Effective fluid mixture of tensor-multi-scalar gravity

It is well known that the scalar-tensor gravity field equations can be written as effective Einstein equations with an effective dissipative fluid in their right-hand side, built out of the Brans-Dicke-like scalar field $\phi$ present in the theory and of its first and second covariant derivatives Madsen:1988ph ; Pimentel89 ; Faraoni:2018qdr ; Quiros:2019gai . The formalism has been generalized to “viable” Horndeski gravity Giusti:2021sku ; Miranda:2022wkz ; Giusti:2022tgq and applied to Friedmann-Lemaître-Robertson-Walker (FLRW) cosmology Giardino:2022sdv , to theories containing non-propagating scalar degrees of freedom Faraoni:2022doe ; Miranda:2022brj , and to specific scalar-tensor solutions Faraoni:2022jyd ; Faraoni:2022fxo . But what is the analogue of a multi-component fluid? Naturally, the simplest multi-fluid equivalent of a theory of gravity is tensor-multi-scalar gravity. Here we extend the effective fluid formalism to this class of theories. The task is much less obvious than it would appear at first sight because all the gravitational scalar fields couple to gravity, which makes them all couple to each other. In general, there can also be direct mutual couplings through their kinetic and potential terms in the action. In the presence of multiple real fluids decoupled from each other, one can describe this mixture in the frame of an observer with timelike four-velocity $u^{\mu}$. This four-velocity can be that of the comoving frame of one of the fluids, or it can be associated with any other observer. In general, it is difficult to define an average fluid EMMacC . This means that the total stress-energy tensor $T_{\mu\nu}$ of the effective fluid mixture, which is a tensor defined unambiguously, can be decomposed in many ways according to the four-velocity $u^{\mu}$ selected. Each of these descriptions is legitimate but the description of the total mixture and its physical interpretation will depend on the observer $u^{\mu}$ selected to decompose $T_{\mu\nu}$. In particular, the density, pressure, heat flux density, and anisotropic stresses of each fluid as “seen” from a particular observer $u^{\mu}$ will differ from those measured in the comoving frame of that fluid. To appreciate the difference between the descriptions of a fluid in different frames, consider a perfect fluid with four-velocity $u_{\mu}^{*}$ that, in its comoving frame, is described by the stress-energy tensor¹¹1We follow the notation and conventions of Ref. Waldbook : the metric signature is ${-}{+}{+}{+}$, $\kappa\equiv 8\pi G$, $G$ is Newton’s constant, and units are used in which the speed of light $c$ is unity.

In the frame of a different observer with timelike four-velocity $u^{\mu}$ related to $u^{\mu}_{*}$ by

this perfect fluid (now “tilted”) will appear dissipative, with the different stress-energy tensor decomposition Maartens:1998xg ; Clarkson:2003ts ; Clarkson:2010uz

where Maartens:1998xg ; Clarkson:2003ts ; Clarkson:2010uz

is the energy density,

is the pressure,

is the energy flux density, and

is the anisotropic stress tensor. It is clear that the (spatial) vector $q^{\mu}$ arises solely due to the relative motion between the two frames, i.e., to the (spatial) vector $v^{\mu}$. In this context it is problematic to interpret this purely convective current as a heat flux according to Eckart’s generalization of Fourier’s law Eckart40

where $T$ is the temperature and $K$ is the thermal conductivity. This law expresses the fact that heat conduction is caused not only by spatial temperature gradients but also by an “inertial” contribution due to the fluid acceleration Eckart40 .

The situation becomes more complicated when multiple fluids are coupled to each other and even more when they are effective fluids and they all couple explicitly with the curvature²²2We do not consider derivative couplings in this work. (more precisely, with the Ricci scalar $R$) and to each other, which is the situation in tensor-multi-scalar gravity. In this work we discuss two possibilities, but other frames may be more convenient for specific problems.

Rather surprisingly, in tensor-multi-scalar gravity formulated in the Jordan conformal frame, one can obtain a particular frame as a sort of fictitious “average” frame, which is generally not possible with real fluids EMMacC . It is obtained by identifying the coupling function of the scalars to $R$ (which depends on all the scalar fields in the theory) with a new field $\psi$ and amounts to a redefinition of the scalar fields. This procedure is routine in tensor-single-scalar gravity, in which the only Brans-Dicke-like field is redefined for convenience, without much consequence or interpretation. In tensor-multi-scalar gravity, instead, this redefinition takes a new meaning. It identifies a four-velocity and a sort of “average” frame because there is only one Ricci scalar $R$ and all the scalar fields in the theory couple to it. This ingredient is missing for real fluids, which do not couple to the curvature and have no “average” frame EMMacC .

In the following we analyze tensor-multi-scalar gravity in its Jordan (conformal) frame formulation. It is possible to discuss it from the point of view of the “average” observer, or from the comoving frame of each fluid, or from that of any other timelike observer $u^{\mu}$. It is important to remember that these descriptions will be different and will provide different physical interpretations of the mechanical and thermal aspects of the fluid mixture, and that these are all legitimate (hence one should not strive to identify the “correct” one). The point is that some of these formulations (originating different decompositions of the total $T_{\mu\nu}$ based on different $u^{\alpha}$) will be more convenient, and some others will be less convenient, for specific physical problems. One should adopt the formulation that is most convenient for the particular problem at hand without prejudice. For example, analyses of the quark-gluon plasma created in heavy ion collisions universally employ the Landau (or energy) frame BRAHMS:2004adc ; PHOBOS:2004zne ; STAR:2005gfr ; PHENIX:2004vcz ; Monnai:2019jkc in which there is no heat flux³³3This frame is found to be non-unique in Ref. Romero-Munoz:2014foa . while in FLRW cosmology, where comoving coordinates are the standard, relativistic fluids are routinely described in their comoving (or Eckart) frame Waldbook ; EMMacC .

Here we are interested in the fluid-mechanical equivalent and in the thermal description of tensor-multi-scalar gravity, where the fluids in the mixture are effective fluids and they all couple explicitly with $R$ and with each other. This is a very specific situation and our choices, although convenient in this problem, are not meant to be recipes with universal convenience (although aspects of our discussion may apply to other situations as well). After this discussion, we present an alternative view of the first-order thermodynamics of tensor-multi-scalar gravity in the Einstein conformal frame, while the last section summarizes our conclusions.

Let us begin with a convenient Jordan frame formulation of tensor-multi-scalar gravity (without derivative couplings). We adopt most of the notations specific to tensor-multi-scalar gravity used in Ref. Hohmann:2016yfd . There are $N$ scalar fields of gravitational nature $\left\{\phi^{A}\right\}$, with $A=1,2,\,...\,,N$, all coupled nonminimally with the Ricci scalar $R$ and between themselves, as described by the action

where capital indices $A,B,J\,,...$ label the scalar fields in the multiplet $\left\{\phi^{1}\,,...\,,\phi^{N}\right\}$, $g$ is the determinant of the spacetime metric $g_{\mu\nu}$, $\nabla_{\mu}$ is the associated covariant derivative, and $V$ is a scalar field potential. The Einstein summation convention is used also on the multiplet indices $J$. The coupling function $F\left(\phi^{1}\,,...\,,\phi^{N}\right)$ depends on all the $\phi^{A}$, i.e., $\partial F/\partial\phi^{I}\neq 0\;\forall I\in\left\{1,\,...\,,N\right\}$, or else some of the scalar fields would not be coupled directly to $R$ and would lose their status of gravitational scalar fields.⁴⁴4The nature of these scalar fields (gravitational or not) depends on the conformal frame Sotiriou:2007zu . Here we refer to the Jordan conformal frame. $F$ is assumed to be positive to keep the effective gravitational coupling $G_{\mathrm{eff}}\simeq 1/F$ positive.

The matrix $Z_{AB}\left(\phi^{1}\,,...\,,\phi^{N}\right)$ acts as a Riemannian metric on the scalar field space of coordinates $\left\{\phi^{1}\,,...\,,\phi^{N}\right\}$. $Z_{AB}$ can be taken to be symmetric without loss of generality because it multiplies the combination of kinetic terms $\nabla^{\alpha}\phi^{A}\nabla_{\alpha}\phi^{B}$ symmetric in $A$ and $B$. The elements of $Z_{AB}$ are all positive to avoid introducing unstable phantom fields. In general, also the potential $V\left(\phi^{1}\,,...\,,\phi^{N}\right)$ depends on multiple fields (although it is not important that it depends on all these fields, which is instead crucial for the coupling function $F$).

Since the matrix $Z_{AB}$ is real and symmetric, it can be diagonalized at each spacetime point $x^{\mu}$ and has positive eigenvalues, turning the sum of kinetic terms appearing in the action (12) into

where a bar denotes fields in the system of principal axes of the matrix $Z_{AB}$ in field space, and

is the diagonal form of $Z_{AB}$. This diagonalization, however, is not crucial and we will not use it explicitly, retaining the non-diagonal form of $Z_{AB}$ in our formulae.

The total stress-energy tensor is obtained by varying the action (12) with respect to $g^{\mu\nu}$. Using $\partial_{A}\equiv\partial/\partial\phi^{A}$ and $D_{AB}\equiv Z_{AB}+\partial_{AB}F$, the associated equation of motion reads

where $G_{\mu\nu}\equiv R_{\mu\nu}-\frac{1}{2}\,g_{\mu\nu}R$ is the Einstein tensor, $T^{\mathrm{(m)}}_{\mu\nu}\equiv-\tfrac{2}{\sqrt{-g}}\,\frac{\delta S^{\mathrm{(m)}}}{\delta g^{\mu\nu}}$ is the matter stress-energy tensor and

The equation of motion obtained by variation of the action with respect to $\phi^{A}$ reads

We can obtain the expression of the Ricci scalar from (15),

where $T^{(m)}\equiv g^{\mu\nu}T^{(m)}_{\mu\nu}$ is the trace of the matter stress-energy tensor. With this expression, Eq. (17) turns into

Assuming $\mathrm{det}\left(3\partial_{A}{}F\partial_{B}{}F+2FZ_{AB}\right)\neq 0$, we use the matrix $M^{AB}\equiv\left(3\partial_{A}{}F\partial_{B}{}F+2FZ_{AB}\right)^{-1}$ to isolate $\hbox{$\hbox to0.0pt{$\sqcup$\hss}\sqcap$}\phi^{A}$, obtaining

The goal of the decomposition given here is to separate $T^{\mu\nu}$ so that each part can be decomposed in the frame of a given fluid. Each fluid then receives an individual stress-energy tensor contribution. The number of purely convective terms is minimised by such a decomposition to allow for a clearer description of the intrinsic dissipative properties of each fluid.

Assuming the gradient of each scalar field to be timelike,

we define the $\phi^{A}$-fluid four-velocity

At this point, in order to avoid ambiguities, all the multiplet summations in this section will be written with an explicit summation symbol. The above identification between a scalar field and an associated effective fluid allows us to rewrite the scalar field derivatives in term of kinematic quantities Miranda:2022wkz . The second derivative $\nabla_{\mu}\nabla_{\nu}\phi^{A}$ in Eq. (16) can be expanded as

where $h^{A}{}_{\mu\nu}\equiv g_{\mu\nu}+u^{A}{}_{\mu}u^{A}{}_{\nu}\,$ is the three-metric of the hypersurface orthogonal to the four-vector $u^{A}{}_{\mu}$, $\Theta^{A}\equiv\nabla_{\mu}u^{A}{}^{\mu}$ is the expansion tensor associated with the $A$-fluid, and $\sigma^{A}{}_{\mu\nu}\equiv\tfrac{1}{2}\left(h^{A}{}_{ac}\nabla^{c}u^{A}{}_{b}+h^{A}{}_{bc}\nabla^{c}u^{A}{}_{a}\right)-\tfrac{1}{3}\,\Theta^{A}\,h^{A}_{ab}$.

With this result, Eq. (16) becomes

where

Since this equation does not depend on four-velocity gradients, we can interpret it as a purely inviscid contribution to the stress-energy tensor mixture.

If we rewrite the metric as

and we define

then, writing explicitly the summations, the stress-energy tensor assumes the form {widetext}

which is interpreted as a mixture of interacting imperfect fluids.

Let us discuss another possible procedure. In the following we redefine the fields $\phi^{A}$ but, before proceeding, it is essential to note (and remember through the rest of this work) that all these fields couple directly with the Ricci scalar $R$ through $F$ and they all play a role of in determining the properties of the effective fluid equivalent to the tensor-multi-scalar theory and the effective gravitational coupling $G_{\mathrm{eff}}\equiv F^{-1}$. (Their role may be different as, in general, $F\left(\phi^{1},\,...\,,\phi^{N}\right)$ is not symmetric in all its arguments.) In particular, the effective temperature of this multi-component fluid is determined by all the fields $\phi^{A}$ and the upcoming redefinition of these fields does not change this fact.

We proceed to redefine the scalar field multiplet as in Ref. Hohmann:2016yfd , which is standard practice in single-scalar-tensor gravity. We can rename the coupling function by electing it to be a Brans-Dicke-like scalar,

and we then have the $N$ scalar fields $\left\{\psi,\phi^{1},\,...\,,\phi^{N-1}\right\}$. This mathematically convenient procedure effectively makes only the field $\psi$ couple explicitly to $R$ but the reader should not be fooled into believing that the remaining fields $\phi^{A}$ do not couple to gravity. In fact, all the fields $\phi^{A}$ are coupled to $\psi$ (and also to each other), which makes them couple also to gravity. Indeed, they were explicitly coupled to gravity before the field redefinition $\psi\equiv F$ and the physics does not change. The action (12) is recast as Hohmann:2016yfd

where

The field equations for $g_{\mu\nu},\psi,\phi^{A}$ obtained by varying the action (28) are Hohmann:2016yfd

where we have used the notation $\partial_{A}\equiv\partial/\partial\phi^{A}$ and $\partial_{\psi}\equiv\partial/\partial\psi$.

Using the metric field equations we can express the Ricci scalar in terms of the matter and effective stress-energy tensors,

where $T^{\mathrm{(m)}}\equiv g^{\mu\nu}T_{\mu\nu}^{\mathrm{(m)}}$. Then, the equation of motion for $\psi$ turns into

Finally, we define the effective stress-energy tensor as

We can now move to the effective fluid picture.

Assume that the gradient of $\psi$ is timelike; using

we define the effective fluid four-velocity

which is normalized, $u^{\mu}u_{\mu}=-1$ (but the sign of the right-hand side of this definition must be adjusted to keep $u^{\mu}$ a future-oriented vector, which is crucial in discussions of dissipation which is time-irreversible). In general, the $\phi^{A}$-fluids are tilted with respect to the $\psi$-fluid, i.e., $u^{A}{}^{\mu}$ and $u^{\mu}$ have different directions. Using the $u^{\mu}$ of the $\psi$-fluid we perform the usual $3+1$ splitting of spacetime into the time direction and the 3-space “seen” by the observer with four-velocity $u^{\mu}$. This 3-space has Riemannian metric

The kinematic quantities (expansion tensor $\Theta_{\mu\nu}$, expansion scalar $\Theta=\nabla_{\mu}u^{\mu}$, shear tensor $\sigma_{\mu\nu}$, shear scalar, and acceleration $\dot{u}^{\mu}$) associated with $u^{\mu}$ are the same as those calculated for single-scalar-tensor gravity in Faraoni:2018qdr . In fact, their definitions are purely kinematic and theory-independent since they do not use the field equations but only the definition (39) of $u^{\mu}$. These kinematic quantities are straightforward, although lengthy to compute. Since they are used here, we report them in A.

The field equations (32) have the form of effective Einstein equations with an effective stress-energy tensor in their right-hand side, which can be seen as the stress-energy tensor of a dissipative multi-component fluid of the form

where

is the effective energy density,

is the effective heat current density describing heat conduction,

is the effective stress tensor,

is the effective isotropic pressure, and the trace-free part of the stress tensor

is the effective anisotropic stress tensor. $q^{\mu}$, $\Pi_{\alpha\beta}$, and $\pi_{\alpha\beta}$ are purely spatial with respect to $u^{\mu}$. The fluid description is obtained by expressing the derivatives of $\psi$ in terms of the relative effective fluid four-velocity (39) and kinematic quantities,

Furthermore, we have

therefore the $\psi$-equation of motion reads

We need these equations to eliminate the dependence of $T_{\mu\nu}$ on $\dot{X}$ and on $\hbox{$\hbox to0.0pt{$\sqcup$\hss}\sqcap$}\psi$. Indeed, prior to using the equation of motion for $\psi$, one obtains

Using the decomposition $\nabla_{\mu}=h_{\mu}{}^{\nu}\nabla_{\nu}-u_{\mu}\,u^{\nu}\nabla_{\nu}$, defining $\dot{\phi}^{A}\equiv u^{\alpha}\nabla_{\alpha}\phi^{A}$, and taking into account the symmetry $Z_{AB}=Z_{BA}$, the interacting terms contribute to the density, pressure, heat flux and anisotropic stress,

and the stress-energy tensor reads

Then, it is straightforward to obtain the effective fluid quantities