IL NUOVO CI.~IENTO

VOL. 60B, N. 1

11 h~ovembre 1980

Gravity and the Tachyon Corridor (')("). L. MA~CHILDO~" D~partement de Physique, Universit~ du Qudbec - Trois-Rivi~res, Qudbec, Canada, G9A-5H7 (rieevuto il 22 Agosto 1980)

- - We investigate how to incorporate the tachyon corridor, that is a preferred spatial direction, in space-time described by a Robertson-Walker metric. We also look at the effects of local gravitational fields on the corridor. The requirement of avoiding causal loops allows us to reach conclusions rather independent of any specific model of the corridor. Summary.

1. -

Introduction.

I t seems now clearer and clearer t h a t one cannot incorporate t a c h y o n s and superluminal frames in the framework of current theoretical physics unless one is prepared to accept at least some i m p o r t a n t conceptual changes. I n a sense, the various approaches to t a c h y o n s and/or superluminal frames t h a t have already been p u t f o r t h can be looked at from the point of view of which substantial modification to previously held ideas is m a d e in each case. The original approach, pioneered b y BILANIUK, DESHPA]~'DE and SUDA~SrrAN (1), considered t a c h y o n s within the framework of special relativity. T a c h y o n s then h a d imaginary rest masses but, since no superluminal reference frames were introduced, this did not cause problems. On the other hand, to m a n y people it seemed unavoidable t h a t such tachyons with unconstrained spatial

(*) To speed up publication, the author of this paper has agreed to not receive the proofs for correction. (**) Work supported in part by the Natural Science and Engineering Research Council of Canada. (1) O. -~'[. P. BILANIUK,V. K. ].)EStIPANDE and F. C. G. SUDARSHAN':Am. J. Phys., 30, 718 (1962); O. 3[. BILANIUKand E. C. G. SUDA~SlrAN: Phys. Today, 22, 43 (1969). 55

~

L. MARCI[ILDON

motions would lead to causal paradoxes, n o t w i t h s t a n d i n g the so-called reinterpretation principle (2). A n o t h e r approach, especially a d v o c a t e d b y Ir and MIG>'A>'r (3), introduces superhuninal frames related to o r d i n a r y subluminal ones b y complex transformations. An extended principle of relativity is t h e n postulated with respect to the set of all inertial frames and t a c h y o n s are ordinary particles as seen b y supcrluminal observers. The main difficulty there lies in the interpretation of the resulting complex co-ordinates a n d vector or tensor fields (4). Still other (or related) approaches invoh-e the use of a sixdimensional, rather t h a n four-dimensional, space-time (5), or of t r a n s f o r m a t i o n laws depending on the p h e n o m e n a being observed (e), to name only a few. I f one wants to introduce, in four-dimensional space-tiine, superluminal frames so t h a t co-ordinate transformations between inertial frames are always one t o one, linear and real, one has to take into account to following result, due to Go~I>'I (7). I n (~ q-1)-dimensional space-time ( n > 3 ) , there are no generalizations of the Lorcntz group (as a group of transformations between equivalent frames) compatible with space isotropy. A n a p p r o a c h along these lines has in fact been p u t f o r t h by ANTtPeA and EVERETT (s). At the outset, space isotropy is broken t h r o u g h the introduction of a preferred direction in space, called the t a c h y o n corridor (~). The set of ~11 inertial frames then includes the preferred ones, whose relative (subluminal of superluminal) velocities are directed along the t a c h y o n corridor and whose co-ordinate tran~sformations were given in ref. (8). The nonpreferred frames, subluminal or superluminal, are then related to the corresponding preferred ones b y ordinary L o r c n t z transformations. The co-ordiuate transformations between a n y two frames (preferred or nonprcferred) are then one to one, linear and real, t a c h y o n s are ordinary particles in superluminal frames, there are no causal loops, b u t space isotropy is broken in processes involving real or virtual tachyons. Once a kinematical framework such as this one is set, it becomes possible (2) O.M. ]~ILANIUI(et al.: Phys. Today, 22, 47 (1969); W. B. ROI,NICK: Phys. Rec., 183, 1105 (1969); P. L. CsoxKa: Nucl. Phys. B, 21, 436 (1970); R. G. NEWTON: Science, 167, 1569 (1970) ; F. A. E. PIICANI: Phys. Rec. D, 1, 3224 (1970) ; G.A. BEN~'ORD, 1). L. BOOK and W. A. NEWCOMB: Phys. Rec. D, 2, 263 (1970). (3) E. RECAMIand R. MIC,.~'A.xI:Rie. Yuoco Cime~lto, 4, 209 (1974). See also references therein. (1) H. C. COJCBEX: .Lett. Nuoco Cime~do, 11, 533 (1974); Nuovo Cimento A, 29, 415 (1975); E. RECAMIand G. D. MACCARI{ONE:Left. N~wvo Cimento, 28, 151 (1980). (s) E. RECA)~I and R. MIG~ANI: Lett. Yuoco Cimento, 16, 449 (1976); P. ])E~IERS: Can. J. Phys., 53, 1687 (1975); E. A. B. COLE: Nuovo Cime~lto A, 40, 171 (1977). (a) R. GOrA)ONI:Letg. 2Vuoco Cimento, 5, 495 (1972); ~uovo Cimento A, 14, 501 (1973). (7) V. GoRI.~I: Commu~,. Math. Phys., 21, 150 (1971). (s) A . F . ANTIPPAand A. E. EVERETT: l)hys. Rev. D, 8, 2352 (1973); ,k. F. A_NTIPPA: Phys. Rec. D, 11, 724 (1975). (9) EVEI~ETT has also considered a class of theories which involve preferred frames and in which rotational invariance remains, but Lorentz invariancc is broken. See A. E. ]':Vs Phys. Rec. D, 13, 785, 795 (1976).

GKAVITY

AND T I I E T A C I I Y O N C O R R I D O R

57

to investigate questions of t a c h y o n b e h a v i o u r and to f o r n m l a t e r a t h e r definite hypotheses concerning t a c h y o n - b r a d y o n interactions. W o r k along these lines has already been done for electromagnetic processes a n d interactions, as observed in a preferred frame. R(fference (no) dealt with electrodynamics i a a Minkowskian space-time, t h a t is in the absence of a n y significant gravitational field. There are m a n y reasons why, in the present approach, it m a y be i m p o r t a n t to t a k e g r a v i t a t i o n a l effects into ,~ccount in t~chyon processes (~). I n t h e large, the structure of space-time is believed to be d e t e r m i n e d b y gravity, according to field equations like those of general relativity. ~Now the t a c h y o n corridor is clearly an object of cosmological signiticance, for the preferred direction should be defined in each a n d every region of space-time. The following question t h e n arises: is there a natural (or, at least, a not too unnatural) w a y of incorporating the concept of a iixed direction in space within realistic cosmological models',~ This p r o b l e m will be investigated in sect. 2, t h e discussion there being restricted to universes with a R o b e r t s o n - W a l k e r metric. Our universe is p r o b a b l y v e r y nearly homogeneous and isotropic, but t h e f a c t remains t h a t there are here a n d there (( small ,) inhomogeneities, like black holes or clusters of galaxies. I t will be interesting to consider the k i n d of effects t h a t :~ local gravitational fiehl can h a v e on the t a c h y o n corridor. This we will do in sect. 3 and we will see, in fact, t h a t the r e q u i r e m e n t of a.voiding causal loops p r e v e n t s the corridor from being m u c h inituenced b y local field configurations. This will imply t h a t , within the present framework, the principle of equivalence i~1 its usual f o r m cannot hold for t a c h y o n processe.~.

2. - T h e t a c h y o n corridor in a R o b e r t s o n - W a l k e r m e t r i c .

At the m o m e n t the m o s t popular models i~ cosmology are the ones based on the so-called cosmological principle, which states t h a t to a v e r y good app r o x i m a t i o n the Universe is spatially homogeneous ~nd isotropic. Such models are described b y a l~obertson-Walker metric which, in t e r m s of co-ordinates x*' = (t, r) ---- (t, x'"), takes the form (~) (1)

(lT 2 ~: - - g ~ v ( l x ' d x " ~ (lt2--R2(t)

{

dr 2 ~

k-'!r" 'lr)-Zl ] - - k r '~] "

(:0) L. ~IARCIIII.DON,~k. ]~. EVERETT aztd A. F. :kNTII'FA: NUOUO Ci~Jte~to B, 53, 253 (1979). (la) Effects of tachyons t~s sources oI the gravitalional tield will not bc dealt with here. Within a. different framework they hav(~ been coz~sidcred by A. PERES: Phys. Lett. A , 31, 36l (1970); L. S. Scnun~A.x': Nuoco (Yime~to B, 2, 38 (1971); P. C. VAII)YA: CIt~'r. Sci., 40, 65l (1971); J. C. FOSTER jr. and J. R. RAY: J. Math. 1)hys. (Y. t".), 13, 979 (1972); J. R. GOTT I I I : Nuoco Cimento B, 22, 49 (1974). (~z) See, for iusta.nce, S. WEIXBERG: Gravitation a~d Cosmology (New York, N. Y., 1972).

~

L. B~iAt~CHILDON

I t e r e k is a fixed number, which can be either 0 or :[:1. The function/~(t) is d e t e r m i n e d b y the m a t t e r content of the Universe t h r o u g h the field equations. So-called F r i e d m a n n models result when one chooses to work with t h e field equations of general relativity. I n a R o b e r t s o n - W a l k e r universe, galaxies (apart f r o m their r a n d o m motions) h a v e fixed spatial co-ordinates a n d at t i m e t t h e distance between t w o nearly galaxies is p r o p o r t i o n a l to R(t). Threedimensional hypersurfaces of constant t i m e are finite a n d positively c u r v e d if k ---- i (in which case t h e y are surfaces of four-dimensional hyperspheres) a n d infinite otherwise. T h e y are flat if k ---- 0 a n d n e g a t i v e l y curved if k ---- - - 1. T h e p r o b l e m we w a n t t o address ourselves to is t h e following one: h o w can a n object like the t a c h y o n corridor be fitted in such universes? L e t us r e m a r k at t h e outset t h a t an object of this t y p e , t h a t is a fixed direction in space, is r a t h e r extraneous t o a R o b e r t s o n - W a l k e r metric, which is spatially isotropic. This means t h a t , in t h e models we are presently considering, the t a c h y o n corridor is not d e t e r m i n e d b y the g r a v i t a t i o n a l field. I t is not our p u r p o s e here to deal in detail with the k i n d of fields or m e c h a n i s m t h a t m i g h t give rise to a preferred direction in space. The g r a v i t a t i o n a l field, however, e v e n t h o u g h it does not single out a preferred direction, does determine t h e overall s t r u c t u r e of space-time, in which the t a c h y o n corridor is to be fitted. H o w can t h e latter be i n c o r p o r a t e d is w h a t we w a n t to investigate. I n space-time described b y a R o b e r t s o n - W a l k e r metric, all spatial points at a given t i m e t are eqnivalent. This means t h a t there exist isometries ( t h a t is co-ordinate t r a n s f o r m a t i o n s t h a t leave the functional f o r m of the m e t r i c unchanged) which m a p a n y given point into a n y other point. These isometries are, in fact, given explicitly b y (12) (2)

t':t,

r'-~r~-a

(1--kr~)~--[1--(1--ka~) 89

r 9a}

- ,

where a is a n y constant triplet (13). The R o b e r t s o n - W a l k e r metric is also m a n i f e s t l y isotropic around t h e origin and, since all points are equivalent, it is isotropic around a n y point. W i t h o u t loss of generality t h e n we can say t h a t a t a given t i m e t the t a c h y o n corridor at r ---- 0 is along t h e x-axis. Thus space isotropy is broken. L e t us focus our a t t e n t i o n to spaces which are either positively or negatively curved, t h a t is for which k - - - - • One can ask: is it possible to fit the t a c h y o n corridor everywhere s m o o t h l y on a three-dimensional hypersurface of c o n s t a n t time, in such a w a y t h a t space homogeneity, if not space isotropy, is retained? The answer to this question is no, for t h e following reason. To keep t r a c k of the corridor direction, one can associate with each spatial point a t h r e e - v e c t o r Vi(r), defined up to a (not necessarily constant) multiplicative factor. The (~3) [ a [ < 1 if k = 1.

G R & V I T Y A N D T I l E TACttYON CORRIDOIr

59

c o n d i t i o n for t h e c o r r i d o r d i r e c t i o n t o p r e s e r v e s p a c e h o m o g e n e i t y is t h e n t h a t t h e L i e d e r i v a t i v e of V ~ w i t h r e s p e c t t o t h e c o - o r d i n a t e t r a n s f o r m a t i o n s g i v e n in (2) v a n i s h e v e r y w h e r e . E q u i v a l e n t l y , one m u s t h a v e (3)

V " ( r ) - - V~(r),

w h i c h for i n f i n i t e s i m a l a gives

0 = ~x---7

1--kr

2

1--kr

2

"

S i n c e aJ is a r b i t r a r y , t h i s i m p l i e s t h a t ?.V i kx, V z o - - ~x~ ~ ~ - - k r ~ ~ '

(~)

w h i c h h a s n o s o l u t i o n s o t h e r t h a n V ~ = 0. T h e u p s h o t is t h a t t h e r e is no w a y t o i n t r o d u c e a fixed d i r e c t i o n e v e r y w h e r e ill t h r e e - s p a c e so t h a t s p a c e h o m o g e n e i t y is p r e s e r v e d (~). G r a n t e d t h i s r e s u l t , i t is n e v e r t h e l e s s p o s s i b l e t o i n t r o d u c e t h e t a c h y o n c o r r i d o r in a r a t h e r s i m p l e w a y . I t c a n b e m a d e t o follow g e o d e s i c s of t h e t h r e e - d i m e n s i o n a l h y p e r s u r f a c e s of c o n s t a n t t i m e . O n t h e s e h y p e r s u r f a c e s t h e m e t r i c t e n s o r a n d Cristoffel s y m b o l a r e g i v e n b y

(5)

g,--

(6)

kx~xJ 1

R~(t) ~, ~ l ~ - r . ~ / , F Jk ~ --kR-~-(t)x~gjk.

C h o o s i n g t h e p r o p e r d i s t a n c e as p a r a m e t e r , one c a n w r i t e t h e g e o d e s i c e q u a t i o n s as (7)

0-

d2x ~ d x j d x ~ __ (12x ~ k ~(ter -f-Fj~da d~ da 2 + ~

xi.

N o w "tt r = 0 t h e t a c h y o n c o r r i d o r is a l o n g t h e x - a x i s , a n d t h e x - a x i s is e a s i l y seen t o b e a geodesic. T h u s , at. all p o i n t s for w h i c h y ~ 0 ---- z, we t a k e t h e x - a x i s a.s t h e d i r e c t i o n of t h e t a c h y o n c o r r i d o r . W e t h e ~ use eqs. (2) w i t h

04) One might argue that homogeneity could be preserved b y merely requiring t h a t V'~(r) ~_A~j(a)VJ(r) with Aii a translation-dependent rotation matrix. If such were the ease, however, those matrices would have to represent the group of isometrics of the space, which is S O 4 when k - 1 and S03,1 when k = - - l . B u t this is impossible, since these groups (over the real numbers) have no normal subgroups.

~0

L. MAI~CtIILDON

--~ (0, a~, a3) to transport the x-axis perpendicularly to itself, t h e r e b y determining the corridor direction everywhere in three-space. I n this w a y the point r ~ (x, 0, 0) goes over into r ' = (x', y', z'), where a

x': x,

y'~-- a2(1 -- kx2) ~ ,

z'~-- a3(i -- kx-~)t ,

and the line y----0 ~ z, i.e. the x-axis, becomes (y,)2

al

§ k(x')~ = 1 ,

(z,)~

a]

+ k(x')~ = 1 .

D r o p p i n g the primes, one can parametrize this as

(Sa)

x = sin~,

y = a2 cos~,

z ~-- a3 cos

for k-----1 and as (8b)

(~ x = sinh ~ , a%

y :

(~ a2 cosh ~ , .t~

z

a3 cosh

(~ A-g

for 1r 1. I t is easily seen t h a t (8a) and (8b) satisfy the geodesic equations (7), for k--~ Jr 1 and - - 1 , respectively. This was to be expected, since eqs. (8) were obtained b y transforming the x-axis b y means of isometrics, and isometrics preserve geodesics. E q u a t i o n s (8a) and (Sb) each represent a t w o - p a r a m e t e r family of lines which fin the three-dimensional hypersurfaces of constant time and give everywhere at a n y time t the direction of the t a c h y o n corridor. Clearly, this is not the only construction t h a t will endow the three-dimensional hypersurfaces with a preferred direction defined everywhere. I n the last analysis, the global structure of the corridor will depend on the precise nature of the m e c h a n i s m t h a t gives rise to it. The construction we have exhibited is p r o b a b l y the simplest one. The corridor direction follows three-geodesics, which in fact correspond to infinite-velocity t a e h y o n four-geodesics. Space homogeneity is preserved in planes of constant x, but, according to the result t h a t it c a n n o t hold everywhere, breaks down from one plane of this kind to the other. The t a c h y o n corridor was originally introduced to extend to three space dimensions, in a consistent way, a previously proposed one-dimensional t h e o r y of t a c h y o n s and superluminal frames (15). I t was t h e n shown that, in the resulting three-dimensional theory, no causality problems could occur (s). These prob-

(15) A. F. ANTIPPA and A. E. EVERETT: Phys. Rev. D, 4, 2198 (1971); A. F. A~TIePA: .Nuovo Cimento A, 10, 389 (1972).

(~ICAVITY A N D T H E T A C ] I Y O N COICRIDOI~.

~1

lems show up when tachyons call t r a n s m i t information both forward and backward in time and in any spatial direction. With the postulate of the t a c h y o n corridor, however, one of these conditions does not hoht. Indeed the fact that bradyons can only transmit information along the positive time direction was shown to i m p l y t h a t tachyons can only do so along the positive direction of the corridor. This in t u r n implies t h a t causal loops c a n n o t occur iu Minkowski space, where in a preferred frame the corridor everywhere follows the x-axis. Let us now see what happens in :Robertson-Walker spaces, given the model we have proposed for the corridor (~). W h e n k =: 1, t h a t is in closed :Robertson-Walker spaces, the construction we have given will avoid causal loops. There the three-dimensional hypersurfaces of constant time are, in fact, surfaces of four-dimensional spheres. The t a c h y o n corridor follows the geodesics of these surfaces. I u order to get an intuitive feeling for what happeus, it is helpful to look at the analogous case in one less dimension, namely the surface of an ordinary three-dimensional sphere. Accordingly let r = 0 correspond to the point on the equator with zero longitude and let the x-axis be the Greenwich meridian. The t a c h y o n corridor then follows meridians everywhere and one can convince oneself t h a t this, like in ~linkowski space, prevents information from being sent to one's own past. The construction for closed spaces, however, exhibits singularities at b o t h the n o r t h and south poles. There the direction of the corridor is not defined b y meridians. I n fact, a t a c h y o n following a meridian n o r t h w a r d from the equator would have to stop sharply at the pole, because the corridor does not go southward. I f one tries to r e m e d y the situation b y letting the meridians (i.e. the corridor) go n o r t h w a r d on the eastern hemisphere a n d s o u t h w a r d on the western hemisphere, then each such line is continuous a n d unbroken. B u t causal loops immediately reappear, since the corridor t h e n closes on itself. There seems to be no way of avoiding causal loops in closed spaces without introducing sharp singularities. When k =1, however, the three-dimensional hypersurfaces of constant time are open. There the t a c h y o n corridor avoids causal loops without the necessary appearance of singularities. This perhaps makes these spaces more natural candidates for the introduction of a t a e h y o n corridor. Up to now we have dealt with R o b e r t s o n - W a l k e r spaces for which k is either ~- 1 or -- 1. The case in which k ----- 0 is m u c h simpler. Three-dimensional hypersrurfaces of constant time are then flat and the corridor direction can everywhere be chosen as parallel to t h e x-axis. Space h o m o g e n e i t y is preserved, which is allowed b y eq. (4) when k ~ 0. Causal loops are forbidden. The case in which the function R(t) is given b y exp [Ht] is especially interesting,

(16) For a very different approach to the causal behaviour of tachyons iu RobertsonWalker spaces, see R. SmAL and A. SI~r.A.~Y: Phys. Rev. D, 10, 2358 (1974).

62

L. M.a~]~CHILDON

for the metric given b y eq. (1) t h e n describes a de Sitter space. These spaces are isometric with respect to t i m e translations as well and t h e y c o r r e s p o n d to s t e a d y - s t a t e models of t h e Universe. The construction we h a v e g i v e n for the t a c h y o n corridor preserves the t i m e translation isometry.

3. -

The tachyon

corridor near

a large mass.

I t is believed t h a t in the large our universe is v e r y nearly h o m o g e n e o u s a n d isotropic. Hence, to a good a p p r o x i m a t i o n , it can be described b y a Robertson-~Valker metric. On a smaller scale, however, inhomogeneities do appear, t h e largest ones p r o b a b l y being clusters of galaxies. H o w will the t a c h y o n corridor b e h a v e in the vicinity of a massive b o d y ? Again no precise a n s w e r can be given to this question unless one knows t h e details of the m e c h a n i s m t h a t establishes t h e preferred direction and, as we h a v e seen, this m e c h a n i s m cannot be purely g r a v i t a t i o n a l However, model-independent a r g u m e n t s can be used to p u t significant restrictions on the corridor. W e will consider a spherically s y m m e t r i c black hole at rest iu a Minkowskian background, i.e. space-time endowed with a Schwarzschild metric. F o r our purposes, this is almost t h e same as a black hole in a R o b e r t s o n - W a l k e r b a c k g r o u n d . This comes f r o m the fact t h a t the radius of t h e Universe is v e r y large c o m p a r e d with dimensions over which the field of a normal-size black hole is appreciable. I n regions far f r o m t h e black hole, one would like the t a c h y o n corridor n o t to be p e r t u r b e d significantly b y the presence of the hole. I n h!inkowskian space-time, the corridor follows t h e x-axis a n d fines parallel to it. Thus, in t h e presence of the hole, one would like the corridor to coincide m o r e or less with lines of constant y a n d z in those regions where t h e metric is essentially ~Iinkowskian. This r e q u i r e m e n t not only seems natural, b u t one can h a r d l y see h o w one would go w i t h o u t it. I f black holes are to m o d i f y substantially t h e direction of the corridor even at large distances f r o m t h e m , the specter of causal loops will again recur. I n Minkowskian space-time, lines parallel to the x-axis coincide with infinite-velocity t a c h y o n geodesics. Hence the corridor everywhere follows t h e s e geodesics. W e h a v e seen t h a t this last s t a t e m e n t also applies to the m o d e l we h a v e given for the t a c h y o n corridor in a l~obertson-Walker space-time. I n the presence of a black hole, however, the corridor can no longer be m a d e to follow infinite-velocity t a c h y o n geodesics. As shown in the appendix, t h e s e geodesics, originating at infinity, m a y revolve a r o u n d t h e black hole as m a n y times as one wishes, p r o v i d e d t h e y come close enough to it. T h e y e v e n t u a l l y again wind up to infinity, b u t in a direction in general completely different f r o m the original one. B y a suitable a d j u s t m e n t of the i m p a c t p a r a m e t e r ,

GRAVITY AND T H E TACItYON CORRIDOR

63

infinite-velocity geodesics can be m a d e to end up in a direction exactly parallel and opposite to the one t h e y came from. Causal loops, therefore, r e a p p e a r if the t a c h y o n corridor is defined b y t h e m . The u p s h o t of the foregoing a r g u m e n t is t h a t , in the presence of a black hole, the t a c h y o n corridor cannot be m a d e to follow infinite-velocity geodesics. I t turns out t h a t other geodesics will not do either. This m e a n s t h a t to an observer lying close to a black hole in a freely falling frame, t h e corridor will look curved. As far ~s t ~ e h y o n processes ~re concerned, t h e n g freely f~lling f r a m e near a black hole is not equivalent to an unaccelerated f r a m e in t h e absence of ~ gravitational field. The a b o v e - m e n t i o n e d conclusion can also be illustrated in ~ r a t h e r direct way. W e h a v e seen t h a t infinite-velocity t a e h y o n geodesics can originate at infinity, go r o u n d ~ black hole a n d come b a c k in the opposite direction. T h e same is true also of some smaller-velocity geodesics. E q u a t i o n (A.5') tells us t h a t this m o t i o n can go b o t h forwgrd ~nd b a c k w a r d in time. T h a t is, in the absence of ~dditional constraints, t h e relative sign of dt a n d dv can be either positive or negative. Clearly t h e n test t a c h y o n s e~nnot follow geodesics, for otherwise ~n observer could use t h e m to t r a n s m i t information to his own p~st. I n a freely falling f r a m e near a b l a c k hole, t a c h y o n trgjectories (not only the t a c h y o n corridor) look c~rved. The general picture we are led to is t h e n the following. I n ~ R o b e r t s o m W~lker (or Minkowskian) b a c k g r o u n d , inhomogeneities will not substantially affect the t a c h y o n corridor. T h a t is t h e presence of large local gravitational fields in an otherwise homogeneous a n d isotropie universe will leave t h e configuration of the corridor more or less unchanged. I n a sense, this picture goes along ~ a c h i ~ n lines. For, if the corridor results f r o m some unisotropic cosmic field, the beh~viour of this field at ~ given point m u s t depend on t h e overall distribution of m a t t e r in the Universe. I f one knows the laws of b r a d y o n physics in Minkowski space, t h e n t h e principle of equivalence tells one h o w to write down t h e corresponding l~ws in ~n a r b i t r a r y gravitational field. N~mely, one should first work in a freely falling frame, ~pply t h e k n o w n laws there ~nd t h e n t r a n s f o r m b~ck to t h e accelerated f r a m e locally equivalent to the g r a v i t a t i o n a l field. F o r t a c h y o n processes, on t h e other hand, knowledge of physic~l l~ws in Minkowski sp~ce, w h e n t h e r e is no gravitational field ~nd t h e corridor is along straight lines, is not sufficient to write down the corresponding laws in an a r b i t r a r y g r a v i t a t i o n a l field. I n d e e d the freely falling f r a m e s defined in ~n a r b i t r a r y g r a v i t a t i o n a l field exhibit ~ curved t ~ c h y o n corridor a n d this is not e q u i v a l e n t to Minkowski space with a straight corridor. I f the t a e h y o n corridor originates f r o m some cosmic field, t h e knowledge of b o t h this field and the gravitational field will be necessary to write down equations of t a c h y o n processes. One could, however, still envisage ~ kind of equivulence principle in the sense t h a t processes in the absence of g r a v i t a t i o n b u t

64

with ,t curved with a straight the cosmic field a viable t h e o r y

L. M A I t C H I L D O N

corridor would be related to in'ocesses in a gravitational field corridor b y a general co-ordinate transformation. At a n y rate, would h a v e to act on tachyons alone, not on bradyons. W h e t h e r of the sort can be developed is at the m o m e n t unknown.

APPENDIX

Tachyon geodesics in a Schwarzschild metric. W e consider s p a c e - t i m e w i t h a Schwarzschild m e t r i c (17) a n d use spherical p o l a r co-ordinates. I n t h e h y p e r p l a n e 0 ~ ~t/2, t h e geodesic equations can be w r i t t e n as (12) (A.t)

dz

(A.2)

d--vd[ l n d- t - ~Ol n B~ } r .

(A.3)

dr'

2-B ~,dz] - - r B

+

\d~/

T

~d~/

= 0,

where B ~ B(r) :

1 --

2MG r

,

dB

,

B F ~

- -

dr

The first a n d secoad equations arc i m m e d i a t e l y i n t e g r a t e d once to give (A.4)

r 2 d~ - - J = c o n s t ,

(A.5)

dt B ~ = const.

L e t t h e t a c h y o n corridor a t infinity be directed along the x-axis a n d motionless with respect to the origin of co-ordinates. F u r t h e r m o r e , let u be t h e veloci t y a t infinity of an incoming t e s t tachyon. F r o m ref. (lo) we t h e n h a v e a t infinity (u, ---- 0) d~

=

# dt {u~ - -

"

u,-l?,

s-

u~

I~,1'

(~7) This result was obtained ia ref. (~s). (~8) Aspects of this problem were treated in the framework of a space-isotropie tachyon theory by R. 0. HETTEL and T. M. HELLIWELL: -~UOVO Cimento B , 13, 82 (1973).

GRAVITY AND TKE

T,kCI[YON C O R R I D O R

65

9~nd thus eq. (A.5) can be r e w r i t t e n a.s dt

(A.5')

2

2

B ~K~ = ~ { u . -

u ~ - - 1}

-~

.

Now, s u b s t i t u t i n g (A.4) a n d (A.5') in (A.3) a n d i n t e g r a t i n g once, we obtain

(h.6)

E == c o n s t .

B u t , at infinity, dr

B(r) =: 1 ,

dv

m

~l~l{uS- u ~ - 1}-~ 2

a n d thus (A.6) becomes

(A.6,)

[d'T _

u~-- I -~ B(r) [u"=~ u~ ~ i

~

"

L e t us now specialize to the case in which u~ = 0, t h a t is in which the motion of t h e t e s t t a c h y o n is originally along the corridor. Making use of eqs. (A.4) a n d (A.6'), we get t h e following differential e q u a t i o n for t h e orbit: j2

dr "

]

u~:- 1 +

1

2

G

i

"}

--ru

.

W e let now r = J o , 2 M G --= a J a n d specialize f u r t h e r to the case in which u.. = c o

a n d :r < 1. We o b t a i n

(,.7)

tdQt \d~]

---- 0 ( e - -

~)(Q2_

1).

The orbit described b y eq. (A.7) consists in o s t a r t i n g out at infinity, decreasing m o n o t o n i c a l l y as a function of ~ till it reaches the value 1 a n d t h e n increasing ~o infinity. T h u s the smallest value of r is equal to J , which coincides with the i m p a c t p a r a m e t e r . The deflection of t h e t e s t t a e h y o n b y the scattering centre is g i v e n b y t h e t o t a l change in ~ m i n u s ~. Explicitly, co

1

F o r :c << 1 (i.e. w h e n t h e i m p a c t para.mcter is m u c h larger t h a n the Schwarzschild radius), t h e deflection is equal to a (17). F o r a close to 1, however, the deflection diverges. The upshot is t h a t infinite-velocity t a c h y o n geodesics will circle a r o u n d a black hole as m a n y t i m e s as one wants, p r o v i d e d the i m p a c t para,metcr is close enough to the Schwarzschild radius. 5 -

ll

N uovo U i m e n t o

]3.

66

L. )IARCHILDON

L e t us n o w l o o k at r a d i a l geodesics a n d see w h a t t h e t i m e d e p e n d e n c e of t h e r a d i a l c o - o r d i n a t e is. L e t t i n g u~ ---- 0 ---- J a n d c o m b i n i n g eqs. (A.5') a n d (A.6')~ we o b t a i n (A.8)

dr : idt

B ( 1 q- B ( u ~ - - 1)}+

L e t 0 < ~ 1 < ~24<2MG a n d let u ~ < < 2 M G . T h e t i m e it t a k e s for a t e s t t a c h y o n t o go f r o m rl ---- 2 M G q- ~ t o r2 ~-- 2 M G q- ~ is g i v e n b y i n t e g r a t i n g (A.8) a n d one o b t a i n s At ~-- 2 M G ln ~~2 q- 3~ - - u~ ( ~ 2 - ~1) q- h i g h e r - o r d e r t e r m s H e n c e , f o r a n y u~, t h e t i m e it t a k e s for t h e t a c h y o n t o escape diverges as ~1 --> 0.

9

RIASSUNT0

(*)

Si studia come inserire il corridoio tachionico, cio~ una descrizione spaziale preferenziale, nello-spazio tempo deseritto da una metrica di Robertson-Walker. Si esaminano anche gli effetti di campi gravitazionali locali sul corridoio. L'esigenza di evitare cappi causali permette di raggiungere conclusioni abbastanza indipendenti da qualsiasi modello specifico del corridoio. (*)

Traduzione a cura della Redazione.

Pe3ioMe He rloay~Ierlo.