Nonlocal effective actions in semiclassical gravity: thermal effects in stationary geometries

We compute the gravitational effective action by integrating out quantum matter fields in a weak gravitational field, using the Schwinger-Keldysh (in-in) formalism. We pay particular attention to the role of the initial quantum state in the structure of the nonlocal terms in the effective action, with an eye to nonlinear completions of the theory that may be relevant in astrophysics and cosmology. In this first paper we consider a quantum scalar field in thermal equilibrium, in a stationary gravitational field. We obtain a covariant expression for the nonlocal effective action, that can be expressed in terms of the curvature tensor, the four-velocity of the thermal bath and the local Tolman temperature. We discuss the connection between the results for ultrastatic and static metrics through conformal transformations, and the main features of the thermal corrections to the semiclassical Einstein equations.


I. INTRODUCTION
The observation of the present accelerated expansion of the universe catalyzed a large number of theoretical speculations, looking for natural explanations for the acceleration.
The aim of many theoretical constructions is to modify general relativity in the infrared, in order to produce an effective cosmological constant at the late stages of the universal evolution. One kind of such modifications are nonlocal cosmological models, which have been the subject of numerous works [1][2][3][4][5][6]. Nonlocality naturally arises when considering the backreaction of quantum fields on the evolution of the universe. Exact calculations for generic spacetimes are very difficult, and therefore there have been several alternative methods to suggest viable modifications. On the one hand, for weak gravitational fields, the effective action resulting from the integration of massless or massive quantum fields is very well known.
Light or massless fields could introduce significant infrared modifications. Nonlinear completions of these effective actions have been considered by several authors (see [1][2][3] and references therein). The nonlocal structure of the effective action is of the form RF (✷)R where R denotes the components of the Riemann tensor and F is a non-analytic function of the D'Alembertian. Related phenomenological proposals consider effective actions of the form RF ( 1 ✷ R), where R is the Ricci scalar [7].
The usual in-out effective actions obtained after the integration of quantum matter fields produce nonlocal and noncausal equations of motion for the metric [9]. This problem is generally avoided in the nonlocal cosmological models by using a procedure suggested many years ago by Barvinsky and Vilkovisky [8], which consists in computing the effective action in Euclidean spacetime, obtaining the field equations, and then replacing there the Euclidean propagators by retarded propagators. It was proved there that this replacement produces the correct field equations when the quantum state of the field is the ground state, and when the metric is, initially, asymptotically flat. In more general situations the use of the so called Schwinger-Keldysh or in-in/CTP formalism [9,10] becomes unavoidable, and it is in general advocated to justify the prescription of Barvinsky and Vilkovisky. Note that, in the phenomenological models, the use of the retarded propagator is just an additional prescription which is crucial to force the reality and causality of the equations of motion.
Another strong argument in favor of using the in-in formalism is that it provides more information than its Euclidean counterpart, i.e. it includes the stochastic effects. Indeed, the resulting field equations contain a stochastic noise source, whose statistical properties are given by the imaginary part of effective action, accounting for fluctuation and dissipation effects [10][11][12].
So far in the literature most of the works regarding nonlocal modifications of general relativity induced by quantum fields consider the field to be in the vacuum state. However, under general conditions the field is expected to be in a different quantum state, the most reasonable of which being that of finite temperature. For this reason, in this paper we will discuss the dependence of the effective action of a massless field with its quantum state, focusing on stationary spacetimes in the limit of weak gravitational fields. Moreover, for massless fields, the most interesting astrophysical and cosmological scenarios lie in a regime where T ≫ L −1 , where T is the temperature and L a characteristic curvature scale.
Therefore, we will pay particular attention to the high temperature expansion of the effective action and the resulting field equations. As we will see, the initial quantum state strongly modifies the nonlocal structure. In addition to its intrinsic interest, we expect the results to be useful for new proposals of in-in nonlinear completions of the effective action that produce real and causal equations of motion.
The dependence of the nonlocal effective action with the initial quantum state of the field has been discussed in previous papers. Indeed, Gusev and Zelnikov [13] computed the Euclidean effective action for a massless quantum field in a thermal state for an ultrastatic gravitational background (ie. in adequate coordinates the metric is g 00 = −1, g 0i = 0 and g ij = g ij (x)), using the covariant perturbation theory [8]. The in-in effective action has been computed by Campos and Hu [11] (see also [14]), in the limit of weak gravitational fields. They obtained the effective action and the corresponding equations of motion for the metric. The technical complexity of the calculation produces cumbersome results, and therefore we found it relevant to compute the effective action for stationary metrics, using an alternative procedure that allows us to find covariant expressions for effective action. As expected, the covariant effective action will involve not only the Riemann tensor but also a vector u µ that describes the four-velocity of the thermal bath, as well as a local redefinition of the temperature. Equivalently, it can be written in terms of the Riemann tensor and the Killing vector κ µ associated to the stationary geometry.
The paper is organized as follows. We present the calculation of the effective action for weak gravitational fields in Section II, where we obtain explicit expressions for the nonlocal kernels. We also write the effective action in a covariant way, adapted to nonlinear completions. In Section III we describe the high temperature expansion of the effective action, showing that the leading nonlocal contribution of the real part is linear in the temperature.
In Section IV we show that, for static metrics, the effective action can be derived from the ultrastatic case, using a conformal transformation. In Section V we write the field equations, and discuss qualitatively some features of the quantum corrections. Section VI contains the conclusions of our work. In the Appendices we include further details of the calculations.

II. EFFECTIVE ACTION
In this section we compute the in-in effective action for a massless, minimally coupled scalar field in a nearly flat spacetime. We follow Ref. [15], generalizing their approach to the case of a thermal state. The classical action for a scalar massless quantum field with minimal coupling is given by We are using the signature (− + ++) for the metric, and natural units = c = 1. For a weak gravitational field we write a perturbation around flat spacetime as and expand the action up to quadratic order in h µν where the perturbative operators in derivatives are andh with h(x) = h α α (x). The complete thermal in-in effective action reads (1) where G ab , (a, b = +, −) are the usual flat space propagators in the vacuum state, and G T the thermal contribution. Note that G T does not depend on the CTP indices a, b [11,14]. and where n(|q 0 |) = (e β|q 0 | − 1) −1 is the bosonic thermal particle distribution and β is the inverse temperature.
In principle, one should compute the effective action in n-dimensions, and absorb the infinities adding appropriate counterterms. However, in what follows we will consider only the temperature-dependent part of the effective action Γ T , which is finite due to the presence of the Bose factor n(|q 0 |). Therefore we can omit the counterterms and the discussion of renormalization. We split the thermal effective action as Γ T = Γ (1) + Γ (2) , with (1) (1) (1) where Γ (1) and Γ (2) are linear and quadratic in the propagator respectively. The zero temperature part can be found elsewhere, for example in Ref. [15]. The main nonlocal contribution Let us now elaborate on each of the two contributions to Γ T .

A. Calculation of Γ (1)
The terms proportional to V (1) in Eq.(8a) can be written in terms ofh µν , and read Analogously, for the terms proportional to V (2) , we have The full action Γ 1 is obtained by adding the above two equations evaluated at h µν → h The calculation of Γ (2) is much more involved. Before presenting the details, it is useful to take into account that, on general grounds [10,11], the quadratic part of the CTP effective action must be of the form for some real and symmetric kernels H(x, y) and N(x, y), and a real and antisymmetric kernel To evaluate the different terms of Eq.(8b) let us first consider the trace The kernel A µναβ (0,p) reads where η ρµτ ν = η ρµ η τ ν − 1 2 η µν η ρτ . The strategy for evaluating this kernel is as follows: we first perform the integration in q 0 , which is trivial due to the Dirac δ-function and the fact that for a stationary metric p 0 = 0.
We then perform the angular integral in the plane perpendicular top, and leave the answer in terms of an integral in k ≡ |k| = |q −p|, and in the remaining angular variable.
To exemplify, we schematically show the calculation of A 0000 (0,p): As mentioned above, computing the integral in q 0 is trivial. Then, changing to thek =q −p variable, we arrive at At this point, we can rely on symmetry properties to simplify the tensorial structure (see Appendix A), and find where the result is now expressed in terms of the scalar integrals I αγ , defined as with χ = cos θ.
In a similar way we can calculate every other contribution of A µναβ (p) (see Appendix B), thus obtaining from Eq.(13) that (1) where P ij = η ij − p i p j p 2 , P i0 = P 0i = 0 and P 00 = η 00 = −1. In this expression we identify two kind of terms: those which are proportional to a pair of projectors P µν = η µν − pµpν p 2 and those which are not. Since the former can be easily written in terms of the Riemann tensor and the four-velocity u µ (see Appendix C), it is necessary to work out the latter, which are all proportional to combinations of the form, (1 + γ) Note that this particular combination of integrals can be exactly evaluated, and that the result vanishes for odd values of γ.
In order to complete the calculation of the ++ part of the effective action, it is neces- Using the explicit form of the propagators, Eqs. (7a) and (7c), it is easy to see that G T (k) = (−2πi)n(|k 0 |)Im(G ++ (k)) then this trace is purely imaginary which can be related with the imaginary part of Eq. (19) replacing n → n 2 .
It is convenient to evaluate separately the real and imaginary parts of the ++ contribution to Γ (2) : the real part can be read from Eq. (19), just taking the real parts of the integrals I αβ .
On the other hand, the imaginary part can be obtained from Eqs.(8b) and (19), replacing I αβ →Î αβ , giving the following result whereÎ C. The complete effective action: covariant form We now collect the previous results. The real and imaginary parts of the ++ effective action read and First, let us notice that throughout the different contributions to Re(Γ ++ ), Eqs. (10), (11) and (19), we find explicit terms proportional to β −4 . However, there will also be implicit contributions coming from the I αγ integrals of Eq. (19) that will surface only after an expansion in high temperatures is performed. This expansion will be discussed in more detail in the next section, but for now we shall borrow these terms in order to treat them in the same standing as the explicit ones. The crucial point is that, when combining all of the contributions that go as β −4 , the resulting term is in fact local. This is also the case for the terms proportional to β −2 , which in this case are all contained within the I αγ 's. For this reason we will separate both types of terms from the other, truly nonlocal, contributions.
All the contributions proportional to β −4 in Γ ++ are which, after comparing with Eq. (C1a), it is found to be the weak field expansion of Under a similar analysis, the terms proportional to β −2 combine to form the weak field expansion of After separating both these contributions, and combining Eqs.(10), (11), and (19), we obtain where Note that the local terms in Eq.(28) are multiplied by powers of the Tolman temperature [16]. These contributions have been correspondingly subtracted to the nonlocal kernels, which now will not contain any terms that go as β −4 or β −2 when expanded at high temperatures.
Thanks to the presence of the projectors P µν we can write the effective action in terms of the Riemann tensor. However, as we are considering a quantum field in a thermal state, this is not enough to obtain a covariant expression, due to the presence of a privileged direction associated to the four-velocity u µ of the thermal bath, as well as the space-dependent Tolman In order to be able to obtain a fully covariant expression, it is necessary to also introduce the timelike Killing vector κ µ (κ 2 < 0) related to the stationary nature of the geometry [17,18]. In terms of this vector, we have u µ = κ µ / √ −κ 2 and g 00 = κ 2 . Indeed, using Eqs. (C4a) and (C2) we obtain where the nonlocal kernels read and p = |p µ p µ |. In order to simplify the notation, we omitted the superscript (+) for the metric in Eq. (30). Notice that these last expressions are valid in any coordinate system, which is evidenced by the promotion ofx → x in Eqs. (30) and (31), and |p| → p in Eqs.
(31). Upon choosing the coordinates such that κ µ = (1, 0, 0, 0) they reduce to the former expressions for which the time-independence is manifest. However, it is worth remarking that we expect additional terms in the effective action for a metric with a generic spacetime dependence.
We can proceed in a similar way with the imaginary part of the effective action. The result is where the imaginary kernelsÂ i (x, y) are obtained from the corresponding expressions in Eq.(31) replacing A, B, C, D, and E bŷ The above Eqs. (30) and (32)  Ref. [13], which are valid for Euclidean ultrastatic geometries (we will describe below an interesting connection between the effective actions for ultrastatic and static situations).
Regarding the work of Ref. [11], we have been able to write the effective action in a fully covariant way, valid for stationary spacetimes.
It is interesting to remark that the local terms proportional to β −4 and β −2 can be interpreted as a cosmological constant and an Einstein-Hilbert action, respectively, modified by a local temperature. The same terms also appear when computing the effective action using the Schwinger-DeWitt expansion for a massive field [18,19]. However, for more general metrics and massless quantum fields nonlocality could arise already at these orders.

III. HIGH TEMPERATURE EXPANSION
As the structure of the nonlocal kernels is rather complex, it is worth analyzing them in the high temperature limit. The main technical point is to expand the integrals I αγ in Eq. (18) in inverse powers of β. After a change of variables we have with and z 0 = pβ/2. The expansion of I αγ for small z 0 is rather involved and is described in For the values of α and γ relevant to the evaluation of the effective action, the structure of the expansion is where a αγ , c αγ , d αγ and e αγ are constants. It is noticeable the absence of a term proportional to β −3 . with µ We now replace the expansions into Eq.(28). The main observation is that there are no contributions proportional to β −4 and β −2 into the kernels, i.e. we separated them on purpose. However the β −3 is not there from the beginning. For these reasons the leading nonlocal contribution is proportional to β −1 . The result for this last term is where the nonlocal kernel is a two-point function defined as Note that the convolution of this nonlocal kernel with a time-independent function is equivalent to the convolution of the function with the kernel δ(t − t ′ )/ √ −∇ 2 . As expected on dimensional grounds, this leading contribution is a linear combination of terms of the form where R denotes components of the Riemann tensor eventually contracted with the four-velocity u µ .
The integrals I αγ also have terms proportional to log(βp), which could potentially be nonlocal subleading corrections to the effective action. An explicit evaluation gives that these terms precisely cancel the nonlocal zero temperature contribution (see Eq. (9)), leaving only a local remainder that goes as log(βµ), with µ an arbitrary renormalization scale.
Following the same steps we find, for the imaginary part whereÂ Note that the nonlocal kernelsÂ i (x, y) are linear combinations of (β √ − ) −n .

SPACETIMES
The structure of the effective action Γ[g µν ] for a static metric g µν (i.e. g 0i = 0) can in fact be recovered from the effective actionΓ[ḡ µν , Ω] for an ultrastatic metricḡ µν (alsoḡ 00 = η 00 ) through an appropriate conformal transformation where Ω 2 = |g 00 |. However, since the action is not conformally invariant, the corresponding effective actionΓ[ḡ µν , Ω] for the ultrastatic metric is associated to a different operator.
Indeed, if we consider a static metric in Euclidean signature, we have for the Euclidean effective action where ξ is the coupling between scalar curvature and the field. Upon applying the transfor- where ∆R is related to the transformation of the scalar curvature and is defined in Eq.(E1).
The Jacobian J[Ω] adds a local contribution to the effective action, that produces the trace anomaly. The effective actions for Euclidean ultrastatic metrics were obtained previously up to quadratic order in the curvatureR, by means of heat kernel methods for a wide class of operators of the form −¯ −P +R 6 , withP (x) a generic potential that may include the curvature as well as interactions. In particular, the effective action at finite temperature has been computed in Ref. [13], where the nonlocal structure is encoded in temperature dependent kernels γ i (β √ −∇ 2 ), with i = 1, .., 4. By choosing the potentialP appropriately (see Appendix E for further details), the effective actionΓ[ḡ µν , Ω] is easily obtained, and after carefully rewriting everything in terms of the static metric g µν , the corresponding Γ[g µν ] arises.
There is an important feature of Γ[g µν ] that can be deduced from this picture. At high temperatures the kernels γ i can be expanded in powers of β −1 , leading to terms in the effective action that are schematically of the form, where we are using a covariant representation that reduces to the one of Ref. and alternative definition for these operators is [20], .
The massive propagator Gm(x, y) = (−¯ 2 +m 2 ) −1 is not conformally invariant. However, up to corrections proportional to the curvature, it can be shown that in four dimensions it and therefore, upon a change of integration variablem → m =m Ω −1 , we can obtain the approximated transformation rule for the nonlocal operator Putting this together with the transformation rules of the other geometrical objects, it can be readily seen that a term like Eq.
in compliance with our result Eq. (30), as well as with other works in the literature [16,19,21,22].
There are limitations to this approach though. On the one hand, only a static metric can be related to an ultrastatic through a conformal transformation, leaving stationary metrics (i.e g 0i = 0) out of this treatment. Indeed, the kernel E(x, y) of Eq. (29e) will never arise from this approach.
On the other hand, it is necessary to know the ultrastatic effective actionΓ[ḡ µν , Ω] for a more general type of operator than the original one whose static effective action Γ[g µν ] we aim to find. This might prove equally or more challenging than a direct calculation of Γ[g µν ], even for more general metrics. In our case, the known results from Ref. [13], which are able to accommodate a generic operator, were obtained for a Euclidean ultrastatic metric and therefore allows us only to obtain the equivalent Euclidean effective action for a static metric, which we can then verify to be equal to the real part of the CTP effective action Re(Γ[g µν ]) we computed in Eq.(30), after the high temperature expansion of Eq. (37). This means that any imaginary contribution Im(Γ[g µν ]), i.e Eq.(39), is absent, and thus we are missing the information on the stochastic properties of the system. In order to obtain this last part from a conformal transformation, we would need to know the CTP equivalent of Ref. [13], that is, the CTP effective action for an ultrastatic metric associated to a generic operator, but this is likely more difficult than to perform the full CTP calculation we presented in the previous sections.

V. FIELD EQUATIONS IN THE HIGH TEMPERATURE LIMIT
In this section we compute the semiclassical Einstein equations by directly taking the variation of S g [g µν ] + S m [g µν ] + Γ[g µν ] with respect to the metric, where S g and S m are the gravitational action and the action for classical matter sources respectively. The semiclassical Einstein equations read where M 2 pl is the Planck mass, G µν is the Einstein tensor, T µν is the stress-energy tensor and j µν a stochastic noise source, i.e. j µν = 0. The stress-energy tensor of the classical matter sources T cl µν is related with S m in the usual way, and analogously, for the expectation value of the stress-energy tensor of the quantum field we have where it has already been taken into account the fact that the in-in formalism ensures the where and u µ are, respectively, the energy density, pressure and four-velocity of an ideal radiation fluid with space-dependent temperature given by the Tolman temperature β(x) −1 . It is worth noting that a source of this form in the Einstein equations does not allow a static solution that also satisfies sensible boundary conditions (such as decaying at spatial infinity). This has been thoroughly studied in the literature, in particular regarding the stability of hot flat space [21]. We will not dwell on this subject, but just note that a full treatment requires to consider a time-dependent scenario.
Similarly, from Eq. (27) for β −2 we obtain, (53) The first parenthesis will have the effect of "renormalizing" Newton's constant by a finite, temperature and space-dependent amount, but in a noncovariant way, i.e. the '00' equation receives a different correction than the others. The second parenthesis survives due to the fact that ∇ µ β(x) = 0.
We now consider the nonlocal part (NL), which starts at β −1 . We take the variation from its expression in terms of h µν , given in Eq. (28) and valid when |h µν | ≪ 1, and then rewrite the result in terms of the components of the Riemann tensor. This naturally separates the different components of the equations, where T µν is explicitly dependent only onx, and Notice that the logarithmic term is local due to the cancellation of all log(p) parts with the usual zero temperature contributions (see Eq. (9)). Also, T 0i (x) (N L) is purely local.
where G µν represents a generic variation of components of the Riemann tensor with respect to the metric.
The full T µν is obtained by adding all the previous contributions, From the explicit expressions for each contribution it can be checked that the full stresstensor is conserved, i.e. ∇ µ T µν = 0.
Let us now discuss the stochastic contribution to Eq. (50). The noise j µν accounts for the fluctuations of the stress-energy tensor of the quantum field around its mean value. Its self-correlation is related to the imaginary part of the effective action, where ∆T µν = T µν − T µν , and the noise kernel N µνρσ (x, y) is defined in Eq. (12). The semiclassical approximation will be a good one as long as the fluctuations of the stressenergy tensor are small compared to its mean value, that is, Note that since j µν (x) is of stochastic origin, it depends both on space and time coordinates.
This can be made compatible with a static metric if the dominant source of Eq. (50) is the semiclassical one and the solution is stable under small time-dependent perturbations.
We close this section by considering the ansatz of a static Newtonian metric, having the form We first assume there is a mechanism by which the effect of the source T µν (4) is stabilized such that a static metric can be a viable solution. The most striking feature is that for this kind of metric the leading nonlocal contribution T µν (N L) proportional to β −1 vanishes exactly, although this can be checked to be a special feature of the minimally coupled case (ξ = 0), and not hold for more general cases. In the absence of both T µν (4) and T µν (N L) , the noise source j µν becomes dominant. This would break the semiclassical approximation for Newtonian metrics at high temperatures.

VI. CONCLUSIONS
We have computed the CTP effective action for a massless quantum field in thermal equilibrium in a stationary background. Our main goal has been to obtain covariant expressions for the real and imaginary parts of the effective action, which may be used to propose nonlinear completions of the theory that are much more general than previous ones, based on generalizations of the zero temperature case. The existence of a thermal bath picks up a particular frame, the one in which the bath is at rest, and therefore the effective action depends not only on the metric but also on the four-velocity of the bath and on the local Tolman temperature. The nonlocal terms involve kernels that can be thought schematically as non-analytic functions of the Laplacian F (∇ 2 ).
The high temperature expansion revealed some interesting properties. On the one hand, and similarly to what happens in the ultrastatic case [13], the leading nonlocal correction to the real part of the effective action is proportional to β −1 . By dimensional analysis, this implies that the nonlocal structure should involve the kernel 1/ √ −∇ 2 . On the other hand, the imaginary part contains nonlocal contributions already at the β −5 -level.
Another interesting aspect of our results is that the leading nonlocal contributions to the semiclassical Einstein equations vanish when evaluated in a Newtonian metric, and therefore do not produce thermal corrections in the large distance limit. However, we have checked that this is true only for minimally coupled scalar fields.
In a forthcoming paper we will address the calculation of the effective action for the case of time-dependent metrics. On general grounds, we expect the nonlocal terms found in the present paper to become more complex functions F (∇ 2 , u µ D µ ), which involve both spatial and temporal derivatives, in a combination that does not necessarily coincide with the D'Alembertian. An interesting issue, with potential applications to the dark energy problem, is whether there are nonlocal contributions at the leading β −4 order or not. We have shown that this is not the case for stationary metrics, since in this case the leading term is local and proportional to the fourth power of the Tolman temperature. The same local term appears when considering massive fields, and the effective action is computed using a Schwinger-DeWitt expansion [18,19], which assumes that the mass m satisfies m 2 ≫ R.
However, it is, in principle, possible that nonlocal terms emerge at leading order in the opposite limit of massless fields, for time dependent metrics.
We have The integral in q 0 is trivial because of the Dirac δ-function. Then, changing variables as k =q −p ≡ k and performing the integral in φ we obtain We split the remaining expressions separating the terms proportional to k 4 , k 3 and k 2 : Hence, using the identities of Appendix A and defining the integrals +p i p j p l p m 1 2

Appendix C: Covariant Formulas
In this Appendix we relate the various expressions valid in the weak field approximation up to quadratic order in h µν with covariant objects.
Some useful expansions of local geometric quantities are For nonlocal kernels that involve the product of projectors P µν we have [15], whereK(p) is the Fourier transform of K(x − y). Schematically, the projectors are replaced following the rules However, as we have shown in the present work, for a thermal state the temporal and spatial components of the projectors appear separately. Therefore, when written in a covariant way, the expressions quadratic in h µν become more complex combinations of contractions of the Riemann tensor with the four-velocity of the bath. The replacements analog to those of Eq.(C3) read h 00 (x)P 00 P 00 h 00 (ȳ)|p| 4 → 4R 00 (x)R 00 (ȳ) = 4R µν (x)u µ u ν R αβ (ȳ)u α u β (C4a) h 00 (x)P 00 P ij h ij (ȳ)|p| 4 → −4R 00 (x)R 00 (ȳ) + 2R 00 (x)R(ȳ) Let us now focus on the first part I αγ . Using Eq. (D2) and the expansion given in Eq. (D4), we have refer to the first and second term respectively. Also, I λ L can be written as a function of z 0 for a generic integer λ > 0 as follows however, notice in Eq. (D5) that we also need I (0) This will force us to treat the casesα > 0 andα = 0 separately.
For the first case,α > 0, using Eq. (D6) we have The first line is the one that needs to be worked out the most. The idea is swap the summation order to be able to then resum the series of the Bernoulli coefficients. Schematically, Recalling that z 0 = β|p|/2, we now show the results for the particular values of α and γ needed, up to O(β 0 ), I 20 = π 2 12β 2 |p| 2 + 1 288 6 log β|p| 2 + 6γ − 2 − 6 log(2π)
With these transformation rules, it is straightforward to check that a more generic, non conformally-invariant field with mass m and coupling to the curvature ξ, will see its inverse propagator change in the following way, − + m 2 + ξR φ = Ω −3/2 −¯ + Ω 2 m 2 + ξR − ξ − 1 6 ∆R φ .
In order to profit from the known results for ultrastatic metrics of Ref. [13], where they find the thermal effective action for operators of the form −¯ −P +R 6 , we choose the potentialP to bē where in the second equality we used the rule (E1b) to find an expression in terms of quantities defined for the static metric g µν . The same tactic is to be applied to the whole ultrastatic effective action Γ[ḡ µν , Ω] in order to find its static counterpart Γ[g µν ].
Since we are working with the weak field approximation, it is useful to consider the linearized versions of some of the rules (E1), R =R + 3∇ 2 h 00 .
For the ultrastatic metric we haveh 00 = 0. However, it is through the Ω's coming from both P and the trasformation rules, that h 00 contributions arise. These must be such that when combined with the rest of the elements they form geometrical objects associated with the static metric g µν .