PHYSICAL REVIEW D VOLUME 49, NUMBER 10 15 MAY 1994

Finite temperature dilaton gravity

Zden6k Kopeck$* Department of Theoretical Physics and Astrophysics, Masaryk University,

Kotlhi;skci 2, 611 37 Bmo, Czech Republic (Received 30 November 1993)

The dilaton free energy density in an external static gravitational field is found. We use the real time formulation of the finite temperature field theory and the free energy density is computed to the first order of the string parameter a'. We obtain the thermal corrections to the a' modified Einstein gravity action.

PACS number(s): 04.60.Ds, ll.lO.Wx, 11.25.Mj

I. INTRODUCTION

The high-temperature properties of quantum gravity are of some interest for their potential cosmological ap- plications. The modifications of the Einstein action have been done in the framework of both finite temperature theory and string theory.

The finite temperature theory yields the thermal- quantum corrections (by gravitons and others particles) to the effective gravity action [l-41.

In standard treatments of string theory, it is shown that a consistent string theory can be formulated from some class of conformally invariant models. The a' cor- rections to the Einstein action are known for strings mov- ing in background fields [5,6].

Some aspects of the a' modified black hole thermody- namics have been recently investigated in Ref. [ 7 ] .

In this paper, we derive the dilaton free energy density in a curved background [8,9] which is considered static. The free energy density with opposite sign is the con- tribution to the total gravity Lagrangian. The action for the dilator1 field is taken from string theory to order 0 ( a 1 ) .

11. DILATON ACTION

Corrections to the external metric field action due to the stringy effect were carried out to O ( a l ) in Ref. [6]. The action has the form, for the bosonic string case,

where D=26 is the dimension which corresponds to the critical bosonic string theory, and gik(x) and m(x) are the metric tensor field and the dilaton field, respectively. We consider the situation without the antisymmetrical back- ground field [5 ,6] in the action Eq. (1). The conformal

'Electronic address: kopecky@elanor.sci.muni.cz

transformation

together with some field redefinitions, changes the action

(1) to [GI

We will assume that all but four space-time dimensions are compactified out in the action (1). In the rest of the article, we consider the four-dimensional manifold with the signature (+, -, -, -) . Related space-time indices and space indices have ranges p , v , . . . = 0 , . . . . 3 and . . z , j , . . . = 1 , . . . , 3 , respectively. The dilaton part of the action Eq. (3) we identify with

We have written out explicitly the gravitational constant G in Eq. (4).

111. DILATON FREE ENERGY TO ORDER O ( a l )

In general, statistical mechanics in static space-time can be developed by constructing the partition function [ l O , l l l

The Hamiltonian H is connected with the T: component of the energy momentum tensor

@ 1994 The American Physical Society

49 FINITE TEMPERATURE DILATON GRAVITY 5195

The parameter 4 corresponds to the temperature 4" vec- tor [Ill in static coordinates:

The local Lorentz-rest-hame inverse (scalar) temperature PR then is

Now, according to the above construction, we consider dilaton field described by Eq. (4). The general path in- tegral formulation for the partition function in the real time finite temperature theory (RTFT) is [12,13]

The action S[#J] is given by Eq. (4). The free energy density F[g] has been introduced in Eq. (9). The time integration dxO in the action S[#J] in (9) is going along the curve in the complex plane [12,13]. The curve is, for the RTFT.

tation of the thermal vacuum diagrams [12,13] to evaluate term (13). The term V(P, g) is the sum of the closed con- nected vacuum graphs, each graph has one vertex with the thermal index fixed to 1 [12,13] and the vertex is com- mon for n = 1 ,2 ,3 , . . . number of the thermal propagator function A[P] ll (x, x) loops. Substituting Eqs. (A8) and (A9) in Eq. (14) yields

We have taken the limit m -+ 0 in Eq. (15). Using the analytic continuation relation

and the function regularization [8] we transform Eq. (15) to

1 az(x) - lim - -

V-1 327r2 u - 1

(17)

The zero-temperature term with the pole is canceled ( lo) against the k = 2 term in Eq. (17) because

and t -+ co. The integration D#J in (9) is over fields - 1 which satisfy the periodicity condition lim r ( u - l)C(O) = lim - .

v-t 1 "-1 2(u - 1) (18)

#J(-t, x) = #J(-t - ip, X) (11) The k > 2 terms in Eq. (17) are finite because of the fol-

a t the end points of the curve (10). lowing relation for the product of the C and I' functions: The interaction part of the action (4) we identify with

the second exponential term in Eq. (4). The free energy density then has the first two terms of the expansion to c(~z)I?(z) = 4 , ~ ~ ~ r(1- 2 4 ((1 - 2z) . 0 (a') : r(l - Z )

(19)

a' F[gI = F o [gI - a Rpu~oRPYru[V(P~ S) + 11 7 (12)

The final form of Eq. (17) using Eq. (19) will be

where

The term Fo[g] is a contribution without string correc- tions:

1 i A [ P l l l ( ~ , 5 ) FO [g] = - 1 dmz - 2 4 r G

The function A[PIll (x, x) is the ( 1 , l ) component of the RTFT causal Green's function (A8) calculated in the A p The Green's function AIP]ll(x, x) for m + 0 can be writ- pendix. We have used the RTFT methods for the compu- ten similarly:

Now we can collect all the terms in Eq. (12) and write the final result for the dilaton free energy density:

1 a3 A(x, is)

647r2

a' 27rG G O" ( 2 k - 2 ) ! 2 ( L - 1 ) --

6 4 ~ G R,,,, R"'" exp

(k - I)!

IV. FINITE TEMPERATURE EFFECTIVE a' GRAVITY ACTION

We identify -F[g] with the effective finite temperature Lagrangian and from Eq. ( 3 ) we find the modification of the a' Einstein action by thermal dilatons:

We have defined the effective cosmological constant X(PR) , the gravitation constant G ( P R ) , and the effective string parameter al(PR) in the high-temperature limit:

We will introduce the Planck mass M p = and abso- & lute temperature T R = (PR)- ' in Eq. (26) for the effec- tive string parameter cul(PR):

V. CONCLUSION

For the first time, we have derived the free energy den- sity [Eq. (22)] for the dilatons in an external static grav- itational field using the RTFT methods to O ( a 1 ) .

We have found a related thermal modification of the a' Einstein action. The temperature dependence of the ef- fective string parameter ot(PR) in the high-temperature region is given by Eq. (27) . It is clear that the temper- ature TR has to be comparable with the Planck mass M p to have some observable changes of the effective pa- rameter a l ( P R ) . The high-temperature behavior of both the effective cosmological constant A ( P R ) and the grav- itational constant G ( P R ) is similar as in the case of the scalar pregeometry which has been discussed in Ref. [I].

ACKNOWLEDGMENTS

The author thanks J. Horskjr, M. Lenc, and M. Pardy for discussions.

APPENDIX: THE REAL TIME GREEN'S FUNCTION

In this appendix, we will derive the RTFT Green's function for the dilatons without the string corrections [a' = 0 in Eq. ( 4 ) ] in an external static gravitational field.

We can rewrite the first term in the action ( 4 ) in the form

S [ ~ I ~ ' = O = --- I J d4x G $ ( ~ ) ( u + m 2 ) 4 ( x ) 8rrG

- - -1 J d4x [$(x)(qYuB,& + nt2)4(r) 87r G

+ 4 ( x ) E ( z i , d Z ) 4 ( ~ ) I . ( A l l

where

E ( z ' , ~ , ) = f i ( O + m 2 ) - ( ~ ~ v a , d , + m 2 ) . (A2)

The second term in Eq. (A2) is calculated with the help of the metric qpu=diag(l,-1, -1, - 1 ) . We have introduced the helping mass term in the action (4). We shall take the

49 FINITE TEMPERATURE DILATON GRAVITY 5197

limit m + 0 in the final equations. It is important that i

the term E(xi , a,) does not depend on the time variable iA(')[p](k) = U(P, ko) !, ) u(P1kO) 1

k2-ma-;, xO because we consider static space-time.

Now we derive from the RTFT diagrammatic technique (A41

[12] RTFT- causal Green's function A[P](x, x') connected where with the action (Al) :

Performing the sum (A3) we obtain the exact expression where ED = -diag(E(xi1 a,), -E(xi, 8,)) and for the Green's function:

xU(P, ko) diag( i A ~ ( k o ) ( x ~ , x ' ~ ) , [iAF(ko)(xi, x '~) ] ' ) U(P, ko)

We see that because the external metric field has been considered static it is possible to diagonalize the thermal dilaton Green's function. The propagator iAF is the Feynman propagator in a curved background which can be expressed in the limit xi + xti using the relevant DeWitt coefficients ak(x) [8,9]:

Now we can express the coincidence limit for A [PI ll (x, x) using Eqs. (A6) and (A7):

where

and

- e x p ( - s m 2 ) ~ ( x l s ) exp (e) n=-oo

1 1 1 aa(x) = -R' + -(-R,,RpV + R ,,,, RPVTu) - - R .

72 180 30 (-410)

We have written out explicitly the first three DeWitt coefficients in Eq. (A10). The inverse temperature PR is the local Lorenz-rest-frame inverse temperature given by Eq. (8).

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