Some recent experiments led to the claim that something can travel faster
than light in vacuum. However, such results do not seem to place relativistic
causality in jeopardy. Actually, it is possible to solve also the known causal
paradoxes, devised for "faster than $c$" motion: even if this is not widely
recognized. Here we want to show, in detail and rigorously, how to solve the
oldest causal paradox, originally proposed by Tolman, which is the kernel of so
many further tachyon paradoxes. The key to the solution is a careful
application of {\em tachyon mechanics}, that can be unambiguously derived from
special relativity

Recami, Erasmo

English

arXiv

Some recent experiments led to the claim that something can travel faster than light in vacuum. However, such results do not seem to place relativistic causality in jeopardy. Actually, it is possible to solve also the known causal paradoxes, devised for ``faster than c" motion: even if this is not widely recognized. Here we want to show, in detail and rigorously, how to solve the oldest causal paradox, originally proposed by Tolman, which is the kernel of so many further tachyon paradoxes. The key to the solution is a careful application of {\em tachyon mechanics}, that can be unambiguously derived from special relativity

Recami, E

CERN Document Server

The Tolman \Antitelephone" Paradox: Its Solution by
Tachyon Mechanics ()
Erasmo RECAMI
Facolta di Ingegneria, Universita Statale di Bergamo, Dalmine (BG), Italy;
INFN{Sezione di Milano, Milan, Italy; and
Dept. Applied Mathematics, State University of Campinas, Campinas, S.P., Brazil.
Abstract { Some recent experiments led to the claim that something can travel faster
than light in vacuum. However, such results do not seem to place relativistic causality
in jeopardy. Actually, it is possible to solve also the known causal paradoxes, devised
for \faster than c " motion: even if this is not widely recognized. Here we want to show,
in detail and rigorously, how to solve the oldest causal paradox, originally proposed by
Tolman, which is the kernel of so many further tachyon paradoxes. The key to the solution
is a careful application of tachyon mechanics, that can be unambiguously derived from
special relativity.
PACS nos.: 03.30 ; 73.40;Gk ; 42.80.L .
() Work partially supported by CNPq (Brazil), and by INFN{Sezione di Catania, MURST and CNR
(Italy).
1
Introduction. { Some recent experiments, performed at Cologne(1), Berkeley(2) and
Florence(3) led to the claim that something can travel with a speed larger than the speed
c of light in vacuum, thus conrming some older predictions.(4) Nevertheless, such results
do not seem to place relativistic causality in jeopardy.
Actually, it is possible to solve also the known causal paradoxes, devised for \faster
than light" motion, even if this is not widely recognized.
In fact, claims exist since long(5) that all the ordinary causal paradoxes proposed for
tachyons can be solved (at least \in microphysics"(6)) on the basis of the \switching pro-
cedure" (swp) introduced by Stu¨ckelberg(7), Feynman(7) and Sudarshan(5), also known
as the reinterpretation principle: a principle which in refs.(8;9) has been given the status of
a fundamental postulate of special relativity (SR), both for bradyons [slower{than{light
particles] and for tachyons. Schwartz,(10) at last, gave the swp a formalization in which
it becomes \automatic".
However, the eectiveness of the swp and of those solutions is often overlooked, or
misunderstood. Here we want therefore to show, in detail and rigorously, how to solve the
oldest \paradox", i.e. the antitelephone one, originally proposed by Tolman(11) and then
reproposed by many authors. We shall refer to its recent formulation by Regge,(12) and
spend some care in solving it, since it is the kernel of many other paradoxes. Let us stress
that: (i) any careful solution of the tachyon causal \paradoxes" has to make recourse
to explicit calculations based on the mechanics of tachyons; (ii) such tachyon mechanics
can be unambiguously and univocally derived from SR, by referring the Superluminal
objects to the class of the ordinary, subluminal observers only (i.e., without any need of
introducing \Superluminal reference frames"); (iii) moreover, the comprehension of the
whole subject will be substantially enhanced if one will refer himself to the (subluminal,
ordinary) SR based on the whole proper Lorentz group L+  L
"
+ [ L
#
+, rather than on
its orthochronous subgroup L"+ only [see refs.(
9), and references therein]. At last, for a
modern approach to the classical theory of tachyons, reference can be made to the review
article(6) as well as to refs.(13;14).
Tachyon mechanics. { In ref.(15) the basic tachyon mechanics can be found exploited
for the processes: a) proper (or \intrinsic") emission of a tachyon T by an ordinary body
A; b) \intrinsic" absorption of a tachyon T by an ordinary body A; c) exchange of
2
a tachyon T between two ordinary bodies A and B. The word \intrinsic" refers to the
fact that those processes (emission, absorption by A) are described as they appear in the
rest-frame of A; particle T can represent both a tachyon and an antitachyon. Let us recall
the following results only.
Let us rst consider a tachyon moving with velocity V in a reference frame s0. If
we pass to a second frame s0, endowed with velocity u w.r.t. (with respect to) frame s0,
then the new observer s0 will see |instead of the initial tachyon T| an antitachyon T
travelling the opposite way in space (due to the swp), if and only if
(1) u  V > c2 :
Recall in particular that, if u V < 0, the \switching" does never come into play.
Now, let us explore some of the unusual and unexpected consequences of the trivial
fact that in the case of tachyons it is
(2) jEj = +
q
p2 −m20 (m0 real; V
2 > 1) ;
where we chose units so that, numerically, c = 1.
Let us, e.g., describe the phenomenon of \intrinsic emission" of a tachyon, as seen in
the rest frame of the emitting body: Namely, let us consider in its rest frame an ordinary
body A, with initial rest mass M , which emits a tachyon (or antitachyon) T endowed with
(real) rest mass m  m0, four-momentum p  (ET;p), and velocity V along the x-axis.
Let M 0 be the nal rest mass of body A. The four-momentum conservation requires
(3) M =
p
p2 −m2 +
p
p2 +M 0 2 (rest frame)
that is to say [V  jV j]:
(4) 2M jpj = [(m2 + )2 + 4m2M2]
1
2 ; V = [1 + 4m2M2=(m+ )2]
1
2 ;
where [calling ET  +
p
p2 −m2 ]:
3
(5)  M 0 2 −M2 = −m2 − 2MET ; (emission)
so that
(6) −M2 <   −jpj2  −m2 : (emission)
It is essential to notice that  is, of course, an invariant quantity, which in a generic
frame s writes
(7)  = −m2 − 2pP  ;
where P  is the initial four-momentum of body A w.r.t. frame s.
Notice that in the generic frame s the process of (intrinsic) emission can appear either
as a T emission or as a T absorption (due to a possible \switching") by body A. The
following theorem, however, holds:(15)
Theorem 1: << A necessary and sucient condition for a process to be a tachyon emission
in the A rest-frame (i.e., to be an intrinsic emission) is that during the process the body
A lowers its rest-mass (invariant statement!) in such a way that −M2 <   −m2. >>
Let us now describe the process of \intrinsic absorption" of a tachyon by body A; i.e.,
let us consider an ordinary body A to absorb in its rest frame a tachyon (or antitachyon)
T, travelling again with speed V along the x-direction. The four-momentum conservation
now requires
(8) M +
p
p2 −m2 =
p
p2 +M 0 2 ; (rest frame)
which corresponds to
(9)  M 0 2 −M2 = −m2 + 2MET ; (absorption)
so that
4
(10) −m2    +1 : (absorption)
In a generic frame s, the quantity  takes the invariant form
(11)  = −m2 + 2pP  :
It results in the following new theorem:
Theorem 2: << A necessary and sucient condition for a process (observed either as the
emission or as the absorption of a tachyon T by an ordinary body A) to be a tachyon ab-
sorption in the A-rest-frame |i.e., to be an intrinsic absorption| is that   −m2. >>
We now have to describe the tachyon exchange between two ordinary bodies A and B.
We have to consider the four-momentum conservation at A and at B; we need to choose
a (single) frame relative to which we describe the whole interaction; let us choose the
rest-frame of A. Let us explicitly remark, however, that |when bodies A and B exchange
one tachyon T| the tachyon mechanics is such that the \intrinsic descriptions" of the
processes at A and at B can a priori correspond to one of the following four cases(15):
8>>>>>>><>>>>>>>:
1) emission|absorption ;
2) absorption|emission ;
3) emission|emission ;
4) absorption|absorption :
(12)
Case 3) can happen, of course, only when the tachyon exchange takes place in the receding
phase (i.e., while A, B are receding from each other); case 4) can happen, by contrast,
only in the approaching phase.
Let us consider here only the particular tachyon exchanges in which we have an
\intrinsic emission" at A, and in which moreover the velocities u of B and V of T w.r.t.
5
body A are such that u  V > 1. Because of the last condition and the consequent
\switching" (cf. Eq.(1)), from the rest-frame of B one will therefore observe the flight of
an antitachyon T emitted by B and absorbed by A (the necessary condition for this to
happen, let us recall, being that A, B recede from each other).
More generally, the kinematical conditions for a tachyon to be exchangeable between
A and B can be shown to be the following:
I) Case of \intrinsic emission" at A:8><>:
if u V < 1 ; then B > −m2 (−! intrinsic absorption at B);
if u V > 1 ; then B < −m2 (−! intrinsic emission at B):
(13)
II) Case of \intrinsic absorption" at A:8><>:
if u V < 1 ; then B < −m2 (−! intrinsic emission at B);
if u V > 1 ; then B > −m2 (−! intrinsic absorption at B):
(14)
Now, let us nally pass to examine the Tolman paradox.
The paradox. { In Figs.1,2 the axes t and t0 are the world-lines of two devices A
and B, respectively, which are able to exchange tachyons and move with constant relative
speed u, [u2 < 1], along the x-axis. According to the terms of the paradox (Fig.1), device
A sends tachyon 1 to B (in other words, tachyon 1 is supposed to move forward in time
w.r.t. device A). The device B is constructed so as to send back tachyon 2 to A as soon
as it receives tachyon 1 from A. If B has to emit (in its rest-frame) tachyon 2, then 2
must move forward in time w.r.t. device B; that is to say, the world-line BA2 must have a
slope lower than the slope BA0 of the x0-axis (where BA0==x0): this means that A2 must
stay above A0. If the speed of tachyon 2 is such that A2 falls between A0 and A1, it seems
that 2 reaches A (event A2) before the emission of 1 (event A1). This appears to realize
an anti-telephone.
The solution. { First of all, since tachyon 2 moves backwards in time w.r.t. body A,
the event A2 will appear to A as the emission of an antitachyon 2. The observer \ t "
will see his own device A (able to exchange tachyons) emit successively towards B the
antitachyon 2 and the tachyon 1.
6
At this point, some supporters of the paradox (overlooking tachyon mechanics, as well
as relations (12)) would say that, well, the description put forth by the observer \t" can be
orthodox, but then the device B is no longer working according to the stated programme,
because B is no longer emitting a tachyon 2 on receipt of tachyon 1. Such a statement
would be wrong, however, since the fact that \ t " observes an \intrinsic emission" at A2
does not mean that \ t0 " will see an \intrinsic absorption" at B! On the contrary, we
are just in the case considered above, between eqs. (12) and (13): intrinsic emission by
A, at A2, with u  V 2 > c
2, where u and V 2 are the velocities of B and 2 w.r.t. body
A, respectively; so that both A and B experience an intrinsic emission (of tachyon 2 or of
antitachyon 2) in their own rest frame.
But the tacit premises underlying the \paradox" (and even the very terms in which
it was formulated) were \cheating" us ab initio. In fact, Fig.1 makes it clear that, if
u V 2 > c
2, then for tachyon 1 a fortiori u V 1 > c2, where u and V 1 are the velocities of
B and 1 w.r.t. body A. Therefore, due to the previous consequences of tachyon mechanics,
observer \t0 " will see B intrinsically emit also tachyon 1 (or, rather, antitachyon 1). In
conclusion, the proposed chain of events does not include any tachyon absorption by B
(in its rest frame).
For body B to absorb (in its own rest frame) tachyon 1, the world-line of 1 ought to
have a slope higher than the slope of the x0-axis (see Fig.2). Moreover, for body B to
emit (\intrinsically") tachyon 2, the slope of the of 2 should be lower than the x0-axis’.
In other words, when the body B, programmed to emit 2 as soon as it receives 1, does
actually do so, the event A2 does happen after A1 (cf. Fig.2), as requested by causality.
The moral. { The moral of the story is twofold: i) one should never mix the descrip-
tions (of one phenomenon) yielded by dierent observers; otherwise |even in ordinary
physics| one would immediately meet contradictions: in Fig.1, e.g., the motion direction
of 1 is assigned by A and the motion-direction of 2 is assigned by B; this is \illegal";
ii) when proposing a problem about tachyons, one must comply(5) with the rules of
tachyon mechanics(15); this is analogous to complying with the laws of ordinary physics
when formulating the text of an ordinary problem (otherwise the problem in itself will be
\wrong").
Most of the paradoxes proposed in the literature suered the above shortcomings.
7
Notice once more that, in the case of Fig.1, neither A nor B regard event A1 as the
cause of event A2 (or vice-versa). In the case of Fig.2, on the other hand, both A and B
consider event A1 to be the cause of event A2: but in this case A1 does chronologically
precede A2 according to both observers, in agreement with the relativistic covariance of
the law of retarded causality.
The author gladly acknowledges stimulating discussions with A.O. Barut, H.C. Cor-
ben, A. Gigli, P.-O. Lo¨wdin, M. Jammer, R. Mignani, E.C.G. Sudarshan and Sir Denys
Wilkinson; and thanks Professor U. Gerlach for a careful reading of the manuscript.
Figure captions:
Fig.1 { The apparent chain of events, according to the terms of the paradox.
Fig.2 { Solution of the paradox: see the text.
8
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10


The Tolman "Antitelephone" Paradox: Its Solution by Tachyon Mechanics

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