White holes in Kaluza-Klein theory: Windows from higher dimensions

Volume 241, number 3

PHYSICS LETTERS B

17 May 1990

W H I T E H O L E S IN K A L U Z A - K L E I N THEORY: WINDOWS FROM HIGHER D I M E N S I O N S V.S. G U R I N and A.P. T R O F I M E N K O Astronomical Section of the Minsk Department of the Astronomical-Geodesical Society of the USSR, Abonent Box 7, SU-220 012 Minsk 12, USSR

Received 8 October 1988; revised manuscript received 6 February 1990

We consider a white hole model within the framework of five-dimensional Kaluza-Klein theory where the additional dimension may be revealed in connection with effects of the event horizon. Such a white hole appears as a window from higher dimensions.

The discussion of gravitation theories (united with other fields) on the background of a space-time with dimensionality higher than four was rather unusual 15-20 years ago, though a number of publications were devoted to the topic. The considerable interest shown by physicists and cosmologists in many-dimensional theories in recent years is connected, mainly, with the hope o f building a unified theory of physical interactions, in particular, on the basis ofsupergravity and superstrings. The most popular theoretical schemes are associated with the development of Kaluza and Klein's ideas [1] on additional compactified dimensions, which are revealed on the microlevel and in the early evolution of the Universe (Metagalaxy) (see, e.g., the last surveys [2,3] ). Among other approaches to the problem of the introduction of higher (additional) dimensions within the framework of general relativity and its generalizations we would like to note the concept of complex space-time under the united consideration of different regions in extended manifolds with event horizons and subluminal and superluminal phenomena [ 4-7 ]. The results o f the present work can be immediately generalised to a complex manifold. It is important for the development of higher-dimensional theories to study different objects which are well investigated within the framework of conventional general relativity. We shall touch on relativistic objects with event horizons, the so called "otons" (from the Russian abbreviation of"general 328

theory of relativity" - " O T O " [8,9] ). Black holes (BHs) are the one representative of such objects inherent to Einstein's theory. There are a number of papers [ 10-14 ] devoted to the analysis of the geometry and physical properties of BHs in Kaluza-Klein theories. It is of interest to note in this context that the presence of horizon-like peculiarities in BH space-times leads to the divergence of the metric coefficient of the additional coordinate at the horizon, and thus these dimensions become "visible" near BHs. In the present work we shall consider objects which are antipodes of BHs - white holes (WHs), i.e. BHs reversed in time [8,9,15,16]. In contrast with BHs called "windows" to higher dimensions" [ 17 ], WHs can be called, as will be shown below, "windows from higher dimensions". In conventional general relativity BHs and WHs appear naturally in the complete space-time manifolds with horizons (e.g., the K e r r - N e w m a n manifold and particular cases). Although the problem of W H stability is a special question, it will be not considered here. The connection between different regions of the global complete manifolds (universes) occurs in the process of the collapse-anticollapse of a relativistic gravitating object. In light o f the manydimensional interpretation, such a process is the connection between different dimensions of global spacetime. We shall generalize the WH model known in the four-dimensional theory [ 15,16 ] ot the five-dimensional Kaluza-Klein version. The further gener-

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alisation to higher dimensions can be easily carried out if the additional dimensions are represented as S"-s here (the most frequent admission). The model of a non-rotating uncharged WH is built using the technique of matching the anticollapsing matter metric with the exterior space-time. In the isotropic form this exterior model solution is described in the following way: ds2= - A 2dT2+B2dXidXi-k C 2 d y 2 ,

(

1)

where

( a r - l ~ k' A=A(r)= \a~ J '

(2)

(ar+ 1 )~
(3)

(ar+ l ~ ~ C=C(r)=Co \ a r - 1J '

(4)

a, e, and k are constants, connected by the correlation e2(k2-k+ 1)= 1 .

17 May 1990

C=Co(ar+ l ) / ( a r - 1 ) =Co(r/rg+ l ) / ( r / r g - 1 ) ,

(6)

where the gravitational radius in isotropic coordinates is rg=a -~. Transforming r to l: l= ( 1 +ar)2r we get the usual form of the Schwarzschild metric coefficient

C=(1-2m/l)

-~,

rg=2M,

(7)

and we omit Co without loss of generality. Thus, the metric coefficient of the additional dimension coincides with that of the radial coordinate, but the Y coordinate itself does not appear in it, and its singularity at rg provides the singularity of this additional dimension and the fact that it begins at the horizon. As the interior metric is similar to those of the canonical WH model [15], we use the FriedmannRobertson-Walker metric, and its five-dimensional generalization [21,22] of the form

(5)

It is the general form of the Davidson and Owen solution [17] with scalar change. We shall not discuss here the physical realisation of this solution, which is rather debatable [18,19]. We choose the particular case when all the metric coefficients of ( 1 ) have some peculiar behaviour at r = 1/a, i.e. at r=rg in the curvature coordinates (see below). Moreover, the solution given by ( 1 ) - ( 5 ) can be immediately generalized to the case of non-zero electrical charge, which is more realistic for the description of higherdimensional BHs [ 18 ]. The physical significance of such a construction (introduced ad hoc) with additional dimensions is seen more clearly from the consideration of the theory of extended manifolds with horizons in higher dimensions [20] when the additional dimensions begin at horizons. There is no need to consider this solution as a Schwarzschild analogue (besides of the formal one), which appears by omitting the new coordinate Y. The latter is not a perturbation of the fourdimensional case and changes the global structure of space-time rather significantly. We shall restrict ourselves to the case when e = 1 k = 1 in ( 1 ) - (5), this is within the range of possible values of ~ in ref. [ 17 ]: l el ~<2/x/~. Then the metric coefficient of the additional dimension

ds2= - d t 2 + R2( t ) ( 1 + ~qx 2) -2dxidx'

+b(t)Zdy z .

(8)

In this geometry the following solution of the Einstein equations is obtained for dust-like matter [21] a t A = 0 :

R2(t)=-qt2+ct+d, b(t)=dR/dt=½(c-2q)(-qtZ+ct+d)

(9)

-'/2,

(10)

where c and d are constants determined by the initial conditions. Evidently, the matching of the four-dimensional part in this case does not differ at all from the usual procedure, which can be performed quite well for the Schwarzschild and Friedmann space-times at the boundary of expanding (contracting) matter: l=lb, =xbS(t), O<~S(t)~< 1. The matching of the fifth dimension can be carried out according to the following correlation:

(ar+ 1 ) ( a r - 1 ) - ' d Y = b ( t ) d y ,

( 11 )

which leads to the connection of the fifth coordinates Y and y:

dY

(c-2q)(ar-1)

dy - 2 ( - q t 2 + c t + d ) ~ / Z ( a r + 1) '

(12) 329

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3.=0

or, after the transformation to the Schwarzschild-like coordinates with l: dY ( c - 2 q ) ( 1 - 2 M / l ) '/2 dy 2 ( - q t 2 + c t + d ) ~/2

(13)

This formula shows that Y does not depend on y at l=2M. Let us analyse now the global space-time structure of such a WH model by means of Penrose conformal diagrams. Since for this investigation a two-dimensional metric is useful (it is rather difficult and unclear in the case of four or more dimensions), when every point on the diagram corresponds to S2, we omit the spatial dimensions of the usual four-dimensional sector and take into account only the temporal and fifth coordinates. This permits us to analyse the causal structure with the additional dimension introduced above, though the metric coefficient of the spatial coordinate is also singular at the horizon. Thus, the initial metric is ds2= - ( 1 - 2 M / l ) d T 2 + ( 1 - 2 M / l ) -ldy2,

(14)

where l is the spatial coordinate of the four-dimensional sector. The outlook of this metric almost coincides with the Schwarzschild one, differing only by the independence of the metric coefficients of Y; l appears here as the parameter. Using the standard transformations u=T-Y*,

v=T+Y*,

(15)

dY*= ( 1 - 2 M / l ) - ~ d Y ,

(16)

leads to u=½[T-2Y/(1-2m/l)]

,

v=½[T+ 2 Y / ( 1 - 2 m / l )

],

(17)

and we get a Kruskal-like space-time with the metric ds 2= - f d u dv.

(18)

Further, the Penrose diagram can be constructed in the usual way [ 23 ], and its outlook coincides with that for the Schwarzschild space-time besides the sense of null lines (fig. 1 ). The surface of the horizon l=2M, as can be concluded, is the time limit ( T = +_~) of usual four-dimensional space-time, since at l= 2M, Y= 0, and the metric (14) will have degenerated into the Schwarzschild one, taking into account also the radial and temporal parts. 330

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l=oo

l=oo

l=0

/5/

Fig. 1. The Penrose diagram for the two-dimensional sector of Kaluza-Klein white hole space-time.

Thus, the event horizon in extended manifolds is the place of the beginning of additional coordinates, and anticollapsing objects (WH), to which the motion along time-like geodesics corresponds (in the usual four-dimensional sense), coming from the region of the variation of the additional Y coordinate. Before concluding, we would like to point out that transitions between different classes of objects (subluminal and superluminal) [24] can be interpreted as a motion in different dimensions (but this is not obligatory in the Kaluza-Klein spirit). We are grateful to Professor G.M. Idlis and Professor E. Recami for discussions and correspondence and to the referee for fruitful remarks.

References [ 1 ] T. Kaluza, Sitzungsber. Preuss. Akad. Wiss. Phys. Math. KI (1921) 966; O. Klein, Z. Phys. 37 (1926) 895. [ 2 ] D. Bailin and A. Love, Rep. Prog. Phys. 50 (1987) 1087. [ 3 ] V.M. Emel'yanov et al., Phys. Rep. 143 ( 1986 ) 1. [4 ] V.S. Gurin and A.P. Trofimenko, Fizika (Yugoslavia) 17 (1985) 101; Rev. Roum. Phys. 31 (1986) 353. [5] E. Recami, Riv. Nuovo Cimento 9 (1986) 1. [ 6 ] M. Pav~i~, J. Phys. A 14 ( 1981 ) 3217; Lett. Nuovo Cimento 30(1981) 111. [7]H. Chandola and B.S. Rajput, J. Math. Phys. 26 (1985) 208; Lett. Nuovo Cimento 40 (1984) 277; Indian J. Pure Appl. Phys. 22 (1984) 1; H.C. Chandola et al., Indian J. Pure Appl. Phys. 24 ( 1986 ) 51. [8]Y.B. Zeldovich and I.D. Novikov, Stars and relativity (Chicago U.P., Chicago, IL, 1971 ). [ 9 ] A.P. Trofimenko and V.S. Gurin, Astrophys. Space Sci. 152 (1989) 105.

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[ 10] G.W. Gibbons, Nucl. Phys. B 207 (1982) 337; G.W. Gibbons and D.L. Wiltshire, Ann. Phys. (NY) 167 (1986) 201. [ 11 ] P.O. Mazur and L. Bombelli, J. Math. Phys. 28 ( 1987 ) 406. [12] R.C. Myers, Phys. Rev. D 35 ( 1987 ) 455; R.C. Myers and N.J. Perry, Ann. Phys. (NY) 172 (1986) 309. [13]V.P. Frolov, A.I. Zelnikov and V. Bleyer, Ann. Phys. (Leipzig) 44 (1987) 371. [ 14] Xu Dianyan, Class. Quantum Grav. 5 (1988) 871. [ 15] J.V. Narlikar and K.M.V. Apparao, Astrophys. Space Sic. 35 (1975) 321. [ 16 ] K. Lake and R.C. Roeder, Astrophys. J. 226 ( 1978 ) 37.

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[ 17] A. Davidson and D.A. Owen, Phys. Lett. B 155 (1985) 247. [18] T. Dereli, Phys. Lett. B 161 (1985) 307. [19] L. Sokolowski and B. Carr, Phys. Lett. B 176 (1986) 334. [20] V.S. Gurin and A.P. Trofimenko, Rev. Roum. Phys. 33 (1988) 533; Hadronic J. (1989), in press. [21 ] A. Davidson, J. Sonnenschein and A.H. Vozmediano, Phys. Rev. D 32 (1985) 1330. [22] M. Gleiser, S. Rajpoot and J.G. Taylor, Ann. Phys. (NY) 160 (1985) 299. [23] S.W. Hawking and G.F.R. Ellis, The large scale structure of space-time (Cambridge U.P., Cambridge, 1973 ). [ 24 ] A. P. Trofimenko and V.S. Gurin, Gen. Rel. Grav. 18 (1986) 58; Pramana 28 (1987) 379.

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