IL NUOVO CIMENTO VOL. 22 B, N. 1 11 Luglio 1974

Tachyon Singularity:

A Spacelike Counterpart o f the Schwarzschild Black Hole (*).

J . 1~. GOTT I I I

Cali]ornia Institute of Technology - Pasadena, Cal.

(ricevuto il 2 Maggio 1973; manoseritto revisionato rieevuto il 15 Dicembre 1973)

Summary. - - A Schwarzschild-like metric is presented for the region around a singularity associated with a spacelike world-line (tachyon singularity). Kruskal-type extensions of the metric are provided. The form of the metric suggests a qualitative explanation of tachyon motion (v > c) tha t does not involve transport of energy or information at speeds greater than the velocity of :light. The extended metrics show both normal and time-reversed gravitational Cerenkov cones of high space curvature. The solutions correspond to half-advanced, half-retarded fields in accord with the VVheeler-Feynman absorber theory.

1. - Introduction.

There has been considerable specula t ion a b o u t the exis tence of t achyons . T h e idea t h a t f a s t e r - than- l igh t par t ic les ( tachyons) conld be a c c o m m o d a t e d b y

special r e l a t iv i ty was ~dvaneed i n d e p e n d e n t l y b y several au tho r s : SCltMiDT (~,~), ~Et~I~ETSI{II (a)~ TANAKA (4)~ and BIALAI~IUK, DEStlPANDE and 8UDARSHAN (5). S u c h super lumina l par t ic les wou ld obey k inema t i c relat ions of the f o r m

E 2 ~ - M~e*-] - P"e ~. FEINBElgG (6) ht~s f o r m u l a t e d a q u a n t u m field t h e o r y of

(*) Supported in part by the National Science Foundation (GP-36687X, ,GP-28027). (1) H. SCHMIDT: Zeits. Phys., 151, 365 (1958). (3) H. SCH~IDT: Zeits. Phys., 151, 408 (1958). {3) YA. P. TERLETSKII: SOY. Phys. Dokl., 5, 782 (1960). (4) S. TANAKA: Progr. Theor. Phys. (Kyoto), 24, 171 (1960). (5) 0. M. P. BILA~IVI(, V. K. D~Sm'ANDE and E. C. G. SU])A~SHA~: Am. Journ. Phys., 30, 718 (1962). (6) G. F~I~m~RG: Phys. Rev., 159, 1089 (1967).

4 - I1 Nuavo Cimento B. 49

50 J . R . CTOTT I I I

tachyons. Fe inberg ' s theory is not Lorentz invar iant , however, see e.g. CAMEN- ZI~D (7). Fur the r research on the q u a n t u m field theory of t achyons has been carried out b y DEAR and SU])AnSHA~ (s.o), A~0~S and SUDAI~S]~A~ (1[}) and ECKER (1~). There have been a num ber of exper imenta l a t t emp t s to observe tachyons. To cite one example, there is the work of DA/~BURG et al. (~2), who examined bubble -chamber pictures for examples of the react ion K - p - ~ A t + t - . To date, all such exper iments have only supplied upper limits on t a chyon product ion and no conclusive t aehyon events have been found. For a more complete b ibl iography of the t aehyon l i terature, see the references listed in DA~]~U~G et al. (~) and ]~ECA~[I ei~ al. (~3). A discussion of the problems and paradoxes associated with the idea of t aehyons has appeared in Physics Today (~4). I f t achyons do appear na tura l ly in re la t iv i ty then it is interest ing to unders tand t h e m and how they fit into the theory as a whole. This should be a useful exercise in t e rms of broadening our unders tanding of the theory, regardless of whether or not we ac tual ly find tachyons exist ing in our Universe.

I f tachyons have appeared in the context of special relat ivi ty , then na tu ra l ly one wishes to unders tand t h e m in the context of general relat ivi ty . The for- mula t ion of t achyons as tes t particles in general re la t iv i ty is s t ra ightforward. Mat te r test particles move along t imelike geodesics of the metric, maximiz ing the proper t ime in terval along the path . Tachyon test particles should m o v e along spaeefike geodesics of the metric, minimizing the proper distance along the path . The more fundamen ta l p rob lem is to determine the t ype of space- t ime curva ture the t aehyon produces a round its world-line. Black-hole physics provides a tool wi th which to a t t a ck this problem. I f we had never seen ma t t e r , t hen a s tudy of the Schwarzsehild black-hole solution wduld tell us someth ing of its properties, for in some sense the Schwarzsehild black hole represents ma t t e r in its most pr imi t ive form. The Schwarzschild s ingular i ty is associated with a center of s y m m e t r y ( r = 0) which defines a t imelike world-line, and so the Sehwarzschild field is the p ro to type for the grav i ta t iona l field of m a t t e r which is also defined by a t imelike world-line. Accordingly, s tudy of a singular- i ty associated with a spaeelike world-line m a y well provide insights into the propert ies of taehyons. Such an approach enables us to work ent i rely wi th

(7) M. CAMENZlND: General Relativity and Gravitation, 1, 41 (1970). (s) I. D~AR and E. C. G. SVDARS~AN: Phys. Bey., 174, 1808 (1968). (0) I. DHAR and E. C. G. SUDARSHAN: Phy8. Rev., 173, 1622 (1968). (10) ~V~. ]~. ARONS a n d ]~. C. CT. SUDARSttAN �9 .~orentz invariance, local field theory and ]aster-than-light particles, Syracuse University report (1968). (lz) G. ECxER: Quantum field theory with spacelike momentum spectra, Vienna University report (1969). (z~) J . S. DANBURC~, CT. 1:~. KALBFL]~ISCH, S. •. BORENSTEIN, R. C. STRAND, V. VAN- DENRU~G, J. W. CHAPMAN and J. LYs: Phys. Rev. D, 4, 53 (1971). (1~) E. RECAMI and R. MIC~NANI: Lett. Nuovo Cimento, 4, 144 (1972). (14) O. M. BILANIUK and E. C. G. SUDAaS~AN: Phys. Today, 22 (No. 5), 43 (1969); 22 (No. 12), 47 (1969); 23 (No. 5), 13 (1970); 24 (No. 3), 14 (1971).

TACI-IYO~I 8:[NGULARITY: A SPACELIKE COUNT]~RPAI~T ETC. 51

Einstein 's vacuum field equations (R~,----0) without having to speculate as to the form of the tachyon energy-momentum tensor.

2 . - D e r i v a t i o n o f t h e e x t e r n a l m e t r i c s .

Let us derive the vacuum field metric for the field of a singularity associated with a spacelike world-line. The simplest co-ordinate system in which to work

is the one in which the t achyon singularity is t ranscendent (v----co). Let X, Y, Z, T be the usual Minkowski space and time co-ordinates (units

G : c ---- 1). The singularity is to be associated with a center of symmet ry which we take to be the Z-axis (at T~-0 ) . The metric must be form invar iant with respect to translations in Z as well as form invariant with respect to all Lorentz transformations preserving the quant i ty X ~ -~ Y ~ - - T ~ (i.e. those

representing velocity shifts in the (X, Y, T)-plane). Divide space-time into three regions: region I, T ~ ( X ~ - Iz2)�89 region I I , -- T ~ ( X 2 ~ Y~)t; re-

gion I I I , T 2 ~ X ~ ~ y2. Figure 1 illustrates the positions of these three regions

X

-Z

X ~ ~ I I

Fig. 1. - Space-time diagram showing the positions of regions I, I I and III . The Z-axis is the space-like world-line associated with the tachyon singularity. In the diagram ~he Y-direction is suppressed.

in the background co-ordinate system; the Y-direction is suppressed in the

diagram. Division of space-time into these three regions is invariant with respect to Lorentz transformations preserving the quant i ty X~-{ - y 2 T2.

I n region I we adopt co-ordinates 7:, 0, % Z defined in terms of the original Minkowskian co-ordinates by the relations T ---- v cosh0, X ~- T sinh 0 cos % Y = ~ s i n h 0 s i n ~ and Z~--Z. I n region I I we adopt co-ordinates T , O , % Z

as defined above except T - - - - - ~ cosh0. In region I I I we adopt co-ordinates 0, ~, % Z defined by T : as inh~, X = a c o s h ~ c o s % y z a c o s h ~ s i n ~ and

Z ~--Z. These co-ordinates are illustrated schematically in Fig. 2. ~, 0 and have a range 0 to ~ 0% ~ and Z have a range -- co to -{- 0% while ~0 is

an angle rood 2~.

W e wish to derive the most general metric which is form invariant with

respect to the Lorentz transformations mentioned above. We shall follow a

~2 ft. /~. GOTT III

procedure analogous to tha t used by MOLT,ER (15) for spherically symmetric line elements. Generally, the co-ordinates X, 17, Z, T mentioned above will

not be Minkowskian because space-time will be curved; nevertheless the metric should become asymptot ical ly Minkowskian far from the singularity and the

TI I

~ . . . . Reg/on I " \ I I . / / /

I / / / / \ \ \ \ - . . I O/f/ / /

. . . r-~ / g / / /

,, \ \ \

--X Regfon ~ .... _ ~ l_( _ _Region X

l / I \ \ ',

Region II I

-T I Fig. 2 . - Tile background co-ordinate system, /7 and Z directions are suppressed.

general line element dS~-- - g~,dX~'dX ~ must be a function only of the form invariants of the group of Lorentz transformations preserving X ~ + 172 - - T 2

in a Minkowskian space-time. The form invariants are:

in regions I and I I :

X~ + I5~ -- T ~ ----- -- ~ ,

d% Z, dZ , (i)

XdX+ 17dY-- TdT: -- TdT,

dX ~ § dY ~ -- dT ~ = -- dr~ + v~(dO ~ + sinh~ 0 d~o~) ;

in region I I I :

(2) X ~ + Y~-- T ~= ~ ,

(3) d~, Z, d Z ,

(4) X d X - ] - 17d l z - T d T : (rda

(5) dX ~ + d:Y 2 -- dT ~-~ d a 2 + a~(cosh ~ dq~ 2 -- do: ~) .

(1~) C. MOLL~R: Theory of Relativity (New York, 1',/. Y., 1952).

TACI-IYON SINGULARITY: A 8PACELIKE COU~TERPAICT ]ETC. 5 3

The m o s t genera l line e lements in each region are quad ra t i c fo rms in t he

f o r m - i n v a r i a n t differentials. The only fo rm - inva r i an t co-ord ina tes ~re Z,

and a. Since we require inva r i ance wi th respect to t r ans la t ion in Z the me t r i c coefficients m u s t be funct ions of ~ on ly (regions I and I I ) or of ~ on ly (re- gion I I I ) . W i t h sui table co-ord ina te t r ans fo rma t ions , we m a y wr i te the most

general metr ics obey ing the t a c h y o n s y m m e t r y condi t ions in the conven ien t f o r m :

regions I and I I

(6) d S 2 = exp [ 2 d ~ ) ] d Z ~ - - exp [-- ,~(r)]dv~ q - r2(dO~ q- sinh20 dqo ~) ,

region I I I

(7) d S ~ : exp [~(a ) ] dZ2-~ exp [-- y2(a)]da~ ~ - a : (cosh : ~ d ~ ~ - d~ 2) .

Le t us n o w subs t i tu te metr ics (6) a nd (7) in to the v a c u u m field equa t ions R ~ = 0 . W e find, af ter s t r a igh t fo rward algebra, t h a t ~I(T) ---- ~(~) --~ In (1 - - 2M/r ) a n d

71(~) = y~(a) ~ In (1 - - 2M/a) , where in each case M is un a r b i t r a r y cons tan t . N a t u r a l l y it makes phys ica l sense t o ~dopt the same c o n s t a n t M for b o t h ~.~

and ~.~. W e also wou ld like to iden t i fy the rea l q u a n t i t y M wi th the absolu te va lue of the (( mass ~) of the t a c h y o n s ingular i ty which wou ld appea r in t he k inema t i c re la t ion E ~ ~- -- M2c 4 ~ p2 c ~. Subs t i t u t i ng these solut ions in to our

metr ics we h a v e :

for regions I and I I

( (8) dS ~ ~ 1 - - dZ 2. l _ 2 M / v ~ ~ ( d 0 2 ~ sinh20d~v ~) , v > 2 M ;

for region I I I

( V) (9) dS ~ - - 1 1 dZ~ ~- 1 - - 2 M / a + a2(e~ ~ d ~ - d~2) ' a > 2 M .

These metr ics (8) a nd (9) t a k e n toge the r f o r m an ex te rna l Schwarzschi ld-

like met r ic for the g rav i t a t i ona l field of a t a c h y o n s ingular i ty . I n the l imi t -~ 00 and a - ~ oo these met r ics become the usua l flat Minkowsk ian me t r i c

in our b a c k g r o u n d co-ord ina te sys tem. The metr ics look s ingular w h e n = 2 M or when a --~ 2M. t t oweve r , this is jus t a s ingular i ty in the co-ord ina te

sy s t em analogous to the s ingula r i ty in the Sehwarzschi ld met r i c a t r--~ 2M. A t the t ime I der ived these met r ics I was unware t h a t e i ther of t h e m h a d

been f o u n d previously . A c t u a l l y b o t h of t h e m h a v e been k n o w n for some t ime

in a l t e rna te fo rms as fo rmal solutions of the field equa t ions w i t h o u t phys ica l

~ J . R . GOTT I I I

in terpreta t ion. Metric (8) is equivalent to the metr ic A2 given b y Eli-LEnS and KUS~DT (.16) with the co-ordinates hav ing a different range. Metric (9) can be derived f rom the analyt ic cont inuat ion of metr ic B1 given b y EI{LEI~S and KU~DT (16) via, a simple co-ordinate t ransformat ion .

PELVES (17) has derived metr ic (8) in connection with determining the gravi- ta t ional field of a tachyon. FER~S proceeded f rom the Schwarzsehild metr ic direct ly to metr ic (8) v ia a complex co-ordinate t r ans format ion wi thout direct ly solving the Einstein field equations. I~ also presents Vwo addit ional in- consistent metrics. These are similar in fo rm to metr ic (9) bu t have • replacing M and, because of an algebraic error in Peres ' calculation, have an erroneous s i n h ~ t e r m replacing the correct c o s h ~ term. These incorrect metr ics presented b y PENES also happen to be solutions to Einstein 's field equations bu t are not the correct metr ics for region I I I since they do not re- duce to the fiat Minkowski metr ic for region I I I as a - + ca. They are of the t y p e B2 given b y :EI{LERS and KIJIw)~ (1~) and their physical significance is not clear. PELVES does not indicate t ha t metr ic (8) is to be applied in region I I as well as in region I .

SCI{ISLMA:~ (~s) has also derived metr ic (8) in connection with de termining the gravi ta t ional field of a tachyon. For causal i ty reasons he believes t h a t the gravi ta t ional field of a t aehyon is confined to region I . His model of a t achyon uses the v a c u u m metr ic (8) for ~ > RQ, uses an interior metr ic (analogous to the Schwarzschild interior metric) with t aehyon mater ia l in it to fill the space 0 < v < Ro while the metr ic in regions I I and I I I is t aken to be fiat. Such a t ime-asymmet r i c model suffers f rom several serious difficulties. First , it would seem to impose unphysica l bounda ry conditions; second, it seems impossible to pass na tura l ly f rom such a t achyon model to a t achyon singular i ty; third, i t leads to t r anspor t of energy a t velocities greater t han the veloci ty of light and leads to t rouble wi th energy conservation. These points will be discussed fur ther below and in subsequent Sections.

In view of Sehulman 's work we should give a t this point a conjecture as to the physical ly correct model to use when t achyon mater ia l is present. The model would use the v a c u u m metr ic (8) in regions I and I I for ~ > Ro and would use the v a c u u m metr ic (9) in region I I I for a > Ro. Then the t a chyon mater ia l with its associated interior metr ic would fill the space 0 < T <, Ro in regions I and I I and the space 0 < a < R0 in region I I I . I t would then be possible for a t imelike geodesic emit ted in region I I to pass f rom region I I into region I I I and finally into region I , t ravers ing the t achyon mater ia l twice in the process.

(16) J. EnLEnS ~nd W. KUNDT: Gravitation: an Introduction to Current t~eseareh, edited by L. WITT~N (New York, •. Y., 1962). (17) A. PENES: Phys. ]Lett., 31A, 361 (1970). (is) L. S. SCnULMAN: NUOVO Cimento, 2 B, 38 (1971).

TACttYON SINC-ULARITR': A SPACELIKE COUNTERPAt~T ETC. ~

The principal difference between the proposed model and Schulman's is tha t this model is t ime symmetr ic while Schulman's is not. I n both models

the tachyon material is n ~ confined to the region of the world-line bu t extends to infinity along Cerenkov cones. The time symmet ry of the proposed model offers advantages which will be discussed in Sect. 3 and 4. I t can also be

urgued tha t a t ime-asymmetr ic model of Sehulman's type suffers f rom boundary condition difficulties tha t the proposed model escapes. By direct appeal to analogy, from the Schwarzschild interior metric SC~V~MAN produces a t achyon

interior metric. There are two free parameters in the interior metric which makes it possible to fit two boundary conditions. SCn-t~L~A~ uses these to fit cont inui ty of the metric with the vacuum solutions at v--~ R0 and T z 0.

However, this is not the procedure one uses for the Schwarzschild interior solu- t ion to which the appeal to analogy is made. There are two free parameters in the Schwarzsehild interior metric~ these are used to fit cont inui ty of the

metric at r--~ R0 and hydrosta t ic pressure P ~ 0 at r--~ Ro. Consequently, gtt at r--~ 0 becomes a eomp]etely determined quant i ty measuring the depth of the potential well at r ~--0. This leads one to believe tha t the t achyon

analogue must also have an additionM boundary condition corresponding to * t aehyon pressure~ P T = 0 at ~--~Ro. This then leaves g , , ~ l at T = 0 . I t will then be impossible to match either continui ty or PT--~ 0 at ~ ~-0. The

proposed model does not suffer from these difficulties. Given M and Ro we use the two free parameters to achieve cont inui ty at v ~--Ro and ensure P T = 0

at T ~ R0. Then g~ ~ 1 at ~--~ 0 and PT has a finite value there. Similarly we use the two free parameters to achieve continui ty at a ~ Ro and PT----0 at a ---- R0, we then find tha t g~ and PT automatical ly are forced to have values at a----0 equal to those found above in the limit z ~ 0. All the boundary conditions from the solutions in regions I, I I and I I I match perfectly at

----v----0 because they are solutions generated via different complex co-ordinate t ransformations (following Peres' technique) from a single Schwarzschild interior

solution. For each and every Schwarzschild interior solution there is an exact ' t aehyon analogue in the proposed model. A t ime-asymmetric mode] such. as tha t presented by SCEULMA~ cannot do this. I f t ime-asymmetr ic t achyon models are unable to fulfil the appropriute (analogue) boundary conditions,

then they cannot be regarded as legitimate solutions to the field equations. A detailed verification Of the proposed conjectural model would require a physical understanding of the tachyon energy-momentum tensor and taehyon

(~ pressure )) as well as a detailed t rea tment of all the boundary conditions involved. This is beyond the scope of the present paper, so let us re turn our a t tent ion to the tachyon singularity.

We note tha t g~,--~ 0 at the surfaces of co-ordinate singularities ~ - 2 M and ~--~ 2M so that space is (~ pinched ~ in the Z-direction at this point. ~PEt~ES has also noted this property. I t is exactly analogous to the (( pinching )) of the t ime co-ordinate in the Sehwarzschild metric.

5(} J . R. GOTT I I I

3 . - E x t e n s i o n s o f t h e m e t r i c s .

Maximal analyt ic extensions of metr ic (8) are known (E~LERS and KU~CDT (16)). These extensions are of the Kruska l type. Le t us begin b y finding the max ima l extension of the metr ic (8) in region I I . I n region I I , ~ > 2M, we replace the co-ordinates T, Z b y the Kruska l - type co-ordinaLes U, V defined b y

~ - - 1 exp [v/4M] sinh (Z/4M), (10)

1

Then, in the new co-ordinates, the m~xim~l extension of the metr ic of region I I takes the fo rm

(ii) d2 2 -~ ~2(dO 2 d- sinh ~ 0 d~ ~) -[- - -

where

t

32M 8

T exp [-- ~/2_d] (d ~Y~-- dW),

< 1 , 0 < ~ , 0 < o o , ~---- m o d 2 ~ .

Reg/on I

I

"--. _ L=~5~ i - -

[ �9 ~=O(slngutdP/ty) I ~ = 0 (singuZa~'/ty)

u

"% IZ=O

V Reg/on. II

Fig. 3. - Kruskal-type diagram for the taehyon singMarity (X and Y directions suppressed).

T A C H Y O N S I N G U L A R I T Y : A S P A C E L I K E C O U N T E R P A R T :ETC. ~ 7

B y set t ing 0 = const, ~v ~ const, we can draw a Kruska l - t ype d iagram of the extended metr ic (see :Fig. 3). This d iagram looks exac t ly like the Kruska l d iagram for the Schwarzschild b lack hole except t h a t i t is r o t a t ed b y 90 ~ Region I I with T > 2M is m a p p e d into the lower quadran t of the diagram. I t is a dynamic region since the metr ic coefficients depend on T, the t imelike co-ordinate. I n the r ight and left quadran t s we find stat ic regions, 5P 1 and ~cz. I n these regions n e w , and Z co-ordinates m a y be defined in te rms of U and V:

u = + l!- j exp cosh (z14]/) ,

(12) Region �89

- t - ~ I - - ~ M ) exp[~/4M] s i n h ( Z / 4 ] / ) ,

0 < T < 2 ] / ,

so t h a t the metr ic in 9~ and 5P2 is in fac t metr ic (8) bu t wi th 0 < ~ < 2 M so t h a t Z becomes the t imelike co-ordinate. Regions ~ and ~ are bounded b y the s ingular i ty a t v = 0 (U ~ - - V ~ = 1). I n the top quadran t of the d iagram we emerge again into a dynamic external region where ~ > 2 M which we shall call region I ' . I t is interest ing t h a t in this d iagram the dynamic regions are the external regions (z > 2 ] / ) while the stat ic regions ( ~ < 2M) are those close to the singularity. This is just the inverse of the behavior of the Schwarz- schild black-hole metric. A t imelike geodesic emi t ted in region I I can ei ther hi t the s ingular i ty a t ~ = 0 or it can pass on into region I ' (~ > 2M). ~ o w region I ' (~ > 2 ] i ) is isometric to region I (v > 2M) and t ime runs in the r ight direction so it is na tu ra l to make the topological identification I ' ~ I . For some purposes, however, we m a y still wish to regard region I ' as being the ~ region I ~> of another universe.

:From the above discussion we see t h a t the extension of the metr ic of region I is tr ivial . We define Krus ka l co-ordinates U, V in t e rms of v and Z in region I (~ > 2M) b y

U = -- ~ - - l exp [~[4M] sinh (Z/~M),

(13) ( )~ V = - - ~ v __1 e x p [ w / 4 M ] e o s h ( Z / 4 M )

I n the new co-ordinates we find a max ima l extension of the metr ic of region I which is again just t h a t given in eq. (11). The Kruska l - t ype d iagram is exac t ly

t ? the same as Fig. 3. We have region I at the top, s ta t ic regions Sz 1 and 5z , a n d a region I I ' a t the b o t t o m which is isometric to region I I . I f we make the topological ~dentification I ~ I ' , I I = I I ' , then regions I and I I are joined and together wi th the stat ic regions 5f~ and 5P2 form a max ima l extension which can be depicted in a single Kruskal- l ike d iagram (Fig. 3). Metric (11)

~ J . 1l. GOT]' I I I

applies throughout the diagram. T and Z co-ordinates can be defined in four

patches via eqs. (10)7 (12), (13) so tha t metric (8) applies in each patch. We note from inspection of the diagram tha t the tachyon singularity does

not have the horizon structure one associates with the term black hole. So Mthough it is a direct counterpar t of the Schwarzschild black hole, the t achyon

singularity is not itself what we would call a black hole. The KruskM-like extension of the metric indicates why Schulman~s t achyon

model cannot be extended to include a t achyon singularity. I f the t achyon

gn'avitational field is confined to the absolute future of the tachyon world- line r then its metric in this region (region I) must be given by (8). The only

complete analytic extension of this metric is tha t given in (11), and depicted in Fig. 3. The analytic extension also includes a region I I which lies in the abso- lute past of the tachyon singularity. Thus, the t achyon singularity inevitably

produces a gravi tat ional field to the past of the singularity as well as to the future and a t ime-asymmetr ic model does not apply.

Since the above extensions are maximal, it is apparent tha t region I I I

and its extensions are disjoint f rom regions I and I I . The metric of region I I I

( a > 2M) does not have a Kruskal - type extension; however, we can extend it to include a second region I I I ' ( a > 2M), isometric to itself. To see this, examine the 2-dimensionM section in region I I I defined by c~ = 0 7 Z--~ const.

The geometry of this sheet is exactly the same as a 2-dimensional spacelike slice through the external Schwarzschild black-hole metric. Thus, when a

spacelike geodesic in the c~----0, Z = const plane reaches ~ = 2M7 it can be extended across an ~ Einstein-Rosen ~; bridge into a second identical sheet in a new region I I I ' . Let us rewrite metric (9) for region I I I ( ~ > 2M) by introducing the following co-ordinate t ransformation: (~=2M/(1--~/4), Z = 4~0M. Metric (9) thus becomes

(14) ass=di [ i cosh d --dq ] ,

The above metric is exactly equivalent to metric (19) for region I I I ( 2 M < a < oc)

and eovers exactly the same co-ordinate range. Metric (14) appears in E~tLEUS and KU~DT (~6) and represents the analytic extension of metrics of class B1.

I f one were to regard ~0 as an angle mod 2~, then one could describe the (~, ~0)-plane as a surface of revolution with center at ~ = 0 which approaches a Euclidean cylinder of circumference 8aM for ~ -+ 2. I n the new co-ordinate

system, the metric is not singular at ~ = 0 (a---- 2M) so we can extend it to the region - - 2 % o < 0. This new region is region I I I ~ which is isometric to region I I I . For metrie (14) with - - 2 < ~ < 2 , the eurvature is every-

where bounded and reaches its largest value at ff = 0. Thus, with this metric, we never meet the singularity but only hover around it. This seems rather

unusual. I t would natural ly be interesting to know whether or not the

metric (14) with -- 2 < ~ < 2 is maximal.

TACH'~01~ SII~CTULARITY: A SI~AC]~LIK~E COUNTJ~RP&RT ETC. 5 9

The Schwarzschild black-hole solution joins two dist inct a sympto t i ca l l y flat universes together . I f we regard regions I and I ' and also I I and I I ' as topological ly distinct, t hen we can regard this t achyon singulari ty solution as also joining two asympto t ica l ly flat universes. The first a sympto t i ca l ly fiat universe is composed of regions I , I I , I I I while the second asympto t i ca l ly fiat universe is composed of regions I ~, I I ' , I I I ' . Geodesics m a y pass be tween I I I and I I I ' while t imelike geodesics emi t ted in region I I m a y emerge in region I ' and t imelike geodesics emi t ted in region I I ~ m a y emerge in region I I .

4 . - T a c h y o n m o t i o n .

The form of the metr ics suggests a qual i ta t ive physical explanat ion of ta- chyon motion. Le t us begin b y examining the asympto t ica l ly fiat background co-ordinate sys tem X, ]7, Z, T. The invar ian t cone T---- ( X ~ + I:~) �89 which divides region I f rom region I I I is a Cerenkov cone, while the invar ian t cone T ~ - - ( X 2 + Iz~)~ which divides region I I f rom region I I I is a t ime-reversed ~erenkov cone. This can be seen clearly b y switching f rom our a sympto t i c co-ordinate sys tem X, Y, Z, 2' to a new co-ordinate sys tem X', Y', Z', T' related to it b y a Lorentz t ransformat ion , wi th T ' : (1 - - ~) -~ T + fi(1 - - fl~)-~Z, Z ' - - - - (1 - - f i2 ) -~Z+/~(1 - - f l~ ) -{T , X ' - - X , I z ' ~ Y, where f i < l . I n the old a sympto t i c co-ordinate sys tem the center of s y m m e t r y of the t a chyon sin- gular i ty was the Z-axis (at T----0) thus represent ing a t ranscendent t a chyon v ~- co. I n the new co-ordinate sys tem X ' , Y'~ Z' , T ' , the center of s y m m e t r y , corresponds to a t achyon moving in the posit ive Z ' -di rect ion with a ve loci ty v--~ (1/fi)c (i.e. c < v < co). See Fig. 4 for the appearance of the t a chyon at some ins tan t of t ime T ' . The t a c h y o n , posit ion >> on its world-line a t t h a t ins tan t is the intersect ion of the two cones. The expanding rear cone in the d iagram is the cone T : ( X 2 + I:~) �89 which divides regions I and I I I . The rear cone is expanding perpendicular to its surface wi th v--~ c. Ins ide this

expcznding r e d r cone (v = c) contrQct ing

y~ / / forward~ cone (v=c )

tachyon " "

(v>c)

Fig. 4. - Schematic diagram of tachyon motion in an otherwise empty universe. The regions of high curvature are associated with the two crossing (Cerenkov) cones. Tachyon motion (v ~ c) is produced through a << scissors ~> effect but no energy or information is transmitted at speeds greater than the velocity of light.

~ 0 J . l~. GOTT I I I

expanding rear cone is region I . Also present in the d iagram is a cont rac t ing forward cone which is the cone T - - - - - ( X : + ]62) ~ dividing regions I I and I I I . I t is contract ing perpendicular to its surface with v z e. Ins ide the forward cone lies region I I and between the forward and rear cones lies re- gion I I I . I n the above remarks space-t ime has been considered to be flat. When the curva ture in t roduced b y the t achyon is considered these invar ian t forward and rear ~erenkov cones will be seen to have a physical significance.

Regions I and I I with the stat ic regions ~ and 92 form a complete, maxi- mal ly extended universe. For an observer in region I with T >> 2 M space- t ime becomes asympto t i ca l ly fiat. The observer m a y then ex t rapola te the local behavior to make the erroneous assumpt ion t ha t he is in a globally flat universe (with the X, ]7, Z, T background co-ordinate system). He then finds t h a t a l though his region of space-t ime is asympto t ica l ly flat, he cannot send any geodesics (spacelike or timelike) beyond the cone T----(X~-~ Iz~) ~. His universe is a sympto t ica l ly flat bu t is confined b y a ~erenkov cone T--~ (X2~ - :Y~)�89 I n the two-dimensional Kruska l - type d iagram (Fig. 3) each point in region !

> 2 M represents a 2-surface t ha t is a single sheet of a two-sheeted hyper- boloid of revolut ion. These hyperboloids of revolut ion all face in the same direc- t ion and asympto t ica l ly become parallel to the Cerenkov cone T ~-- (X2-[ - :Y~)�89 as a l imiting surface. (Note: A similar s i tuat ion occurs for observers in an open F r i edmann cosmology. As the expansion continues, space-t ime becomes asymp- tot ical ly fiat so t hey m a y make the erroneous assumpt ion t h a t t hey are in a globally fiat Milne-type special-relat ivi ty cosmology. They find however t h a t because of the curva tu re t hey cannot send any geodesics beyond the l ight-cone T ~ (X2~ - :Y~- Z~) "~ associated with the singularity. Al though this space- t ime is asympto t ica l ly fiat, it is l imited within the light-cone T--~ (X2~ - :Y~-Z~)t . ) I f an observer tries to send a geodesic toward the cone T = (X~-~ lr~) ~, the geodesic mus t cross into regions of higher and higher curvature , t ha t is, regions of lower and lower values of v. Before reaching the l imiting cone the geodesic crosses v--~ 2M, enters the region ~ or Jr2 and hits the s ingulari ty (or perhaps emerges in region I I ) . The regions of high curva ture associated with the singulari ty (corresponding to small values of ~) are not confined near the t achyon world-line bu t ra ther extend to infinity along the inside edge of the expanding Cerenkov cone T--~ (X2-~ :y2)�89 I f we wish to avoid regions of high curvature , we mere ly s tay far away f rom the cone; if we wish to enter regions of higher curva ture and mee t the s ingulari ty itself, we mus t approach the cone. The fact tha t the regions of high curva ture in region I lie along a ~erenkov cone can be seen f rom an examina t ion of the external metr ic (8) alone. PERES and SCgVL~CA~ have called a t ten t ion to this p r o p e r t y and have noted tha t one would like to th ink of these regions of high cu rva tu re as a wake of gravi ta t ional ~e renkov radiation. This is cer ta inly an appea l ing physical idea a l though one should caution tha t s trong fields are involved here for the t a chyon singtflarity (geodesics m a y disappear into new regions, for example) so this ~erenkov

TACHYOI~ SII'~GULA:EITY: A SPACELIKE COUNTERPART :ETC. 6 1

gravi ta t ional shock wave has just a qual i ta t ive kinship with the gravi ta t ional waves of the linearized theory.

Similar arguments apply for region I I . For large values of ~, region I I becomes asymptot ical ly flat bu t the flat region is confined within the contract ing ~erenkov cone T - - - - - (X2-} - ~)�89 Regions of high curvature ex tend to in - f ini ty along the inside edge of this limiting cone. Jus t as the Kruska l - type diagram (Fig. 3) gives a good description of the geometry near the singularity, Fig. 4 gives a good description of the movement of the regions of high curvature . The asymptot ical ly flat parts of the universe (I, I I , 5f~, 5P) lie within the hourglass-shaped region, bounded by the two gravi tat ional Cerenkov shock fronts. The actual geometry near the center point a t a scale of 2 M is of course more complicated than can be depicted in Fig. 4, bu t this is of no importance to our arguments. A mat t e r test part icle (timelike geodesic) in region I I is t r apped by the forward gravi tat ional shock front impinging upon it with v-----c. As the shock f ront sweeps over it, i t ei ther hits the singulari ty or is expelled into region I. I t then finds itself in the other half of the hourglass region, behind the shock front, which it sees expanding away f rom it with , ~- c. Thus, the taehyon singularity and the regions of high curvature around i t are not associated merely with a point moving with v > c. Rather , the sin- gular i ty and the regions of high curvature arc in t imate ly connected with the two ~erenkov cones which propagate with v----c.

The whole hourglass configuration moves along the world-line with a phase veloci ty corresponding to v > c. However, the ~erenkov cones with their shock fronts of high space curvature which form the boundary of the hourglass region propagate perpendicular to their surfaces with only v-~ c. The regions of high curvature form an extended source whose local propagat ion (at v----c) is not parallel to the motion of the system as a whole. I t is this geometrical configuration of the fields tha t allows product ion of a phase veloci ty of the system at v > c. An essential factor is tha t the singulari ty itself (at z----0) is an extended source whose location is not merely the world-line (Z-axis) bu t ra ther the entire pair of ~erenkov cones (T~= X ~ + ]z~). W h a t is hap- pening here is analogous to the scissors effect. We can make the intersection point of the blades of a pair of scissors move faster t han the speed of light even though the velocity of the blades never exceeds the velocity of light. Construct

pair of infinitely long, straight blades. :By pre-arrangemcnt set the blades in uniform motion perpendicular to their lengths with velocity Vb < c. Le t the blades make an angle r with each other at their crossing point. The inter- section point of the blades will then move with velocity V~----V.o/sin (r B y choosing r small enough we can make V / > c. (Note: we cannot, of course, produce such a uniform blade motion by merely shaking the blades at one end, because the mechanical force is t ransmi t ted down the blades with only V~oun d < c. We must exert a force all along the length of the blade at a given instant of time. I f we plan ahead and have cohorts with synchronized watches s tat ioned

~'~ J . R. GOTT I I I

all along the blade there is no difficulty in doing this. So in this fashion we produce two blades moving with velocity V~.) While the intersection point

(and thereby the entire cross-shaped blade configuration) does move with v > c no energy or information is t ransmit ted with v > c. The blades form an extended source whose propagat ion is not parallel to the displacement

of the system as a whole. The ~ crossing ~> Cerenkov cones of the tachyon and the curvatures associated with them (the blades of the taehyon) propagate with v ---- c just as one expects gravitat ional effects to do. :No energy or informa-

t ion is in fact t ransmit ted here faster than the velocity of light. For the scis- sors effect to operate it is not essential for an actual crossing to occur. One could imagine, for example, a moving scissors configuration formed by mem-

bers of a marching band where the marchers reverse direction when they reach the intersection point so tha t no crossing takes place. The pat tern still moves with a phase velocity greater than the speed of the individual marchers, who

fall further and fur ther back in the moving pat tern with time. This is relevant for the tachyon solutions since the geometry near the center point at a scale of 2 M is more complicated than a simple crossing. Thus, the scissors effect

is still applicable to these taehyon solutions. We notice tha t the presence of the two ~erenkov cones of curvature is essential for the scissors effect argument.

I f only the backward Cerenkov wake were present (as in Schulman's model) the whole configuration would plow into virgin flat space ahead and could i a principle t ransmit energy or information at v > c.

I f we restrict our a t tent ion to the entirely separate universe of regions I I I and I I I ' , then a similar picture emerges. For a large, region I I I becomes asymptot ical ly fiat. Yet the flat region is bounded by the cones T--~ (X 2 + 1~) �89

and T z - (X2-L y2)~ for geodesics m a y not be sent beyond them. As we approach the cones, this time from the outside, we enter regions of higher and higher curvature. At a = 2M the curvature reaches its max imum (finite)

value. The surface a = 2M is a hyperbolo~d of revolution of one sheet then

stretched out in the Z-direction; it encloses the cones T ~-- J= ( X 2 ~ y2)~ and becomes asymptot ical ly parallel to them. Region I I I may thus be mapped also

in Fig. 4. I t takes up the space in between the two cones. I t s regions of high curvature lie along the outside edges of the moving cones. So an observer

in region I I I also sees expanding and contract ing gravitat ional Cerenkov shock fronts which move with v ~ c. Again, no energy or information is t ransmit ted faster than the velocity of light. The universe of regions I I I

and I I I ' and the universe of regions I, I I , ~ , 5~ are complementary in t h a t they are interlocking and can be fitted together back to back (as in Fig. r

without touching. Yet the two universes are completely disjoint. One won- ders in what sense one may speak of them as both being the result of the ~( same )> taehyon singularity.

I t is perhaps worth-while here to mention some additional relevant points

concerning these vacuum metrics. The vacuum metrics as presented would

TAC/- ITON S I N G U L A R I T Y : A S P A C E L I K E C O U N T : E R P A R T :ETC. 6 ~

involve topological changes in the universe as a whole. I n par t icular the uni- verse is cut into two pieces (the universe I , I I , ~1 , 5f~ and the universe of I I I ) along the l imiting Cerenkov cones T ~ - - - X ~ - Y~. I t should be r emembered t h a t s imi la r ly unpleasant topological changes are associated with near ly all the pure v a c u u m field solutions. For example, the pure v a c u u m Schwarzschild black-hole solution includes a topology change t ha t joins our Universe to ano the r one just like it. Similar examples can be found in the v a c u u m Ker r solution. When m a t t e r is used to (( cover )> these solutions, however, the topological abnormali t ies disappear and we are left wi th <( reasonable ~> solutions. The same effect is expec ted to occur when the t achyon v a c u u m solutions are ~ covered ~> with t achyon mater ia l . I n fact , the proposed model including t achyon mate r ia l presented earlier represents an example of this. I n t h a t model, the universe is not divided into separa te par ts . Another point wor th ment ioning in this connection is t ha t the serious exper imenta l efforts toward finding t achyons are aimed at the e lementary-par t ic le level. A t aehyon e lementa ry part icle might have a Schwarzsehild radius much smaller t han the Planok-Whceler length (10 -33 era). At such small scales, the topology of space- t ime is cer ta inly

quite complicated. I n this f r amework the topological difficulties of the t achyon v a c u u m metrics do not seem too serious. The only impor t an t th ing would be the overall t achyon field. Another question raised b y the metr ics is whe ther tachyons are (( small ~>. These studies indicate t h a t the t aehyon is not confined near a world-line but ra ther t h a t the t achyon extends indefinitely along, and is in t imate ly connected with, the two associated ~erenkov cones (lone migh t view these gravi ta t ional Cerenkov cones as actual ly const i tut ing the tachyon) . I n this sense the t achyon is not small, bu t of course nei ther is a p lane wave of e lectromagnet ic radia t ion (it also extends to infinity along a light-cone).

The present picture involving a forward Cerenkov shock wave as well as the rear Cerenkov wake resolves one of the paradoxes associated with tachyons . Recent ly JoIvEs (19) has provided for tachyons a classical formula t ion of electro- magnet ic Cerenkov emission which is Lorentz invar iant . Applying Jones ' formulat ion to the gravi ta t ional case, we see t ha t energy and m o m e n t u m con- servat ion require t ha t the emission of grav i ta t iona l Cerenkov radia t ion resul ts in a nongeodesic world-line. The pa radox arises when one then considers the v a c u u m field solutions of PENES and SCHU~MA~ (region I , T > 2M). These are solutions tha t represent a t achyon t ravel ing along a geodesic and ye t t h e y appear to be re ta rded solutions and certainly show the presence of a g ray i ta - t ional Cerenkov wake. JO~ES himself remarks t ha t his results and Sehulman ' s are inconsistent. According to the usual ideas concerning ~erenkov rad ia t ion a Schulman- type solution should violate energy conservation. I t represents a t achyon emit t ing Cerenkov radia t ion wi thout suffering any c h a n g e in its own energy. (We can see d i rec t ly t ha t Sehulman 's model violates energy conserva-

(19) F. C. Jo~:Es: Phys. Rev. D, 6, 2727 (1972).

64 J.R. GOTT HI

tion. For his t ranscendent t achyon we have an e m p t y space- t ime for all t imes before t ~ 0, bu t af ter t - - 0 we have the gravi ta t ional ~erenkov radia t ion as- sociated with the tachyon.) On the other hand, the Peres and Schulman v a c u u m field metrics are exact solutions to Einste in 's field equations and should con- s t i tu te a legi t imate solution to the problem.

The ex tended metr ics presented here resolve this paradox. The max ima l extensions of the Peres and Sehulman metrics (region I , ~ ~ 2M) are found to be t ime symmetr ic . Thus the Peres and Schulman metrics are not pure ly re ta rded field solutions, bu t ra ther are pa r t of a t ime-symmet r i c solution. All the t aehyon v a c u u m field metr ics considered here are t ime symmetr ic and show the presence of a forward cone of gravi ta t ional Ceren]~ov radia t ion (advanced potenti~I) as well as the backward cone of gravi ta t ional ~erenkov radia t ion (retarded potential) . The usual ideas of ~erenkov radia t ion can apply. The energy ~nd m o m e n t u m emi t ted by the expanding rear cone is just equal to t h a t supplied b y the contract ing forward cone. Ene rgy and m o m e n t u m are conserved and the t aehyon proceeds with the same energy and m o m e n t u m along its inertial pa th .

The t ime-symmet r i c v a c u u m field metrics presented here are in complete accord with the Whee le r -Feynman (~o) absorber theory. These metr ics represent a t achyon in otherwise e m p t y space. I n such a s i tuat ion with no absorbers present the Whee le r -Feynman theory would predict the field to be given b y 1 the advanced plus �89 the re ta rded field of the taehyon. This is consistent wi th our finding of bo th the forward and rear ~erenkov cones of high space curva- ture. The impor tance of bo th the re ta rded and advanced potent ia ls can be seen direct ly f rom the Kruskal- l ike d iagram (Fig. 3). The singulari ty (at U 2 - - V ~ = 1) produces curva ture in region I via the re ta rded potent ia l while it produces curva ture in region I I via the advanced potential . I n the Wheeler- F e y n m a n theory radia t ion react ion forces are due to the induced fields of the absorbers so in an e m p t y universe with no absorbers present there should be no radia t ion react ion forces. Thus, there should be no force on the t aehyon and its pa th should be a geodesic. The Whee le r -Feynman theory is fo rmula ted for electrodynamies. I t is interesting, therefore, t h a t we have in the present case a solution ~rom general re la t iv i ty where the (gravitational) rad ia t ion reac- t ion is expected to be impor t an t (~erenkov radiat ion) and where there is agree- men t with the Whee le r -Feynman picture.

The t aehyon world-line should be geodesic in any space- t ime where the configurat ion of absorbers is t ime symmetr ic . This is of interest in relat ion to the solutions to the field equations obtained b y ~FoSTER and RAY (2~) and GOTT (22). They find ( t ime-symmetr ic) space-t imes where the energy-momen-

.(2o) j . A. WH~L~R and R. P. FEYNMAN: ~ev. Mod. Phys., 17, 157 (1945). {21) j . C. FOSTER jr. and J. R. R~y: Journ. Math. Phys., 13, 979 (1972). (~2) j . R. GOTT [H: Astrophys. Journ., 187, 1 (1974).

TACHu SINGULARITY: A SPACELIK:E COUNTERPAR, T ETC. ~ 5

turn tensor corresponds to tachyons traveling along geodesics of the metric. I n the Wheeler-Feynman theory the configuration of the absorber in our

Universe is time asymmetr ic due to initial conditions. The absorber thus pro- duces advanced effects which, evaluated in the neighborhood of the tachyon, combine with the half-advanced, half-retarded field of the t achyon to give the

full retarded field. The t achyon is subject to a radiation reaction force due to these advanced effects and should follow a nongeodesie path. The puth should

be one in which the energy loss of the taehyon is equal to the energy emit ted in (retarded-field) Cerenkov radiation. I n such a situation the formulat ion of JONES (~s} should be applicable. JO~ES finds the taehyon world-line in Min-

kowski space (X, ~, Z, T) to be given by T ~ - Z ~ - - fl-2. This represents a t achyon and an an t i tachyon approaching each other and annihilating at the

point (0, 0, 0,--fi-~). For a classical t reatment , take M for the t achyon equal to the electron mass and e for the t achyon equal to the electron

charge and take the size of the charge distribution to be the classical elec- t ron radius r . Then, using Jones ' formula, one calculates f l - ~ r o . Far from the annihilation point the t achyon and an t i tachyon world-lines asymptot ica l ly

approach the null lines T : Z and T z - - Z respectively. Thus, if one is permit ted to view the t achyon gnd ant i tachyon as independent, there is no

transmission of energy or information faster than the velocity of light, except over distances of the order of to. Transmission of information with v ~ c over a distance of the order of r is also found in situations involving ordinary mat te r

where radiation reaction forces are important . An example of this is a delta- function electromagnetic plane wave impinging on an electron as discussed by DIRAC (2s) and WHEELER and F E Y ~ A ~ (s0). Jones ' results can be converted

to the gravitat ional case by analogy. Taking the size of the mass distribution

to be the tachyon Schwgrzschild radius rs----2M we find tha t f i - ~ r s. I n this ease energy or information is not t ransmit ted faster than the speed of

light except over a distance of the order of r s : 2M. This behavior is similar to tha t encountered in our t ime-symmetric metrics where we also found no trgnsmission of energy or information faster than the speed of light over scales larger tha~l 2M.

Thus, the behavior of tachyons is found to be int imately connected with Cerenkov radiation. I t is illuminating to compare the gravitat ional results

with some for the electromagnetic ease. The electromagnetic field of a charged

tachyon can be calculated by applying a generglized Lorentz t ransformgtion (MIG~A~I and I~ECAMI (~4}) with v > e to the static Coulomb field of a point charge as done by CmAS'G (~5) and GLt3cx (2s). Since the static point churge

(~3) p. A. M. DInAC: Proc. Roy. Soc. London., A167, 148 (1938). (2a) R. MIGNANI und E. R~CAMI: ~Tuovo Cimento, 14 B, 169 (1973). (25) C. C, CHIA~G: preprint CPT/ll7 (Austin, Tex.) (1971). (26) M. GL?JCX: ~:uovo Cimento, 1 A, 467 (197]).

5 - Y1 N u o v o C i r a e n t o B .

66 z . i t . G O T T I I I

experiences no forces, the t rans formed solution should show the t achyon ex- periencing no forces. Thus, the t achyon world-line should be geodesic. This has been t aken to indicate t ha t charged tachyons do not have any net emis- sion of ~erenkov radia t ion ( C m A ~ (~) and MIGNANI and RECA~I (~7)). The reason for this was correct ly apprehended b y Cn~ANG (~5): the t r ans formed solution yields a t ime-symmet r i c field (�89 ~ ~Fd~) ; any radia t ion emi t ted b y the re ta rded cone is made up for b y radia t ion absorbed f rom the advanced cone. This is exac t ly in accord with the presenta t ion in this paper for the grav- i ta t ional case. The fur ther conclusion of CHIA~G (ss) and others tha t t achyons in our Universe do not rad ia te in v a c u u m is not justified however. The par- tieles of the Whee l e r -Feynman ~bsorber add a field (�89 o _ �9 ~/~ ~) which brings the observed field up to F ~ and creates a force on the tachyons . Thus, charged tachyons in our Universe will be observed to emit retarded-field electromagnet ic Cerenkov radia t ion in v a c u u m and will have nongeodesic world-lines because of it. The amoun t of retarded-field Cerenkov radia t ion emit ted by a charged point t achyon is infinite (see CmANG (~) and JACKSON (~s)). (Iqote: the result of zero-retarded-f ield emission b y GL~2CK (~) is incorrect due to in tegrat ing the flux over an incorrect surface; the correct t r e a t m e n t is given in JACKSON (~s)). The configuration of fields produced b y a charged t achyon in v a c u u m is exac t ly the same as t ha t produced b y a fast electron (v ~ c/~/[) in a perfect, nonpermeable , nondispersive dielectric. One mere ly makes the replacements c ' = c/~/s, e ' = e/~re, E ' = ~r B' = B to conver t the v a c u u m to the dielectric case, where s is the dielectric constant (cf. JACKSON (~s) and T A ~ (ss)). I n the dielectric we know the Cerenkov radia t ion is produced b y induced dipoles along the Cerenkov cone. Thus, the currents is associated with the charged point t achyon cannot be confined to the world-line alone bu t mus t lie along the entire Cerenkov cone. (The same localization prob lem is to be found in the gravi ta t ional case.) I n the classical case, then, the ra te of Cerenkov emis- sion f rom a charged point t achyon is infinite. FEYI~IVIAI~ and K A U ~ A ~ N (~o) repeat the prob lem of a point t a chyon using q u a n t u m elect rodynamics and also find an infinite ~erenkov flux. As we have ment ioned earlier, Jo~Es (~) proposes a finite ~erenkov flux produced b y a t a chyon of finite size.

Finally, there has been considerable discussion of causal i ty paradoxes as- sociated with tachyons. I t has been shown tha t co-operat ing observers can send posi t ive-energy tachyons between each other in such a way tha t one can send a signal into his own past , thus leading to causal i ty paradoxes (P~]~A~I (~)). I t has been argued (PA~nNTO~A and YEE (a~)) tha t rigorous appl icat ion of

(27) R. MIGt~AI~I and E. R]~CAMI: Lett. Nuovo Cimento, 7, 388 (1973). (:s) j . D. JACKSON: Classical Electrodynamics (New York, N. Y., 1962). (ep) I. G. TA~ilvI: Zur. ~iz. SSR, 1, 439 (1939). (3o) S. K. KAUFF~IANI~: Cerenkov ef]ects in tachyon theory (preprint) (1970). (al) F. A. E. PIRA~I: Phys. ]~ev. D, 1, 3224 (1970). (~) J. A. PA~I:~TOLA and D. D. H. Y]~: Phys. ]~ev. D, 4, 1912 (]970).

TACH~rOlq SINGULARITY: A SPACELIKE COUNT~ERPA:RT ETC. 6 7

the re in terpre ta t ion principle (i.e. t ha t a negat ive-energy t achyon t ravel ing toward the pas t is regarded as a posi t ive-energy an t i t achyon t ravel ing toward the future) avoids such paradoxes. This a rgument rests essentially on the pic ture t ha t space-t ime is a fixed tab le t with various world-lines inscribed on it; and t h a t events can happen only one way which mus t be self-consistent. Wi th this v iewpoint one avoids causal i ty paradoxes a priori . We are left however with some constraints on the abi l i ty of all observers to emit pos i t ive-energy tachyons as they choose. The problems inherent in the s i tuat ion are well sum- marized b y R o o t and T~EFIL (~). Wi th the results of this pape r concerning the inabi l i ty of tachyons to t r ansmi t energy or informat ion over large distances a t speeds grea ter than the speed of light, we see t ha t m a n y of the causal i ty questions concerning tachyons become moot . Not all problems are resolved however. As JO~TES (~9) points out, in the retarded-field ease wi th a hyperbol ic t achyon world-line our abi l i ty to emit a t achyon is con t ingent on the presence of the appropr ia te an t i t achyon which will eventual ly annihi late with the one we have emit ted. The impor tance of t achyons not t r ansmi t t ing energy or informat ion fas ter t han light goes beyond just causal i ty questions however. As Sehulman 's (~s) model shows, tachyons t r ansmi t t ing energy fas ter t han the speed of light would create problems with energy conservation. The idea of not t r ansmi t t ing energy or informat ion fas ter t han the speed of light fits well with the w o r k of BERS et al. (~), who show tha t particles obeying the Klein- Gordon equat ion ([N ~ M ~) r ~ 0 would not give superluminal signal prop- agat ion and also with the work of I~O~A~ and SANTILLI (~5), who show t h a t for a Lorentz - invar ian t wave opera tor H (as a polynomial in the d 'A lember t i an operator) with H e ( x ) = 0 the speed of light represents an upper l imit to the signal velocity. I t is also consistent with the work of B~ATIA and ~PANDE (36), who find t ha t in a degenerate t achyon gas the veloci ty of sound is still a lways less t han the speed of light.

5 . - C o n c l u s i o n .

Vacuum field metr ics are presented for the region around a s ingular i ty as- sociated with a spacelike world-line. I t seems appropr ia te to call this a t a c h y o n singularity. The metrics are of the Sehwarzschi ld- type (classes A2 and B1) and have been known for some t ime as formal solutions to the v a c u u m field equations. I t is now possible to give bo th metrics (and their extensions) a

(3~) R. G. ROOT and J. S. TRE~IL: Lett. Nuovo Cimento, 3, 415 (1970). (84) A. B~RS, R. Fox, C. G. KUPER and S. G. LIPSON: The impossibility o I free tachyons, Technion-Israel Instilute of Technology preprint (Haifa, 1969). (35) p. ROMA~ and R. M. SANTILLI: Derivation o/ Poincarg covariance from causality requirements in field theories Boston, University preprint (1969). (~6) M. S. BnATIA and L. K. PANDE" Phys. Rev. D, 5, 2936 (1972).

6 8 J , F~. GOTT I I I

physical interpretation. The form of the metrics suggests a qualitative descrip-

tion of the gravitat ional field of a t aehyon singularity as two crossing Cerenkov cones of high space curvature. The taehyon singularity is thus able to move

with a phase velocity v ~ e al though the cones of curvature associated with it only propagate with v ~ c. ~ o energy or information is t ransmit ted at

velocities greater than the speed of light. The extended metrics are time sym- metric representing half-advanced, half-retarded fields and the tachyon world-

line is geodesic in accord with the Wheeler-Feynman theory. These solutions represent a t achyon propagat ing in an otherwise empty universe. When a

t ime-asymmetr ic absorber is present, the advanced effects of the absorber combine with the half-advanced, half-retarded taehyon field to produce the full retarded field. I n such a situation, the formulation of JO~ES (~9) should

apply and the world-line should be nongeodesic. I n this case the world-line is bent in such a fashion tha t transmission of energy or information over large

distances is also limited to v z e. The behavior of tachyons is int imately connected with ~erenkov radiation and the nature of the radiation reaction

force. The metrics presented here constitute an example from genera] relat ivi ty where the Wheeler -Feynman conception of the radiation reaction force is ap-

plicable. The fact tha t it was possible to find a Schwarzsehild-like metric cor- responding to a t achyon singularity testifies to the deep symmetries between space and time in Einstein's theory.

I t is a pleasure to thank Dr. K. Kuc~A~ for many stimulating discussions

and Drs. J. G. ~I]~LEn, J. E. G v : ~ and R. P. FEu for helpful comments. This work was begun while the author was at Princeton Universi ty and was

supported there by a ~:ational Science Foundat ion predoctoral fellowship. The work has been completed at the California Ins t i tu te of Technology where

it has been supported in par t by ~a t iona l Science Founda t ion grants

GP-36687X and GP-28027.

@ R I A S S U N T O (*)

Si presenta ann metrica del tipo di Sehwarzschfld per l'intorao di una singolarits asso- ciata con una linen d'universo spaziale (singolarir ~achioniea). Si forniscono delle cstensioni della metrica di tipo di Kl'uskal. La forma di detta metriea suggerisce una spiegazione qualitativa del moto dei taehioni (v> c) che non coinvolge trasporto d'energia o d'informazione a veloeits pi~ elevate della veloeit~ della luce. L~ metriea estesa presenta dei coni di Cerenkov gravitazionali di elevuta eurvatura spaziule sin normali che invertiti rispetto al tempo. Le soluzioni corrispondono a campi semi- antieipati, semi-ritardati seeondo la teoria dell'~ssorbimento di Wheeler e Feynman.

(*) Traduz ione a cura della Redazione.

TACttYO:N SINGULARITY: A SI~ACELIK~E COUNTERPART ETC. 69

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