We discuss the two-dimensional dilaton gravity with a scalar field as the
source matter. The coupling between the gravity and the scalar, massless, field
is presented in an unusual form. We work out two examples of these couplings,
and solutions with black-hole behaviour are discussed and compared with those
found in the literature.Comment: Latex, 11 page

Alves, Marcelo

English

arXiv

We discuss the two-dimensional dilaton gravity with a scalar field as the source matter. The coupling between the gravity and the scalar, massless, field is presented in an unusual form. We work out two examples of these couplings, and solutions with black-hole behaviour are discussed and compared with those found in the literature.We discuss the two-dimensional dilaton gravity with a scalar field as the source matter. The coupling between the gravity and the scalar, massless, field is presented in an unusual form. We work out two examples of these couplings, and solutions with black-hole behaviour are discussed and compared with those found in the literature

Alves, M.

CERN Document Server

CERN-TH/95-259DILATON GRAVITYWITH A NON-MINIMALLY COUPLEDSCALAR FIELDM. ALVES?Theory Division - CERNCH-1211 GENEVE 23SwitzerlandABSTRACTWe discuss the two-dimensional dilaton gravity with a scalar eld as the source matter.The coupling between the gravity and the scalar, massless, eld is presented in an un-usual form. We work out two examples of these couplings, and solutions with black-holebehaviour are discussed and compared with those found in the literature.PACS: 04.60.+n; 11.17.+y; 97.60.LfCERN-TH/95-259October 1995?e-mail: ALVES@VXCERN.CERN.CHOn leave from Instituto de Fisica, Universidade Federal do Rio de Janeiro, Brazil.11 INTRODUCTIONIt is widely recognized that two-dimensional models of gravity can give us a betterunderstanding of the gravitational quantum eects. These models, derived from a stringmotivated eective action [1] or from some low-dimensional version of the Einstein equa-tions [2], have a rich structure in spite of their relative simplicity. Gravitational collapse,black holes and quantum eects are examples of subjects whose description is rather com-plicated in four-dimensional gravity while their lower-dimensional version turns out to bemore treatable, sometimes completely solved.This is the case of the seminal work of Callan, Giddings, Harvey and Strominger(CGHS)[1], where the Hawking eect is analysed semiclassically. Black-hole solutions arefound and used to obtain an expression for the Hawking radiation. Some improvementshave been done [3] to circumvent diculties that arise from its quantum version, butleaving the initial purpose unchanged.In the CGHS model, one starts with the four-dimensional Einstein-Hilbert action inthe spherically symmetric metric. Then, the assumption of dependence in two variablesof all the elds is done. With a suitable form of the metric, it is possible to integratethe angular dependence. The resulting equations can be derived from a two-dimensionalaction, the dilaton being the relic of the integrated coordinates. Then this action ismodied, rendering a simpler one, which is called the two-dimensional dilaton gravity.With this modied action, black-hole solutions are found to be formed from non-singular initial conditions, namely a source of scalar matter, coupled minimally with thetwo-dimensional gravity sector (the terminology will be claried later). It is shown thatthere is also a linear vacuum region, which imposes conditions to the expression for theHawking radiation.In this model, the scalar matter is coupled only with the two-dimensional gravitysector, without interaction with the dilaton eld. Since this eld is part of the formerlyfour-dimensional world and, as pointed out in [1], some results obtained are model- de-pendent, it could be interesting to analyse some consequences in considering the coupling2between the scalar matter and the dilaton eld, as originally derived from the dimensionalreduction. This is the main purpose of the present work. It is organized as follows: webegin with the construction of the 2d dilaton gravity, but in an unusual way, and thenmodications are introduced with the respective solutions. Discussion and nal remarksare in the conclusion.2 THE 2d DILATON GRAVITY AND THE DIMENSIONAL REDUCTIONIntending to compare results, we present in this section the CGHS model. However,this will be done in a slightly dierent way, closer to the dimensional reduction, givingmore motivations for the modications we will work out.The starting point is the Einstein action on D dimensions coupled with a dilaton eld [4]S =ZdDxp−g e−g@ @ + R; (1)where R is the curvature scalar in D dimensions and D = D + d. When D = 10, this isthe bosonic part of the heterotic string eective action in the critical dimension [4]. Theantisymmetric part was omitted, since our interest is in the 2d case.The theory is considered in a spacetime M K, where M and K are D and d dimen-sional spacetime, respectively. If the elds are independent of the d coordinates, the Dmetric can be decomposed as [4]g = g +AγAγ AA G!: (2)The resulting dimensionally reduced action isS =ZdDxp−g e−R+ g@@+14g@G@G −14ggGFF(3)with  = − 12log (detG) being the shifted dilaton and F  = @A − @A.3Now we make the assumptions to reach the action we are interested in, namely takingthe simple case where the original dilaton eld  is zero and writing the metric in themost general spherically symmetric form:ds2 = gdxdx +12e−2d2Ω: (4)So, with D = d = 2 and for Aγ = 0 we have the 2d dimensionally reduced actionS =12Zd2xp−g e−2R+ 2(r)2 + 22e2: (5)It is worth to mentioning that this is the simplest example of the dimensional reductiongiving an action with O(d; d)d=2 symmetry. If one starts with a scalar eld, action (1)changes toS =ZdDxp−g e−g@ @ + R −12g@ f@ f(6)and we have, with the same assumptions:S =Zd2xp−g e−2R+ 2(r)2 + 22e2 −12(rf)2(7)for the reduced version. Note that in (7), the scalar matter couples with both the 2d-gravity sector and the dilatonic eld. We will consider this coupling as the rst modi-cation to the CGHS action. The second one concerns the semiclassical quantization andsome explanation is needed.The semiclassical quantization via functional method requires integrating over elds(the scalar one in this particular case). On the other hand, we are interested in theoriesthat have the full quantized version free of anomalies. This paradigm can be worked outvia the BRST analysis of the theory by using the Fujikawa’s technique [6]. In this frame-work, the quantized theory is anomaly-free if the functional measure is BRST-invariant.It follows that this invariance requires some redenition of the eld variables, the so-calledgravitational dressing, and we must consider it as the eld variable of the model that we4are studying. It is instructive to show how the gravitational dressing works in a verysimple example, the 2d conformal scalar eld, minimally coupled with gravity.The rst-order quantum-corrected version for the 2d scalar eld gives a traceless vac-uum expected value (VEV) for the energy momentum tensorhT  i = ghTi = 0 (8)but non covariantly conservedrhTi = h@fr@fi (9)since (9) is an ill-dened quantity. However, most of the quantities of interest arise fromnon-classical eects (e.g. the Hawking radiation) it could be useful to consider the theorywithin this scope. At the same time, we hope that the model (in the semiclassical version)agrees with some basic principles, one of them being the (semiclassical) EMT covariantlyconserved [5], unlike the one given by (9).This undesired result can be avoided by a redenition of the eld variable [7]:f ! ~f = g(14)f: (10)It is straightforward to see that, considering ~f as the variable,h ~T  i = hfr@fi =124R (11)andrh ~Ti = 0; (12)to renormalized vacuum expected values. The classical action for the new eld variableis obtained from the old one, using the denition (10):S[ ~f ] =Zd2x(r ~f)2: (13)5In the four-dimensional case, the redenition of the eld variable is not sucient to giveus the conservation of the VEV of the energy-momentum tensor. Instead of this, we mustimpose (12) as a condition to be satised. Actually, it is used to eliminate ambiguitiesfrom the denition of some numerical factors in the expression of the renormalized VEVof the four-dimensional energy-momentum tensor [5].From the denition (10) and the coupling given by (7), these follows a modied actionto the dilaton gravityS =12Zd2x e−2p−g[R+ 4(r)2 + 42]−12(r ~f)2; (14)where we have done the same modications, regarding the numerical factors and thecoupling between the dilaton and the constant , as those in the CGHS model. Sincethere are couplings other than the original dilaton gravity action, we call the theory givenby (14) the non-minimally coupled one.3 BLACK-HOLE SOLUTIONS FROM SCALAR MATTERIn this section, we show that there are black-hole-type solutions arising from a pulseof scalar matter, as in the CGHS model. However, due to the new couplings with boththe dilaton eld and the two-dimensional gravity sector, these solutions have new featureswith respect to those from the usual dilatonic gravity model.The equations of motion, derived from (14), aree−2[rr+ g((r)2 −r2− 2)−12rfrf ] = 0; (15)e−2[R + 42 + 4r2− 4(r)2] = 0 (16)andr2f = 2rfr ; (17)6for the dilaton, the metric, and the scalar eld, respectively. The rst two equations implye−2[R+ 2r2] = 0: (18)It is useful to work with light-cone coordinatesx = x0  x1 (19)and choose the conformal gauge g = e2 that, in the light-cone coordinates, givesg+− = −12e2 ; g++ = g−− = 0: (20)Now, using (17), the (+−) component of the equation for the dilaton (15) becomes@+@−− 2@+@− =122e2 −12@+f@−f (21)or@+@−(e−2) = −2 + @+f@−fe−2 (22)and eq.(18)@+@− = @+@−: (23)We can use the residual gauge freedom [1] to make = : (24)This relation, besides rendering calculations easier, has some implications on the originalfour-dimensional metric (4). We will comment on this point later.The equations of motion for the gauge-xed components g++ and g−− must be imposedas constraints:7r+r+ =12r+fr+f and r−r− =12r−fr−f: (25)Using (24) and Γ+++ = 2@+ and Γ−−− = 2@− these relations can be written as@2+− 2(@+)2 =12(@+f)2 (26)and@2−− 2(@−)2 =12(@−f)2: (27)Let us consider, as a source of matter, a shock-wave travelling in the x− direction. Asin [1], this will be described by the stress tensor12@+f@+f = a(x+ − x+0 ); (28)where a is its magnitude.Dening h(x+; x−) = e−2 eqs. (26) and (27) can be written as@2+h = −a(x+− x+0 )h (29)and@2−h = 0: (30)We note that there is a dierence between eq.(28) and the corresponding one from thedilaton gravity. This dierence is due to the coupling we have introduced in the action(14). The solution of this new equation has the following general form:h(x+; x−) = −ah(x+0 ; x−)x+ +A(x−)x+ +K; (31)where A(x−) is a function of only x− and K, a constant. The quantity h(x+0 ; x−) canbe evaluated taking (31) at x+ = x+0 , giving8h(x+0 ; x−) =K1 + ax+0+x+01 + ax+0A(x−): (32)Using the eq.(22), the nal expression for the solution ish(x+; x−) = −2x+x−0 +M; (33)with x−0 = x− + a c12and c1 =M(1+ax+0 ).Comparing this expression with the one from the dilaton gravity, we see that (33)corresponds to a black-hole solution with mass given by M = K and the coordinate x−shifted tox−0 = x− + aM3(1 + ax+0 ): (34)When a = 0, which means no source, we have a situation called the linear vacuum solution,also present in the CGHS model, in the same situation.Another situation can be easily analysed, namely when there are two matter sources,one of them being like (28) and the other given by@−f@−f = b(x− − x−0 ) (35)The only modication with respect to the previous case is in the constraint equation(27), for the g−− component, written as@2+h(x+; x−) = −b(x− − x−0 ): (36)It is straightforward to show that the solution ish(x+; x−) = −2x+x− +K; (37)that is the eternal black-hole solution, with K = M, found in the CGHS model, whenthere are no sources. This result is not completely unexpected: as seen at the rst case,the non-minimal coupling introduces solutions with the same behaviour in both x+ < x+09and x+ > x+0 regions. When the source (35) is present, the regions are separated by thisfront wave that turns to be unchanged, and no horizon is presented. We can see thisdirectly in eq.(37).4 CONCLUSIONS AND FINAL REMARKSIn the last section, we have analysed the solutions generated from the non-minimalcoupling between the scalar matter and both the dilaton eld and the 2-d gravity sector.This coupling was derived from the dimensional reduction of the string motivated models,discussed without details in section 2, and gives us, in a more natural way, the dilaton-scalar matter coupling. The second modication was motivated by the o-shell relations,which can be used in a semiclassical version of the model. The resulting theory is solvedin a way very similar to the CGHS model, showing in the two studied cases black-holetype solutions. We mention that the relation (24), present in both the CGHS and themodied version presented here, it is important to make the solutions in this simple way.In the new version, however, this happens only when we consider the two modicationstogether.When there are two matter sources, as given by (28) and (35) the situation is theone in which the solution is called eternal black-hole, already discussed at the end of thesection 3. However, it is the rst case, the one with the same source as CGHS model, thatseems to have more interesting features. As mentioned before, there is also an horizon,dened by the shift in the x− coordinate, but all the spacetime has a black-hole behaviour,in contrast to the CGHS model, where only in the x+ > x+0 region has this solution. Thelinear vacuum region x+ < x+0 is important in the CGHS model because this condition isused to determine the Hawking radiation. It seems interesting to calculate this eect inthe picture given by this new solution. This work is in progress.10ACKNOWLEDGEMENTSThe author is grateful to E.Abdalla for useful comments. This work was supportedby CNPQ (Brazil).REFERENCES[1] C.G. Callan, S.B. Giddings and J.A. Strominger, Phys. Rev. D 45 (1992) R1005; J.A.Strominger, in Les Houches Lectures on Black Holes, Les Houches (1994)[2] R. Mann, A. Shiekm and L. Tarasov, Nucl. Phys. B341 (1992) 134; R. Jackiw, inQuantum Theory of Gravity, ed. S.Christensen (Adam Hilger,Bristol, 1984), p.403; C.Teitelboim, ibid, p. 327.[3] J. Russo, L. Susskind and L. Thorlacius, Phys. Rev. D 45 (1992) 3444; 47 (1993) 533[4] J. Maharana and J.H. Schwarz, Nucl. Phys. B390 (1993) 3; J.Scherk and J.H. Schwarz,Nucl. Phys. B153 (1979) 61; J. Maharana, Phys. Rev. Lett. 75,2 (1995) 205.[5] N.D. Birrel and P.C. Davies, in Quantum Fields in Curved Spacetime (CambridgeUniversity Press, Cambridge, 1984)[6] K. Fujikawa in Quantum Gravity and Cosmology, ed. H.Sato and T.Inami (Singapore:World scientic); Phys. Rev. D 25 (1982) 2584.[7] K. Fujikawa, U. Lindstrom, N.K. Rocek and P.van Nieuwenhuizen, Phys. Rev. D 37(1988) 391.11

Dilaton Gravity with a Non-Minimally Coupled Scalar Field

Abstract

Similar works

Full text

Available Versions

CERN Document Server