Duality between static spherically or hyperbolically symmetric solutions and cosmological solutions in scalar-tensor gravity
Alexander Yu. Kamenshchik, Ekaterina O. Pozdeeva, Alexei A. Starobinsky, Alessandro Tronconi, Tereza Vardanyan, Giovanni Venturi, Sergey Yu. Vernov
aa r X i v : . [ g r- q c ] D ec Duality between static spherically or hyperbolically symmetric solutions andcosmological solutions in scalar-tensor gravity
Alexander Yu. Kamenshchik ∗ Dipartimento di Fisica e Astronomia, Universit`a di Bolognaand INFN, Via Irnerio 46, 40126 Bologna, Italy,L.D. Landau Institute for Theoretical Physics of the Russian Academy of Sciences,Kosygin str. 2, 119334 Moscow, Russia
Ekaterina O. Pozdeeva † Skobeltsyn Institute of Nuclear Physics,Lomonosov Moscow State University,Leninskie Gory 1, 119991 Moscow, Russia
Alexei A. Starobinsky ‡ L.D. Landau Institute for Theoretical Physics of the Russian Academy of Sciences,Kosygin str. 2, 119334 Moscow, Russia,National Research University Higher School of Economics,101000 Moscow, Russia
Alessandro Tronconi, § Tereza Vardanyan, ¶ and Giovanni Venturi ∗∗ Dipartimento di Fisica e Astronomia, Universit`a di Bolognaand INFN, Via Irnerio 46, 40126 Bologna, Italy
Sergey Yu. Vernov †† Skobeltsyn Institute of Nuclear Physics,Lomonosov Moscow State University,Leninskie Gory 1, 119991, Moscow, Russia
We study static spherically and hyperbolically symmetric solutions of the Einstein equations inthe presence of a conformally coupled scalar field and compare them with those in the space filledwith a minimally coupled scalar field. We then study the Kantowski-Sachs cosmological solutions,which are connected with the static solutions by the duality relations. The main ingredient of theserelations is an exchange of roles between the radial and the temporal coordinates, combined withthe exchange between the spherical and hyperbolical two-dimensional geometries. A brief discussionof questions such as the relation between the Jordan and the Einstein frames and the description ofthe singularity crossing is also presented.
I. INTRODUCTION
The exploration of the exact solutions of Einstein equa-tions has been attracting attention of researchers fromthe dawn of General Relativity. Exact solutions possess-ing spherical symmetry were one of the main branchesof this activity since the time of the classical works bySchwarzschild [1], Tolman [2], Oppenheimer and Volkoff[3]. The study of static spherically symmetric solutionsof the Einstein equations in the presence of a masslessscalar field has rather a long history [4–16] (see also [17]as a review). In particular, in paper [13], a duality be- ∗ [email protected] † ‡ [email protected] § [email protected] ¶ [email protected] ∗∗ [email protected] †† [email protected] tween spherically symmetric static solutions in the pres-ence of a massless scalar field and the Kantowski-Sachscosmological models [18], which instead possess hyper-bolic symmetry was studied. It was noticed also thatthe spherically symmetric Kantowski-Sachs universes areconnected by a duality transformation to the static so-lutions possessing hyperbolic symmetry. In the limitingcase of the absence of the scalar field, the correspondingstatic solution represents some hyperbolic analogue of theSchwarzschild geometry. While such a hyperbolic solu-tion was mentioned already in paper [19], its propertieswere studied in detail in papers [13, 20]. Let us emphasizethat the main ingredient of this duality is the exchangeof roles between the radial coordinate and the tempo-ral coordinate combined with the exchange between thespherical two-dimensional geometry and the hyperbolicaltwo-dimensional geometry.The study of gravity models, where a scalar field isnon-minimally coupled to the scalar curvature has a longhistory, too [21–23]. Recently, the actuality of such mod-els has grown due to the study of inflation models basedon the Higgs scalar field non-minimally coupled to grav-ity [24]. Models wherein a non-minimal coupling betweengravity and a scalar field is conformal were also largelystudied [25–36]. These are interesting for two reasons:firstly, it is relatively easy to find exact solutions and toestablish the relations between these solutions and thoseobtained in models with a minimally coupled scalar field;secondly, there is a possibility of a change of sign of theeffective gravitational constant and of the construction ofa singularity-free isotropic cosmological model includinga scalar field conformally coupled to the scalar curvature.We would also like to mention the papers [12, 14, 37–40]where the relation between the exact static solutions inthe Jordan frame and in the Einstein frame was studiedin detail. In particular, it was noticed that a singular-ity in one frame can correspond to the regular geome-try in another frame. In the paper [41] a very detailedcompendium of the conformal transformations of suchgeometrical quantities as the metric, Christoffel connec-tion coefficients, the Riemann tensor and different cur-vature invariants was presented. Further, the conformaltransformations of the matter energy-momentum tensorwere also given. The connection between the conformaltransformations and the duality transformations in su-perstring theories was explained in the paper [41] also.Here, we wish to stress that what we call “duality” be-tween static and cosmological solutions in General Rela-tivity is quite different from the duality in the superstringtheories, because our duality involves the exchanges ofthe coordinates and not the field variables.In the present paper we study static spherically andhyperbolically symmetric geometries in the presence of amassless scalar field conformally coupled to gravity andtheir relations with Kantowski-Sachs cosmologies. Wecompare the solutions found with those obtained in atheory with the minimally coupled massless scalar field.The structure of the paper is as follows: in the sec-ond section we present some general formulas for gravitywith a conformally coupled scalar field. The third sec-tion is devoted to the study of static spherically symmet-ric solutions while in the forth section we obtain statichyperbolically symmetric solutions. In the fifth sectionwe discuss the duality relations between static and cos-mological solutions and present some details concerningtime evolution of the Kantowski-Sachs universe in thismodel. The last section includes some concluding re-marks about the relations between different frames andabout the problem of the singularity crossing. II. SOME GENERAL FORMULAS FORGRAVITY WITH A CONFORMALLY COUPLEDMASSLESS SCALAR FIELD
Let us consider an action S = Z d x √− g (cid:18) U ( σ ) R − g µν σ ,µ σ ,ν (cid:19) . (1) The Einstein equations are U (cid:18) R µν − g µν R (cid:19) + g µν (cid:3) U − ∇ µ ∇ ν U = 12 σ ,µ σ ,ν − g µν σ ,α σ ,α . (2)The variation with respect to σ gives the Klein-Gordonequation (cid:3) σ + dUdσ R = 0 . (3)The Einstein equations (2) can be rewritten as U (cid:18) R µν − g µν R (cid:19) + g µν d Udσ σ ,α σ ,α + g µν dUdσ (cid:3) σ − d Udσ σ ,µ σ ,ν − dUdσ ∇ ν ∇ µ σ = 12 σ ,µ σ ,ν − g µν σ ,α σ ,α . (4)On contracting Eq. (4) with the contravariant metric, weget − U R + 3 d Udσ σ ,µ σ ,µ + 3 dUdσ (cid:3) σ + 12 σ ,µ σ ,µ = 0 . (5)For the case of a conformal coupling U c = U − σ
12 (6)one easily finds that R = 0 (7)and (cid:3) σ = 0 . (8) III. STATIC SPHERICALLY SYMMETRICSOLUTIONS
We shall consider a static spherically symmetric metricin the form ds = b ( r ) dt − a ( r ) (cid:0) dr + dθ + sin θdφ (cid:1) . (9)For the metric (9) and for the scalar field σ , that de-pends only on the radial variable r we obtain (cid:3) σ ( r ) = − a σ ′′ − ab ′ + a ′ ba b σ ′ , (10)where primes mean derivatives with respect to r .The Ricci scalar is R = 2 a b (cid:16) b ′′ a + b ′ a ′ a − ba ′ − ba + 2 baa ′′ (cid:17) . (11)For the case of the conformal coupling (6) it followsfrom Eqs. (7) and (11) that b ′′ a + b ′ a ′ a − ba ′ − ba + 2 baa ′′ = 0 . (12)Equation (8) with (10) can be easily integrated, giving σ ′ = Cab , (13)where C is an integration constant.The Einstein equations are now6 U c h a ′ + a − aa ′′ i = C b + Cσab ′ b , (14)6 U c h a b − a ′ b − a ′ ab ′ i = − Caσ (cid:20) b ′ b + 2 a ′ a (cid:21) − C b , (15)6 U c h a ′ b − aba ′′ − a b ′′ i = C b + σCa ′ . (16)In order to simplify the equations obtained above weintroduce new functions a e ( r ) = s − σ ( r ) U a ( r ) , b e ( r ) = s − σ ( r ) U b ( r ) . (17)In terms of these functions, Eqs. (14)–(16) take the fol-lowing form a e b e − b e a e a ′′ e + b e a ′ e = C U , (18) a e b e − b e a ′ e − b e a e a ′ e b ′ e = − C U , (19) b e a e a ′′ e − b e a ′ e + b e a e b ′′ e = − C U (20)Equation (12) in terms of the new variables is2 b e a e a ′′ e − a e b e − b e a ′ e + b e a e b ′′ e + b e a e a ′ e b ′ e = − C U . (21)The introduction of the new functions a e and b e usedtogether with Eq. (13) allows us to obtain equations in-dependent of the scalar field σ and its derivatives.On introducing A = a ′ e a e , B ≡ b ′ e b e , we can rewrite Eqs. (18)–(20) in the following form:1 − A ′ − A = C U b e a e , (22) 1 − AB − A = − C U b e a e , (23) A ′ + B ′ + B = − C U b e a e . (24)The resulting equations are quite similar to the Einsteinequations for the model with minimally coupled masslessscalar field, considered in [13].On summing Eqs. (22) and (23) and Eqs. (22) and (24),we obtain the following equations:1 − A − A ′ − AB − A = 0 , (25)1 − A ′ + B ′ + B − A = 0 . (26)From this pair of equations one can obtain another: A = − B ′ B − B, (27) (cid:18) B (cid:19) ′′ − B = 0 . (28)There are two independent solutions of Eq. (28). One ofthese solutions is proportional to the hyperbolic cosineand the other is proportional to the hyperbolic sine. Letus choose as a solution B = γ cosh r , (29)where γ is a constant. Then from Eq. (27) it follows that A = tanh r − γ cosh r . (30)On substituting the expression (30) into the left-handside of Eq. (22), we see that it is equal to − (1+ γ )cosh r <
0, while the right-hand side of this equation is positive.Thus, we should discard the solution (29)–(30).Let us now consider B = γ sinh r , (31)then A = coth r − γ sinh r . (32)On substituting the expression (32) into Eq. (22), weobtain 1 − γ sinh r = C U b e a e , (33)which tells us that γ . (34)On now integrating Eqs. (31) and (32), we obtain b e = b (cid:16) tanh r (cid:17) γ (35)and a e = a sinh r (cid:0) tanh r (cid:1) γ , (36)where a and b are constants.On substituting the expressions (35) and (36) intoEq. (33), we obtain C = ± a b p U p − γ . (37)In what follows we shall choose the “plus” sign on theright-hand side of Eq. (37) without loss of generality.On substituting (37) together with (36) and (35) and(17) into Eq. (13), we obtain σ ′ − σ U = 2 √ U p − γ sinh r . (38)On integrating this equation, we obtain σ = p U A (cid:0) tanh r (cid:1) q − γ − A (cid:0) tanh r (cid:1) q − γ + 1 , (39)where A > a and b : a = a (cid:18) A (cid:0) tanh r (cid:1) q − γ + 1 (cid:19) sinh r √ A (cid:0) tanh r (cid:1) γ + q − γ , (40) b = b (cid:18) A (cid:0) tanh r (cid:1) q − γ + 1 (cid:19) (cid:0) tanh r (cid:1) γ √ A (cid:0) tanh r (cid:1) q − γ . (41)Let us first look at the particular cases, when γ = ± σ = √ U A − A +1 implies simply some changesof the Newton constant. Thus, this case coincides withthat considered for a minimally coupled scalar field in pa-per [13]. Let us give here some details for completeness.In the case γ = 1, the metric (9) has the form ds = b ( A + 1) tanh r A dt − a ( A + 1) cosh r A ( dr + dθ + sin θdφ ) . (42) On introducing a new “Schwarzschild” radial variable˜ r = a ( A + 1) √ A cosh r , (43)we can rewrite the metric (42) in the familiarSchwarzschild form ds = b ( A + 1) A (cid:18) − a ( A + 1) √ A r (cid:19) dt − d ˜ r (cid:16) − a ( A +1) √ A r (cid:17) − ˜ r ( dθ + sin θdφ ) , (44)where the quantity a ( A +1) √ A plays the role of theSchwarzschild radius and the constant b ( A +1) A can beabsorbed in the definition of the time parameter.For the case γ = −
1, the metric (9) has the form ds = b ( A + 1) coth r A dt − a ( A + 1) sinh r A ( dr + dθ + sin θdφ ) . (45)On introducing a variable ˆ r byˆ r = a ( A + 1) √ A sinh r , (46)we can rewrite the metric (45) as ds = b ( A + 1) A (cid:18) a ( A + 1) √ A r (cid:19) dt − d ˆ r (cid:16) a ( A +1) √ A r (cid:17) − ˜ r ( dθ + sin θdφ ) , (47)where on choosing a <
0, we again have the standardSchwarzschild metric, where the Schwarzschild radius isproportional to some positive point-like mass.On now, looking at the expressions (40) and (41), weget b ( r ) ∼ r γ − q − γ , a ( r ) ∼ r − γ − q − γ , (48)when r → γ , it is γ = 12 . (49)Indeed, if γ = 1 /
2, then at r = 0 both factors a ( r ) and b ( r ) and, hence, the corresponding metric coefficients arefinite. This regime does not have a counterpart in thecase of a minimally coupled scalar field [13] and we shalldiscuss it in detail later.It is easy to see that for γ > / b ( r ) → , a ( r ) → ∞ , when r → . If γ < /
2, then the behavior of the functions a and b isthe opposite: b ( r ) → ∞ , while a ( r ) →
0, when r →
0. Asimple calculation shows that if γ = 1 /
2, then at r → R µν R µν ∼ B r − (cid:18) − γ − q − γ (cid:19) → ∞ , (50) where B is a positive constant. Thus, the solutions with γ = 1 / r = 0. In this casewe consider a ( r ) and b ( r ) for r > γ = 1 / a and b are finite at r = 0. Theexplicit expression for the metric is now ds = b (cid:0) A tanh r + 1 (cid:1) A dt − a (cid:2) A tanh r + 1 (cid:3) cosh r A ( dr + dθ + sin θdφ ) . (51)The scalar field is given by σ = p U A tanh r − A tanh r + 1 . (52)One can see that both the expressions (51) and (52) arequite regular at r = 0 and can be smoothly continued inthe region r <
0. The expression for the scalar field inthis region is such that σ > U and, hence, U c < U c <
0, is accompanied bythe appearance of the cosmological singularity. Thus, oneshould expect that removing the assumption of sphericalsymmetry or, more generally, axial symmetry, will re-sult in formation of general curvature singularities juston spatial hypersurfaces beyond which gravity becomesrepulsive.Let us see what happens at r <
0. There are twooptions. If the integration constant A <
1, then thegeometry is regular for all values of the variable r . Theasymptotic expression for the metric (51) at r → −∞ is ds = b (1 − A ) A dt − a (1 − A ) A e − r ( dr + dθ + sin θdφ ) . (53)On introducing a variable¯ r = a (1 − A ) e − r √ A , (54)we can rewrite the metric (53) as ds = b (1 − A ) A dt − d ¯ r − ¯ r ( dθ + sin θdφ ) , (55)and it describes the Minkowski spacetime. It is easy tosee that at r → ∞ we again encounter an asymptotically flat Minkowski spacetime. Let us note that for a value ofthe radial variable r = − arctanh A , (56)the factor a ( r ) has a minimum value. Thus, we can imag-ine that this value of the variable r corresponds to athroat of some wormhole-like configuration. Let us again emphasize that the absence of the singu-larity at r = 0 provided γ = 1 / r = 0. Therefore, inthis case the conformal continuation, described in [14],is possible. A similar phenomenon for the Friedmann-Lemaˆıtre-Robertson-Walker cosmology was described indetail in paper [34].Let us now consider a more interesting case whereinthe integration constant A ≥
1. When r → r = − A , (57)both scale factors tend to zero as ( r − r ) and we stumbleupon the singularity, characterized by the invariant R µν R µν ∼ r − r ) . (58)Nevertheless, for r < r the metric and the scalar fieldare well defined and one can construct the continuationof the solution into this region. Then, for r → −∞ weagain have an asymptotically flat Minkowski spacetime.We wish to note that in contrast with the case of r = 0the singularity at r = r and A ≥ Note that the fact that this configuration requires the ghostbehavior of graviton in the antigravity regime for r <
Let end this section by observing that for the case γ = 1 / r enters into the solutions in pow-ers of γ and q − γ and is ill-defined at r < r is negative. IV. STATIC HYPERBOLICALLY SYMMETRICSOLUTIONS FOR THE CASE OF ACONFORMALLY COUPLED SCALAR FIELD
We shall consider a static hyperbolically symmetricmetric of the form [13, 20]. ds = b ( r ) dt − a ( r ) (cid:0) dr + dχ + sinh χdφ (cid:1) , (59)where the hyperbolic angle χ runs from 0 to ∞ . Allconsiderations are analogous to those presented in thepreceding section.We then obtain the general solution in the followingform; ds = b (cid:18) A (cid:0) tan r (cid:1) q − γ + 1 (cid:19) (cid:0) tan r (cid:1) γ A (cid:0) tan r (cid:1) q − γ dt − a (cid:18) A (cid:0) tan r (cid:1) q − γ + 1 (cid:19) sin r A (cid:0) tan r (cid:1) q − γ +2 γ ( dr + dχ + sinh χdφ ) , (60)while σ = p U A (cid:0) tan r (cid:1) q − γ − A (cid:0) tan r (cid:1) q − γ + 1 . (61)For the cases γ = ± γ = 1 /
2, at the point r = 0 one has the singularity of the same kind as that studied in the preceding section.However, another singularity arises at r = π , if γ = − / R µν R µν behaves as R µν R µν ∼ ( π − r ) − (cid:18) γ − q − γ (cid:19) . (62)Note that the right-hand sides of Eqs. (58) and (62) havethe standard ρ − behavior if expressed in terms of theproper distance ρ . Thus, if γ = ± / r =0 and r = π . Let us now consider two particular cases. If γ = 1 /
2, the solution (60), (61) has the following form: ds = b (cid:0) A tan r + 1 (cid:1) A dt − a (cid:2) A tan r + 1 (cid:3) sin rA tan r (cid:2) dr + dχ + sinh χdφ (cid:3) . (63) σ = p U A tan r − A tan r + 1 . (64)The value r = 0 is now regular and we can construct acontinuation of the solution into the region r <
0. How-ever, in this region we encounter a singularity at r = − A . (65) One can construct a continuation through this singular-ity because the expressions (63) and (64) are well definedat r < r . Finally, we encounter the singularity, whichwas already described above at r = − π . Thus, one cansay that the solutions (63), (64) are defined between thetwo singularities at r = − π and r = + π with an inter-mediate singularity at r = r = − A , which canbe continued through.Let us consider another particular case γ = − /
2. Thesolution is now ds = b (cid:0) A tan r + 1 (cid:1) A tan r dt − a (cid:0) A tan r + 1 (cid:1) sin r A ( dr + dχ + sinh χdφ ) . (66)The expression for the scalar σ is given by Eq. (64). Thesolution (66) is defined between two singularities at r = 0and r = 2 π and is nonsingular at r = π . There is also anintermediate singularity at r = 2 π − A . V. RELATION BETWEEN STATIC ANDCOSMOLOGICAL SOLUTIONS
In this section we shall use the method for the construc-tion of cosmological solutions, starting from the dualityrelations described in the paper [13]. The correspond-ing transformations can be considered as a special kindof complex transformations used for the construction ofnew solutions of the Einstein equations (see e.g. [43]).As an example one can mention also the complex trans-formations connecting cosmological Kasner solutions [44]for a Bianchi-I universe with the static Kasner solutions(see, e.g. [45]). However, the particular form of the com- plex transformations, exchanging hyperbolic and spheri-cal symmetry in the form implemented in paper [13] andin the present paper does not appear to be widely used.Let us consider the static spherically symmetric space-time. If we make the substitution r ↔ t, (67)followed by a change of the sign of all the metric compo-nents g µν → − g µν , (68)and by the substitution θ → iχ, (69)we obtain a Kantowski-Sachs cosmological solution,where the spherical symmetry is replaced by the hyper-bolic one: ds = a (cid:18) A (cid:0) tanh t (cid:1) q − γ + 1 (cid:19) sinh t A (cid:0) tanh t (cid:1) γ +2 q − γ ( dt − dχ − sinh χdφ ) − b (cid:18) A (cid:0) tanh t (cid:1) q − γ + 1 (cid:19) (cid:0) tanh t (cid:1) γ − q − γ A dr . (70)It is curious to look at the particular solution for the case γ = 1 / A <
1. Now ds = a (cid:0) A tanh t + 1 (cid:1) cosh t A ( dt − dχ − sinh χdφ ) − b (cid:0) A tanh t + 1 (cid:1) A dr . (71)One can see that the evolution of the universe is a non-singular one. The scale corresponding to the variable r (which can be both compact or non-compact) is al-most constant. Let us look at the evolution of the two-dimensional hyperboloid with the metric dχ + sinh χdφ . When t → −∞ the metric of the universe can be repre-sented as ds = d ˜ t − ˜ t ( dχ + sinh χdφ ) − b (1 − A ) A dr , (72)where a cosmic time parameter ˜ t is defined by˜ t = − a (1 − A ) e − t √ A . (73) It is easy to see that the metric (72) describes the directproduct of the line or circle by the 2 + 1 dimensionalMilne universe, which is equivalent to the Minkowskispacetime. An analogous expression can be written for t → + ∞ . Thus, the universe begins its evolution in thedistant past from the asymptotically Minkowski space-time, represented in the Milne form, then it contractsuntil the moment t = − A and the it begins anexpansion, which ends again in the asymptotically flatMinkowski spacetime.Another interesting case arises if we start from thestatic hyperbolically symmetric metric (60) and make theduality transformations presented above with the differ-ence that now χ → iθ. Then we arrive at the Kantowski-Sachs universe, wherethe spatial sections are direct products of the one-dimensional submanifold (the r variable) and a two-dimensional sphere: ds = a (cid:18) A (cid:0) tan t (cid:1) q − γ + 1 (cid:19) sin t A (cid:0) tan t (cid:1) q − γ +2 γ ( dt − dθ − sin θdφ ) − b (cid:18) A (cid:0) tan t (cid:1) q − γ + 1 (cid:19) (cid:0) tan t (cid:1) γ − q − γ A dr . (74)As before the cases γ = ± < γ <
1, then one has the singularities at t = 0and t = π and it is not clear if the continuation throughthese singularities makes sense. When t → a ( t ) → ∞ ,while b ( t ) →
0. At t → π , the scale factor a tends tozero, while b → ∞ . All the evolution takes place at the gravity regime ( U c ≥ − < γ < /
2, then at t = 0 we have a singularitysuch that a → b → ∞ . Then at t → π the scalefactor a again vanishes while b grows infinitely.If − < γ < − /
2, then at t → a → b → ∞ .When t → π , a → ∞ and b → γ = ± / γ = 1 / ds = a (cid:0) A tan t + 1 (cid:1) cos t A ( dt − dθ − sin θdφ ) − b (cid:0) A tan t + 1 (cid:1) A dr . (75)This metric is regular at t = 0 and has singularities at t = ± π and at t = t = − A . At t → ± π thescale factor a → b → ∞ . At t → t both scale fac-tors vanish. At t <
0, we find ourselves in the region withantigravity because U c <
0. We see that the expression(75) contains only integer powers of the trigonometricalfunctions and one can describe the crossing of the singu-larities in a unique way. Thus, we can imagine an infiniteperiodic evolution of the universe. Let us consider a pe-riod between − π and π . At t = − π , the universe goes outof the singularity with the vanishing value of the scale fac-tor a and an infinite value of the scale factor b . Then, thescale factor b decreases and vanishes when the universeapproaches to the singularity at t = t = − A .Meanwhile the scale factor a increases and reaches itsmaximal value at t = t = − π +arctan A and then beginsdecreasing and vanishes at the singularity at t = t . Afterthat b increases reaching an infinite value at the singu-larity at t = π , while a increases until t = t = arctan A ,where it achieves its maximum value and then decreasesand vanishes at t = π . Then, the evolution repeats it-self. Let us now also look more carefully to the structureof the anisotropy of these cosmological singularities. Inthe vicinity of the moment t → π , the asymptotic ex-pressions for the metric coefficients become simpler andwe can introduce a cosmic time parameter T →
0. Themetric now has the following form: ds = dT − c T dθ − c T sin θdφ − c T dr . (76) This form has a structure similar to that of the Kasnersolution for a Bianchi-I universe [44, 46], where the Kas-ner indices have the values p = 12 , p = 12 , p = − . (77)Let us note that while these indices do not satisfy thestandard Kasner relations [44, 46] p + p + p = p + p + p = 1 , (78)they satisfy the generalized relation X i =1 p i = 2 X i =1 p i − X i =1 p i ! , (79)discussed in our preceding paper [36]. We can find asimilar asymptotic representation of the metric (75) inthe vicinity of the singularity at t = t . It is ds = dT − c T dθ − c T sin θdφ − c T dr . (80)Thus, this behavior is isotropic and the Kasner indices p = p = p = 12 (81)again satisfy the relation (79).Lets us consider another particular case where γ = − /
2. The metric is now ds = a (cid:0) A tan t + 1 (cid:1) sin t A ( dt − dθ − sin θdφ ) − b (cid:0) A tan t + 1 (cid:1) A (cid:0) tan t (cid:1) dr . (82)In this case also we can also consider a periodic evolutionof the universe, which crosses the singularities. It beginsat the singularity at t = 0 when the scale factor a isequal to zero and the scale factor b is infinite, then b begins decreasing and arrives to a value equal to zero atthe singularity at t = t = 2 π − A . Meanwhilethe scale factor a increases, arriving to a maximum valueat t = π − arctan A , then it decreases and vanish at t = t . After that the scale factor a grows infinitelyuntil arriving to the singularity at t = 2 π while the scalefactor b reaches its maximal value at t = 2 π − arctan A and vanishes at t = 2 π . Then the evolution repeats itself.We can add here that in the vicinity of the singularityat t = 0,the Kasner indices are given by Eq. (77) whilein the vicinity of the singularity at t = t , the Kasnerindices are given by Eq. (81). VI. CONCLUDING REMARKS
It is well known that on combining the conformal trans-formation of the metric with the reparametrization of thescalar field, one can rewrite the action of a model witha non-minimally coupled scalar field in a form where itbecomes minimally coupled. Such a procedure is calledthe transformation from the Jordan frame to the Einsteinframe. For the first time this transformation was used inpaper [23].Many papers discuss this topic, which sometimes isdescribed as a study of the equivalence between frames[34, 47]. In a way, one can say that mathematically theprocedure of the transition between the frames is well de-fined and can be used in different contexts. We wish toemphasize that the physical cosmological evolutions arethose seen by an observer using the cosmic (synchronous)time, which is different in different frames. Thus, evolu-tions in the Einstein and Jordan frames, connected by aconformal transformation and by the reparametrizationof the scalar field can be qualitatively different. In thepresent paper we have shown that the the static spher-ically or hyperbolically symmetric solutions of the Ein-stein equations and their Kantowski-Sachs counterpartsin the presence of the conformally coupled scalar fieldpossess some special regimes, which are absent for thecase of a minimally coupled scalar field [13]. Moreover,one can see that for the models considered here there ex-ist situations when a transition from the Einstein frame to the Jordan frame or viceversa can remove or createa singularity. Similar effects were studied in detail forFriedmann models in paper [34]. It was shown that whenthe universe encounter the singularity in the Einsteinframe, it is absent in the Jordan frame, because this sin-gularity is reabsorbed by the conformal transformationfactor. Such effect is however absent in the Bianchi-Imodels and the singularities arise simultaneously in bothframes [31, 35, 36]. Let us add that the fact that theconformal transformations can essentially change the ge-ometry of the spacetime due to an effective creation ofsome additional matter was discussed in the paper [41].In recent years there has been an intensive discussionon the possibility of the crossing of the Big Bang —Big Crunch type singularities in cosmology [48]. Themain point here is that one can describe the singular-ity crossing if in spite of the presence of some divergentinvariants at the singularity, it is possible to establishsome well-defined prescription for matching some non-singular quantities before and after the singularity. Inpaper [34] such a procedure was based on the Jordan-Einstein frame transitions. In papers [35, 36] other fieldreparametrizations were used. In the present paper wehave used the fact that for some special choices of the pa-rameters, the expressions for the metric and the field arewell-defined (contain only integer degrees of some simplefunctions) and hence, the matching between regions sepa-rated by a singularity arises naturally. We think that thequestion concerning possible generality of such a proce-dure deserves further investigations. On the other handthe finding some exact solutions of Einstein equationswhich have more complicated structure than solutionssuch as Friedmann-Lemaˆıtre universes or Schwarzschildblack holes, can be useful for both cosmology and blackhole physics. In particular, it concerns the questions con-nected with the general relativistic singularities. Let usnote that spherically symmetric solutions for models witha minimally coupled scalar field and nonzero potentialhave been studied in [14, 17, 49, 50]. We plan to studysimilar solutions in the models with non-minimal cou-pling in the further investigations.
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