Casimir energy and the cosmological constant Текст научной статьи по специальности «Математика»

2. Nojiri S„ Odintsov S.D. // Phys. Letters. 2003. V, B576. P. 5; S. Nojiri S„ Odintsov S.D. II Mod. Phys. Letters. 2004. V. A19. P. 627.

3. Chiba T. II astro-ph/0307338; Dolgov A.D., Kawasaki M, II astro-ph/0307285; Soussa M.E., Woodard R.P. II astro-ph/0308114; Nunez A, Salganik S. // hep-th/0403159.

4. Lue A., Scoccimarro R„ Starkman G.D. II asfro-ph/0401515.

5. Nojiri S„ Odintsov S.D. II Phys.Rev. 2003. V. D68. P. 123512, bep-tb/0307288; GRG. to appear, hep-th/0308176.

8. Vollick II II Phys.Rev. 2003. V. D68. P. 063510, astro-ph/0306630; Meng X., Wang P. II astro-ph/0307354, astro-ph/0308031, astro-ph/0308284; Flanagan E. II Phys.Rev.Lett. 2004. V. 92. P. 071101, astro-ph/0308111, gr-qc/0403063; Ezawa Y. et al. II gr-qc/0309010; Olmo G„ Komp W. II gr-qc/0403092.

7. Allemandi G„ Borowiec A., Francaviglia M. II hep-th/0403264;

8. Allemandi G., Borowiec A., Francaviglia M. II hep-th/0407090;

9. Brevik l„ Nojiri S„ Odintsov S.D., Vanzo L. II hep-th/0401073,

10. Barrow J.D., Otiewili A.C. II J. Phys. A: Math. Gen. 1983. V. 16. P. 2757.

11. Faraoni V, De Sitter attractor in generalized garvity.// arXiv: gr-qc/0407021.

12. Pagani E„ Tecchiol G„ Zerbini S. II Letters in Mathematical Phys. 1987. V. 14. P, 311.

13. Vilenkin A. // Phys. Rev. 1985. V. D32. P. 2511.

14. Sfarobinski A. // Phys.Lett. 1980. V. B91. P. 99; Marnaev S.G., Mostepaneriko V.M, //JETP. 1980. V. 51. P. 9.

15. Geyer B„ Odintsov S.D., Zerbini S. II Phys. Letters 1999. V. B460. P. 58.

16. Nojiri $., Odintsov S.D., Zerbini S. II Phys. Rev. 2000. V. D62. P. 064006.

Gamttmi R.1

cmmm ENERGY ANP THE- ' OSMOLOGICAL CONSTANT

Université ctegli Studi di Bergamo, Facoità di Ingegneria, Viaie Marconi 5, 24044 Dalmine (Bergamo) ITALY.

I. Introduction ;.d as the bare etymological

V LU U» i i„>;. ",ds!. . .. " '

:!. - , ' 1 1 .rtG(p). C6)

(I)

where At, is the cosmological constant, G is the gravitational constant and T is the energy-momentum tensor. By redefining A.,

7' lot __ nr

1 M-V ~= *■ [IV "

BnG

(2)

one can regain the original form of the field equations

(3)

^v --g^R=8itcr;:' = 8kG(I;1v +'0,

at the prize of introducing a vacuum energy density and vacuum stress-energy tensor

A.,

Pa =-

, » hIV "PA<?U.V

(4)

EnG *

Alternatively, Eq. (1 ) can be cast into the form, 2'

where we have included the contribution of the vacuum energy density in the form Tm = -<p)#(JV. In this case

^RV ^ + ^eff 8,,v = 0'

(5)

') i) OfT^

(2%)

2 + »i2 =

(7)

■~\0n GeV4

16k2

This gives a difference of about 118 orders [1]. The approach to quantization of general relativity based on the following set of equations

2k

jfV -&(R-2AC)

2k

(8)

and

-2V(.Jtff»F[ftf] = 0, (9)

where R is the three-scalar curvature, Ac is the bare cosmological constant and k = 8jiG , is known as Wheeler-De Witt equation (WDW) [2]. Eqs. (8) and (9) describe the wave function of the universe. The WDW equation represents invarianee under time reparametrization in an operatorial form, while Eq. (9)

' .Email: Remo.Garattini@unibg.it

represents invariance under diffeomorphism. GijU is

the supermetric defined as

Gyu = \(-SikSji + g„8jt ~8ii8u)- (10)

Note that the WDW equation can be cast into the form

_ fs

(11)

which formally looks like an eigenvalue equation. In this paper, we would like to use the Wheeler-De Witt (WDW) equation to estimate (p) . In particular, we will compute the gravitons ZPE propagating on the Schwarzschild background. This choice is dictated, by considering that the Schwarzschild solution represents the only non-trivial static spherical symmetric solution of the Vacuum Einstein equations. Therefore, in this context the ZPE can be attributed only to quantum fluctuations. The used method will be a variational approach applied on gaussian wave functional. Hie rest of the paper is structured as follows, in section II, we • -,v i,. ,.<!.r .1 approach to the . ".'; , i - , • ) „. "ive some of the v approximated to .. ■ _ <n section 111, we

serator acting on . • • UIA, we analyze »ve use the zeta coining from the

, j , , . •'!• _ 1 a -Y, ¡: . . and we write the »normalization group equation, in section IVA we use the same procedure of section IV, but for the trace part. We summarize and conclude in section V.

II. The Wheeler-De Witt equation and the cosmological constant

The WDW equation (8), written as an eigenvalue equation, can be cast into the form

(12)

where

A ' =

(13)

.......... (i4)

V <¥|>F)

The formal eigenvalue equation is a simple manipulation of Eq. (8). However, we gain more information if we consider a separation of the spatial part of the metric into a background term, gtj, and a

perturbation, hy,

gy^gy+hy. (15)

Thus eq. (14) becomes

where Air" represents the i* order of perturbation in hi}. By observing that the kinetic part of A,; is quadratic in the momenta, we only need to expand the three-scalar curvature jd3x^gR'J> up to quadratic

order and we get

--A^fe,,

(16)

j^'WF

+-hV,V h" —h"R,J'i

+-hR..h" + — h(R{"')h

2 J 4

4 4 * 2 J

(17)

where h is the trace of /.:,, and R'u' is the three

dimensional scalar curvature. To explicitly make calculations, we need an orthogonal decomposition for both kv and hy to disentangle gauge modes from physical deformations. We define the inner product (h,k) |;^Gvl'hv (x)ku (x)d\ (18)

by means of the inverse WDW metric GjJk,, to have a metric on the space of deformations, i.e. a quadratic form on the tangent space at hi}, with

Gi'kl=(g>kgJ,+gugJk~2g,Jgk'). (19)

The inverse metric is defined on cotangent space and it assumes the form

<P. 4> l4gGmp* (x)qu (x)d3x, (20)

so that

We, now multiply Eq. (12) by and we

functionally integrate over the three spatial metric ,

then after an integration over the hypersurface I, one can formally re-write the WDW equation as

(21)

Note that in this scheme the "inverse metric" is actually the WDW metric defined on phase space. The desired decomposition on the tangent space of 3-metric deformations [3-6] is:

A,

(22)

where the operator L maps into symmetric tracefree tensors

(23)

Thus the inner product between three-geometries

becomes

(A, A) := J= №

(24)

With the orthogonal decomposition in hand we can define the trial wave functional as

¥[/% =m>i^ am.4 ®p 't at* a% (25)

where

^(jt^expj-^WT'ft^J

¥r/if«'(í)I = e)tp —{hK~[h)T™ i 4

(26)

and K~' is the inverse "propagator". We will fix our attention to the TT tensor sector of the perturbation representing the graviton and the scalar sector. Therefore, representation (25) reduces to

¥f/%(i)] = JVexp j-^Wr1^.

(27)

xexpj-^AJT1/!)^ '

Actually there is no reason to neglect longitudinal perturbations. However, following the analysis of Mazur and Mottola of Ref. [5 on the perturbation decomposition, we can discover that the relevant components can be restricted to the TT modes and to the trace modes. Moreover, for certain backgrounds 'IT tensors can be a source of instability as shown in Refs. j7]. Even die trace part can be regarded as a source of instability. Indeed this is usually termed conformed instability. The appearance of an instability on the TT modes is known as non conformal instability. This means that does not exist a gauge choice that can eliminate negative modes. To proceed with Eq. (16), we need to know the action of some basic operators on ¥[ft6J. The action of the operator hy on |¥)=¥IA#] is realized by [8]

hiJiW) = hij(xWlh,jl (28)

The action of the operator ntj on [ *F>, in general, is

<29)

while the inner product is defined by the functional integration:

<¥,(¥,)= IVhyWlihyW^]. (30)

We demand that

1 (¥[|ti3xAi:|¥).......

V 0F|*F>

, f , A (31)

_ 1 fPtggTr-^jpxVFfy]

be stationary against arbitrary variations of ¥[/i(>].

Note that Eq. (31) can be considered as the variational analog of a Sturm-Liouville problem with the cosmological constant regarded as the associated eigenvalue. Therefore the solution of Eq. (14) corresponds to the minimum of Eq. (31). The form of

i;¥|A,:|¥\ can be computed with the help of the

wave functional (27) and with the help of

<4MMi)M.v)|4'> „ ...... '

- = K,ji< y)

(32)

<kP I ¥>

Since the wave functional (27) separates the degrees of freedom, we assume that

A' = A'1 +A'№C", then Eq. (31) becomes

(33)

1 <¥|Â^|¥)=a,x

V <¥¡¥> i CP IÂJ" I

(34)

= A

vV OF IVF)

Extracting the TT tensor contribution, we get

ij ld3xJfGiJ" [(2k)K-ll(x,x\u

1

-(A 2y-KL(x,x)k

(35)

(2k)

The propagator KL(x,x)mU can be represented as

2\(t)

(36)

where h^1 (it) are the eigenfunctions of A2. x

denotes a complete set of indices and X(x) are a set of variational parameters to be determined by the minimization of Eq. (35). The expectation value of A^

is easily obtained by inserting the form of the propagator into Eq. (35)

cof(x)

X 1=1

(2x)Xi{x) + -

h! = A;' - - 6/A + -hih = (hr V + - 5Ui,

, 3 , 3 . ■ " 3 '

(41

"V,(}/}{ = 0. Thus

-(ArA")/=-Av(A7T)/+4il-—

r" I r

f, 2MG^\ d2 (2r~:

-3MC'Vl £

(43)

liir r

and R" is the mixed Ricci tensor whose components are:

2 MG MG MG1

with

(37)

(2k)Xj(T)_

By minimizing with respect to the variational function X,.(x), we obtain the total one loop energy density for TT tensors

(38)

4 x

where

An (X.) = A'1 (X,) = -AL(Xj)/K. (39)

The above expression makes sense only for of (x) > 0.

111. The transverse traceless (TT) spin 2 operator for the Schwarzschild metric and the W.K.B. approximation

The Spin-two operator for the Schwarzschild metric is defined by

(Ahrr )j := ~(AThrr)! + 2(Rh'T)/, (40)

where the transverse-traceless (TT) tensor for the quantum fluctuation is obtained by the following decomposition

H(r) = hl(r)--h(r) K(r) = h*(r)-U(r).

(47)

From the transversality condition we obtain h%(r) = h%(r). Then K(r) = l(r)., For a generic value of the angular momentum L, representation (46) joined to Eq. (42) lead to the following system of PDE's

—As +— 1 -

2MG) 4MG,

—j- \H(r)^wuH(r)

-As+- 1-

r t r

2MG) 2 MG + ——

. (48)

K(r} = m;jK(r)

Defining reduced fields

H(r) = M!lt = (49)

r r

and passing to the proper geodesic distance from the

throat of the bridge

dr

dx = ±

1-

2 MG

(50)

This implies that (h* )/ 8'f = 0. The transversality condition is applied on (hr )f and becomes

the system (48) becomes

f1ix) = m;jlix)

(42)

dx

d'l A2

+ V2(r)

(51)

/2W = <,/2(jc)

where. As is the scalar curved Laplacian, whose form is

with

This implies that the scalar curvature is traceless. We are therefore led to study the following eigenvalue equation

(A2/i TT)]=(dh) (45)

where or is the eigenvalue of the corresponding equation. In doing so, we follow Regge and Wheeler in analyzing the equation as modes of definite frequency, angular momentum and parity [9]. In particular, our choice for the three-dimensional gravitational perturbation is represented by its even-parity form

(h"m)'j(r,m = dmg[H(r),K(r),L(rWlmm), (46)

r"

V^^ + UJr)'

where we have defined r = r(x) and l/,(r):

(52)

6 f 2MG) 3MG À r j" r3

6 ( 2MG) 3MG h +

U2(r) = Note that U. (r) > 0 when r>

(53)

5 MG

Ul{r)<0 when 2MG<r< (J2(r)> 0 Vre [2MG,+<»)

5 MG

The functions Ui (r) and U2 (r) play the role of two r-dependent effective masses m,2(r) and mi(r), respectively. In order to use the WKB approximation, we define two r-dependent radial wave numbers k{{r,l,(S}Uli) and k2(r,l,minl)

,2, , , 1(1+1) ,, , k; (r, /, mUl! ) = 0), - - m; (r) r

, V , . 2 1(1 + 1) 2. ,

K 0"> I, ®2,„i ) = - - «2 (f)

, , 5MG „., ^ 5MG

for r>----. When 2MG<r<--,

2 2

becomes

, V i x ? 1(1 + 1) 2/ ,

C A ) = ©Ult,--—— + fft, (r)

' 'if} trace part contribution

The trace part of the perturbation can be extracted from Eq. (17) to give

—Mft + ~ft.Y,VA 4 2

2 ' 4 '

(57)

(58)

,U).. ll.JiU. V-

1 i... i .. ' » k"V I uadik 1; I; * , i'l 4;> 11,ICC : I '1 ••

curvature vanishes. Therefore the only piece of the quadratic order coming from the scalar curvature expansion is

—AAA 4

(59)

Moreover, if we follow the orthogonal decomposition of Eq. (22), we can write

^=^+<"+4,(60)

then the trace part of Az becomes 2k f ,, it 1 r ,, r=( 1

■

(61)

-f fc/vf -Ifrf'WF \--Mh

6 k j= 2K & Vi( 4

By repeating the scheme of calculation of section II to the trace part in Eq. (61), we get the scalar part contribution to the ZPE

-(2f£)irW)

Är-^WF

i

(2k)

(A )K(x,x)

(62)

The propagator K(x, x) can'be represented as Jiw(x)h{t)(y)

2X(t)

(63)

(55)

ktirJ^J

(56)

where h's}(x) are the eigenfunctions of A . % denotes a complete set of indices and X(t) are a set of variational parameters to be determined by the minimization of Eq. (62). The expectation value of

A|BtT is easily obtained by inserting the form of the propagator into Eq. (62)

6 (2k)A,(T)

(64)

4 -

By minimizing with respect to the variational function k(%), we obtain the total one loop energy density for the scalarcomponent 1 ¡2*

(65)

where A'mc' (X) = (A')""" (X) = - A"™" (A.) / k . The above expression makes sense only for o)3(t)>0. In the Schwarzschild background, the operator A can be identified with the operator As of Eq. (43). If we repeat the same steps leading to Eq. (51), we get

r'

dx1

~+V(r)

where we have defined

(66)

(67)

and used the proper distance from the throat defined by Eq. (50). In order to use a W.K.B. approximation, we define a r-dependent radial wave number Mr,I,&,,,") 1(1 + 1) MG

3 .

(68)

IV. One loop energy Regularization and Renormalization

In this section, we proceed to evaluate Eq. (38). The method is equivalent to the scattering phase shift method and to the same method used to compute the entropy in the brick wall model. We begin by counting the number of modes with frequency less than co,, ¿ = 1,2. This is given approximately by

g(m,)= |v,(i, c0i)(2/ + l), (69)

where v¡(1,(01), ¡=1,2 is the number of nodes in the mode with (/,«,.), such that (r s r(x))

(70)

Here it is understood that the integration with respect to x and I is taken over those values which satisfy

kf (r, I, to, ) > 0, i = 1,2. With the help of Eqs. (69,70), constant reads

we obtain the one loop total energy for TT tensors

By extracting the energy density contributing to the cosmological constant, we get

An = -i- f 0)? ^T^(r)dmi, (72)

Ion

87t G

+m24(r)

mÎ(r)

In

+ 21n2-

+ 21n2-

1

(79)

where we have included an additional An coming from the angular integration. We use the zeta function regularization method to compute the energy densities p, and p2. Note that this procedure is completely equivalent to the subtraction procedure of the Casimir energy computation where the zero point energy (ZPE) in different backgrounds with the same asymptotic properties is involved. To this purpose, we introduce the additional mass parameter (X in order to restore the correct dimension for the regularized quantities. Such an arbitrary mass scale emerges unavoidably in any regularization scheme. Then we have

p,(e) =

where

1

36jf

'i>r

co

(73)

1

HriT 1

p2 =„_ Cml/ml +ml(r)dm2 1 OK

(74)

The integration has to be meant in the range where eo? + mf{r)t0 One gets

p,-(e) = -

mj(r)

256k

+ ln

+ 2M2-

(75)

_e {mf(r)j i = 1,2 . In order to renormalize the divergent ZPE, we observe that from Eq. (72), after reinserting the gravitational constant, we write

A = -8îtG(pj (e) + p2 (e)).

(76)

To handle with the divergent energy density we extract the divergent part of A" , in the limit e -» 0 and we set

G

, TTJif _

-(rrh4(r) + mî(r)).

(77)

The quantity in Eq. (79) depends on the arbitrary mass scale p. It is appropriate to use the «normalization

group equation to eliminate such a dependence. To this aim, we impose that [10]

1 dAl' Qi.) d Tr/ .

8 KG

(80)

Solving it we find that the renormalized constant Ajf should be treated as a running one in the sense that it varies provided that the scale ji is changing

Af (p, r) = Af (|t0 + (hi,4 (r) + ml (r)) In -t. (81)

Substituting Eq, (81) into Eq. (79) we find

A,',' Ot0,r)

'0 » J — .

1

8rcG

+/n24(r)

111

256K2

M-o

111 (m Î(r)^

I. Mi ;

- 2 In 2 + -

-21n2 + —

(82)

(83)

In order to fix the dependence of A on r and M , we find the minimum of A"(p,0,r). To this aim, last equation can be cast into the form2

Aor^l«.f) _ ¡4

8re G

256jc

x2(r)

lnUW]+I

4 2

+y2(r)

1

M2?)-*,

(84)

(85)

- < r and we get

32îte

Thus, the renormalization is performed via the absorption of the divergent part into the re-definition of the bare classical constant Arr ATl" -» Aj*f - A7™'. (78)

The remaining finite value for the cosmological

where x(r) = ±m,2(r)/|4 and y(r) = ±m^(r)/\il. Now we find the extrema of A0(|i.0,x(r), >'(/•)) in the range

5MG 2

x(r) = 0 y(r)=0' which is never satisfied and x(r) = 4/e U*(r) = 4^/«

(86)

_y(r) = 4/e [TO:;(r) = 4f4/e

(87)

which implies M = 0 and F = / 2|i,0. On the other

hand, in the range 2MG <r<

5 MG

. s We get again

1 Details of the calculation car. be found in the Appendix.

2 Recall Eqs. (55,58), showing a change of sign in ml (r). Even if this is not the most appropriate notation to indicate a change of sign in a quantity looking like a "square effective mass", this reveals useful in the zeta function regularization and in the serch for extrema.

h:(r) = 0 |y(r) = 0'

which has no solution and

which implies f M = 4|ijF3 / 3eG

[7 = yf6e/4\i0

Eq. (85) evaluated on the minimum, now becomes A = ^

2el%

(88)

(89)

(90)

(91)

It is interesting to note that thanks to the renormalization group equation (80), we can directly compute A" at the scale ji0 and only with the help of Eq. (81), we have access at the scale p..

f f^ ioiip ent'tgy Regulc*:iz<fr>on and

•'i;cii->ai'?xtion 'or tfv,\-;e

as lor the iicj. (/2), the energy density associated with A"m is

[2 1

J^iract:___

3 64it'

17

3 64K'

where we have defined

. MG m-(r)=—r-.

CO" J 00'

MG

dm

oí^Jm1 - iri2(r}dù\

(92)

(93)

The use of the zeta function régularisation leads to

(e) =

2 m4(r)

2 1 -

---Ll

3 64k2

tier

3 1024n*

• + ln

m2(r)

(or -m2(r))~ 2

+ 21m2--~ .

(94)

In order to renormalize the divergent trace contribution to 2PE, we observe that from Eq. (72), we get A""'™ -SiiGp"'0"'. (95)

To handle with the divergent energy density we extract the divergent part of A'ma, in the limit e 0 and we set

A"

2 G

3 128œ

mUj).

(96)

-»Aï**- A'

trace.div

(97)

Therefore, the remaining finite value for the eosmological constant reads

Ar = 2 m4(r)

8KG i 3 1024n2

x In

m2(r)

+ 21n2 —

2

(98)

As for the quantity in Eq. (79), we have a dependence

on the arbitrary mass scale p.. The use of the renormalization group equation gives

1

8%G

dA T

Solving Eq. (99), we find a running A™" that the scale p is changing

Ar(H.r) = Ar(Vv,r) + S—^GlaA

V 3 647C (i0

Substituting Eq. (100) into Eq. (98) we find (Ho,r)_ ¡2 m4(r)

8nG

(99) provided

(100)

a trace J\0

X 111

m\r)

V-l

3 1024T12 1

-21n2 +

(101)

Thus, the renormalization is performed via the absorption of the divergent part into the re-definition of the bare classical constant A"*"'*

If we adopt the same procedure of finding the minimum for A^^.r) we discover that the only consistent solution is for M = 0. This leads to the conclusion that the trace part of the perturbation does not contribute to the cosmological constant.

¥, Summary and Conclusions

In this paper, we have considered how to extract information on the cosmological constant using the Wheeler-De Witt equation. In particular, by means of a variational approach and a orthogonal decomposition of the modes, we have studied the contribution of the transverse-traceless tensors and the trace in a Schwarzschild background. The use of the zeta function and a renormalization group equation have led to Eq. (91) and recalling Eq. (39), we have obtained

A'™ct(M,7) = 0

If we choose to fix the renormalization point p,0 = m,

we obtain approximately Axn {M,r)~ 1037 GeV~ which, in terms of energy density is in agreement with the estimate of Eq. (7). Once fixed the scale p0, we can see what happens at the cosmological constant at the scale p., by means of Eq. (81). What we see is that the cosmological constant is vanishing at the sub-

planckian scale p. = »j expl -—

, but unfortunately is

a scale which is very far from the nowadays observations. However, the analysis is not complete.

Indeed, we have studied the spectrum in a W.K.B. approximation with the following condition kf(r,l,io;), ¿ = 1,2. Thus to complete the analysis, we need to consider the possible existence of nonconformal unstable modes, like the ones discovered in Refs. [7]. If such an instability appears, this does not mean that we have to reject the solution. In fact in. Ref. [11], we have shown how to cure such a problem. In that context, a model of "space-time foam" has been introduced in a large N wormhole approach reproducing a correct decreasing of the cosmological constant and simultaneously a stabilization, of the system under examination. Unfortunately in that approach a renormalization scheme was missing and a W.K.B. approximation on the wave function has been used to recover a Schrodinger-like equation. The possible next step is to repeat the scheme we have adopted here in a large N context, to recover the correct vanishing behavior of the cosmological constant.

Appendix A: The zeta function regularization

In this appendix, we report details on computation leading to expression (73). We begin with the following integral

p(e) =

(!)

(®2+m2(r)) 2 CO2

(Al)

Wf dm- ,

(rn2-m2(r)f2

where we have used the following identities involving the beta function.

- Rex > 0, Rey > 0 (A3)

B(x,y) = 2^ dt-related to (

B(x,y)=--

t

(tl + l)x+>'

related to the gamma function by means of . T(x)T(y)

(A4)

T(x+y)

Taking into account the following relations for the F function

T(e-2) =

e , =

rp + e) e(e-l)(e-2)

Fl E + '

(A5)

1

and the expansion for small e

r(l + e) = 1 ~7E + 0(e2)

r|e + I|-r

-I-eF] ~ |(y+21n2) + 0(E2)

(Â6)

/ =1 + elnx + 0(e ),

where y is the Euler's constant, we find

P(£) = -

r«4(r)

16

+ ln

f 1 \

¥

m\r)

+ 2 In 2 - -

(A.7)

2. /. computation

If we define t = to/-JnFlr), the integral /_ in Bq.(Al) becomes

with irr (r) > 0.

1.1+ computation

If we define t = m/^jrn2(r), the integral I+ in Eq. (Al)

becomes

p(e) = \iumA-u(r) ¡~dt-—r

, E--

(i2+l) 2

=—(J,26m4^2e (r)fif—, e - 2

1 F 2 r(E~2) r £ — •

2 >

•s/tc 4 f ¡1

=-m (r) :—

4 • \m2(r)

r(e-2) ri.-:

P(e) = p. in (#■) J dt

2 -iV

(t ~ t)

= (i2Ew4^2e (r)B | e - 2, - e

rl —-e jr(e-2)

(A8)

4-Jn

m\r)

f ¿J m\r)

r|--e|r(e-2),

where we have used the following identity involving the beta function

1

B

[-v-ii,vj= jTdtf"1 (tp -l)v~1

(A9)

(A2)

p> 0, Rev > 0, Rep < p - pRev

and the reflection formula

r(z)F(l - z) = -zr(~z)r(z) (A 10)

From the first of Eqs. (A5) and from the expansion for

small £

we find

(1 - £(~y - 2 In 2)) + 0(e2 )

(AH)

+ 21n2-

2

(AI 2)

x£ =l + elnjc + 0(e2),

References

1. For a pioneering review on this problem see Weinberg S, II Rev. Mod. Phys, 1989, V. 61. P. t. For more recent and detailed reviews see Sahni V., Starobinsky A. Int. J. II Mod. Phys, 2000. V. D9. P. 373, astro-ph/9904398; Straumann N. The history of the cosmological constant problem//gr-qc/0208027; Padmanabhan T. II Phys.Rept. 2003. V. 380, P, 235, hep-th/0212290.

2. DeWitt B.S. II Phys. Rev. 1967. V. 160, P. 1113.

3. Berger M., Ebin D. // j. Diff. Geom. 1969. V. 3. P. 379.

4. York Jr. j.W. Ill Math. Phys, 1373. V. 14. P. 4: Ann. Inst. Henri Poincaré, 1974, V. A21. P. 319.

5. Mazur P.O., Mottola E. II Nucl. Phys. 1990. V. B341. P. 187.

6. Vassilevich D.V. II Int. J. Mod. Phys. 1993. V. A8. P. 1637. Vassilevich D.V. II Phys. Rev. 1995. V, D52. P. 999, gr-qc/9411036.

7. Gross D.J,, Perry M.J., Yaffe LG. II Phys, Rev. 1982. V. D25. P. 330; Allen B. II Phys. Rev. 1984. V. D30. P. 1153; Witten E. II Nucl Phys. 1982. V. B195. P. 481; Ginsparg P., Perry M.J, II Nucl. Phys. 1983. V. B222. P. 245; Young R.E. // Phys, Rev. 1983. V. D28. P. 2436; Young R.E, II Phys. Rev. 1983. V. D28. P. 2420; Hawking S.W., Page D.N. Il Commun. Math, Phys. 1983. V. 87. P. 577; Gregory R„ Latlamme R. II Phys. Rev. 1988. V. D37. P. 305; Garattini R. II Int. J. Mod. Phys. 1999. V. A14. P. 2905, gr-qc/9805096; Elizalde E„ Nojiri S„ Odintsov S.D. II Phys, Rev. 1999. V. D59. P. 061501, hep-th 9901026; Volkov M.S., Wipf A. II Nucl. Phys. 2000. V. B582. P. 313, hep-th/0003081 ; Garattini R. II Class. Quant, Grav, 2000, V. 17, P, 3335, gr-qc/0006076; PrestitJge T. II Phys, Rev. 2000. V. D61. P. 084002, hep-th/9907163; Gubser S.S., Mitra i. Instability of charged black holes in Anti-de Sitter space II hep-th/0009126; Garattini R. II Ciass. Quant. Grav, 2001. V. 18. P. 571, gr-qc/0012078; Gubser S.S., Mitra I. //JHEP. 2001. V. 8. P. 18; Gregory J.P., Ross S.F. II Phys, Rev, 2001. V. D64. P. 124006, hep-th/0106220; Real! H.S. II Phys. Rev. 2001, V. D64. P. 044005, hep-th/0104071; Gibbons G„ Hartnoll S.A. II Phys, Rev, 2001. V. D66. P. 084024, hep-th/0206202.

8. Ksrman A.K., Vautherin D. (/ Ann. Phys. 1989. V, 192. P. 408; Cornwall J.M., Jackiw R„ Tomboulis E. II Phys, Rev. 1974. V. D8. P. 2428;

Jackîw R; II in Séminaire de Mathématiques Supérieures, Montréal, Québec, Canada- June 1988 - Notes by P. de Sousa Gerbert; M, Consoli

and G. Preparafa II Phys. Lett. 1985. V, 6154, P. 411.

9. Regge T., Wheeler J.A. Il Phys. Rev, 1957. V. 108. P. 1083.

10. Perez-Msrcader J„ Odintsov S.D. Il Int. J. Mod. Phys. 1992. V. Di, P. 401; Cherednikov I.O. Il Acta Physica Slovaca. 2002. ¥, 52. P. 221; Cherednikov |,0. // Acta Php. Polon. 2004. V. B35. P. 1607; BoreJag M., Mohideen U„ Mostepanenko V.M. Il Phys, Rep. 2001, V. 353. P, 1; Inclusion of non-perturbative effects, namelv bevond one-loop, in de Sitter Quantum Gravity have been discussed in Falkenberg S„ Odintsov S.D. H Int. J. Mod. Phys. 1998. V, A13. P. 807, hei>th 9812019.

11. Garattini R inî. II J. Mod. Phys. 2002. V, D4, P. 635, gr-qc/0003030.

12. Gradshteyn l,S„ Ryzhik i,M. Table of Integrals, Series, and Products (corrected and enlarged edition), edited b y Â. jeffrey(Aoademic Press, Inc.).

• ■'■ t'rri

í:í:ThOPV f;C Î1. WiZCHS I < THEC^SEC» 0»c GRAVITmííON

Dipartimento di Matematica, Universitá di Torino Via C. Alberto 10,10123 TORINO (Italy)

There are a number of similarities between black-hole physics and thermodynamics. Most striking is the similarity in the behaviors of black-hole area and of entropy [...] It is natural to introduce, the concept of black-hole entropy as the measure of information about a black-hole interior which is inaccessible to an exterior observer [...]

I. Black Holes Entropy: an overview

The first law in Classical Thermodynamics was first introduced by Clausius for isolated macroscópica!

systems under the form: W = TbS - 8W,

where U denotes the internal energy, T the temperature, W the work done by the system and S the entropy of the system. One can use the principle (1) to define S classically, provided the other quantities

(1)

J. D. Bekenstein, Phys. Rev. D7, (1973)

1 email allemandiQdm.unlto.il

2 email fatibeneidm.unito.it

3 email francaviglia § dm. unto .it

4 email ralteri@dm.unito.it