Classical tachyons and possible applications

21

CLASSICAL TACYIYO~-S

5. - A model theory for tachyons: an ~ e x t e n d e d r e l a t i v i t y ~ dimensions.

(ER) in two

Till now we have not t a k e n account of tachyons. L e t us finally t a k e t h e m into consideration, starting from a model theory, i.e. from (, e x t e n d e d relativity )~ (Ell) ( M A C c ~ o ~ and I g v . c ~ , 1982a; M)~CCAR]tO~ et al., 1983; BARge et al., 1982; review I) in two dimensions.

5"1. A duality principle. - W e got from experience t h a t the invariant speed is w = c. Once an inertial f r a m e so is chosen, t h e invariant character of t h e light speed allows an c ~ m u s t i v e p a r t i t i o n of the set {/} of all inertial frames ] (of. sect. 4) into the two disjoint, c o m p l e m e n t a r y subsets {s}, {S} of the frames having speeds ]u] < c and [U[ > c relative to so, respectively. I n the following, for simplicity, we shall consider oltrselves as ((the observer so*. At the present time we neglect Em luminal frames (u--~ U = 0) as (( unphysical )). The first postula.te reqltires frames s and S to be equivalent (for such an extension of t h e criterion of (~equivalence ~> see CALDIIlOLA and IIECA3I:[, 1980; t~ECA~MI,1979a), and in particular observers S - - i f t h e y e x i s t - to have at their disposal the same physical objects (rods, clocks, nucleons, electrons, mesons, ...) as observers s. Using the language of mnltidimensionM space-times for fut~tre convenience, we can say the first two postulates to req]~ire t h a t even observers S m u s t be able to fill their space (as seen b y themselves) with a (( lattice work )) of m e t e r sticks and synchronized clocks (TAYLOR and WI-IEELEll, 1966). I t folh)ws t h a t objects must exist which are at rest relatively t~o S and faster t h ' m light relatively to frames s; this, together with the fact t h a t luxons l show the same speed to a n y observers s or S, implies t h a t the objects which are bradyons B(S) with respect to a frame S must appear as tachyons T(s) with respect to a n y frame s, and vice versa: (26)

]/(S) .... T(s),

T(S) = B(s),

[(S) = l(s).

The s t a t e m e n t t h a t the terms B, T, s, S do not h a v e an absolute, b u t only a relative meaning, and eqs. (26) constitute the so-called duality principle (OLKIIOVSKY and ]:~ECA_MI,] 971 ; 14].:CA~ and ~M_IGNA~I, 1972, 1973a; M[GNANI et al., 1972; /k~TrePA, 1972; MIGNA5I and ~]gCA~MI,1973a). This means t h a t the relative speed of two frames sl, s2 (or $1, S~) will always be smaller t h a n c; an(i t h e relative speed b e t w e e n two frames s, S will be always larger t h a n c. Moreover, t h e above exhaustive 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 to v a r y inside {s} (or inside {S}), whilst the subsets {s}, {S} get on the c o n t r a r y interchanged when we pass from % e {s} to a n y frame So e {S}. The main problem is finding oat how objects t h a t are subluminM w.r.t. ( - with respect to) observers S apt)e~r to observers s (i.e. to tls). I t is, there-

22

~. ~ E C A ~

fore, finding out t h e (Superluminal) Lorentz t r a n s f o r m a t i o n s - - i f t h e y e x i s t ~ connecting the observations made b y S with the observations b y s. 5"2. Sub- and Super-~uminal Zoren$z ~rans]ormations: preliminaries. W e neglect space-time translations, i.e. consider only restricted Lorentz transformations. All frames are supposed to h a v e the same event as their origin. L e t us also recall t h a t in the chronotopical space Bs are characterized b y timelike, is b y lightlike, and Ts b y spacelike world-lines. The ordinary, subluminal Lorentz transformations (LT) from s~ to s~, or from S~ to S~, are known to preserve the four-vector type. After subscct. 5"1, on the c o n t r a r y , it is clear t h a t the ~ Superluminal Lorentz transformations ~> (SLT) from s to S, or from S to s, must t r a n s f o r m timelike into spacelike quantities, and vice versa. W i t h assumption (25) it follows t h a t in eq. (15) the plus sign has to hold for LTs and the minus sign for SLTs: (15)

ds ' 2 = ~: ds ~

(u~l);

therefore, in (~e x t e n d e d relativity ~>(ER), with assumption (25), the quadratic form d s 2 ~-- d x . d x .

is a scalar u n d e r LTs, b u t is a pseudoscalar u n d e r SLTs. I n t h e present case, we shall write t h a t LTs are such t h a t (279)

dt ' ~ - d x ' 2 = -~ (dr 2 - dx-')

( 9 2 < 1),

dt '* -- dx '2 = -- (dr ~-- dx 2)

(u: > 1).

whilc for SLTs it m u s t be (27b)

5"3. Energy-momentum space. - Since tachyons are just usual particles w.r.t, t h e i r own rest frames ], where t h e ]'s are Superluminal w.r.t, us, t h e y will possess real rest masses mo ( l ~ c A ~ x and I'VlXG~A~I, 1972; L ~ T ~ , 1971a; PA~R, 1969). F r o m eq. (27b) applied to the e n e r g y - m o m e n t u m vector p , , one derives for free tachyons the relation (28)

E 2 -- p~ = -- ml < 0

(too real),

provided t h a t p~ is so defined to be ~ G-vector (see the following); so t h a t one

has (cf. fig. 5) (29a)

(29b) (290)

PUP"=-

+ m o 2> O

for bradyons (timelike case),

0

for luxons

--m] 1),

which can be assumed t~s the canonical ]orm of t h e SLTs in t w o dimensions. L e t us observe t h a t eqs. (39') or (39") yield for t h e speed of so w.r.t. S' dx' x--0-~----

(42)

~: --

===F U

(u~~l)'

and all of t h e m preserve the quadratic form, its sign included: d~or d~p' -~ d~ d V. The point to be emphasized is t h a t eqs. (48) in the Superha-ninal case yield directly eq. (39"), i.e. t h e y automatically include the ((reinterpretation ~) of eqs. (39). Moreover, eqs. (48) yield --

6--1

U--~-

(u2r

(49) =

qa,

o2~

O c. I n such a particular conformal mapping (inversion) the speed e is the one, and the speeds zero, infinite correst)ond to each other. This clarifies the meaning of observation ii), subsect. 3"2, b y EInSTEIn. Of. also fig. 9, which illustrates the i m p o r t a n t eq. (32). I n fact (review I), the relative speed of two (( dual ,) frames s, S (frames dual one to the other are characterized in fig. 9 b y 2~B being orthogonal to the u-~xis) is infinite; t h e figure geometrically del)icts, therefore, the circumsta.nce t h a t (so--~ S) = (so -> s). (s -> S), i.e. the f m l d a m e n t a l t h e o r e m of t h e (bidimensional) (~extended rel~tivity ~): (( The SLT: So -* S(U) is the p r o d u c t P

B1 -- C

-)

so

Fig. 9.

tt r

32

~. R ~ C A M ]

of the LT: so-->s(u), where u------1/U, by the transcenden$ SLT ~>: of. subseet. 5"5, eq. (32) (MIGNAm and RECAMX, 1973a). Even in more dimensions, we shall call > two objects (or frames) moving along the same line with speeds satisfying eq. (51): (51')

v]7 = c =,

i.e. with infinite relative speed. Let us notice that, if p , a n d / ~ , are the energym o m e n t u m vectors of the two objects, t h e n the condition of in/inite relative speed can be written in G-invariant way as

(51")

p , P , = O.

5"12. The /or tachyons. - The problem of the double sign in eq. (50) has been already taken care of in sect. 2 for the ease of bradyons (eq. (9)). Inspection of fig. 50) shows that, in the case of tachyons, it is enough a (suitable) ordinary subluminal orthochronous Lorentz transformation JLt to transform a positive-energy t a c h y o n T into a negative-energy t a c h y o n T'. For simplicity let us here confine ourselves, therefore, to transformations 15 --/~+ e~'~+, acting on free taehyons (see also, e.g., MARx, 1970). On the other hand, it is well known in SR t h a t the chronological order along a spaeelike p a t h is not ~t+-invariant. However, in the ease of Ts it is even clearer t h a n in the b r a d y o n case t h a t the same transformation /~ which inverts the energy sign will also reverse the motion direction in time (review I; R~,CxMI, 1973, 1975, 1979a; C~])n~OL~t and R~,CXM~, 1978; see also G ~ u c c I o et al., 1980). In fact, from fig. ]0 we can see t h a t for going from a positive-energy state T~ to a negative-energy state T: it is necessary to by-pass the state T~ (with V----oo). F r o m fig. 11a) we see moreover that, given in the initial frame so a t a c h y o n T travelling, e.g., along the positive x-axis with speed Vo, the 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