the tetravectors' fourth
t h e s y m b o l T ) a n d first t h r e e c o m p o n e n t s
(e.g. P),
a n d so on.
( W e a l r e a d y specified t h a t C ~ ~ m e a n s t h e sign i n v e r s i o n of all t h e a d d i t i v e (~ c h a r g e s ~.) I t h o l d s of c o u r s e t h a t
(49ter)
{ T ~ : T~"'"p~(R~IP,) .
-~ =~-~P "" '
8"2. Case o/tachyons. - L e t us veri]y e x p l i c i t l y i n t h e case of T ' s t h e c o n c l u sions a b o v e ( S u b s e c t . 8"1). F i r s t of all, f r o m T a b l e I I i t c a n b e d e d u c e d t h a t , b y m e a n s of s u i t a b l e t r a n s f o r m a t i o n s of t h e t y p e - - A < , of t h e i r k i n e m a t i c a l effects a n d R I P , w e
TABL~ II. - E]lect o] GLT's on the sign o] various ]our-vector components o/an observed object, in the case o] coil±near motion aTong the x-axis. Both subluminal (u ~ % ) and Superluminal (U ~ U~) relative velocities are considered. Analogously, both bradyons (having velocities v relative to the ]irst. unprimed frame) and taehyons (having velocities V relative to the unprimed frame) are as well considered. For simplicity, only the cases + A ( f l 2 > c2) are considered, as well as only the eases v > 0 and V > 0. Notice explicitly t h a t only x-components of v or V are effective, in this context. Lastly, for compactness' sake, it is assumed V~ > c. tce]~-
lull ~ M < c,
l u l l ~ IUI > c
rive V= > c velocity
luxl ~ lul < c,
0 < v, < c 0 < v= < c
V~ > c
u>0
u0
U0
U< v~
+
--
+
+
+
--
± for U>< V~
--
± for u > c~')) on various physical quantities' signs: see Table I I . 8"3. Case o/ luxons. - The G L T ' s m a p p i n g the u p p e r light-cone into the lower light-cone, and vice versa, arc all the (subluminal, nonm%hochronous) L T ' s of the tyl)e - A < ( f l X 0 ) and all the S L T ' s of the t y p e d-iA>(D>c) or c). Let us consider in particular the tot.d inversion ~ / 5 ~ = _ A (as neutrinos, that bear (~leptonic charge ~), since 0/55P changes the helicity sign, then a luxon (neutrino) with helicity 2 = -- 1 will be t r a n s f o r m e d into its ((antiluxon )) [46] (antineutrino) with hclicity 2 = @ 1. Cf. Fig. 12 b). I n the ease when luxons do not seem to bear a n y (fi formation (or of (( annihilation )>). Both 4escriptions are consistent with specialrelativity theory. In particular, the total electric charge Q is conserw~d during the reaction, in every /tame of reference (but it is not necessarily Lorentz invariant: see the text; in these examples, Q changes in fact, its value under the GLT considered). Notice that the total number of particles (initial ones pbts firml ones) particip:lting in the reaction is Lorcntz invariant.
I t is instructive to analyse :m explicit example (Fig. 18). L e t us consider a positively charged particle a t h a t , with respect to a first, observer 01, decays into a neutral particle c and ~mother positively (;harged particle b having in general different velocity (Fig. 18 a), where the t i m e arrow is represented too). I t is possible to find a n o t h e r observer, 02, with respect to whom e.g. the outgoing particle b behaves as an incoming antiparticlc b bearing a negative charge. Observer O~ will judge the process as a process.
252
E. RECAMIand R.
MIGNANI
Both observers, however, will agree t h a t the electric charge conservation law is verified in the observed processes. Moreover, before interaction, 01 will see one particle, while 02 sees two particles. Therefore, the very number of particles (e.g. of taehyons), at a certain instant of time, is not Lorentz invariant [16, 9]. However, the total number o/particles (e.g. of tachyons) participating in the reaction (in both initial and final states) is Lorentz invariant, due to the very features of R I P [9]. Again, we are encouraged to build the physical theory in terms of rather t h a n of , app!icd to an interaction among T's (and/or luxons), acts as in point 1), while simulCa,l~.eously transforming T's into B's (i.e. changing -- s, -- t, ... into -~ s, ~- t, ...). 4) A Superluminal boost, L = =J=iA>, applied to an interaction among B's (and/or luxons), allows transition from a certain process p either to p itself or to i) any scattering p' obtained by CT-ing [57] one or more B's with Vx > 0 and 0PS~-ing all B's with v~ ( / ? ) as the SLT from us to the rest frame of T. Notice, moreover, that m----mo for fi----~/2, consistently with Sect. 6 and Fig. 10. Let us underline t h a t the imaginary unit i entering ect. (76) is due to the SLT, and not at all to the fact t h a t tachyons must be actually attributed '~ (( pure imaginary ~>[15] proper mass (see eqs. (43)) ! One m a y well ~voi(l completely the use of any imaginary unit, as in eq. (77). The same holds of course for the other 1)roper quantities, such as proper time and proper length. For example, since (lro ~ ~ - ( 1 T ~ / l l - fl21, when passing from the rest system of the above-considered t a c h y o n T to our frame So, one h~s
(78)
dvo
i(lro
d~= + V l l _ ~ I ~/1-~
where obviously dr, d~o are both real.
dro
~/~-.L
(~>~)'
Again, it is d~----d~o for ~ = ~/2.
14"2. Generalized relativistic Newton's law. - As is well known, in the bradyonic case the f u n d a m e n t a l equation of dynamics reads [40]
(79a)
d(
d~'.~
T , ~- c (i s m o c - d ~ ]
(fl: < 1) .
:By applying a SLT (in pnrticular the (( transcendent transformation ~> K+ of Subsect. 4"3) ~o eq. (79a), we get i m m e d i a t e l y - - a c c o r d i n g to Sect. 9 - - t h e ]undamental equation o/ tachyon dynamics:
(79b)
d ( dx,\ .F, ~- --('(is m°C-ds ]
(f12> 1),
where, as expected, in the case of reetiline~r motion the '~eceleration direction is opposite to the force direction (cf. Fig. 13). In fact, tachyons must decelerate in order to incre~se their energy, and vice versa. B y remembering t h a t for tachyons d s = J= icdr0, where of course (lr0 is a G-eovariant quantity, it is immediate to write the relativistic Newton's law in G-covariant ]orm :
(80) valid for both bradyons and tachyons. Notice explicitly t h a t dx/ds is a fourvector only with respect to the group ~fl of usual LT's, whilst dx/dTo is a /ourvector with respect to the whole group G (and the latter represents tile actual, e x t e n d e d /our-velocity).
264
v.. RECAMI a n d R. M I G N A ~
:Explicit calculations have been done also b y YEZHOV [81], who, in the particular case of particle velocities along the x-axis, found (+~ ~: dv=/dt) 2'= ~
(Sl) /~ ~
dp:
m~): -- ÷ - -
d~o
for b r a d y o n s ,
tl --~:l 2
dp: d~o
--
m~): I1 --fl:l ~
for t a e h y o n s .
14"3. A n application. - The fact t h a t for a free t a c h y o n (see eqs. (42), (43)) (42b)
E ~----p~-- m~
(f12;> 1, mo real)
has some unusual consequences [55, 16]. For instance, a p r o t o n p - - w h e n absorbing, e.g., a t a e h y o n t or an antit a e h y o n t (coming from a n e a r b y emitter or from cosmic r a d i a t i o n ) - - m a y transform into itsel]: (82)
p-+-t-+p ,
p-+- t - + p ,
as can be kinematically verified in the global c.m.s. In the same frame (the global c.m.s.) it is similarly straightforward to verify that, on the contrary, the decay of a proton is not kinematically allowed. However, if we pass from the global e.m.s, to another (subluminal) inertial frame, moving collinearly, e.g., with positive speed u = u x ~ c2/V~: (where V: is the velocity x-component of ¢ or t, and where it is assumed t h a t V: ~ c), we know from Subsect. 8"5 t h a t the t a c h y o n t entering the first reaction of (82) will appear as an escaping antitachyon; and the antitachyon t entering the second reaction of (82) will appear as an outgoing tachyon. I n the new ]tame, therefore, the following reactions are kinematically allowed, according to our crossing relations, Subsect. 11"3: (83)
p --+ p~-~,
p -+ p-+-t
(only in ]light).
In other words, a proton in ]light (but not at rest!) m a y a priori [10] be seen to decay into itself plus a t a c h y o n or autitachyon (which will either be absorbed n e a r b y or proceed out to cosmic distances). B y comparing the couples of reactions (82) and (83), we can see t h a t - - s t r i c t l y s p e a k i n g - - t h e second of eqs. (83) is not the time-reversed of the first of eqs. (82), owing to the intervening kinematical boost. However, on the other hand, extended relativity requires nothing but that, for tachyonie kinematics (and not ((pure ~ T-reversibility). I n the case of tachyons, therefore, we m a y still speak of (kinematical) T-reversal, b u t with ~he above-mentioned clarification.
CL:&SSICA.L T H E O R Y
OF
TACItYONS
26~
14"4. (, Virtual particles ~>and tachyons. - Let us now briefly invade a field usually reserved to q u a n t u m mechanics, i.e. t h a t of models for elementaryparticle strong interactions, confining o u r s e l v e s - - h o w e v e r - - t o their relativistic (high-energy) beh,uviour. Since 1968, essentially within the framework of (( peripheral models ~ (e.g. one-particle exchange models), RECAMI [79] has suggested the possibility t h a t the usual ((virtual particles ))[37b] actually be considered as tachyons. In fact (in the s-channel) t h e y generally bear the negative squared four-moment u m [64~ 82,451 (8~)
p~
t < 0.
(Tile results in ref. [79] appeared also in ref. [46, 47].) Let us remember t h a t in a t w o - b o d y - t o - t w o - b o d y scattering the above q u a n t i t y t is known both to become positive and ~o change its meaning (e.g. from ~(m o m e n t u m transfer squared ~>to (( total energy squared ~>) when passing from the s-channel to the t-channel. This accords with the fact (Sect. 11"2) t h a t a SLT m a y transform a reactiou (among bradyons) into the crossed one (among tachyons). These points help clarify as well why passing through two SLT's (e.g. one SLT changing channel s for B's into channel t for T's, and one SLT only changing T's into B's, without affecting the channel) is needed for going from an interaction among bradyons to the crossed interaction, still among bradyons (cf. Sec~. 11"4). I n the previously mentioned framework of the peripheral model (with ~( ~bsorption )>[8311), a v e r y rough t e s t - - m e a n i n g f u l only within t h a t m o d e l - has been hE,de [79, 46, 47] for virtual particlc velocities~ which just supported the Superluminal velocity hypothesis. In model ref. [83a], to calculate the effect of ~,absorptive ~ chammls on the (( one-particle exchange model ~), one cuts out t,he low 1)artial wa-~-es from the Born amplitude. Precisely, an impact parameter (Fourier-Bessel) expansion of the Born amplitudes is used, and a sharI) cut-off (step function) is adopted at a radius R, which is wLried to fit the experimental data. Now, while considering--for example--different cases of pD reactions with K-meson exchange, in ref. [83a] values of R ranging from 0.9 to 1.1 fm have been found, i.e. radius values much greater t h a n the K-meson (( Compton wavelength ~>. In ref. [83cl, the previous model (with form factors, at a few GeV/e) has been also applied to pion-nueleon reactions with ~ production, via p exchange, on the basis of ref. [83b]; and analogously a value has been found even for 1he p-meson of R ~ 0.8 fro, much greater than the p Compton wavelength, h t t h a t model, therefore, one shouhl d e e m - - f r o m [84] the Heisenberg uncertainty prineil)le--that (,-,~irtual ~ K and p mesons of the proton (( c l o u d , [841 travel faster than light [851. For instance [79, 46, 471, in the 2 first case, for t - -- m K, one would find appears meaningless [96, 98], unless one simultaneously provides an improbable, suitable t h e o r y about a possible . Secondly, let us consider a particle t h a t appears to us (frame So) as a T moving at a constant velocity in vacuum. Such a particle will appear as a B with respect (e.g.) t o its rest frame S; according to frame S, therefore, the (bradyonie) differential energy loss through ~erenkov radiation in v a c u u m is obviously zero [90]: (89)
dE d--~= 0
(f12< 1 ) .
If we transform such a law b y means of the SLT from S to so (and remembering the vaeuum-covariance postulate), we get the differential energy loss through (~erenkov radiation for a t a c h y o n in v a c u u m [23, 96]: dE' (90)
(*) See references in ref. [23, 96, 97].
ds' - - 0
(/~ > 1 ) .
CLASSICAL
THEORY
OF
271
TACtIYONS
We can therefore conclude that, irr extended relativity theory, tachyons are not expected at all to emit ECR in vacuum [23, 96-98]. Moreover, a particle uniformly moving in vacuum does not emit any radiation both in the subluminal case and in the Superluminal one, as required by (extended) relativity. 16"2. The general problem. - The principle of duality, together with the tuchyonization rule, allows us to solve also the general problem of ECR from tachyons. Let us first generalize (Fig. 22) the (, drag effect ~)[27] for Superluminal velocities, i.e. extend the calculation of the apparent velocity v of light in a moving medium, with respect to a given observer 0, for tachyonic media. Figure 22 just illustrates the (extended) , N, for moving media, vs. the medium speed u, derived [23] from mere consideration of the generalized velocity composition law (Sect. 5): (91)
N(u) =-- N =
2~oC -~ U
c+Nou
(u 2 ~
c2);
the quantity v = c/5r will give the light speed in the medium considered, with respect, to observer 0. As anticipated in Subsect. 15"3, it is immediate to observe that i) the light speed v in a bradyonic medium appears to a bradyonic observer 0 always slower than c; ii) the light speed v in an hypothetical, ideal I]
where No is still the proper refraction index of the medium, and w the tachyon speed (relative to 0). In particular, for N o > l (i.e. for the vacuum and for bradyonic media), one gets (93b)
(dW)
~
o
( w 2 > e 2)
=0
"
The (~Cerenkov relation ~>, giving the cone angle in the bradyonic case [95] (94a)
1 cos0 ---- - flNo
(f12< 1),
transforms under the same SLT into the following relation, giving the cone angle in the tachyonic case: (945)
cos0 ---- ----fl No
(f12> 1).
All previous experiments [17, 99-102] looking for tachyons, or better looking for ECI~ from tachyons, failcd, being based upon a wrong assumption, originally due to SUDAICUItANhimself [15]. We have seen, on the eontrary~ t h a t tachyons do not emit ECI~ in bradyonic media.
17. - D o p p l e r effect for S u p e r l u m i n a l
astrophysical
sources.
Extended relativity allows in particular generalizing [23] the Doppler-effect formula [27]. In fi~ct, from the time interval extended transformation law
273
CLASSICAL TH:BOR¥ OF TACttYON8
(Sect. 6), it is immedi~te t,o get the Doppler-effect formula for Superluminal sources [103]. Namely, in the c~se of rel~tive motion parallel to the x-~xis, we have in b o t h subluminal and Superluminal eases [23, 103l
(95)
~
~°1 ~-/?cos~
(u~X c~),
where u ~ u, =:/3c is the relative speed and c¢ ~ ul, the vector l being directed from the observer to the source. In the particular c~se of relatiw~ motion strictly along the observation r~y, since [sign (u)[ × [sign (cos ~)] = / ~ - - corresponds to approach, ÷ corresponds to recession, we obtain the behaviour shown in Fig. 23, where the dashed curve refers to components. See also Subseet. 4" 2. (**) As usual, we shall indicate the magnetic field by H, even if it actually is a threedimensional, second-rank antisymmetric tensor.
276
E. I~CAMI and R. MIGNAN~
Of course the generalized electromagnetism must satisfy the extended relativity requirements: in particulur the duality principle (which ruled our Sect. 16). 18"1. Autodual electromagnetic Maxwell equations read [29a]
tensor. - Let us remember t h a t standard
(97) ~_P~---- O, where _P, is the dual tensor of / ~ , and Jr ~ (G,J) is the electric current density four-vector. :Notice t h a t we defined (*)[105-107]
(98)
~
_]_ 1
(a, fl, y, ~ = O, 1, 2, 3) ,
where e ~ is the real, completely antisymmetric Ricci tensor (normalized so t h a t s o a s = 1); definition (98) should not be confused with some previous ones (cf., e.g., ref. [107]). :Note t h a t
(98 bis)
G. = G~ "
Moreover, the present 1) where, in Gaussian units, H~ = E~ and Ex = H~. By adopting a di]]erent formalism, one can for example write (for f12> l)
E" ,s~T)>--H=, H;
(SLT)).E ~ ,
where now, in Gaussian units, H~ = E: and E~ : H:. These considerations do agree with the well-known invariance of the electromagnetic tensor T~ under the (~duality exchanges ,. (**) By the way, the transformation K+ works so that E~=H~, H~----E~, Es=-H~ = Ey, as heuristically forecast in ref. [18].
281
CLASSICAL THEORY OF TACHYONS
where j,(s) and g~,(S) are the subluminal electric current and the Superluminal magnetic current respectively, and so on. I f (subluminal) magnetic monopoles are assumed not to exist, t h e n eqs. (106) will read [106]
{ ~ T,,, = j,,(s) -- ij,,(S) , ~?~ = T,,~
(]07)
(v~ c~).
:Notice t h a t the q u a n t i t y T,,, which is a tensor u n d e r the s t a n d a r d Lorentz group, is no longer a usual tensor u n d e r the group G, since T under a SLT behaves like a s t a n d a r d tensor, except--however--lot a f a c t o r i. T h a t is to say, T is no longer supposed to be a s t a n d a r d tensor in eqs. (106), (107). However, we are assuming j~ Qou~, where ~o is the p r o p e r charge density and u : d x , / d v o is the G-four-velocity, so t h a t j, is a G-four-vector. L e t us now for simplicity reduce to sublmninal L T ' s : in this case eqs. (107) are equivalent to the following ones:
V. D V.B
O(s) , :
--~(S) , ~B
(]08)
VAE=j(S)--
VAH=j(s)
c~t '
+
[v2~ c~; s = subluminal; S = Superluminal],
~D
whose physical meaning is clear. All such problems will be dealt with b y the present authors in other papers, with more details and attention to the experimental consequences too. Let us here observe the following, with reference to eqs. (107). Since Superluminal has been shown. Since Superluminal (( electric charges ~>(as predicted by relativity) are enough for getting the (generalized) Maxwell equations in a completely symmetric form, the theoretical basis for expecting on the contrary existence of (snbluminal) magnetic monopoles becomes very much weaker [106]. :New experiments are needed, since almost all previous searches for tachyons were based on theoretical assumptions not consistent with (extended) special relativity. Our philosophy is synthesized in Sudarshan's known statement that (( if tachyons exist, they ought to be found. If they do not exist, we ought to be able to say why ~>. And till now no serious objection has been found against tachyon existence; on the contrary, all of relativity theory, both classical and possibly quantistic, does suggest ~heir existence, as has been shown.
CLASSIC&L TIIEORY OF TACIIYONS
2~
I n a n y case, we w a n t to stress t h a t ~he present generalization of speciM r e l a t i v i t y is interesting both in itself, and for the fact ~h~t it has already ~llowed progress in the physical u n d e r s t a n d i n g t~lso o] usual (bradyonic) matter. F o r instance, it even shed m o r e light o a the m~tter/~mtim~tter connection, it yielded a l)hysie~l meaning to ~he well-known C/~T eovari~mce and to the operation, ~md it p e r m i t t e d the d e m o n s t r a t i o n of the (~crossing relations ~ in usual elementary-particle physics.
The authors wish ~o t h a n k Profs. A. AGODI, M. BALDO, IV[. CIN[ and V. S. OLKI~OVSKY for interest a n d maI~y discussions. ThaI~ks are also due to Profs. E~ A~ALDI, C. BERNAgDINI~ ~ . CA:BIBBO~ P. CALDIROLA~ A. GIGLI~ I. F. QUERCIA~ G. SALVINI, G. SCHIFFI~ER, G. TAGLIAFEP~RI~M. VEgDE~ G. V. WATAGHIN for useful talks, ~ad to Drs. g . BALDINI~ ]~. MACORINI~ S. R0DONb for kind collaboral~iou. For the rem~ining ~cknowledgements, see ref. [23] ,~nd references therein. The authors ~re gr~teful as well to Mrs. G. GIUFFRIDA for her p~tient, perseverating tyl)ing, :rod to Mr. F. ARRIV~ for his skilful figure drawing. At last, one of the ,~uthors (E.R.) th~nks his own relatives L. ]~ECAlV[ISA~S0~I, T. RO]]EgTO RECA~[I and Dr. U. RECA:~I, who r~ised b y their lo~ns the university ~lms-p~y ~s ~ professor of physics from the Italian E d u c a t i o n Ministry (MPI), thus helping him to survive ,~nd, therefore, indirectly ~llowing the present work to be done.
REFERENCES
[1]
J.J.
[2] [3] [4] [5] [6] [7] [8] [9] [10]
O. HEAVISIDE: Electrical Papers, Vo]. 2 (London, 1892), p. 497.
[11] [12] [13] [14]
THOMSON: Phil. Mag., 28, 13 (1889).
TH. DES COUDRES: Arch. N~erland Sci. ([I), 5, 652 (1900). A. SOMMERFJ~I,D: K. Akad. Wet. Amsterdam Prec., 8, 346 (1904). A. SOM~F]~LD: Nachr. Ges. Wiss. GSttingea, Feb. 25 (1905), p. 201. A. EINSTEIN: Anat. der Phys., 17, 891 (1905). R . G . CAWL:~Y: Ann. o/ Phys., 54, 132 (1969). E. ]~ECAMI: Acead. Naz. Lit~cei Rend@. Sci., 49, 77 (Reran, 1970). M. BALDO, G. FON'n~ and E. R n c ~ I : Lett. Nuovo Cimento, 4, 241 (1970). E . C . G . SC;DARSHAN: Particles travelli~g /aster than light, preprint CPT-166 (Austin, Tex., 1972); Tachyons, prcprint NYO-3399-191/SU-1206-191 (Syr~cuse, N. Y., 1968). R . C . TOI,~AN: The Theory o/Relativity o] Motion (Berkeley, Cal., 1917), p. 54. See also D. BOHM: The Special Theory o/ Relativity (New York, N. Y., 1965). H. Scnmm': Zeits. Phys., 151, 365, 408 (1958). See a,lso E. WIGNJ~R: ref. [42]. S. T~tNA~A: Progr. Theor. Phys. (Kyoto), 24, 171 (1960). This thcory is not Lorentz-cow~rian$, nor is (e.g.) Fv,IN~:aRC,'S, ref. [16]. YA. F. Tnl~L]~TSKY: Soy. Phys. Dokl., 5, 782 (1960).
286
v.. R:ECAMIand R. MmNANI
[15]
O . M . P . BILANIUK, V. K. DESHPANIIE and E. C. G. SUDARSHAN: Am. Journ. Phys., 30, 718 (1962). G. F]~INmmG: Phys. Rev., 159, 1089 (1967). See also E. :RECAMI: Giornale eli .Fisica, 10, 195 (1969); V. S. 0LKHOVSKI and E. ]~ECAMI: NUOVO Cimento, 63 A, 814 (1969). T. ALVXG~R, J. BLOMQVIST and P. ERMANN: 1963 Annual Report o] Nobel Research Institute, Stockholm (unpublished); T. ALVXGE~, P. ERMAI~N and A. KEREK: 1965 Annual Report o/ Nobel Research Institute, Stockholm (unpublished); Nobel Institute preprint (Stockholm, 1966). See also B. M•GLIC et al.: Bull. Am. Phys. Soe., 14, 840 (1969). L. PARXER: Phys. Rev., 1BB, 2287 (1969). This paper refers to a bidimensionM space-time. E. REC-~MI and. ]{. MIGNANI: .~ett. Nuovo Cimento, 4, 144 (1972). See also E. RECAMI and. R. MIGN. at A (when u.V and its solution. 9"3. Solution of the Edmonds paradox. 9"3.1. The paradox. 9"3.2. The solution. 9"3.3. Comment. 9"4. Causality (~in micro-~ and ((in macro-physics ~. 9"5. The Bell paradox and its solution. 9"5.1. The paradox. 9"5.2. The solution; and comments. 9"6. Signals b y modulated taehyon beams: discussion of a paradox. 9"6.1. The paradox. 9"6.2. Discussion. 9"6.3. F u r t h e r comments. 9"7. On the advanced solutions. 5"5.
5"6.
CLASSICAL TACHYONS
77 78 79 80 81 82 84 84 84 85 88 89 91 91 91 92 92 92 93 95 95 97 98 99 99 101 102 104 104 105 105 106 106 109 110 111 114 115 116 118 118 120 121 124 127 127 127 128 128 130 137 138 138 141 144 147 148 149 150 151
3
1O. Taehyon classical physics (results indepcndent of the SLTs' explicit form). 10"1. Tachyon mechanics. 10"2. Gravitational interactions of tachyons. 10"3. About (derenkov radiation. 10"4. About Doppler effect. 10"5. Electromagnetism for tachyons: preliminaries. 11. Some ordinary physics in the light of ER. 11"1. Introduction. Again about CPT. 1 l"2. Again about the . 11"3. Charge conjugation and internal space-time reflection. 11"4. Crossing relations. 11"5. F u r t h e r results and remarks. PART I I I . - General relativity and ~cachyons. 12. About tachyons in general relativity (GR). 12"1. Foreword, and some bibliography. 12"2. Black-holes and taehyons. 12"2.1. Foreword. 12"2.2. Connections between BIIs and Ts. 12"2.3. On pscudo-Riemannian geometry. 12"2.4. A reformulation. 12"3. The apparent Superluminal expansions in astrophysics. 12"4. The model with a single (Superlnminal) source. 12"4.1. The model. 12"4.2. Corrections due to the curvature. 12"4.3. Comments. 12"5. The models with more t h a n one radio source. 12"5.1. The case ii). 12"5.2. The cases i) and iii). 12"6. Are
