Conformal invariance and apparent universality of semiclassical gravity
aa r X i v : . [ h e p - t h ] J u l Conformal invariance and apparent universality of semiclassical gravity
A. Garbarz, G. Giribet, and F. D. Mazzitelli
Departamento de F´ısica Juan Jos´e Giambiagi,FCEyN UBA, Facultad de Ciencias Exactas y Naturales,Ciudad Universitaria, Pabell´on I, 1428 Buenos Aires, Argentina. (Dated: today)In a recent work, it has been pointed out that certain observables of the massless scalar fieldtheory in a static spherically symmetric background exhibit a universal behavior at large distances.More precisely, it was shown that, unlike what happens in the case the coupling to the curvature ξ isgeneric, for the special cases ξ = 0 and ξ = 1 / h T µν i turns out to be independent of the internal structure of the gravitational source. Here, weaddress a higher dimensional generalization of this result: We first compute the difference between ablack hole and a static spherically symmetric star for the observables ˙ φ ¸ and h T µν i in the far fieldlimit.Thus, we show that the conformally invariant massless scalar field theory in a static sphericallysymmetric background exhibits such universality phenomenon in D ≥ h T µν i for a weakly gravitating object. These results lead tothe explicit expression of the expectation value h T µν i for a Schwarzschild-Tangherlini black hole inthe far field limit. As an application, we obtain quantum corrections to the gravitational potentialin D dimensions, which for D = 4 are shown to agree with the one-loop correction to the gravitonpropagator previously found in the literature. PACS numbers: 04.62+v, 04.70Dy
I. INTRODUCTION
In quantum field theory in curved spaces, vacuum polarization effects exhibit, in general, a non-localdependence on the spacetime metric. For example, particle production in Robertson Walker metrics dependon the whole evolution of the scale factor [1]. More closely to the present work, in static and sphericallysymmetric geometries, the expectation value of the energy momentum tensor evaluated outside a weaklygravitating object depends on its inner structure [2]. More generally, for arbitrary metrics, a covariantexpansion of the effective action in powers of the curvature tensor is explicitly non-local [3, 4].In a recent work, Anderson and Fabbri [5] studied what they called “apparent universality in semiclassicalgravity”, which is exhibited by certain observables corresponding to the theory of a massless quantum scalarfield on static spherically symmetric backgrounds. More specifically, they have shown that, far from theclassical gravitational source, the mean value (cid:10) φ (cid:11) in the Boulware state, does not depend on the internalstructure of the source when the scalar field is minimally coupled to the curvature, i.e. the result is the samefor a black hole, a neutron star, or a weakly gravitational object, as long as they are static and sphericallysymmetric. The situation for h T µν i is different, because the universal behavior holds both for minimal andconformal couplings.In this paper, we will be concerned with the computation of the expectation values (cid:10) φ (cid:11) and h T µν i ,corresponding to a massless scalar field φ formulated on a D -dimensional spherically symmetric background,being such expectation values defined with respect to the Boulware state. In D dimensions, and in the largedistance limit, these observables are typically given by (cid:10) φ ( x ) (cid:11) ≃ aMr D − , h T µν ( x ) i ≃ bMr D − , (1)where M is the mass of the gravitational background, while a and b are two numerical coefficients thatdepend on D , the coupling ξ , and may also depend on the internal structure of the gravitational source.In the case the gravitational object is a star [23], these coefficients are obtained by reading the largedistance behavior of non-local terms arising in the one-loop computation. On the other hand, in the case ofa black hole, these coefficients may be obtained by using the method of [6, 7]. In fact, in a generic case, theprecise values of a and b do depend on whether a horizon exists or not. Nevertheless, as it was pointed outby Anderson and Fabbri in Ref. [5], there exist very special cases where (1) exhibit some kind of universality,so that the large distance limit of the expectation values turn out to be independent on the nature of thegravitational object. Here, we will study this universality phenomenon, which can be seen to occur in theminimally coupled and conformally coupled scalar field theories.In [5], it was shown that in the four-dimensional conformally coupled case ( ξ = 1 / D = 4) the largedistance behavior of h T µν i results independent on whether the gravitational object is either a black hole ora star. This also occurs for the minimally coupled case ( ξ = 0), for both h T µν i and (cid:10) φ (cid:11) . We can expressthese agreements by saying that in the large distance limit it happens that∆ h T µν ( x ) i = h T µν ( x ) i Star − h T µν ( x ) i BH ∼ ξ ( ξ − / Mr + O ( M /r ) , (2)and ∆ h φ ( x ) i = h φ ( x ) i Star − h φ ( x ) i BH ∼ ξMr + O ( M /r ) (3)As already pointed out in [5], the coincidence of the results for minimal coupling can be traced back tothe fact that the large distance behavior of the observables is determined by the s -wave in the low frequencylimit. The field modes turn out to be independent of the metric in this limit, so the differences ∆ h T µν i and∆ (cid:10) φ (cid:11) vanish.In the absence of a simple physical explanation for the intriguing universality of h T µν i in the conformallycoupled theory, one may wonder whether the vanishing of ∆ h T µν i in the case ξ = 1 / ξ − /
6) usually arises in semiclassical computations in dimensions D ≥ a of the Schwinger-De Witt expansion is a = ( ξ − / R in all dimensions [1]. In thispaper we work out a dimensional extension of the computation of [5] and show that conformal invariance isactually playing a crucial role in this phenomenon.We will perform the explicit computations of the observables ∆ h T µν i and ∆ (cid:10) φ (cid:11) in the large distancelimit of a spherically symmetric static space-time in arbitrary number of dimensions D , and with arbitrarycoupling ξ between the scalar field and the curvature. In particular, we will show that the following expressionholds ∆ h T µν ( x ) i = h T µν ( x ) i Star − h T µν ( x ) i BH ∼ ξ ( ξ − ξ D ) Mr D − + O ( M /r D − ) (4)with ξ D = ( D − D − , i.e. the conformal coupling in D dimensions. This implies that the large distance behaviorof the semiclassical correction to the stress tensor of a conformally invariant scalar field is independent ofthe nature of the gravitational source. This manifestly shows that conformal invariance plays an importantrole in this universality phenomenon.An additional motivation to extend the computation of [5] to higher dimensions would come from the con-jectured correspondence between quantum corrected black holes in D -dimensional braneworlds and classicalextended objects in D + 1-dimensional bulks [8, 9]. Typically, the number of gravitational solutions with agiven asymptotic symmetry is known to grow as the dimensionality of space-time increases, and, therefore,it would be natural to ask whether the universality in the computation of the backreaction effects inducedby h T µν i is maintained when D becomes larger. Speculatively, studying the universality of h T µν i in the D -dimensional conformally coupled theory might be useful to indirectly learn about the unicity of extendedsolutions representing localized objects in D + 1-dimensions. We derive the explicit expression of h T µν i of aSchwarzschild-Tangherlini black hole in the far field limit in Section 3.The explicit computation of h T µν i in the D -dimensional conformal theory would be also important withinthe context of AdS/CFT correspondence [10]. It is well known that the so-called Randall-Sundrum Malda-cena complementarity [11] yields a remarkably numerical agreement between boundary and bulk computa-tions of the corrected graviton propagator. This agreement is relatively well understood for D = 4 where, bymeans of the introduction of a IR cut-off, the boundary theory corresponds to the N = 4 SYM theory coupledto gravity, and non-renormalization theorems are available. In general, performing such a bulk-boundarycomparison is a highly non-trivial problem, and one has no hope of having an explicit D -dimensional ana-logue of the computation of [11]. Nevertheless, even in this case, having achieved to explicitly compute h T µν i is important, as this quantity gives the one-loop scalar matter correction to the graviton propagator in D dimensions [12]. This provides important information about the functional form of both bulk and boundaryquantities.The paper is organized as follows. In Section 2 we will compute the differences ∆ (cid:10) φ (cid:11) and ∆ h T µν i fora massless scalar field in D dimensions, showing explicitly that both vanish for minimal coupling and that∆ h T µν i vanishes also for conformal coupling. In Section 3 we compute explicitly (cid:10) φ (cid:11) and h T µν i in the weakfield approximation. These results, combined with the differences computed in Section 2, allow us to computethe large distance behavior of the vacuum polarization around a D dimensional Schwarszchild-Tangherliniblack hole. As another application of the results for weak gravitational fields, we compute the quantumcorrections to the Newtonian potential in D dimensions. Section 4 contains the conclusions of our work. II. UNIVERSALITY IN THE CONFORMALLY INVARIANT THEORY
In this section we will compute the quantities ∆ (cid:10) φ (cid:11) and ∆ h T µν i , as defined in (2)-(3). This allows tocompare the vacuum polarization effect produced by a star and that produced by a black hole, both in thelarge distance limit. First, we will compute the differentce ∆ h φ i = h φ i Star − h φ i BH for a massless scalarfield in D dimensions and with arbitrary coupling to the curvature. Then, we will address the computation of∆ h T µν i in the large distance limit. To compute these expectation values we resort to a dimensional extensionof the method developed in [5], which we will follow closely. Let us briefly review the main steps.First, consider the Euclidean static spherically symmetric space in D dimensions, with metric ds = f ( r ) dτ + 1 k ( r ) dr + r d Ω n , (5)where f ( r ) and k ( r ) are two positive functions, and where d Ω n is the line element of the unit n -sphere, with n = D −
2. In the absence of matter, the metric (5) is given by the Schwarzschild-Tangherlini [13] solution f ( r ) = k ( r ) = 1 − (cid:0) r h r (cid:1) n − , and for the black hole case it develops a horizon at r = r h .To compute the expectation value h φ i , let us be reminded of the fact that the unrenormalized value of h φ i is given by the real part of the Euclidean Green function G E ( x, x ′ ) in the coincidence limit x → x ′ .Namely h φ ( x ) i = lim x ′ → x Re( G E ( x, x ′ )) . (6)The differential equation to be obeyed by the Euclidean Green function is [5]( ✷ x − ξR ) G E ( x, x ′ ) = − δ ( D ) ( x − x ′ ) √ g . (7)To solve this equation, it is convenient to consider the form G E ( x, x ′ ) = 1 π Z ∞ dω cos ( ω ( τ − τ ′ )) X l, { m } Y ( n ) l { m } (Ω) Y ( n ) ∗ l { m } (Ω ′ ) R lω ( r, r ′ ) , (8)where Y ( n ) l { m } (Ω) are the harmonic functions on the n -sphere, S n , satisfying [14] △ Y ( n ) l { m } (Ω) = − l ( l + n − r Y ( n ) l { m } (Ω) , (9)being ∆ the Laplacian on S n . Then, in the vacuum region, (7) takes the form ∂ r R lω ( r, r ′ ) + (cid:16) nr + ( ∂ r log f ) (cid:17) ∂ r R lω ( r, r ′ ) − (cid:18) ω f + l ( l + n − f r (cid:19) R lω ( r ) = − δ ( r − r ′ ) r n , (10)where f ( r ) = k ( r ) = 1 − (cid:0) r h r (cid:1) n − .It is also convenient to factorize R lω ( r ) as follows R lω ( r, r ′ ) = C ωl p ωl ( r < ) q ωl ( r > ) , (11)where r > (and r < ) means the grater (resp. the smaller) between r and r ′ , and where p ωl ( r ) and q ωl ( r ) aretwo independent homogeneous solutions to (10).In addition, p ωl and q ωl satisfy the Wronskian condition C ωl ( q ′ ωl ( r ) p ωl ( r ) − q ωl ( r ) p ′ ωl ( r )) = − f ( r ) r n , (12)where the prime denotes the derivative with respect to r . This expression (12) follows from integrating Eq.(10) over an infinitesimal region around the point r ′ .Now, let us compute the quantity ∆ h φ i ≡ h φ i Star − h φ i BH . From the expressions above, we can write∆ h φ ( x ) i = Re π Z ∞ dω X l, { m } Y ( n ) l { m } (Ω) Y ( n ) ∗ l { m } (Ω) (cid:0) C Star ωl p Star ωl ( r ) q Star ωl ( r ) − C BH ωl p BH ωl ( r ) q BH ωl ( r ) (cid:1) , (13)where the superscripts Star and BH label the modes corresponding to the star and the black hole, respectively.Note that, although h φ i Star and h φ i BH are both divergent quantities, their difference must be finite outsidethe star, since the covariant renormalization involves the subtraction of the Schwinger-DeWitt expansion ofthe Green function [1, 7], which is local in the metric.The reason why the modes for the star and those for the black hole differ from each other, is that theymust satisfy different boundary conditions. More precisely, the modes q ωl must be regular at infinity, forboth star and black hole, so q BH ωl = q Star ωl = q ωl . On the other hand, the modes p BH ωl must be regular atthe horizon, while p Star ωl must be regular at the origin. Such are the boundary conditions for the two-pointfunction to be well defined in the region where the Schwarzschild metric holds.Outside the star, we can write p Star ωl as a linear combination of two independent solutions p BH ωl and q ωl , p Star ωl ( r ) = α ωl p BH ωl ( r ) + β ωl q ωl ( r ) . (14)In turn, coefficients β ωl mix the modes in the star background. The reader may refer to Ref. [5] for furtherdetails.By evaluating Eq. (12) for both the case of the star and the case of the black hole, and using (14), we getthe relation α ωl C Star ωl = C BH ωl , so we get∆ h φ ( x ) i = Re π Z ∞ dω X l, { m } Y ( n ) l { m } (Ω) Y ( n ) ∗ l { m } (Ω) C BH ωl β ωl α ωl ( q ωl ) . (15)As we are interested in the region far from the gravitational bodies, we consider the leading contributionin the 1 /r expansion. Consequently, we are interested in the flat space modes q flat ωl ( r ) = r − n ω a +1 k a ( ωr ) , p flat ωl ( r ) = r − n ω − a i a ( ωr ) , (16)where k a and i a are the modified spherical Bessel functions with a = l + n −
1. In turn, the Wronskiancondition reads C BH ωl = π .Dimensional analysis, combined with the mean value theorem, leads to the conclusion that only the ω = l = 0 contribution is relevant in the 1 /r expansion, yielding the result∆ h φ ( x ) i = ( n − π n +12 r n − Γ (cid:18) n − (cid:19) Re (cid:18) β ω =0 ,l =0 α ω =0 ,l =0 (cid:19) Γ (cid:0) n (cid:1) Γ (cid:0) n − (cid:1) Γ (cid:0) n +12 (cid:1) . (17)This is valid for any static spherically symmetric star. It is worth noticing that this quantity vanishes for ξ = 0. This is because, when ω = l = 0 and ξ = 0, the homogeneous solutions to (10) that have to be regularat the black hole horizon, or regular at the center of the star, are constant. Then, because of the relation(14) and because q ω =0 ,l =0 is not a constant, β ω =0 ,l =0 must be zero. Actually, it would be convenient to keepin mind that β ω =0 ,l =0 is proportional to ξ .The result for ∆ h φ i depends on the inner structure of the star through the factor Re (cid:16) β ω =0 ,l =0 α ω =0 ,l =0 (cid:17) . Now,let us compute this factor explicitly for the case of a weakly gravitating star. First, we can perturbe themodes as follows p ω =0 ,l =0 ( r ) = p flat ω =0 ,l =0 ( r ) + δp ( r ) , q ω =0 ,l =0 ( r ) = q flat ω =0 ,l =0 ( r ) + δq ( r ) (18)being δp and δq small perturbations around flat solutions p flat ω =0 ,l =0 = √ π − n Γ (cid:0) n +12 (cid:1) , q flat ω =0 ,l =0 ( r ) = √ π n − r n − Γ (cid:18) n − (cid:19) . (19)By writing α ωl and β ωl in terms of the modes and their first derivatives, and keeping only first order terms,one gets β ω =0 ,l =0 α ω =0 ,l =0 = ( δp Star ′ − δp BH ′ ) q flat ′ ω =0 ,l =0 (cid:12)(cid:12)(cid:12) r = r ∗ (20)Where, again, the prime means the derivative with respect to r . Then, it remains to compute δp Star ′ and δp BH ′ evaluated at the radius of the star r ∗ . The latter is exactly zero, as it turns out that p BH ω =0 ,l =0 = p flat ω =0 ,l =0 .On the other hand, by solving the linearized differential equation for δp Star , and demanding regular behaviorat the origin, we find ddr δp
Star ( r ) = ξ √ π − n Γ (cid:0) n +12 (cid:1) r n Z r dr r n R ( r ) . (21)Now, it is possible to evaluate expression (20) as a function of D . Using (17), we eventually find∆ h φ ( x ) i = − ξ − D π − D Mr D − Γ (cid:0) D − (cid:1) Γ (cid:0) D − (cid:1) ( D − (cid:0) D − (cid:1) . (22)Here we additionally used the identity R d D − xR ( x ) = πMD − which holds for any static mass distribution.This allows us to claim that (22) is independent of the internal structure of the weakly gravitating star.Expression (22) is the difference between h φ i computed for a weakly gravitating star and the same quantitycomputed for a black hole of the same mass in the region far from these objects. It is worth mentioning thatthis result agrees with that of [5] for the case D = 4.Now, we move on to compute the quantity ∆ h T µν i = h T µν i Star − h T µν i BH , which corresponds to the farfield limit of the difference between the expectation value h T µν i for a static spherically symmetric star andthat for a Schwarzschild-Tangherlini black hole. Since the computation of ∆ h T µν i is quite similar to that of∆ h φ i we discussed above, and in order to avoid redundancies, we will limit ourself to present the results.The reader can find the details in [5].To compute h T µν i , it is convenient to write this quantity as the coincidence limit of the Euclidean Greenfunction G E ( x, x ′ ) and of its covariant derivatives G E ; µ ′ ν = ∇ µ ′ ∇ ν G E ( x, x ′ ). Namely [6], h T µν ( x ) i = lim x ′ → x (cid:18)(cid:18) − ξ (cid:19) (cid:16) g α ′ µ G E ; α ′ ν + g α ′ ν G E ; α ′ µ (cid:17) + (cid:18) ξ − (cid:19) g µν g α ′ σ G E ; α ′ σ − ξ (cid:16) G E ; µν + g α ′ µ g β ′ ν G E ; α ′ β ′ (cid:17)(cid:17) . (23)Then, following similar steps to those described above, and after some lengthy calculations, we find thefollowing results for the differences ∆ h T µν i ,∆ h T νµ ( x ) i = ( D − − D π − D − r D − Γ(2 D − D − ) Re (cid:18) β ω =0 ,l =0 α ω =0 ,l =0 (cid:19) ( ξ − ξ D ) diag (cid:18) , − , D − D − , ..., D − D − (cid:19) . (24)As in the case of ∆ h φ i , this quantity is found to vanish in the minimally coupled theory, because β ω =0 ,l =0 is proportional to ξ . Then, replacing in (24) the value of β ω =0 ,l =0 α ω =0 ,l =0 that corresponds to a weakly gravitatingstar, we find∆ h T νµ ( x ) i = − ξ ( ξ − ξ D ) 2 − D π − D Mr D − Γ(2 D − D − ) × diag (cid:18) , − , D − D − , ..., D − D − (cid:19) . (25)As expected, this expression agrees with that of [5] in the particular case D = 4. III. EXPECTATION VALUES ON A WEAKLY GRAVITATING BACKGROUND
In this section we will make use of the results of [3] to calculate the expectation values h φ i and h T µν i fora weakly gravitating object.Let us start by considering equation (23) in [3], from which we can write the one-loop effective action Γ (1) for D > (1) = 12 (4 π ) − D/ Z d D x √ g (cid:16) ξ Rβ (1) D/ ( ✷ ) R − ξRβ (3) D/ ( ✷ ) R + R µν β (4) D/ ( ✷ ) R µν + Rβ (5) D/ ( ✷ ) R + O ( R ) (cid:17) , (26)where the functions β ( i ) D/ ( ✷ ) are given by β ( i ) D/ ( ✷ ) = √ π − D/ Γ(( D − / f ( i ) D/ (cid:16) − ✷ (cid:17) D/ − ln − ✷ µ for even D , while β ( i ) D/ ( ✷ ) = 14 π / ( − ( D − / Γ(( D − / f ( i ) D/ (cid:16) − ✷ (cid:17) D/ − (27)for odd D . The factors f ( i ) D/ in (26) are given by f (1) D/ = 1 , f (3) D/ = ξ D = D − D − ,f (4) D/ = 12( D − D + 1) , f (5) D/ = ( D/ − D/ − D − D + 1) . (28)In order to compute h φ i in D dimensions, one could address the calculation by using a resumation of theSchwinger-DeWitt expansion [15], or by computing perturbatively the two point function, along the lines ofRef. [2]. However, even when these methods lead to the right expression, here we prefer to take a shortcutby exploiting the fact that varying the effective action with respect to ξ yields ddξ e − Γ (1) = Z [ D φ ] ddξ e − S [ g µν ,φ ] = 12 Z d D x √ gR h φ i , (29)so that one can read h φ i form this expression directly. Varying (26) with respect to ξ and then performinga Wick rotation, we find h φ ( x ) i = − − π D +12 (2 π ) D ( − D − D Γ( D − ) ( ξ − ξ D )( − ✷ ) D − ln − ✷ µ R ( x ) . (30)On the other hand, the analogous expression for odd dimensions reads h φ ( x ) i = − π D +32 (2 π ) D ( − D +12 − D Γ( D − ) ( ξ − ξ D )( − ✷ ) D − R ( x ) . (31)It is worth noticing that for a weakly gravitating object the expectation value h φ i vanishes in the confor-mally coupled case. The reason is the following: Being a scalar, on general grounds we expect h φ ( x ) i = ( F ( − ✷ ) + ξF ( − ✷ )) R (32)for adequate form factors F i ( − ✷ ). As this equation must be valid for any metric, we can specialize it for ametric which is conformally flat and asymptotically flat in the past. In this situation, it is clear that h φ ( x ) i must vanish for conformal coupling, since the conformal vacuum coincides with the IN vacuum. Thereforewe conclude that F ( − ✷ ) = − ξ D F ( − ✷ ), i.e. h φ ( x ) i is proportional to ( ξ − ξ D ).From expressions (30) and (31) we can obtain the explicit form of h φ i for a static spherically symmetricstar in the far field limit. So, imposing these conditions we get h φ ( x ) i Star = − ( ξ − ξ D )2 − D π − D Mr D − Γ (cid:0) D − (cid:1) Γ (cid:0) D − (cid:1) ( D − (cid:0) D − (cid:1) , (33)which is valid for arbitrary number of dimensions D > ξ inboth h φ i Star and ∆ h φ i . Since no dependence on ξ appears in the mode equation for p BH ω,l and q BH ω,l , thenthe quantity h φ i BH turns out to be independent of that coupling constant. In other words, we verify∆ h φ i | O ( ξ ) = h φ i Star | O ( ξ ) .Notice also that expression (33) permits to obtain h φ i in the black hole background in the region far fromthe horizon. In fact, using (22) we find that in D dimensions this quantity is given by h φ ( x ) i BH = ξ D − D π − D Mr D − Γ (cid:0) D − (cid:1) Γ (cid:0) D − (cid:1) ( D − (cid:0) D − (cid:1) . (34)On the other hand, the expectation value h T µν i in a weak field background is given by varying the effectiveaction with respect to the metric, and writing the result up to second order in the curvature; namely h T µν ( x ) i = − √ g δ Γ (1) δg µν + O ( R ) , (35)This expression can be written down in the following way h T µν ( x ) i = ( ξ − ξ D ) A µν + B µν , (36)where A µν = f D F ( ✷ ) H (1) µν , (37) B µν = f D (cid:16) ( f (5) D/ − ξ D ) F ( ✷ ) H (1) µν + f (4) D/ F ( ✷ ) H (2) µν (cid:17) (38)and, for even dimensions, F ( ✷ ) = ( − ✷ ) D − ln − ✷ µ (39) f D = π D/ D/ ( − D/ (2 π ) D ( D − H (1) µν = 4 ∇ µ ∇ ν R − g µν ✷ R + O ( R ) (41) H (2) µν = 2 ∇ µ ∇ ν R − g µν ✷ R − ✷ R µν + O ( R ) , (42)The term B µν in (36) is the only one that contributes in the conformal invariant case ξ = ξ D . Suchcontribution can be seen to be traceless, so it does not appear in the trace anomaly, and h T µµ i vanishes. Thisis because the anomaly is of higher order in the curvature.The case of odd dimension D is similar. In fact, it follows from (36)-(42) by replacing f D and F ( ✷ ) in theexpressions above by ˜ f D = 2 − D π / (4 π ) − D/ ( − D +12 Γ( D − ) (43)˜ F ( ✷ ) = ( − ✷ ) D/ − , (44)which come from (27).Once spherical symmetry and staticity are imposed, expression (36) yields the following result for theexpectation value of the stress tensor in the region far away from the star, h T νµ ( x ) i Star = − − D π − D Mr D − Γ(2 D − D − ) (cid:16) (cid:16) ( ξ − ξ D ) + f (5) D/ − ξ D (cid:17) × diag (cid:18) , − , D − D − , ..., D − D − (cid:19) + f (4) D/ diag (cid:18) − D, − , D − D − , ..., D − D − (cid:19)(cid:19) . (45)which is valid in arbitrary number of dimensions D ≥ h T µν i for the case of a Schwarzschild-Tangherlini black hole back-ground in the region far from the horizon; namely h T νµ ( x ) i BH = − − D π − D Mr D − Γ(2 D − D − ) (cid:16) (cid:16) ξ D ( ξ D − ξ ) + f (5) D/ − ξ D (cid:17) × diag (cid:18) , − , D − D − , ..., D − D − (cid:19) + f (4) D/ diag (cid:18) − D, − , D − D − , ..., D − D − (cid:19)(cid:19) . (46)It is important to emphasize that this last result, together with h φ i (see (34)), are vacuum expectationvalues for the black hole background in the far field limit computed entirely with analytical methods, i.e.without the aid of numerical computations.As an application of (45) we can address the calculation of the semiclassical correction to the Newtoniangravitational potential [16]. To do this, we write the semiclassical Einstein equations using h T µν i Star as asource. In the Lorentz gauge, the quantum corrections to the metric satisfy ✷ h µν ( x ) = − π (cid:18) h T µν ( x ) i Star + η µν D − h T λλ ( x ) i Star (cid:19) . (47)Then, by making use of (45), we getΦ( r ) = − − D π − D Γ(2 D − D − Γ( D − ) Mr D − (cid:18) ( ξ − ξ D ) + ( D − D − ( D + 1) (cid:19) . (48)It is worth pointing out that this expression, in the special case D = 4 and ξ = 1 /
6, agrees with thesemiclassical correction to the gravitational potential [2, 11], namely V ( r ) = − M Gr (cid:18) π Gr (cid:19) , (49)where we have reintroduced the four dimensional Newton constant G for major clarity. This also agrees withthe one-loop correction to the graviton propagator in the conformally coupled theory [12, 17, 18]. IV. DISCUSSION
Motivated by the question about the connection between conformal invariance and the universality phe-nomenon discussed in [5], we addressed the explicit computation of the observables ∆ h T νµ i and ∆ h φ i , definedas in (2)-(3), in an arbitrary number of dimensions. These observables gather the vacuum polarization effectsfor the case of a massless scalar field in a static spherically symmetric background. We have shown that inthe D -dimensional theory both observables vanish for minimal coupling, and that ∆ h T νµ i also vanishes inthe conformally coupled theory. This result extends the results of [5] to D ≥ h T νµ i for a weakly gravitating object. This, togetherwith the expression for ∆ h T νµ i , enabled us to write down the explicit expression of the expectation value h T νµ i for a Schwarzschild-Tangherlini black hole. As an application of our results, we obtained the quantumcorrection to the gravitational potential in D dimensions, which for D = 4 are seen to agree with the one-loop correction to the graviton propagator previously found in the literature. It is worth mentioning thatthe functional form of the quantum correction to the D -dimensional gravitational potential we obtained,agrees with the classical correction induced by an extra dimension in the Randall-Sundrum scenario [19, 20],both yielding a 1 /r D − dependence in the (corrected) Newtonian potential. This is to be expected, as the0classical action in this scenario reproduces the nonlocal effective action given in (26) when restricted to thebrane [21].Even though the explicit computation we carried out in Section 2 can be regarded as a proof of thevanishing of ∆ h T νµ i in both the minimally and conformally coupled theory, one might still wonder whetheran intuitive physical explanation for this phenomenon exists. Actually, there is a particular case in which theuniversality can be demonstrated using simple arguments. Let us consider a massless field in D = 2, where ξ = 0 corresponds both for minimal and conformal coupling. For a two-dimensional metric of the form ds = f ( r ) dτ + 1 k ( r ) dr , (50)it is well known [22] that the conservation law ∇ ν h T νµ i = 0 together with the trace anomaly determinethe expectation value h T νµ i (in particular for the Boulware state, when chosen the appropriate boundaryconditions). Therefore, since the trace anomaly h T µµ i = R/ π depends locally on the metric, one can showthat all the components of h T νµ i are determined by the local values of f ( r ) and k ( r ). Probably, a similarintuitive explanation for the universality in D > s -wave sector of the quantum scalar field. However, inabsence of such an intuitive explanation, and given the fact that in D > h T νµ i are not fully determined by the trace anomaly, one has to resort to the computations of Section 2 toexplain the so called apparent universality. Acknowledgements
This work was supported by Universidad de Buenos Aires, ANPCyT, and CONICET. Conversations withP. Anderson are acknowledged. A.G. and G.G. thank the members of the Centro de Estudios Cient´ıficosCECS for their hospitality. G.G. also thanks the hospitality of the members of CCPP during his stay atNew York University. [1] N.D. Birrell, P.C.W. Davies,
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