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Prami~a, Vol. 8, No. 2, 1977, pp. 133-143. @ Printed in h~dia.

Taehyon emission from white-holes

S V D H U R A N D H A R

Tata Institute of Fundamental Research, Bombay 400005

MS received 25 Jtme 1976

Abstract. Investigations are made about the motion of a radially outward propa- gating tachyon which is created in the singularity with the white-hole. The problem of confinement or escape of such a tachyon from a white-hole is discussed. It is shown that the confinement or escape of the tachyen depends ou the maximum radius of the white-hole and also on a parameter k (defined in the text) associated with the momentum of the tachyon. Also it is shown that when a tachyon escapes it always escapes before the white-ttole has expanded to half its Schwarzschild radius.

Keywords. General relativity; tachyons; white-holes.

1. |ntroduction

Recently considerable amount o f research has been directed towards the in'~olve- ment o f taehyons in astrophysical and cosmological phenomena (Narlikar and Sudarshan 1976, Narlikar and Dhurandhar 1976, Davies 1975, Raychaudhari 1974, Honig e t a l 1975). Experimentally the attempts to produce or detect, tachyons have till now yielaed null results. But as far as production is concerned o n e may look to high-energy astrophysics where phenomena are found to take place on a much grander scale, than can e'~er be achievea in terrestrial settings, One such large-scale phenomenon is the big-bang. Narlikal and Sudharshan (1976) have already discussed the behaviour o f a primorctial tachyon in 1he big- bang universe, whose sole interaction with the sunounding matter was gJa~ita- tion. Such tachyons are shown to encounter a time-barrier, and the epoch o f the time-battier depends o n the initial enelgy o f the tachyon and ~Iso on the Friedmann model considered.

I n this paper the propagation of a tachyon inside a white-hole is discussed.

The geometry inside a homogeneous dust type o f white-hole is the same as that o f a big-bang Friodmann model, So to some extent we expect the situation to be similar to the above.memioned problem discussed by Narlikar and Sudarshan (!976). There is, however, one essential difference. Here, we are also concerned with the pioblem o f confinement or the escape o f a tachyon from a white-hole.

Sach a problem did not arise, when a tachyon in the expanding uni~'erse was considered.

133

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The role o f white-holes in high energy astrophysics has been discussed by Narlikar and Apparao (1975). Exploding galactic nuclei, transient x-rays, gamma- ray bursts are some of the examples of likely white-hole phenomena. There have also been arguments (Eardley 1974, Zeldovich

et al

1975) to show that white- holes cannot exist for a long enough time to be physically relevant. These objec- tions have been successfully countered by Lake and Rocder (1976). We will not enter here into the discussion o f these arguments or of the other implications o f white-holes for astrophysics. There is already sufficient observational evidence for exploding objects in astrophysics (Hoyle 1975), apart from the varying degrees of faith among different astronomers, for the big-bang origin of the universe.

The, model of the white-hole considered here may be regm ded as a simplified version o f such exploding phenomena.

2. Geometry in the interior o f a white-hole

We shall consider the homogeneous dust model of a white-hole, that is, a spherical object with uniform density and zero pressure. The object emerges fi'om a singular state and subsequently obeys Einstein's field equations

R,~ ~ ½ R g ~ = - - 87r 7~. (1)

We have chosen units in which C ---- 1, G = 1. We shall consider the white-hole ia tile co-moving frame o t reference of outward moving particles. In this frame the interior of the white-hole has the line-element,

ds z--dr z - S g(t)[ dr2-

kl ~ ar 2 + r'z (dO" + sin e 0 c/~ z) ,

]

r < rb (2) where

(r, O, ~)

are the constant co-ordinates of a co-moving pal ticle and t the proper time o f a co-moving observer, r = r~ is the coordinate of a particle on the boundary of the white-hole. S ( t ) is the expansion fhctor

,,nd

it satisfies the

differential equation

[dS,"

~ ( 1 - S)

t, ) 7 ) = S (3)

where,

2m 8~po

m ~ - ~ p o rb 3 4~

oo is the lowest density attain{d by the white-hole when the expansion Iactor S = 1. m is the mass of the white-hole. It may be remarked that the line element (2) resembles the big-bang Friedmann line-elements for the closed universe.

Outside the wlfite-hole the metric is Schwarzschild. The radial Schwarzschild co-ordinate is

rS(t),

inside the white-hole.

3. Tachyon propagation

We assume the tachyon to be created when the white-hole is in the singular state given by S---0 and its motion being directed radially outward. The tachyon

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T a c h y o n e m i s s i o n f r o m w h i t e - h o l e s 135 interacts with the surrounding matter of the white-hole through the relativistic law of gravitation, but is otherwise free from any other interaction.

Under the above assumptions the tachyon motion is along a space-like geodesic starting at r ---- 0 and t = 0. Initially the tachyon momentum is radial, directed along (, = 0 . , q~ =~b0, say. The integration of the 0, ~ geodesic eqlmtions lead to the result, that the tachyon motion continues to be radial and given by 0 = 0 o,

~ - - + 0 .

For a tachyon trajectory we have ds ~ < 0, so that ds is imaginary. We define a real affine parameter ~ by the relation,

d , , 2 = - - , i s 2. (4)

For a radially moving tachyon we have,

S ~ ( 0 a r~ a t - (5)

de2 = - - - ~ " ~ 1 ~ ar "

The geodesic equations corresponding to the radial motion given by (5), w~en integrated, result in the relation,

S ~ dr

xl(1 __~r-r2 ) ~ = k (a constant) (6)

where k is a real constant. Using (5) and (6) we get,

( d t ~ k 2 - - S ~

d ~ ] - - S 2 (7)

The interpretation of the constant k may be sought by considering the 3-velocity o f the tachyon in the rest-frame of an outward moving particle o f the white-hole, which coincides instantaneously with the tachyon at (r, t ) . Th~s velocity v (t) is given by

S ( t ) dr

v (t) -- "V'(1 - - a r ~) d t " (8)

Using (6), (7) and (8) we get, S (t) v (t)

k = V,[v 2 (t) - - II " (9)

The momentum per unit meta-mass ol the tachyon may be definea by the relation

v ( 0

P (t) = V,[v ~ ( t ) - - 1]" (1O)

From (9) and (I0),

k = S ( 0 P (t) (! 1)

(11) clearly gives the physical interpretation of the constanl k, 4. Tachyon tra|eetories

It is cmwenient to investigate the radical trajectory of a tachyon in terms o f the expansion factor S instead of t. Defining a parameter ~ by S = s i n 2 " q and then integrating (3) we get,

t = ~-~ ('1/-- cos ~ sin ~). 1 (12)

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In the expansion phase of the white-hole, as Sincreases f r o m 0 to 1, ,/ ranges from 0 to ~r/2. In this range o f 71, it is seen that S is a monotonic furaztion of t, so that our choice o f S instead o f t as the independent variable is not unjustified. (3), (6) a n d (7) give the differential equation o f the trajectory in terms o f r and S as,

(1 - - S )

dr

I = k 2 (1 - - ar 2) (l-Z)

:l: 3/a dr kdS

- - = ( l - - s ) ( k s - s 2 ) ] " ( 1 4 )

It is convenient to define a new variable ~ by the relation,

R = sin -1 v ' a r. (1~)

The values over which R ranges are determinea by the values over which r ranges.

Since 0 ~ r < r t , we have

O ~ R ~ R b

where,

R~ = sin -1 V'a-r~. ([6)

In view o f a =

2mitt ~,

(16) becomes

R, = sin -1 4 2 " ~ m (17)

We shall considel only those white-holes for which rb > 2m, that is, those white- holes whose boundary crosses their Schwarzschild radius. ( 1 7 ) immediately implies that R, < ~r/2. Henceforth we shall consider Rb : ,-r/2 as the upper limit of the permitted range of R,.

It can be remarked here that in (13), r is the co-moving rauial coordinate of the tachyon, and hence the equation is valid physically only when the taehyon is inside the white-hole. In the region exterior to the white-hole, the geometry is aifferent and consequently the tachyon would obey a different equation. It is meaningful to consider (13) as describing the trajectory of the tachyon only for r _< rb. The condition r ~< rb, in our newly defined coordinate is equivalent to R ~ R,. Equation (14) with t;le help of (15) becomes,

- v ' [ S ( l - - (18)

Solving (18) we shall have R as a function of S. Initially, when the white-hole is i,n its singular state S = 0, we shall have R = 0. With the aid of (18~ and the abov,: initial condition, we can plot the trajectory of tee tachyon in the R-S plane. However, since the maximum value attained by R~ is ,r/2 and R <_ Rt, w¢

shall consider only the part of the plane described by 0 _< R _<< ~r,'2.

5. Equations of trajectories

For a physically acceptable solution one would require

dR/dS

in (18) to be non- negative in the neighbourhood of R = 0, S = 0. Since we have the choice of sign in (18) we can choose the positive sign and hence restrict k to non-negative values.

We note that

dS/dR

= 0 at S--- 0 and at S = k (the other roots of

dS[dR

are irrelevant) and

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Tachyon emission .from white.holes

d,S d?#l =

dRe e-o =

½' dRe ~e'*

-- (! + k).

The trajectory reacheb a maximum at S ~ k, when R : - R , say.

gration of (18) two cases arise according as 1. k < l

2. k ~ l For R-~/~

f x ( S , k), k < 1 R (S, k) = fa (S, k), k ~ 1 wh~r¢~

S

f

A (s, k) = V [ s (l-s) (k' -

s g ]

0 S

A (s, k) = f

o

R , , = {

A ( k , k ) ,

k < l A ( 1 , k), k ~ 1 For R / > / ~ ,

O ~ S ~<k

kdS 0 ~ S < I

V[S (t

-

s) (k ~

-

sol

137

In the into-

09)

(20)

R (S, k) --- { 2 R . -- A (S, k), k < 1 (21) 2R. - - A (s, k), k >_ 1

(19) and (21) comprise the equations of the trajectory. A typical trajectory is shown in figure 1, corresponding to k = 0" 086.

(19) and (21) show that dS/dR > 0 for R < / ~ and dS/dR < 0 for R > R.,.

Since S is a monotonic function of time, the trajectory of 2~ < R . ~epIesents a tachyon moving forward in time, while for 2~ > / ~ , the tachyon mo~:,s backward in time, which may be interpreted as an antitachyon moving forward in time (see Sttdarshan 1970). Both the ~achyon and ,the antiltachyem am'hil~t, each o f f e r at R = R m.

One may remark at this sta.ge that the trajectory is mathematically defined for

~) < R ~ 2Rm. But if R~ < 2 / ~ , then the physically relevant portion o f the trajectory would be that for which its R-coordinate is less than or equal to R,.

6. Main problem

We ask the following question: Under what circumstances does the tachyon escape from the white-hole ? The question is oquivalcnt to the choice between the two conditions:

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0 . S t - s

0.3

0.2

0.1 0 - 0 8 0 - 0 7

k =O-Oe,6

~ l I I I I I - " - - ~ I

0.1 0,2 0.3 0.4 71"/4 1-0 1.5 rr/2

Figure 1. Tachyon trajectories with various values of k are shown. In particular the trajectories with k = k~= 0.306 and k = k2= 0.086 are also shown.

In case (a) the trajectory of file tachyon does n o t cross the bour, dmy coordinate R = Rb which implies that the tachyon is c o n f i n e d to the white-hole. Case (b) represents the trajectory o f a tachyon, which crosses the b o u n d a r y coordinate

R = R~, which means that the tachyon escapes. T h e two situations are shown in

figure 2.

In case (b) when the tachyon escapes two cases arise according as, (i) R ~ < Rb < 2R.,

(ii) Rb < R,..

The two cases are. shown in figure 3.

In case (i) the trajectory bends back in time near R ~ R~, so that one can inter- pret this situation as the annihilation o f the tachyon and the antitachyon taking place inside the white-hole, with the antitachyon originating outside the white- hole. Or, one can say that the tachyon escapes f r o m the white-hole, whil~ moving

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0"06 0.05 0'04 0"03 0"02 O'O1 $J 0'06 0"0! 0"0, 0"02 0'0~ 0"01

0.1 0.2 0.3 0"4 0.5

k -'06

"X

Rm 2 R m R b W'/2

(a)

)- R

j•,

, I 1 -~ C1 L:,.;" 03 0.4 0.5 R m

k - -06 Ri} 1"0 2R m

(b] "rr/2 R Figure 2. The two cases (a) 2Rm < Rb and (b) 2P~, > R~ are sbo~n sepa- rately. In the first case the tachyon is confirmed, while in the second case the tachyon escapes.

0.06

o.os

0.0, 0-0: O.O2 0.01

W .06 Rb 2R m 0.1 0.2 0,3 0.4 0-5 R m Wl2 ) R (1i) 0-06 0 O5 0.04~ ~- 0"03t-

/ °-° T /

L-<~____L.__.L_ ._..J~ :. ['9.1 0'20-3R O'40,b R m ZR m 1£/2 I~" Figure 3. The two cases (i) R= < R~ < 2R m and (ii) R~ < R,~ are shown. In case(i)the tachyon escapes moving backward in time, while case (ii) represents a tachyon'which escapes moving fo~waTd in time•

k - .06

? r~ kO

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backward in time. In case (ii) the tachyon always moves forward in time for R < Rb, reaches R =

Rb

and then escapes. There is no annihilation in this ca~e.

From the-foregoing one can make the following statements:

(i) If R~ > rr/4, then the tachyon surely escapes either going back in time or going forward in time.

(ii) If P~ > ~-/2, the tachyon surely escapes moving forward in time.

(iii) If

Rb

> R,, > rr/4, the tachyon surely escapes moving forward in time.

In the light of the above statements it is necessary to investigate the behaviour of R~ as a function of k. (20) gives the required relations in terms o f elliptic functions. For k < 1,

R,, (k) =

f

( l - - S ) (kS _ S2)] k a s

0

For k > 1,

(22)

2

--k --2

V [ S (1-- S) (k = - - S")]

e

(23) Th~ plot of R , (k) vs. k is shown in figure 4. As k increases from 0 to 1, P,. (k) inoreas~ monotonically from 0 to oo; further increase in k makes R,, decrease monotonically from oo until it asymptotically tends to ft.

In view of statement (ii) the case for k >_ 1 becomes exceedingly simple, tmcause then R ~ > rr > ~r/2 and the tachyon sarely escapes mo~ing fcrwmd in time, what- ever be the value'of R~ for the white-hole. The case of interest arises when k < 1.

":[

4

2

0"5 I 1-0 , 1.5 ! 210- k L

Fisure 4. Curve showing the dependence of R~, on k. For large values of k, R,. (k) asymptotically tends to ~.

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Tachyon emission from white-holes

141 When 0 < k ~ 1, Rm ( k ) i s m o n o t o n i c increasing; we can use this fact to translate the three statements pertaining to R,,, into equivalent statements regarding k. Define kl and k2 by,

Rm (kO = rr/4

a n d

kb

by,

Now we (ii) and

F r o m

We can

(24)

R. (/,3

= •

(25)

can replace

rr/4, ~/2, Rb, R,,,

by

kl, k2, kb, k

respectively in statements (i) (iii) and obtain equivalent statements.

(22) it is possible to evaluate kx and k2, lq = 0" 086

k2 = 0" 306.

get an u p p e r - b o u n d o n

kb

using the fact that R~ < zr/2

t h a t is,

kb < 0"306.

I f a taohyon has

k > kb,

the tachyon escapes moving f o r w a r d in time.

list in table 1 a few values o f k~ corresponding to various values o f R~.

7. Epoch o f escape o f a tachyon

T h e next problem o f c o n c e r n to us is summarised by the following

We shall

queslion:

Assuming t h a t a t a c h y o n does escape, when does it escape ? In what state o f expansion o f the white-hole does the tachyon e~cape ? T h e p r o b l e m is to find the point o f intersection o f the t a c h y o n trajectory R = R (S, k) a n d the line R =- Rb a n d to observe h o w this intersection point behaves for different k's or R~'s. T h e S-coordinate o f this point o f intersection will determine the phase o f expztt, ion o f the white-hole, when the t a c h y o n escapes. To this end, ~'e first plopo~,e to

Table 1.

SI. No. Rb 2-m c°sec2 Rb

1. 15 ~ 14.93 0-01

2. 20 ° 4.00 0. b4

3. 45 ° 2.bO 0.086

~. 60 ° 1-33 0-148

5. 75 ~ 1.072 0. 222

6. 90 ° 1.00 0.306

P - - 6

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find a n upper-limit o n this S-coordinate o f the point o f intersection. Only the p o r t i o n o f the trajectories for which R ~___ Rm will be needed. We have,

S

f

R (S, k) : - ~/[x ( l - - x ) (k 2 - - x2)] ' 0 < S _< min (1, k)

o

where m i n (1, k) denotes the m i n i m u m o f the numbers 1 a n d k.

N o w

k / v ' k 2 - - x ~

is a decreasing f u n c t i o n o f k for each x, hence R (S, k) is a decreasing function o f k for a fixed S. In figure 1 the Uajectoiies are plotted for different k's. As k increases the trajectories move ' higher u p ' in the R-S plane. Finally the trajectory for which k - + c~ will have the least R fcr a given S or equivalently a greatest S for a given R, when c o m p a r e d wflh oll~er trajectories with finite k. So i f we consider the extreme case, that is, a white- hole with R~ :--7r/2 a n d a t a c h y o n with k--> c% the t a c h y o n will escatx in the greatest e x p a n s i o n phase of the white-hole, as c o m p a r e d with other wbite-l~ oles or

with t a c h y o n s with finite k. F o r k -+ c~, we lzave,

lim R (S, k) =- R0, (S) = 2 sin -1 ~/S. (26)

k - - ~ o o

F r o m (26) we have, R.o ( s = 0 . 5 ) = ,,/2.

Hence in the extreme case tl=e t a c h y o n escapes wl-cn S = 0"5.

On this basis, one can now make a general ~lat~m~nl, that wl-cn a ltxl)3cn does escape, it always escapes before the whitc-1;ole r¢zchcs h a l f its u l l i m ~ e lin¢z,r size.

Incidently it m a y be remarked t h a t the t a c h y o n trajectory with k -+ c~ corres- ponds to the trajectory o f a p h o t o n .

It is seen t h a t a tachyon having R,, = R~ escapes at an epoch

to

given by S (to) -~ k. F r o m (11), it is n o t e d that such a t a c h y o n has

P (to)

: 1, that is, in the f r a m e o f reference of the surface o f the white-hole the tachyon has m o m e n t u m unity or zero energy. In general the m o m e n t u m o f escape o f the tact.yon at an epoch to is given by (11) to be P ( t o ) =

k/S(to).

Silzce S ( t o ) < 0 " 5 ,

P(to)--->cx~

as k--> c~. In the particular case o f

Rb

= rr/2, we have

S(to)

: 0"306, a n d in the limit k >~ 1, P (to) ' ~ 2k.

A n o t h e r aspect one can investigate is the radial distance at which the tachyon escapes f r o m the white-hole. We intend to get an upper limit on this radial distance o f escape of a tachyon.

F o r a given rb, the maximum radial Schwarzschild coordinate o f escape rb S (to) will b e o b t a i n e d by considering the trajectory k ---> c~, as is easily seen f r o m figure 1 T o this e n d we solve the equation R,o (S) = R~ tor

rb,

that is

2 sin -1 v/So = sin -1 ~ (27)

where

So = S (to).

(11)

Tachyon emission f r o m white-holes after simplification o f (27) one gets,

143

m (28)

rb So - - 2 ( 1 - - So)

Since So is t h e e p o c h o f escape, So ~ O. 5, (27) immediately gives the relation,

rb So < m. (29)

Hence if a t a c h y o n escapes, it always escapes before the white-hole h a s e x p a n d e d to h a l f its Schwarzschild radius.

In the limit rb---> c~, a n d k sufficiently large, we shall h a v e f r o m (28), So--->-0 in which case, R (S, k) ,-~ 2 sin -1 ~/S. O n e can apply co.. (28) in this case, which in the limit So---> 0 yields,

r~ So = m/2. (30)

Hence it is seen t h a t the t a c h y o n always escapes, when the white-hole is entirely inside the Schwarzschild radius. After the escape o f a t a c h y o n its m o t i o n is governed by Schwarzschild geometry. Such a m o t i o n h a s already been discussed b y N a r l i k a r a n d D h u r a n d h a r (1976).

8. Conclusion

F r o m the foregoing o n e can conclude t h a t n o t all t a c h y o n s escape f r o m white- holes; s o m e a r e confined, b u t generally the energetic ones escape. H e n c e o n e

m a y l o o k for t a c h y o n s which have velocities near t h e speed o f light.

Acknowledgement

T h e guidance o f J V N a r l i k a r is gratefully acknowledged.

References

Davies P C W 1975 Nuovo Cimento !!25 571 Eardley D M 1974 Phys. Rev. Lett. 33 442

Hawking S W and Ellis G F g 1973 The Large Scale Structure o f Space-time I-Io~ig E, Lake K and Roeder R C 1974 Phys. Rev. DI0 3155

Hoyle F 1975 Astronomy and Cosmology (W. I-L Freeman: San Francisco) Lake K and Roeder R C 1976 Lett. Nuovo Cimento 16 N1

Narlikar J V and Apparao IC M V 1976 Astrophys. Space Sci. 35 321 Narlikar J V and Dhurandhar S V 1976 Pramana 6 388

Narlikar J V and Sudarshan E C G 1976 Mon. Not. Roy. Astron. Soe. 175 110 Raychaudhari A K 1974 J. Math. Phys. 15 856

Sudarshan E C G 1970 Symp. on Theor. Phys. and Maths.

Weinberg S 1972 Gravitation and Cosmology (John Wiley).

7eldovich, Ya B, Novikov I D and Starobinskii A A December 1974 Soy. Phys. JETP 39 6

References

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