New derivation of the Lagrangian of a perfect fluid with a Title barotropic equation of state Author(s) Minazzoli, O; Harko, TC Citation Physical Review D, 2012, v. 86, p. 087502:1-4 Issued Date 2012 URL http://hdl.handle.net/10722/181697 Rights Physical Review D. Copyright © American Physical Society. PHYSICAL REVIEW D 86, 087502 (2012) New derivation of the Lagrangian of a perfect fluid with a barotropic equation of state Olivier Minazzoli Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Drive, Pasadena, California 91109-0899, USA Tiberiu Harko Department of Physics and Center for Theoretical and Computational Physics, The University of Hong Kong, Pok Fu Road, Hong Kong, People’s Republic of China (Received 9 August 2012; published 9 October 2012) In this paper we give a simple proof thatR when the particle number is conserved, the Lagrangian of a 2 2 barotropic perfect fluid is Lm ¼½c þ PðÞ= d, where is the rest mass density and PðÞ is the pressure. To prove this result, neither additional fields nor Lagrange multipliers are needed. Besides, the result is applicable to a wide range of theories of gravitation. The only assumptions used in the derivation are: 1) the matter part of the Lagrangian does not depend on the derivatives of the metric, and 2) the particle number of the fluid is conserved (rðu Þ¼0). DOI: 10.1103/PhysRevD.86.087502 PACS numbers: 04.20.Cv, 04.20.Fy, 04.50.Kd I. INTRODUCTION field equations. Therefore, the field equations of those theories seem to be different whether one considers In order to obtain solvable equations of motion, recently Lm ¼ or Lm ¼ P. Hence, in these models the physics developed alternative theories of gravitation use some seem to be different depending on this choice, which is not specific forms of the perfect fluid Lagrangian [1–5]. Most satisfactory with respect to the monistic view of modern of these approaches start from the work of Brown [6], physics, which requires a unique mathematical description where the on-shell perfect fluid Lagrangian Lm, without of the natural phenomena. elastic energy, is shown to reduce in general relativity (GR) The first obvious thing one can claim about this situation to Lm ¼, where is the energy density of the fluid. is that there is absolutely no reason why the results ob- This result is obtained by introducing various additional tained in GR by Brown [6] should be applicable in the fields, as well as Lagrange multipliers in order to effec- theories where the matter Lagrangian enters directly into tively be able to reconstruct a perfect fluid stress-energy the field equations. Even more than that, the simple fact tensor concordant with the laws of thermodynamics, that the laws of physics depend on the considered such as the matter current conservation. However, because Lagrangian should be viewed as a proof of the nonapplic- of the additional fields, it turns out that the on-shell ability of Brown’s results in theories where the degeneracy Lagrangian can also be Lm ¼ P, where P is the pressure of the matter Lagrangian leads to a variety of different field of the fluid. This degeneracy of the Lagrangian has no equations. Therefore, all works [2–5,11] considering that consequences in GR, since both Lagrangians lead to the one can write Lm ¼ P for their on-shell perfect fluid same equations of motion. In Ref. [7] an alternative perfect Lagrangian—or any linear combination of À and P— fluid Lagrangian was proposed which is a function of the may be incorrect as long as they deal with theories where hydrodynamic variables u , , and T, where u is the fluid the matter Lagrangian enters directly into the field equa- four-velocity, and T the rest temperature of the fluid, and of tions (unless one can prove the opposite in some specific the gravitational field variables g. Also, the equations of situation). hydrodynamics for a perfect fluid in general relativity have To be more specific, two main theories have been con- been cast in Eulerian form, with the four-velocity being sidered where Lm enters directly into the field equations: expressed in terms of six velocity potentials in Ref. [8]. fðRÞ theories with some nonminimal matter/curvature cou- The velocity-potential description leads to a variational pling [1–4,12], or Brans-Dicke-like scalar-tensor theories principle whose Lagrangian density for the perfect fluid with some nonminimal matter/scalar coupling [5,11]. is the pressure P. Let us also note that a matter Lagrangian In a recent work [12], instead of using the results of of the form Lm ¼ð1 þ Þ, where is the elastic po- Brown [6] and trying to apply them to the theory under tential (or the internal energy), was considered in Ref. [9] consideration, the Lagrangian for a barotropic perfect fluid to derive the equations of motion of the perfect fluid from a was derived from the equations of motion induced by the variational principle. Otherwise, variational principles for adopted action. In addition, the conditions of the conser- perfect and imperfect general relativistic fluids were con- vation of the matter fluid current [rðu Þ¼0, where u sidered in Ref. [10]. is the four-velocity of the fluid, and is the rest mass However, it turns out that in some alternative theories energy density], as well as of the nondependency of the of gravity, the matter Lagrangian appears explicitly in the matter Lagrangian with respect to the derivatives of the 1550-7998=2012=86(8)=087502(4) 087502-1 Ó 2012 American Physical Society BRIEF REPORTS PHYSICAL REVIEW D 86, 087502 (2012) metric were also imposed. By using these assumptions one dL ðÞþPðÞ m ¼ : (7) can show that for the consideredR modified gravity model d one has L ¼½1 þ PðÞ=2d. m Using Eqs. (6)and(7), we obtain the following first-order In the following, we extend the result of Ref. [12]toany linear differential equation for the energy density of the fluid: gravitational theory that satisfies the conditions of the conservation of the matter fluid current as well as the non- dðÞ ðÞþPðÞ dependency of the matter Lagrangian with respect to the ¼ : (8) d derivatives of the metric. Besides, the present demonstra- tion is actually much simpler than in Ref. [12]. Also, in the The general solution of this equation is Appendix we show that in the case of scalar-tensor theories Z with scalar field/matter coupling the result is compatible PðÞ ðÞ¼C þ 2 d; (9) with a Brown-like way of deriving the Lagrangian. In the present paper we use the Misner-Thorne-Wheeler where C is an arbitrary integration constant. Therefore, we conventions [13]. Also, while c ¼ 1 has been used in the have shown that the Lagrangian Introduction in order to match previous studies’ notation, Z we will explicitly keep c in the rest of the paper. PðÞ L ¼C À d (10) m 2 II. THE LAGRANGIAN OF A BAROTROPIC leads to an energy-momentum tensor of the form PERFECT FLUID T ¼f½C þ ÅðÞ þ PðÞguu þ PðÞg; (11) We start with the usual definition of the stress-energy where tensor T, given by ffiffiffiffiffiffiffi Z Z p L PðÞ dP P 2 ð Àg mÞ Å T ¼pffiffiffiffiffiffiffi : (1) ðÞ¼ 2 d ¼ À (12) Àg g is the elastic compression potential energy per unit mass of Considering the usual assumption that the matter part of 2 L the fluid [14]. The integration constant is given by C ¼ c .A the Lagrangian m does not depend on the derivatives of simple way to figure this out is to take the point-particle limit the metric, we obtain of the considered action, andR to equate it with the usual point- pffiffiffiffiffiffiffi 2 2 @ð ÀgL Þ @L particle action (Sm ¼ mc ds,wherem is the rest mass of T ¼pffiffiffiffiffiffiffi m ¼ L g À 2 m : (2) Àg @g m @g the point particle). Therefore, the Lagrangian of a barotropic perfect fluid is given by Now, by considering a fluid with a barotropic equation Z of state PðÞ, we can assume that L depends on only. PðÞ m L ¼ c2 þ d : (13) If one considers that the matter current is conserved m 2 r ð Þ¼0 ( u ), then one can prove that [12,14] The corresponding stress-energy tensor can be written as 1 ¼ ð À Þ 2 2 g uu g ; (3) T ¼f½c þ ÅðÞ þ PðÞgu u þ PðÞg ; (14) where u is the four-velocity of the fluid, defined in a or, equivalently, system of coordinates x as u ¼ dx =ds, where ds is 2 T ¼f½c þ ÅðÞ þ PðÞgUU þ PðÞg; (15) such that ds2 ¼c2d2, with the proper time of the fluid 1 particles, and with c2 the rest mass energy density. Using where U ¼ cÀ dx =d is the four-velocity of the fluid 1 Eqs. (2) and (3) we obtain [12] divided by the speed of light. One can verify that this stress-energy tensor is indeed of the form assumed, for L L d m d m instance, in celestial relativistic mechanics [16–20]. T ¼ u u þ Lm À g : (4) d d Also, from the conservation of the rest mass density 0 Now, since we want to obtain the Lagrangian of a baro- [rðu Þ¼ ] and using Eqs. (9)and(12), one derives the tropic perfect fluid, we have to equate this equation with the usual nonconservation equation for the total energy density: usual stress-energy tensor of a barotropic perfect fluid, rðu Þ¼ÀPrðu Þ: (16) T ¼½ðÞþPðÞuu þ PðÞg; (5) Let us note that, for a fluid satisfying a linear barotropic 2 where ðÞ is the total energy density of the fluid.