Was the tachyon overlooked?

Volume 74A, number 3,4

PHYSICS LETTERS

12 November 1979

WAS THE TACHYON OVERLOOKED? J.K. KOWALCZYNSKI Institute of Physics, Polish Academy of Sciences, 02-668 Warsaw, Poland Received 10 July 1979

This paper describes the shape of the singularities of space—time on which the creation point of the tachyon lies. The physical interpretation of the singularities is presented., It is suggested that the tachyon might have been recorded in nuclear emulsions but that such events have been overlooked.

2(2

+ m~1)] = constant,

i~

0.

1. Let us consider the space—time characterized in the nonorthogonal frames ~ [labelled by a continuous parameter v E (—1, 1)] of real coordinates R, Z, T, by the metric:

a

ds2

space—times, one convex (ar~<0) and the other concave (ai~>0). Figs. 1 and 2 concern the case a = —1.

~,

=

+

R 2d~2+ dR2

+ ~2

~p1 {ln[q2p8(T

—

—

—

=

(1

—

u2y’12

(la)

R cos Øy8]

pwhere a = ± a(T2 1, T R2)112, ~ R ~ 0, q (T2 = Z=y(Z—uT),

dT2

—

—

p, R2)112

constant >0,

i~=

hypersurfaces p = [2]) 0 and q =essentially 0 are tangent to eachThe other (proof in ref. and singular

~ 0,

R I~I.U’PT and qit because R~LVPTRPTKRK~V space—time 0. Thus curvature these hypersurfaces tend ainvariants space—time tocan infinity essentially when thatfor surrounds psimplicity isolate 0 or the (figs. 1the and(1) 2).from Thus we assume -~

constant. (ib)

This metric is a solution of the Einstein—Maxwell equations (under the condition ~q > 0) and has been derived from solution (6) given in ref. [1] (for X 0) by the coordinate transformation Y, ?~p, q + R, Z, T which is given by the relations (ib) and the following relations (see ref. [1] )~ ‘: ~,

Y = [—(1+ a2)R cos ~+ (1

—

o2)~ —

Thus this space—time is specific for a tachyon [1]. The metric (1) represents two essentially different

4Zp1} dq2,

T=7(T—VZ), =

±exp[—2

2ioR sin 0]

-~

that the surrounding space—time is flat. The frames are also well defined in this flat space—time (as inertial frames) ~2 where we can observe the motion of the tachyon and its shock surface ~ (see fig. 2 and the text below). Consequently, there the fe’s are orthogonal laboratory frames. In the following, since we shall only be concerned with the singularities, we shall use the as laboratory frames [no singularities exist in the interior of space—time (1)]. The intersections of the hypersurface T = constant ~

with the hypersurfacesp = 0 and q = 0 form the singular surfaces ~‘ and respectively, in an appropriate ~

X [(a2—l)RcosØ-i-(l +a2)T+2a(T2

R2)~2]~

We can assume here that the argument of the logarithm in

This will be justified in detail in ref. [2] , among others by the fact that the metric (1) tends to the form R2d~2

eq. (la) is dimensionless because we can put lb expressions in ref. [1]. We have here 4t~ = b where 0 I =b1 isin the the constant that appeared in ref. [1].

2 + dZ2 — dT2 when IpI—~=. Note that we use only thedRsubluminal frames where the tachyon is always faster + than light.

*2 *1

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Volume 74A, number 3,4

PHYSICS LETTERS

~ISDTWL CAP~CP

(a)

12 November 1979

~ R

~

P~q~o

CAP~AP R

‘IWL

Tk

(b)

~

R

CAP

f5~o

(C)

Fig. 1. The singular hypersurfaces p = 0 and q = 0 in the section 4> = constant, constant + ir in various frames ~ for the convex (a), (b), and the concave (c) cases. The hypersurface p = 0 is a half-infinite light wedge and the hypersurface q = 0 is a fragment of a light cone. The section T = constant > 0 in case (a) corresponds to fig. 2a; in case (b) it corresponds to fig. 2c; and in case (c) to fig. 2f. More explanation in the text. TWL is the tachyon’s world line. SD is the surface enveloping the space region where is detectable by detectors belonging to set D (see section 2).

,

(a)

f~>o

(e)

f~
(b)

f~-o

(c)

L~o

(d)

f~
~I~f: ~

(f)

f~>o

(g)

fu~o

IT

~-~°:~ (h) f~>o

Fig. 2. Shock surface ~ in the section 4~= constant, constant + in various frames f0 for the convex (a)—(d) and the concave (e)—(h) cases. Shaded regions are curved spaces and blank exterior regions are flat spaces. ~ is finite and closed in cases (a, e), and infinite and open in the other cases. Curved spaces do not exist during T < 0 in the cases (a, b, e, f) and are created at the moment T 0 in the cp ‘s (a, e) or on the half-axes Z, i.e. V ± (b, f). In the cases (d, h) tachyons enter from infinity ~ = during T < 0 and disappear at the moment T = 0 in the ap’s producing Es’s and leaving their Ee’s; for T> 0 tachyons do not exist (c, g). More explanation in the text.

f~, ~~. The surface E~is a fragment of the sphere with radius T at the moments T >0 in each f0 (speed of light = 1), and ~~is a finite cone in certain fe’s (fig. 2a, e), in f~0(critical frame) it is a semi-infinite cylinder with radius Tat the moment T>0 (fig. 2b, 1), *3 The 2gsurfaces and Eem. Ef~and~are The nomenclature denotedused in ref. in ref. [1], [1] respectively, is unfortunate, because each surface may be formed by both fields as i.e. gravitational and electromagnetic (see appropnate discussion in ref. [2]). Note that the solution (1) represents the space—time of the charged tachyon [1].

158

in some other fr’s it is an infinite cone (fig. 2c, d, g, h). Thus the curved space enveloped by the singular surface , where = ~ U

U

U

expands in the flat space (~expands towards a curved space in the concave case). Each point of ~, moves withvertex the the velocity of the cone of light moves alongalong a normal the Z to axis with , while the —1 velocity V v (thus I J1I > 1, i.e. tachyon). Hence, is the shock surface produced by the tachyon. The surfaces ~ and ~ are tangent to each other, conse.

-

Volume 74A, number 3,4

PHYSICS LETrERS

quently ~U is smooth everywhere except at the vertex of the cone, and except at the circumference ~p ~ ~q in the concave case. The concave case is thereby hard to accept intuitively, since ~U is self-structured by the field. The problem disappears, however, in situations like that presented in fig. 3 Solution (1) invariantly determines the geometrically favoured singular space—time point R = z = T = 0. This point has the general meaning of the creation— annihilation point (CAP) of the tachyon. In certain fU’s it is a creation point (CP), in some other ~ it is an annihilation point (AP), and it is a CAP in (fig. 1). The space point R = Z = 0 will be denoted, respectively, by cp, ap, cap. The semantic meaning of this nomenclature should be evident from figs. 1 and 2. Note that cp or ap or cap is the centre of the spherical surface ~ in each ~ at any moment T>0. Thus ~ is a spherical (partially annected by ~ field signal of an event that occurred in this space point at the moment T 0 (thus in CAP). World lines of cp, ap, cap are identical with the Taxes in appropriate fU’s, thus they are different in different ~U’~• The CAP may be also looked upon as the point of interaction between the tachyon and bradyons, luxons or other tachyons [2] . The last case will be called a star of tachyons (fig. 3). The star consisting of our tachyons must contam one and only one convex space— time and may contain an arbitrary number of concave ones. The exterior singular hypersurface (surface in space) enveloping each star is everywhere convex in each All the tachyons in the star must have a cornmon CAP ~

~

2. The infinite curvature (infinite concentration of the field on the hypersurface p = 0 U q = 0 is of course

Ti

(a)

(b)

Fig. 3. A star formed by three tachyons presented in the convention of fig. 2 in two different reference frames. Tachyons Ti and T2 generate concave spaces while T3 generates a convex space. Ti and T3 move in the plane of the figure while T2 does not. C is a fragment ofa conic curve being an intersection of the plane of the figure with EP of T2. All the cones are tangent to ~q of T3. All the tachyons have a common cp [ap for Ti in case (b)] -

12 November 1979

a mathematical idealization being the consequence of the singular-field-source model in the case of solution (1) which is an exact solution of the Einstein equations without matter tensor and of the Maxwell equations without currents [1]. The situation is the same as that described by the, e.g., Levi-Civita (LC), Schwarzschild, Reissner—NordstrOm or Kerr solutions in general relativity or the laws of Newton or Coulomb in classical We can assume that a real physical system characterizes itself by finite values in each laboratory reference frame. Thus we can assume that in some physical reality the hypersurface p = 0 U q = 0 is the hypersurface of the high but finite concentration of the field (considerably higher than that inside the curved space— time) that increases towards the CAP. Let us consider a set D of identical real detectors, i.e. havingfinite sizes and sensitivities, all motionless in a certain arbitrarily chosen ~U The set D determines the surfaces SD (see fig. 1) for the space—time (1). Our physical interpretation of solution (1) implies that SD exists and is finite and closed in space. The trajectory of the tachyon intersects SD there. The detectors lying on SD need *4 Note that in the spatial representation the singularities of certain bradyonic solutions are points while the singularities of the tachyonic solution [e.g. LC LC—Reissner—Nordström solution (1)] are shock surfaces that expand with the speed of light. Their tachyonic character results from the existence of the faster-than-light vertices of shock cones [singular space-like world lines in space—time (TWL’s)]. The question now arises if the techyonic manifest themselves only through appropriately formed expanding fields. If so, then the following important question arises: do those vertices transport energy—momentum from cp’s or are they only causes extracting energy—momentum from space, e.g., from the Dirac sea (maybe only over a limited distance)?

~

~

(a)

.~::-

(b)

Fig. 4. Presumable black patches produced by single tachyons in a nuclear emulsion as viewed by an observer looking perpendicularly to the trajectories of the tachyons, (a) in the convex case, (b) in the concave case (if it can exist independently). The observer sees the projection of the space region enveloped by SD on the plane of observation. The density of grains might be larger or smaller.

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Volume 74A, number 3,4

PHYSICS LETTERS

not detect ~U at the same moment Tin any ~U (fig. 1; thus the shape shown in fig. 4a need not be identical with the shape of in fig. 2a). Simultaneous detection would occur under certain special conditions in the convex case and in one and only ~ in which among others CAP = CP. The existence of SD means that CAP has the physical meaning of CP (cp in space) in each ~ Then AP (ap in space) has a purely kinematical meaning which results from the relativity of simultaneousness in the theory of relativity (the EU’s are formal intersections for T = constant, see section 1). The kinematical meaning ofAP is that the nonphysical effect of the tachyon’s entering from infinity in some ~ (figs. ib, c and 2b, c, d, f, g, h) is unreal, The concept of CAP is crucial for our physical interpretation of solution (1). There are reasons for which solution (1) should be the simplest description of the real tachyon (if the tachyon exists at all) in general relativity [2] * ~. 3. Single grains in nuclear emulsions can be approximately treated as detectors of the set D. If EU is strong enough to blacken the grains in a volume of at least

12 November 1979

a few mean distances between the grains, then we could see pictures like those presented in fig. 4. If we look at the trajectory from a different angle the patterns would, of course, be different and generally less regular. The star of tachyons would create a much less regular black patch. It seems that it is most plausible that cp’s should coincide with the centres of stars of trajectories of charged baryons in the emulsions, or at least to be on such trajectories. However, the tachyonic black patches may appear in the blank parts of emulsions, since the tachyons could be produced by interactions between neutral particles. The analysis of the distribution of the electromagnetic field in the interior of the space— time (1) [2] shows that such a production does not necessarily violate the law of conservation of charge. The question now arises: were the finger prints of tachyons in the emulsions treated as artifacts? Would it not be worth while then to look through the stacks of old emulsions (or bubble chamber photographs) once again? References [1] J.K. Kowalczyuiski, Phys. Lett. 65A (1978) 269.

*

Solution (5)

in ref.

[1] has identical singular hypersurfaces

p = 0, q = 0, surfaces E~,CAP, CP,AP, cap, cp, and ap as solution (1).

160

[2] J.K. Kowalczyiiski, in preparation. This paper is the same as that cited in ref. [i] as ref. [4]. It is still in preparation.