The matter Lagrangian of an ideal fluid.

Since the pioneering work by Taub [1] and by Fock & Kemmer in their 1955 monograph about gravitation (with English translation versions in Fock and Kemmer [2] and Fock [3]) it has become important to calculate the value of the matter Lagrangian used in relativistic theories of gravity, particularly when dealing with extended theories of gravity with curvature-matter couplings as first explored by Goenner [4], later by Allemandi et al. [5] and Bertolami et al. [6] and the constructions of Harko and Lobo [7, 8] deriving in couplings of the trace of the energy momentum tensor with the curvature of space-time [9, 10].

The value of the matter Lagrangian has been known to have values (up to a sign depending on the sign definition of the matter action and thus the energy-momentum tensor and the chosen signature of the metric of space-time) of [e.g 3, 11, 12, 13, 14] $e$ or [e.g 15, 16, 17] $p$ and even the trace of the energy-momentum tensor [e.g. 18], where $e$ is the total energy matter density, i.e. mass energy density plus internal energy density -which for the referred cases were obtained only for an isentropic or a barotropic fluid, and $p$ the pressure of the fluid . For the case of the Hilbert action, both values for the matter Lagrangian are commonly claimed to yield the same Einstein field equations, but as explained by Harko [19] the choice $p=0$ yields a null matter Lagrangian and this appears to be an inconsistency for extended theories of gravity where non-null matter Lagrangians are to be selected. More general studies using multifluids are beginning to shed some light on the construction of a matter Lagrangian using general thermodynamical arguments [20].

Using well known modern techniques of calculus of variations on the definition of the matter Lagrangian, we show in the present work that for the case of an ideal fluid the matter Lagrangian equals (up to a sign) the total energy density of the fluid which consists of its rest mass energy density plus its internal energy density. In Section II we set up the matter action and define the energy-momentum tensor as a function of the variations of the matter Lagrangian. In Section III we write down some basic thermodynamic relations which will be used in Section IV in order to obtain the final value of the matter Lagrangian through the use of the mass continuity equation of general relativistic hydrodynamics. Finally in Section VI we conclude and give examples of the matter Lagrangian useful for many astrophysical and cosmological applications useful in extended theories of gravity with curvature-matter couplings.

The matter action is given by [see e.g. 21]:

The metric, Hilbert or Belifante-Rosenfeld energy-momentum tensor $T_{\alpha\beta}$ is then defined through the variations $\delta S$ of the matter action (1) with respect to $\delta g^{\alpha\beta}$ and is given by [see e.g. 21, 8, 22, 23]:

Using the fact that [21]:

The appropriate form of the first law of thermodynamics for relativistic fluids can be written as [see e.g. 24]:

An ideal fluid is that for which no heat is exchanged between its components and so, the fluid moves adiabatically [see e.g. 24, 25, 26, 27, 28], i.e. $\mathrm{d}\left(\sigma/\rho\right)=0$. In other words, the first law of thermodynamics for an ideal fluid is given by:

For the case of an ideal fluid, the energy-momentum tensor takes the form [see e.g. 21, 24]:

If there are no (baryonic) mass sources, the fluid satisfies a continuity equation given by [24]:

In other words:

Using the fact that $\delta\left(g_{\alpha\beta}u^{\alpha}u^{\beta}\right)=0$ and with the help of equation (5) it follows that:

and so:

Substitution of this last result in equation (19) and with the help of relation (3) yields:

From equations (14) and (23) it follows that:

In order to find a differential equation for the matter Lagrangian $\mathcal{L}_{\text{matt}}$, we proceed in the following four alternative ways:

1. (i)

   Equate terms with $u_{\alpha}u_{\beta}$ and the ones with $g_{\alpha\beta}$ in equation (24), to obtain:

   	$$\displaystyle e+p=\mp\rho\frac{\mathrm{d}\mathcal{L}_{\text{matt}}}{\mathrm{d}\rho},$$		(25)
   	$$\displaystyle-p=\pm\rho\frac{\mathrm{d}\mathcal{L}_{\text{matt}}}{\mathrm{d}\rho}\mp\mathcal{L}_{\text{matt}}.$$		(26)

2. (ii)

   Multiply equation (24) by the metric tensor $g^{\alpha\beta}$ to obtain:

   $$\displaystyle e-3p=\pm 3\rho\frac{\mathrm{d}\mathcal{L}_{\text{matt}}}{\mathrm{d}\rho}\mp 4\mathcal{L}_{\text{matt}}.$$

   (27)

3. (iii)

   Multiply equation (24) by the four velocity $u^{\alpha}$ to obtain:

   $$\mathcal{L}_{\text{matt}}=\mp e.$$

   (28)

4. (iv)

   Multiply equation (24) by the projection tensor $P^{\alpha}{}_{\lambda}:=u^{\alpha}u_{\lambda}-\delta^{\alpha}{}_{\lambda}$, which is orthogonal to the four velocity $u_{\alpha}$, to obtain exactly equation (26)

The sets of equations (25)-(26), relation (27) and (28) seem all contradictory at first sight and as mentioned in the introduction, their study has caused quite a lot of confusion for at least 65 years, even for the simple case of dust, for which $p=0$ and so $e=\rho c^{2}$. However, note that according to equation (28), $\mathcal{L}_{\text{matt}}=\mp e$ and the same result is also obtained by the system of equations (25)-(26). With the value of $\mathcal{L}_{\text{matt}}=\mp e$ equations (25)-(27) become:

As mentioned in the introduction, there are many works in which the matter Lagrangian $\mathcal{L}_{\text{matt}}=\pm p$. In general terms the analysis is based on the following procedure [see e.g. 29, 18, and references therein], which uses auxiliary scalar fields to find the value for the matter Lagrangian of an ideal fluid.

Assume that the matter Lagrangian

Direct comparison of the previous equation with the value for the energy-momentum tensor for an ideal fluid given in equation (14) means that:

Note that the four velocity $u_{\alpha}$ defined above is normalised such that $u_{\alpha}u^{\alpha}=1$. This procedure is to be taken with care since the four velocity needs to be time-like or else the perfect fluid approximation breaks down. This condition cannot be fully guaranteed if $u_{\alpha}\propto\nabla_{\alpha}\phi$. Also note that using the first law of thermodynamics (13) for a perfect fluid, which by definition is adiabatic, and the fact that $\mathcal{L}_{\text{matt}}=\pm p$, the second differential equation in the previous set of equations can be manipulated so that:

Since $\zeta=|\boldsymbol{\nabla}\phi|^{2}/2$, then the scalar field $\phi$ is a purely thermodynamical function and the same occurs for the four velocity $u^{\alpha}$. In other words, the previous relation constitutes a general “constraint” for the four velocity of any ideal fluid in any space-time regardless of the constitutive field equations and so, proposition (30) must be incorrect. Any proposed Lagrangian as a function of a scalar field for which its derivatives represent a four velocity will also be incoherent. The beauty of the combined results of equations (25)-(27) with $\mathcal{L}_{\text{matt}}=\mp e$ is the purely thermodynamic relation (29), which is the first law of thermodynamics (13), and is directly obtained from the metric definition of the energy-momentum tensor (2) with no auxiliary fields needed to find the correct matter Lagrangian for an ideal fluid.

In this work we have shown that the value of the matter Lagrangian

In the present work we showed that the general result of the matter Lagrangian for an ideal fluid in equation (35) is in perfect agreement with the conservation of the matter current through the continuity equation (16) and coherent with the fact that an ideal fluid moves adiabatically, according to the first law of thermodynamics (13).

For the case of dust, i.e. $p=0$, it follows that:

The total energy density of a barotropic fluid, one for which the pressure $p(\rho)$ is a function of the mass density $\rho$ only, is given by [see e.g. 24]:

The energy density equation (40) becomes the Bondi-Wheeler equation of state [28] for the case in which the pressure of the fluid is much greater than its rest energy density $\rho c^{2}$, i.e. $p\gg\rho c^{2}$ and so:

In cosmological applications the previous two relations are valid for the cases of radiation with $\kappa=4/3$, a monoatomic gas with $\kappa=5/3$ and for cosmological vacuum with $\kappa=0$. The case of cosmological dust is represented by a matter Lagrangian given by equation (36).²²2For the choice of a ($-,+,+,+$) signature, since the right-hand side of equation (15) equals $-1$, and the energy-momentum tensor in equation (14) is given by: $$T_{\alpha\beta}=\left(e+p\right)u_{\alpha}u_{\beta}+pg_{\alpha\beta},$$ it follows that equation (21) turns into: $$\delta\rho=\frac{1}{2}\rho\left(g_{\alpha\beta}+u_{\alpha}u_{\beta}\right)\delta g^{\alpha\beta}.$$ With all this, the $\pm$ and $\mp$ signs that appear in equations (25)-(27) are inverted so that: $$\mathcal{L}_{\text{matt}}=\pm e,$$ (44) which leaves equation (29) unchanged. This means that for this metric signature, the $\pm$ and $\mp$ signs on equations (36)-(38) and (41)-(43) are also inverted.

This work was supported by DGAPA-UNAM (IN112019) and Consejo Nacional de Ciencia y Teconología (CONACyT), México (CB-2014-01 No. 240512) grants. SM acknowledges economic support from CONACyT (26344).