7. R. K. Tavakol and N. van den Bergh, Phys. Lett., I I2A, 23-25 (1985). 8. C. W. Misner, K. S. Torne, and J. A. Wheeler, Gravitation, Freeman, San Francisco (1973). 9. S. Watanabe and S. Ikeda, Tensor N. S., 40, 97-102 (1983).

i0. S. Watanabe and S. Ikeda, Tensor N. S., 39, 37-41 (1982). ii. C. M. Will, Theory and Experiment in Gravitational Physics, Cambridge University Press,

Cambridge (1981).

ARE BLACK HOLES BLACK?

A. A. Grib UDC 530.145

This work presents a resolution of the causality paradox formulated by T. D. Lee in the theory of the Unruh effect and in the theory of black holes. The basis of the resolution is to take into account the transformation of a pure state into a mixed state in a measurement, which leads to a corresponding modification of the Bogolyubov transformations, so that "black holes remain black."

An article with a similar title by T. D. Lee appeared recently [I], in which attention was drawn to a paradox arising in the quantum theory of black holes and in the system of a noninertial observer. The essence of the paradox is that, in applying the often used Bogolyu- bov transformations [1-3] taking into account the restructuring of the vacuum with quantiza- tion in a noninertial system (the Fulling--Unruh effect) or in the gravitational field of a black hole (the Hawking effect), one is easily led to the conclusion that "because of the glo- bal nature of the quantum state inevitably extending beyond the event horizon, the observer can obtain information from beyond the event horizon."

In fact, in quantum theory in Rindler coordinates describing an observer moving with a constant acceleration w, the vacuum of an inertial observer, as is well known [3], is per- ceived as a mixed state with a density matrix describing the temperature distribution so that kT = hw/2~c.

The existence of the event horizon for an observer moving in the right (left) Rindler an- gle leads to the introduction of the concept of "right" and "left" particles, such that the observer moving to the right has no information about the "left" particles, and conversely. The creation and annihilation operators for the right (left) particles

r(+) c-~ /+) i(~)

in terms of the creation and annihilation operators a (+) , a (-) for which are usually expressed

the Fock vacuum is the vacuum of the inertial observer in Minkowski space, according to the formula [3] :

~, r (--7 T

~ ( o , , q , # ) l exp 4.o, l ~o--[- ,r )'~ . . . . . (a (q -- q'). (1) (2~XoUl-' (exp 2~w -- 1)1, '~ z

~(m, q, # ) 1 I (~'o+~h '' . . . . . . . g ( q _ q ' ) , (2=Ko) I,':' (exp 2r.o~ -- 1)1;2 x k /

~,, = (m'-' + ~'-')~ 2, , = ( m -~ .+. q-~)~J~

If [OM> is the vacuum of the inertial observer, so that

a~7~lOM > = O. (2)

N. A. Voznesenskii Institute of Finance and Economic's, Leningrad. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 5, pp. 95-97, May, 1989. Original arti- cle submitted October 15, 1986.

0038-5697/89/3205-0401512.50 �9 Plenum Publishing corporation 401

then, in view of (i), the operators r(-), l(-) annihilate the new vacuum [ORi >, and not ]OM>. ~q ~q The observation of Lee leading to the casuality paradox (reception of information from beyond the event horizon) consists of the following. Let us consider the superposition

'<+~] I OM > (3) 0

If one assumes that [i> differs from [OM> only when something (a particle has been created) has changed to the left of an observer moving in the right angle (his particles are described by the operators r(+) r~ )) then, by measuring mq '

< 1 [rg~l 1 > + o, (4)

the observer to the right "knows" about the happening beyond the event horizon by comparing with

< OM I r ~ l oM > = o.

In this article we resolve this paradox by analogy to the well-known Einstein--Podolsky- Rosen paradox in which, as is well known, violation of causality (transmission of information at hyperlight velocity) does not occur [5].

First, it is seen in Eq. (i) that the effect of the operator Z(+) on I OM> corresponds,

after transition to the operator a~t ) , to an excitation affecting n~ only the left region,

but also the right. So it is obvious that the observer in the right angle notes a change re- lated to the transition from [OM> to Ii>, but this will not be the reception of information from beyond the event horizon.

Second, in order to answer the question as to what occurs when a change in state actu- ally takes place beyond the event horizon, one must rewrite the transformation (i) in another form explicitly taking into account the conversion of the pure state [OM > for the right ob- server to a mixed state. This transformation was derived previously in [6].

By analogy to the Einstein--Podolsky--Rosen paradox, one must argue in the following man- ner. If a system consisting of two subsystems is described by a pure state, then the subsys- tem (when the wave function of the system is not the product of the wave functions of the sub- systems) is described by the density matrix. Moreover, the observer measuring observables of the subsystem will say that the entire system, if he knows of the existence of the other sub- system, is described by a density matrix which is the product of the density matrices for the subsystems. This fact is the "complementarity between the whole and the part," well known in quantum mechanics.

So the creation and annihilation operators for right (left) particles must be written with the Bogolyubov transformation in terms of the annihilation and creation operators for which the corresponding density matrix, and not the vacuum [OM> , plays the role of the Fock vacuum,

Thus, we must write:

-'-~ , (5) r~ =~d~ ' [~* (~ , q, # l a ~ l | 1 7 4 q , #)a~r

7~ ' = ; a ~ ' [** ( - ~, q, ~ ' ) a ~ ' | 1 - 1 | ~ (o,, - q, ~') &*~l.

The operators a~ ) | I, 1 | a~ > have the vacuum (cyclic vector) 0>'@ lO >,

a i : ) | 1 . 1 0 > | = * | | = o. This vacuum corresponds to the previously mentioned temperature density matrix. A struc-

ture of the form (5) -- the "reducible representation of the commutation relations" --was first described in the theory of the Bose gas in [4].

We now obtain zero in calculating the quantity <l]r~)Ii>~- where

> = [Co+ a qc, > | 0

By analogy t o (5), the Bogolyubov transformations must also be written for the Hawking radiation of black holes. Correlations in the observations of observers separated by the

402

event horizon, as in the case of the Einstein--Podolsky-Rosen paradox, cannot be observed by one observer. It is necessary that they "meet" and compare the results of their individual observations.

The results of calculations unrelated to correlations between the observations of observ- ers separated by the event horizon are the same, both with (I) as well as with (5), i.e., the use of (i) does not then lead to any error, but

The quantity

i(-)~(-) "~ OM[ ~,~q --.u--q [ OM ~" =/= O.

~(-)-(--) ,

< O[ | < Olt,,,q r~_q lU > | 0 > = O,

which indicates the inobservability of correlations from the point of view of the subsystem, and so there are no contradictions with causality -- black holes remain black.

The author thanks N. Sh. Urusova for discussion of the work.

LITERATURE CITED

i. T. D. Lee, Nucl. Phys. B, ~64, 437 (1986). 2. W. Unruh, Phys. Rev. D, 14, 870 (1976). 3. N. Birrel and P. Davis, Quantized Fields in Curved Space--Time [Russian translation], Mir,

Moscow (1984), p. 619. 4. H. Araki and G. Woods, J. Math. Phys., ~, 637 (1963). 5. A. A. Grib, Usp. Fiz. Nauk, 142, 619 (1984). 6. A. A. Grib and N. Sh. Urusova, in: Abstracts, 10th International Conference on General Rel-

ativity of Gravitation, Vol. 2 (1983), p. 1081.

ROTATING COSMOLOGICAL MODELS OF BIANCHI TYPE Vlll

V. F. Panov UDC 530.12:531.51

We find cosmological solutions with rotation of Bianchi type VIII for the energy-mo- mentum tensor of a perfect fluid with heat flow.

i. Introduction

In recent years interest has intensified in the theoretical investigation and experimental detection of the possible rotation of the Universe ([1-14], etc~ In this connection, with regard to observational properties of the Metagalaxy (spatial homogeneity, its expansion, and directional preference associated with rotation) we can assume that the geometric structure of the Universe is described by a more complicated metric than the stationary Godel metric.

In [ll] some rotating time-dependent cosmological models of Bianchi type VIII with heat flow are obtained. In [ii] the coordinates (t, x ~, x 2, x s) are introduced such that the time lines are world lines of matter. In this connection the metric of the model is considered in the form

~s ~ = ( a t + A ~ ) ~ - - (B~') ~ - - C 2 ( ( ~ y + (~)~) , (1)

where A, B, and C a r e f u n c t i o n s o f t , and ~ ~= and ~3 , , are 1-forms of the following type:

~ = dx~ § ll + (x ' )=ldx2 + [ x ~ - xa -- (x')2x=l dg a, m = = 2 x ' d x 2 + (1 - - 2 x ' x 2) d x 3, (2)

~ = d x ~ + [ - - 1 + (x ' ) ~] ax~ + I x ~ + x~ - ( x f F x q a x ~.

In the Lorentz tetrad the components of the Ricci tensor are

Perm State University. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 5, pp. 98-103, May, 1989. Original article submitted December 24, 1986.

0038-5697/89/3205-0403512.50 �9 1989 Plenum Publishing Corporation 403