Earliest stages of the non-equilibrium in axially symmetric, self-gravitating, dissipative fluids
aa r X i v : . [ g r- q c ] S e p Earliest stages of the non–equilibrium in axially symmetric, self-gravitating,dissipative fluids
L. Herrera ∗ Escuela de F´ısica, Facultad de Ciencias, Universidad Central de Venezuela, Caracas,Venezuela and Instituto Universitario de F´ısica Fundamental y Matem´aticas, Universidad de Salamanca, Salamanca, Spain
A. Di Prisco † Escuela de F´ısica, Facultad de Ciencias, Universidad Central de Venezuela, Caracas, Venezuela
J. Ospino ‡ Departamento de Matem´atica Aplicada and Instituto Universitario de F´ısicaFundamental y Matem´aticas, Universidad de Salamanca, Salamanca, Spain
J.Carot § Departament de F´ısica, Universitat Illes Balears, E-07122 Palma de Mallorca, Spain
We report a study on axially and reflection symmetric dissipative fluids, just after its departurefrom hydrostatic and thermal equilibrium, at the smallest time scale at which the first signs ofdynamic evolution appear. Such a time scale is smaller than the thermal relaxation time, thethermal adjustment time and the hydrostatic time. It is obtained that the onset of non–equilibriumwill critically depend on a single function directly related to the time derivative of the vorticity.Among all fluid variables (at the time scale under consideration), only the tetrad component of theanisotropic tensor in the subspace orthogonal to the four–velocity and the Killing vector of axialsymmetry, shows signs of dynamic evolution. Also, the first step toward a dissipative regime beginswith a non–vanishing time derivative of the heat flux component along the meridional direction. Themagnetic part of the Weyl tensor vanishes (not so its time derivative), indicating that the emissionof gravitational radiation will occur at later times. Finally, the decreasing of the effective inertialmass density, associated to thermal effects, is clearly illustrated.
PACS numbers: 04.40.-b, 04.40.Nr, 04.40.DgKeywords: Relativistic Fluids, nonspherical sources, interior solutions.
I. INTRODUCTION
Many issues related with the structure of self–gravitating fluids may be addressed within the staticregime. In this case, the spacetime admits a timelike,hypersurface orthogonal, Killing vector. Thus, a coordi-nate system can always be chosen, such that all metricand physical variables are independent on the time likecoordinate. The static case, for axially and reflectionsymmetric spacetimes, was studied in [1]. In such a casethe fluid is in equilibrium, implying that the hydrostaticequilibrium equations (Eqs.(21,22) in [1]) are satisfied.If, instead, the system evolves with time, we have toconsider the full dynamic case where the system is outof equilibrium (thermal and dynamic), the general for-malism to analyze this situation, for axially and reflec-tion symmetric spacetimes was developed in [2] using aframework based on the 1 + 3 formalism [3–6].However, some part of the life of stars (at any stage ofevolution), may be described on the basis of the quasi- ∗ [email protected] † [email protected] ‡ [email protected] § [email protected] static approximation (slowly evolving regime). This isso, because many relevant processes in star interiors takeplace on time scales that are usually, much larger thanthe hydrostatic time scale [7],[8]. In this case, the systemis assumed to evolve, although slowly enough, so that thehydrostatic equilibrium equations (Eqs.(21,22) in [1]) areassumed to be satisfied, all along the evolution.This regime has been recently described in detail,within the context of the 1 + 3 formalism [9].Nevertheless, during their evolution, self-gravitatingobjects may pass through phases of intense dynamical ac-tivity for which the quasi-static approximation is clearlynot reliable (e.g., the quick collapse phase preceding neu-tron star formation).It is worth mentioning that both regimes (“quick” and“quasi–static”), may be present, at different phases ofthe collapse of massive stars. Indeed, after the corebounce, leading to a supernova, the hydrostatic equilib-rium is reached within few milliseconds, while the subse-quent, Kelvin–Helmholtz phase, lasts for about 20 sec-onds, during which the system is in the quasi–staticregime, thereby satisfying the hydrostatic equilibriumequations [10]. We recall, that the hydrostatic time for aneutron star is of the order of 10 − seconds, while the or-der of magnitude of the relaxation time for neutron starmatter range from 10 − to 10 − seconds.All these phases of star evolution (“slow” and “quick”)are generally accompanied by intense dissipative pro-cesses, usually described in the diffusion approximation.This assumption, in its turn, is justified by the fact thatfrequently, the mean free path of particles responsible forthe propagation of energy in stellar interiors is very smallas compared with the typical length of the star.Here we shall focus on the “quick” phase, with the inclu-sion of all the dissipative processes.However, instead of following the evolution of the systemfor a long time after its departure from equilibrium, weshall analyze its behaviour immediately after such depar-ture.In this work “immediately” means at the smallest timescale, at which we can observe the first signs of dynamicalevolution. Such a time scale is assumed to be smallerthan the thermal relaxation time, the hydrostatic time,and the thermal adjustment time.Doing so we shall be able to extract important conclu-sions about the very early stages of non–equilibrium,avoiding the introduction of numerical procedures whichmight lead to model dependent conclusions.The price to pay for such a simplification, is that we shalldescribe only the very early stages of the evolution. Thereward is that we shall be able to answer to the followingquestions:1. what are the first signs of non–equilibrium?2. what physical variables do exhibit such signs?3. what does control the onset of the dynamic regime,from an equilibrium initial configuration?Our approach may be summarized as follows: We ob-serve a system which is initially static, and leaves theequilibrium for unknown causes which are not relevantfor the discussion. At this moment we put the clock towork, and watch the system until the first signs of non–equilibrium appear. At this very moment, we stop theclock. It is during this time scale that we describe thebehaviour of the systemAs we shall see, a specific function related with thetime derivative of the vorticity vector, appears as the fun-damental variable, controlling the departure from equi-librium and the ensuing evolution. By analogy (in itsphysical meaning) with the Bondi’s news function [11],we shall refer to this quantity as the fluid news function.From the analysis of the transport equation we shallsee that the time derivative of one of the heat flux compo-nents (“radial”) vanishes at the time scale under consid-eration, whereas the time derivative of the other (“merid-ional”) component, is controlled by the fluid news func-tion.Also we shall see that, at the time scale under con-sideration, the only fluid variable which exhibits devia-tion from the equilibrium is the tetrad component of theanisotropic tensor in the subspace spanned by the twospace–like vectors orthogonal to the four–velocity and theKilling vector of axial symmetry. At this same time scale, the magnetic part of the Weyltensor vanishes, implying that no emission of gravita-tional radiation is produced at this stage of evolution.However, the time derivative of the magnetic part of theWeyl tensor does not vanish and depends upon the fluidnews function, in such a way, that the vanishing of thelatter imply the vanishing of the former. In other wordsthe emission of gravitational process occurs at a timescale larger than the one considered here, and is tightlyrelated to the fluid news function.Finally, by using the transport equations together withthe “conservation” laws, we put in evidence the decreas-ing of the effective inertial mass density, associated withthermal effects.In this work we shall heavily rely on the formalismdeveloped in [2], thus in order to avoid the rewriting ofsome of the equations we shall frequently refer to [2],however we warn the reader of some important changesin the notation. II. BASIC DEFINITIONS AND NOTATION
In this section we shall deploy all the variables requiredfor our study, some details of the calculations are givenin [2], and therefore we shall omit them here.
A. The metric, the source, and the kinematicalvariables
We shall consider, axially (and reflection) symmetricsources. For such a system the line element may be writ-ten in “Weyl spherical coordinates” as: ds = − A dt + B (cid:0) dr + r dθ (cid:1) + C dφ +2 Gdθdt, (1)where
A, B, C, G are positive functions of t , r and θ . Wenumber the coordinates x = t, x = r, x = θ, x = φ .We shall assume that our source is filled with ananisotropic and dissipative fluid. We are concerned witheither bounded or unbounded configurations. In theformer case we should further assume that the fluid isbounded by a timelike surface S , and junction (Darmois)conditions should be imposed there.The energy momentum tensor may be written in the“canonical” form, as T αβ = ( µ + P ) V α V β + P g αβ + Π αβ + q α V β + q β V α . (2)The above is the canonical, algebraic decompositionof a second order symmetric tensor with respect to unittimelike vector, which has the standard physical mean-ing when T αβ is the energy-momentum tensor describingsome energy distribution, and V µ the four-velocity as-signed by certain observer.With the above definitions it is clear that µ is the en-ergy density (the eigenvalue of T αβ for eigenvector V α ), q α is the heat flux, whereas P is the isotropic pressure,and Π αβ is the anisotropic tensor. We emphasize thatwe are considering an Eckart frame where fluid elementsare at rest.Since we choose the fluid to be comoving in our coor-dinates, then V α = (cid:18) A , , , (cid:19) ; V α = (cid:18) − A, , GA , (cid:19) . (3)We shall next define a canonical orthonormal tetrad(say e ( a ) α ), by adding to the four–velocity vector e (0) α = V α , three spacelike unitary vectors (these correspond tothe vectors K , L , S in [2]) e (1) α = (0 , B, , e (2) α = , , √ A B r + G A , ! , (4) e (3) α (0 , , , C ) , (5)with a = 0 , , , e α ( a ) is easily computed from thecondition η ( a )( b ) = g αβ e α ( a ) e β ( b ) , e α ( a ) e ( b ) α = δ ( b )( a ) , (6)where η ( a )( b ) denotes the Minkowski metric.In the above, the tetrad vector e α (3) = (1 /C ) δ αφ is par-allel to the only admitted Killing vector (it is the unittangent to the orbits of the group of 1–dimensional rota-tions that defines axial symmetry). The other two basisvectors e α (1) , e α (2) define the two unique directions that areorthogonal to the 4–velocity and to the Killing vector.For the energy density and the isotropic pressure, wehave µ = T αβ e α (0) e β (0) , P = 13 h αβ T αβ , (7)where h αβ = δ αβ + V α V β , (8)whereas the anisotropic tensor may be expressed throughthree scalar functions defined as (see [2], but notice thechange of notation):Π (2)(1) = e α (2) e β (1) T αβ , , (9)Π (1)(1) = 13 (cid:16) e α (1) e β (1) − e α (2) e β (2) − e α (3) e β (3) (cid:17) T αβ , (10)Π (2)(2) = 13 (cid:16) e α (2) e β (2) − e α (3) e β (3) − e α (1) e β (1) (cid:17) T αβ . (11) This specific choice of these scalars is justified by thefact, that the relevant equations used to carry out thisstudy, become more compact and easier to handle, whenexpressed in terms of them.Finally, we may write the heat flux vector in terms ofthe two tetrad components q (1) and q (2) : q µ = q (1) e (1) µ + q (2) e (2) µ (12)or, in coordinate components (see [2]) q µ = (cid:18) q (2) GA √ A B r + G , q (1) B , Aq (2) √ A B r + G , (cid:19) , (13) q µ = , Bq (1) , √ A B r + G q (2) A , ! . (14)Of course, all the above quantities depend, in general,on t, r, θ .The expressions for the kinematical variables are (see[2]).For the four acceleration we have a α = V β V α ; β = a (1) e (1) µ + a (2) e (2) µ , (15)with a (1) = A ′ AB ; a (2) = A √ A B r + G " A ,θ A + GA ˙ GG − ˙ AA ! , (16)where the dot and the prime denote derivatives with re-spect to t and r respectively.For the expansion scalarΘ = V α ; α = 1 A BB + ˙ CC ! + G A ( A B r + G ) − ˙ AA − ˙ BB + ˙ GG ! . (17)Next, the shear tensor σ αβ = σ ( a )( b ) e ( a ) α e ( b ) β = V ( α ; β ) + a ( α V β ) −
13 Θ h αβ , (18)may be defined through two independent tetrad com-ponents (scalars) σ (1)(1) and σ (2)(2) , which may be writ-ten in terms of the metric functions and their derivativesas (see [2]): σ (1)(1) = 13 A ˙ BB − ˙ CC ! + G A ( A B r + G ) ˙ AA + ˙ BB − ˙ GG ! , (19) σ (2)(2) = 13 A ˙ BB − ˙ CC ! + 2 G A ( A B r + G ) − ˙ AA − ˙ BB + ˙ GG ! . (20)It is worth noticing that the shear tensor has no pro-jection in the subspace e (1) α e (2) β .Finally, for the vorticity tensorΩ βµ = Ω ( a )( b ) e ( a ) β e ( b ) µ , (21)we find that it is determined by a single basis component:Ω (1)(2) = − Ω (2)(1) = − Ω , (22)where the scalar function Ω is given byΩ = G ( G ′ G − A ′ A )2 B √ A B r + G . (23)Now, from the regularity conditions, necessary to en-sure elementary flatness in the vicinity of the axis of sym-metry, and in particular at the center (see [12], [13], [14]),we should require that as r ≈
0Ω = X n ≥ Ω ( n ) ( t, θ ) r n , (24)implying, because of (23) that in the neighborhood of thecenter G = X n ≥ G ( n ) ( t, θ ) r n . (25)Beside the kinematical variables defined above, itwould be convenient for our discussion to introduce the“specific velocities”, defined in [9] (with the change ofnotation already mentioned): V (1)(1) = e α (1) e β (1) ( σ αβ + 13 Θ h αβ + Ω αβ ) , (26) V (2)(2) = e α (2) e β (2) ( σ αβ + 13 Θ h αβ + Ω αβ ) , (27) V (3)(3) = e α (3) e β (3) ( σ αβ + 13 Θ h αβ + Ω αβ ) , (28) V (1)(2) = e α (1) e β (2) ( σ αβ + 13 Θ h αβ + Ω αβ ) , (29)which become, using (17), (19), (20) and (22) V (1)(1) = 13 (cid:0) σ (1)(1) + Θ (cid:1) , V (2)(2) = 13 (cid:0) σ (2)(2) + Θ (cid:1) , (30) V (3)(3) = 13 (cid:0) Θ − σ (1)(1) − σ (2)(2) (cid:1) , V (1)(2) = − Ω , (31)satisfying V (1)(1) + V (2)(2) + V (3)(3) = Θ . (32)The physical meaning of the above expressions be-comes intelligible when we recall that the tensor σ αβ + Θ h αβ + Ω αβ defines the proper time variationof the infinitesimal distance δl between two neighboringpoints on the three-dimensional hypersurface (say Σ),orthogonal to the four velocity, divided by δl (see [9])for details). B. The electric and magnetic part of the Weyltensor and the super–Poynting vector
Let us now introduce the electric ( E αβ ) and magnetic( H αβ ) parts of the Weyl tensor ( C αβγδ ), defined as usualby E αβ = C ανβδ V ν V δ ,H αβ = 12 η ανǫρ C ǫρβδ V ν V δ . (33)The electric part of the Weyl tensor has only three in-dependent non-vanishing components, whereas only twocomponents define the magnetic part. Thus we may writethese two tensors, in terms of five tetrad components( E (1)(1) , E (2)(2) , E (1)(2) , H (1)(3) , H (3)(2) ), respectively as: E αβ = (cid:20)(cid:0) E (1)(1) + E (2)(2) (cid:1) (cid:18) e (1) α e (1) β − h αβ (cid:19)(cid:21) + (cid:20)(cid:0) E (2)(2) + E (1)(1) (cid:1) (cid:18) e (2) α e (2) β − h αβ (cid:19)(cid:21) + E (2)(1) (cid:16) e (1) α e (2) β + e (1) β e (2) α (cid:17) , (34)and H αβ = H (1)(3) (cid:16) e (1) β e (3) α + e (1) α e (3) β (cid:17) + H (2)(3) (cid:16) e (3) α e (2) β + e (2) α e (3) β (cid:17) . (35) Also, from the Riemann tensor we may define threetensors Y αβ , X αβ and Z αβ as Y αβ = R ανβδ V ν V δ , (36) X αβ = 12 η ǫραν R ⋆ǫρβδ V ν V δ , (37)and Z αβ = 12 ǫ αǫρ R ǫρδβ V δ , (38) where R ⋆αβνδ = η ǫρνδ R ǫραβ and ǫ αβρ = η ναβρ V ν .The above tensors in turn, may be decomposed, so thateach of them is described through four scalar functionsknown as structure scalars [15]. These are (see [2] fordetails) Y T = 4 π ( µ + 3 P ) , X T = 8 πµ, (39) Y I = 3 E (1)(1) − π Π (1)(1) , X I = − E (1)(1) − π Π (1)(1) Y II = 3 E (2)(2) − π Π (2)(2) , X II = − E (2)(2) − π Π (2)(2) ,Y III = E (2)(1) − π Π (2)(1) , X III = −E (2)(1) − π Π (2)(1) . and Z I = ( H (1)(3) − πq (2) ); Z II = ( H (1)(3) + 4 πq (2) ); Z III = ( H (2)(3) − πq (1) ); Z IV = ( H (2)(3) + 4 πq (1) ) . (40)From the above tensors, we may define the super–Poynting vector by P α = ǫ αβγ (cid:0) Y γδ Z βδ − X γδ Z δβ (cid:1) , (41)where ǫ αβρ = η ναβρ V ν . In our case, we may write: P α = P (1) e (1) α + P (2) e (2) α , (42)with P (1) = 2 H (2)(3) (cid:0) E (2)(2) + E (1)(1) (cid:1) + 2 H (1)(3) E (2)(1) + 32 π q (1) (cid:2) ( µ + P ) + Π (1)(1) (cid:3) + 32 π q (2) Π (2)(1) ,P (2) = − H (1)(3) (cid:0) E (1)(1) + E (2)(2) (cid:1) − H (2)(3) E (2)(1) + 32 π q (2) (cid:2) ( µ + P ) + Π (2)(2) (cid:3) + 32 π q (1) Π (2)(1) . (43)In the theory of the super–Poynting vector, a state ofgravitational radiation is associated to a non–vanishingcomponent of the latter (see [16–18]). This is inagreement with the established link between the super–Poynting vector and the news functions [19], in the con-text of the Bondi–Sachs approach [11, 20].We can identify two different contributions in (43). Onthe one hand we have contributions from the heat trans-port process. These are in principle independent of themagnetic part of the Weyl tensor, which explains whythey remain in the spherically symmetric limit. Next wehave contributions related to the gravitational radiation.These require, both, the electric and the magnetic partof the Weyl tensor to be different from zero. III. THE HEAT TRANSPORT EQUATION
In order to avoid the drawbacks generated by the stan-dard (Landau–Eckart) irreversible thermodynamics [21],[22], (see [23]-[26] and references therein) we shall needa transport equation derived from a causal dissipativetheory [27–32]. In this work we shall resort to M¨uller-Israel-Stewart second order phenomenological theory fordissipative fluids [27–30]). However, as we shall see, themain conclusions generated by our study are not depen-dent on the transport equation chosen, as far as it is acausal one, i.e that it leads to a Cattaneo type [33] equa-tion, leading thereby to a hyperbolic equation for thepropagation of thermal perturbations.Thus, the transport equation for the heat flux reads[24, 28–30], τ h µν q ν ; β V β + q µ = − κh µν ( T ,ν + T a ν ) − κT (cid:18) τ V α κT (cid:19) ; α q µ , (44)where τ , κ , T denote the relaxation time, the thermalconductivity and the temperature, respectively.Contracting (44) with e (2) µ we obtain τA (cid:0) ˙ q (2) + Aq (1) Ω (cid:1) + q (2) = − κA G ˙ T + A T ,θ √ A B r + G + AT a (2) ! − κT q (2) (cid:18) τ V α κT (cid:19) ; α , (45)where (23), has been usedOn the other hand, contracting (44) with e (1) µ , we find τA (cid:0) ˙ q (1) − Aq (2) Ω (cid:1) + q (1) = − κB (cid:0) T ′ + BT a (1) (cid:1) − κT q (1) (cid:18) τ V α κT (cid:19) ; α . (46)It is worth noticing that the two equations above arecoupled through the vorticity. IV. LEAVING THE EQUILIBRIUM
We shall now take a snapshot of the system, just after ithas abandoned the equilibrium. As mentioned before, by“just after” we mean on the smallest time scale, at whichwe can detect the first signs of dynamical evolution.The general “philosophy” of our approach consists ofconsidering a fluid distribution which is in equilibrium(in the sense exposed in the Introduction), and assumethat, for a reason which is not relevant for the discussion,at some initial time (say t ) the system abandons such astate. Thus, at t the clock is put to measure time, andwe stop it as soon as we detect the first sign of dynamicevolution. The scale time under consideration is definedby the time interval measured by our clock. This is, soto speak, the “philosophy” of the approach.However, in practice we shall proceed slightly differ-ently. Indeed, we are going to choose a given time scale,which we shall specify below. Two possible results maythen appear: • No signs of dynamic evolution are observed withinthe choosen time scale • Such signs do appear, at such time scale.Of course in the case of the first result, we should haveto enlarge our time scale.Now, in the study of dissipative fluids, there are threefundamental time scales, each of which endowed with adistinct physical meaning, namely: the hydrostatic time(sometimes also called the hydrodynamic time), the ther-mal relaxation time and the thermal adjustment time(see [7, 8] for details).The hydrostatic time is the typical time in which afluid element reacts on a slight perturbation of hydro-static equilibrium, it is basically of the order of magni-tude of the time taken by a sound wave to propagatethrough the whole fluid distribution.The thermal relaxation time is the time taken by thesystem to return to the steady state in the heat flux (whether of thermodynamic equlibrium or not), after ithas been removed from it.Finally, the thermal adjustment time is the time ittakes a fluid element to adjust thermally to its surround-ings. It is, essentially, of the order of magnitude of thetime required for a significant change in the temperaturegradients. From the above it is evident that the thermaladjustment time is, generally, larger than the thermalrelaxation time.We shall evaluate the system at a time scale whichis smaller than the three time scales described above.It should be emphasized that such a time scale is cho-sen heuristically. Thus, as mentioned before, if no signof evolution could be detected within this time scale, itshould be enlarged until these signs appear. However, aswe shall see below, such signs do appear within the timescale under consideration.The above comments imply that: • At the time scale at which we are observ-ing the system, which is smaller than the hy-drostatic time scale, the kinematical quantitiesΩ( G ) , Θ , σ (1)(1) , σ (2)(2) as well as the “velocities” V (1)(1) , V (2)(2) , V (3)(3) , V (1)(2) keep the same valuesthey have in equilibirum, i.e. they are neglected(of course not so their time derivatives which areassumed to be small, say of order O ( ǫ ), where ǫ << • From (A5) (A6) (Eqs. B6, B7 in [2]), it followsat once that the heat flux vector should also beneglected (once again, not so its time derivative).The vanishing of the flux vector also follows at oncefrom the fact that the time scale under considera-tion is smaller than the relaxation time. • From the above conditions it follows at once thatfirst order time derivatives of the metric variables
A, B, C can be neglected.Then, we have for the four acceleration a (1) = A ′ AB ; a (2) = 1 Br A ,θ A + ˙ GA ! . (47)Also, from the conditions above and (17, 19, 20, 23,30, 31), it follows that˙Θ = 1 A BB + ¨ CC ! , ˙ σ (1)(1) = ˙ σ (2)(2) ≡ ˙¯ σ = 13 A ¨ BB − ¨ CC ! , (48)˙Ω = 1 AB r ˙ G ′ − ˙ GA ′ A ! , (49)and V (1)(1) = V (2)(2) ≡ V, ˙ V = ¨ BAB , ˙ V (3)(3) = ¨ CAC . (50)Now, at thermal equilibrium, when the heat flux van-ishes, the Tolman conditions for thermal equilibrium [34](
T A ) ′ = ( T A ) ,θ = 0 , (51)are valid.Therefore just after the system leaves the equilibrium,at a time scale which is smaller than the thermal adjust-ment time and the thermal relaxation time, the equa-tions (51) are still valid, even though the system startsto leave the thermal equilibrium. This is so because ofthe fact that our time scale is smaller than the relaxationtime, and therefore the temperature gradients have thesame values they had in equilibrium. However, the ful-fillment of (51) is not enough to ensure the vanishing of˙ q (2) , due to the appearance of a ˙ G term in (45) (through a (2) ), which eventually would lead to the breaking of thethermal equilibrium in the meridional direction (at latertime).Thus, the evaluation of (46) and (45) just after leavingthe equilibrium, produces respectively˙ q (1) = 0 , (52)and τ ˙ q (2) = − κAT ,θ Br − κAT a (2) , (53)or, using (51) τ ˙ q (2) = − κT ˙ GABr . (54)Therefore, at the very beginning of the evolution, thedissipative process starts with contributions along the e (2) µ (meridional) direction.We shall now turn to fluid variables( µ, P, Π (1)(1) , Π (2)(2) , Π (2)(1) ). Using MAPLE weshall calculate the components of the Einstein tensor G αβ and evaluate them just after the system leaves theequilibrium. At this time scale, this tensor have threetypes of terms: On the one hand, terms with first timederivatives of the metric functions A, B, C , which are areset to zero, next, there are terms that neither contain G , nor first time derivatives of A, B, C , these correspondto the expression in equilibrium, finally, there are termswith first time derivatives of G and/or second timederivatives of A, B, C , which of course are not neglected.Then using (7, 9, 10, 11) and the Einstein equations , G αβ = − πT αβ , (55) we obtain 8 πµ = 8 πµ ( eq ) , (56)8 πP = 8 πP ( eq ) − A ˙Θ+ 23 A B r (cid:18) ˙ G ,θ + ˙ G C ,θ C (cid:19) , (57)8 π Π (1)(1) = 8 π Π (1)(1)( eq ) + ˙¯ σA + 13 A B r (cid:20) ˙ G ,θ − ˙ G (cid:18) B ,θ B − C ,θ C (cid:19)(cid:21) , (58)8 π Π (2)(2) = 8 π Π (2)(2)( eq ) + ˙¯ σA + 13 A B r (cid:20) − G ,θ + ˙ G (cid:18) B ,θ B + C ,θ C (cid:19)(cid:21) , (59)8 π Π (2)(1) = 8 π Π (1)(1)( eq ) − ˙Ω A + ˙ GA B r (cid:20) ( Br ) ′ Br − A ′ A (cid:21) , (60)where eq stands for the value of the quantity at equilib-rium.Now, from (56) it follows at once that the energy den-sity, after leaving the equilibrium, at the time scale con-sidered here, has the same value it had in equilibrium.Then since there should be a generic equation of staterelating the energy density with the isotropic pressure,it is reasonable to assume that at the time scale un-der consideration we have P = P ( eq ) , and following thisline of arguments it would be also reasonable to assumeΠ (1)(1) = Π (1)(1)( eq ) , Π (2)(2) = Π (2)(2)( eq ) .Once again, it is important to remark that such as-sumptions are purely heuristic. Therefore if it wouldhappen that as a consequence of their imposition, wedetect no signs of evolution (at the time scale under con-sideration), we should relax them and enlarge our timescale, until these signs become observable. However thisis not the case. Indeed, from these latter conditions and(23, 57, 58, 59), it follows at once that:˙ G = B f ( t, r ) , (61)˙¯ σ = − f ( t, r )3 Ar (cid:18) C ,θ C − B ,θ B (cid:19) , (62)˙Θ = f ( t, r ) Ar (cid:18) B ,θ B + C ,θ C (cid:19) , (63)˙Ω = f ( t, r ) Ar (cid:18) ln B √ fA (cid:19) ′ , (64)where f ( t, r ) is an arbitrary function of its arguments.Two comments are in order at this point: • Because of (25) it is obvious that f = P n ≥ f ( n ) ( t ) r n in the neighborhood of the center. • Observe that f controls the evolution of G (Ω) , Θand ¯ σ .The situation is quite different for the scalar Π (2)(1) .In fact, as we shall see, we cannot assume that Π (2)(1) =Π (2)(1)( eq ) .Indeed, because of (60), to assume that Π (2)(1) =Π (2)(1)( eq ) , amounts to impose the condition˙Ω A = ˙ GA B r (cid:20) ( Br ) ′ Br − A ′ A (cid:21) , (65)which together with (23) produces˙ G = B r g ( t, θ ) , (66) where g is an arbitrary function of its arguments. But,(66) clearly violates the regularity condition (25), closeto the center. Accordingly, at the time scale under con-sideration we have Π (2)(1) = Π (2)(1)( eq ) , more precisely8 π Π (2)(1) = 8 π Π (2)(1)( eq ) + f ( t, r )2 A r (cid:18) ln r f (cid:19) ′ . (67)Thus we see that, after leaving the equlibrium, at thetime scale under consideration, the energy density, theisotropic pressure and the (1)(1) and the (2)(2) tetradcomponents of the anisotropic tensor may be assumed tokeep the values they have in equilibrium. However for thetransverse tension Π (2)(1) the situation is different, andthe first signs of the dynamic regime are already presentin this tetrad component of the anisotropic tensor, at ourtime scale.Using MAPLE we can also easily calculate the scalarsdefining the electric part of the Weyl tensor, after thesystem leaves the equlibrium, we obtain: E (1)(1) = E (1)(1)( eq ) − ˙¯ σ A − A B r (cid:20) ˙ G ,θ − ˙ G (cid:18) B ,θ B − C ,θ C (cid:19)(cid:21) , (68) E (2)(2) = E (2)(2)( eq ) − ˙¯ σ A + 16 A B r (cid:20) G ,θ − ˙ G (cid:18) B ,θ B + C ,θ C (cid:19)(cid:21) (69) E (2)(1) = E (2)(1)( eq ) + ˙Ω2 A − ˙ G A B r (cid:20) ( Br ) ′ Br − A ′ A (cid:21) . (70)Using (60), (61) and (63) in (68), (69) and (70), it followsat once that E (1)(1) = E (1)(1)( eq ) , E (2)(2) = E (2)(2)( eq ) , E oeq (2)(1) = − π Π oeq (2)(1) , (71)which impliy, because of (39) X I = X I ( eq ) , X II = X II ( eq ) , X III = X III ( eq ) , (72)and Y I = Y I ( eq ) , Y II = Y II ( eq ) , Y oeq.III = − π Π oeq. (2)(1) , (73)where oeq stands for the value of the quantity “out ofequilibrium”, and as it follows at once from (67)8 π Π oeq (2)(1) = f ( t, r )2 A r (cid:18) ln r f (cid:19) ′ . (74)Let us now analyze the “generalized Euler equations“(A2) (Eq. A7 in [2]), derived from the “conservationlaws“ ( T µν ; ν = 0). Evaluated within the time scale under consideration, these are the equations (A3) and (A4) inthe Appendix:Observe that these two equations have the “Newto-nian” form M ass density × Acceleration = F orce, (75)and where we can clearly identify the “effective inertialmass density” as the factors multiplying ˙ V and ˙ V (3)(3) .Also, it is worth noticing that the first term in the righthand side of (A3), and the first term in the right handside of (A4), represent the “gravitational force”. Thisis in agreement with the equivalence principle, accordingto which, the “effective inertial mass density” equals the“passive gravitational mass density” (the factor multi-plying the square brackets in (A3) and (A4)).We observe that, according to (A3) and (A4) thereare two different “effective inertial mass densities”, de-pending on the anisotropy of the fluid. This is a clearreminiscence of the situation appearing in relativistic dy-namics, where a moving particle offers different inertialresistances to the same force, according to whether it issubjected to that force longitudinally or transversely. Finally, replacing ˙ q (2) , by its expression from (53), into(A4) we obtain (cid:0) µ + P + Π (2)(2) (cid:1) (cid:20) − κTτ ( µ + P + Π (2)(2) ) (cid:21) ˙ V (3)(3) = − (cid:0) µ + P + Π (2)(2) (cid:1) (cid:20) − κTτ ( µ + P + Π (2)(2) ) (cid:21) (cid:2) πA (cid:0) P + 2Π (1)(1) + 2Π (2)(2) (cid:1) − AC ′ a (1) BC (cid:21) + force and dissipative terms . (76)This last equation illustrates the well known decreas-ing of the inertial mass density (and consequently, of thepassive gravitational mass density) associated to thermaleffects, which was discovered in [35], and that has beenshown to appear in a great variety of scenarios (see [36–48] and references therein).Next, observe that by evaluating the physical variablesout of equlibrium, we may obtain˙ V = B ,θ ˙ GAB r = B ,θ fAr B , (77)˙ V (3)(3) = C ,θ ˙ GACB r = C ,θ fAr C , (78)from where it is apparent that f controls the evolutionof the different “velocities”.We can now turn to the equations (B1, B3) and (B5)in [2].They describe the evolution of Θ, ¯ σ and Ω, andusing (62, 63, 64) they become identities. On the otherhand (B4) becomes an identity when using (61, 67, 71).Next we have the equations (A5), (A6) (B6, B7 in [2]),which from the all results obtained above become identi-ties, whereas the equations (B8) and (B9) imply H (1)(3) = H (3)(2) = 0 , (79)of course their time derivatives do not vanish, as we shallsee below.Equations (B10–B13) in [2] describe the evolution ofthe structure scalars X I , X II , X III . It is a simple matterto check that within the time scale considered here ˙ X I =˙ X II = ˙ X III = 0. Also, it is a simple matter to see thatequations (B14–B16) in [2] do not provide any additionalinformation.Finally, the equations (A7) and (A8) (B17, B18 in [2])describe the evolution of the magnetic part of the Weyltensor in terms of the function f ( t, r ), more specifically,these equations become:˙ H (1)(3) = f ABr (cid:20) f ′ rf − f ′′ f − (cid:18) r − f ′ f (cid:19) (cid:18) A ′ A − B ′ B + C ′ C (cid:19)(cid:21) + f B (cid:0) E (1)(1) + E (2)(2) (cid:1) Ar , (80)˙ H (3)(2) = f ABr (cid:18) r − f ′ f (cid:19) (cid:18) A ,θ A − B ,θ B + C ,θ C (cid:19) + f B E (2)(1)( eq ) Ar , (81) from which it is evident that the evolution of the mag-netic part of the Weyl tensor is fully controlled by thefunction f . V. CONCLUSIONS
We have carried out an exhaustive analysis of axi-ally symmetric fluid distributions, just after its departurefrom equilibrium, at the smallest time scale at which wecan detect signs of dynamical evolution.As our main result, we have found that the evolution ofall variables is controlled by a single function f , which wecall the fluid news function, in analogy with the Bondi’snews function. Indeed, if anything happens at all at thesource leading to changes in the field, it can only doso through the function f , and viceversa, exactly as itappears from the analysis of the spacetime outside thesource (Bondi). However, an important difference be-tween these two functions must be emphasized, namely:our function f controls the evolution only within the timescale considered here, a limitation which does not applyto the Bondi’s news function (see below for a deeper dis-cussion on this point).Among all the physical variables, there are two, whichplay a significant role in the departure from equilibrium.On the one hand, it is the heat flow along the e µ (2) direc-tion, the one which shall appear first. On the other hand,it is also remarkable that it is the tetrad component ofthe anisotropic tensor, in the subspace spanned by thetensor e µ (2) e ν (1) , the one which shows the first indicationsof the departure from equilibrium.It is worth mentioning, that at the time scale used here,there is not gravitational radiation, as it follows at oncefrom (43). Thus, the emission of gravitational waves isan event which occurs at later times. This fact becomesintelligible at the light of the following comments.For a second order phenomenological theory for dissi-pative fluids we obtain from Gibbs equation and conser-vation equations (see [24, 43] for details):0 T S α ; α = − q α " h µα (ln T ) ,µ + V α ; µ V µ + β q α ; µ V µ + T (cid:18) β T V µ (cid:19) ; µ q α , (82)where S α is the entropy four–current, and β = τκT .From which it becomes evident that at the time scaleunder consideration S α ; α = 0.We recall that in the above expression, terms involv-ing couplings of heat flux to the vorticity, vanish at thetime scale under consideration. Also, we have excludedshear and bulk viscosity contributions in (82). The factis that these absent terms are proportional to the sheartensor, the expansion scalar, terms quadratic in the bulkviscosity pressure, terms proportional to the bulk vis-cosity pressure multiplied by its time derivative, andterms proportional to the anisotropic stress tensor as-sociated to the shear viscosity multiplied by itself, or byits time derivative (see Eq.(2.20) in [24]), (we recall thatthe anisotropic stress tensor may, but does not need to,be related to viscosity effects, since it may be sourced bymany other physical phenomena. Thus, for example itmay be different from zero for a static configuration). Ofcourse, within the time scale used here, all these termsvanish. However, it should be clear that in the studyof any specific astrophysical scenario, these dissipativephenomena may be present and might play an importantrole in the detailed description of the structure and evo-lution of the object (at a time scale larger than the oneconsidered here).Thus, within our time scale, our observers do not de-tect a real (entropy producing) dissipative process. Butas it was already pointed out in the seminal Bondi’s pa-per on gravitational radiation(see section 6 in [11]), inthe absence of dissipation, the system is not expected toradiate (gravitationally) due to the reversibility of theequation of state, at variance with the fact that radia-tion is an irreversible process (see also [49] for a furtherdiscussion on this point).Therefore, it is obvious that, in the presence of gravita-tional radiation, an entropy generator factor should alsobe present in the description of the source. But as wehave just seen, such a factor does not appear within thetime scale under consideration. Accordingly it is reason-able, not to detect gravitational radiation at that sametime scale.The reversibility of the evolution, at the time scale un-der consideration, implied by the above comments, couldalso be inferred from a simple inspection of (54), (61),(62), (63), (64), (77), (78), (80), (81).Indeed, it results at once from these equations, that ifthe function f is different from zero until some time, andvanishes afterwards (always within the time scale underconsideration), the system will turn back to equilibrium,without “remembering” to have been out of it previously.In other words, the fluid news function, unlike theBondi’s news function, is the precursor of, (appears be-fore), the dissipative process related to the emission of gravitational radiation, and should be different from zerountil such emission starts.In relation with the point above, another comment isin order: in [19] the link between radiation and vorticitywas put in evidence (see also [50]), more specifically it wasexplicitly assumed that such a link was a causal one (thetitle of [19] is: “Why does gravitational radiation producevorticity?”), i.e. it was assumed that radiation precedesthe appearance of vorticity. However as we have justshown, both the magnetic part of the Weyl tensor, andΩ vanish at the time scale under consideration, whereastheir first time derivatives do not vanish at that sametime scale, suggesting that both phenomena (radiationand vorticity) occur essentially simultaneously.An interesting particular case is represented by thesituation appearing if we impose that the system wasinitially spherically symmetric (besides of being in equi-librium), and assume that it remains spherically sym-metric afterwards. In such a case, it is obvious that wemust have ˙ G = 0, implying that departures from equi-librium (dynamic and thermal) only occur if Tolman’sconditions (51), are violated. However, since the sys-tem was initially at equilibrium, such a violation mayonly happen at time scales larger that the thermal ad-justment time. In other words, departures from equi-librium, keeping the spherical symmetry, take place attime scales larger than the corresponding to the, gen-eral, non–spherical case. Observe that in the purelyspherically symmetric case the assumptions P = P ( eq ) ,Π (1)(1) = Π (1)(1)( eq ) , Π (2)(2) = Π (2)(2)( eq ) do not hold(since we have to enlarge the time scale in order to ob-serve the first signs of evolution), and of course the onsetof evolution is not controlled by the function f as definedby (61).We would like to emphasize the appearance of the ther-mal effect leading to a decreasing of the effective iner-tial mass density. In this respect, it is worth stressingthat the first term on the left, and the T a ν term on theright, of (44), are directly responsible for the decreasingin the effective inertial mass density. The former shouldbe present in any causal theory of dissipation, whereasthe latter is just an expression of the “inertia” of heatalready pointed out by Tolman [34].Therefore any hyperbolic, relativistic dissipative the-ory yielding a Cattaneo-type equation in the non-relativistic limit, is expected to give a result similar tothe one obtained here. The possible consequences of thiseffect on the outcome of gravitational collapse have beendiscussed in some detail in [40, 41]. It is also worthnoticing that such an effect appears already at the earli-est stages of the non–equilibrium (though only along the V (3)(3) direction).Finally we would like to conclude with the following re-1mark: In the stationary case one may have a steady rota-tion around the symmetry axis, leading to non vanishing(time independent) vorticity Ω µν = 0, which of coursemay be compatible with thermal equilibrium. In this casethe spacetime outside the source is described by a metricof the Lewis-Papapetrou family (e.g. Kerr) which as weknow admits vorticity in the congruence of the world lineof observers (the line element is non-diagonal). The vor-ticity of the source produces the vorticity in the exteriorspacetime. However, in the static situation (the one con-sidered here) you have no vorticity at the outside, whichis described by a metric of the Weyl family (e.g.Curzon,Erez-Rosen, etc). In this latter case (non stationary) wemust have Ω µν = G = 0 since the metric is diagonal.Since you have no vorticity outside (no frame dragging),you should not expect to have vorticity in the source (see[1] for a discussion on this case).This last result may be obtained in a more rigourousway, by evaluating (A5) and (A6) in the static case andthermal equilibrium (assuming Ω = 0). Then after somelengthy but simple calculations, and using the regularitycondition (25), one obtains Ω = 0. Thus there is novorticity associated to the static case. This brings out thedifference between the steady vorticity of the stationary case and the vorticity considered here.Also, the result above, shows that vorticity and heatflux are inherently coupled. This fact was already em-phasized in [2]. ACKNOWLEDGMENTS
L.H. thanks Departament de F´ısica at the Universitatde les Illes Balears, for financial support and hospitality.ADP acknowledges hospitality of the Departament deF´ısica at the Universitat de les Illes Balears. L.H and J.O.acknowledge financial support from the Spanish Ministryof Science and Innovation (grant FIS2009-07238) andFondo Europeo de Desarrollo Regional (FEDER) (grantFIS2015-65140-P) (MINECO/FEDER) .
Appendix A: Some basic equations
In what follows we shall deploy only those equations ofthe formalism which are required for our discussion. Thewhole set of the equations can be found in [2].The conservation law T αβ ; α = 0 leads to the followingequations (Eqs. A6, A7 in [2]): µ ; α V α + ( µ + P )Θ + (cid:0) σ (1)(1) + σ (2)(2) (cid:1) Π (1)(1) + (cid:0) σ (2)(2) + σ (1)(1) (cid:1) Π (2)(2) + q α ; α + q α a α = 0 , (A1)( µ + P ) a α + h βα (cid:16) P ; β + Π µβ ; µ + q β ; µ V µ (cid:17) + (cid:18)
43 Θ h αβ + σ αβ + Ω αβ (cid:19) q β = 0 . (A2)The first of these equations is the “continuity” equation,whereas the second one is the “generalized Euler” equa- tion.This last equation has two components, which, withinthe time scale under consideration may be written as: (cid:0) µ + P + Π (1)(1) (cid:1) ˙ V = − (cid:0) µ + P + Π (1)(1) (cid:1) (cid:20) πA (cid:0) P − (1)(1) (cid:1) − AB ,θ a (2) B r (cid:21) + “ f orce terms ′′ , (A3)and (cid:0) µ + P + Π (2)(2) (cid:1) ˙ V (3)(3) = − (cid:0) µ + P + Π (2)(2) (cid:1) (cid:20) πA (cid:0) P + 2Π (1)(1) + 2Π (2)(2) (cid:1) − AC ′ a (1) BC (cid:21) − AC ,θ BCr (cid:20) ˙ q (2) A (cid:21) , + “ f orce terms ′′ , (A4)where by “force terms” we denote different terms con-taining pressure gradients and anisotropic stresses. Next, from the Ricci identities we have (Eqs. B6, B7in [2])223 B Θ ,r − Ω ; µ e µ (2) + Ω (cid:16) e (2) β ; µ e µ (1) e β (1) − e µ (2); µ (cid:17) + σ (1)(1) a (1) − Ω a (2) − σ (1)(1); µ e µ (1) − (cid:0) σ (1)(1) + σ (2)(2) (cid:1) (cid:16) e µ (1); µ − a (1) (cid:17) − (cid:0) σ (2)(2) + σ (1)(1) (cid:1) (cid:16) e (2) β ; µ e µ (2) e β (1) − a (1) (cid:17) = 8 πq (1) , (A5)13 √ A B r + G (cid:18) GA Θ ,t + 2 A Θ ,θ (cid:19) + a (2) σ (2)(2) + Ω ; µ e µ (1) + Ω (cid:16) e µ (1); µ + e µ (2) e β (1) e (2) β ; µ (cid:17) + Ω a (1) − σ (2)(2); µ e µ (2) + (cid:0) σ (1)(1) + σ (2)(2) (cid:1) (cid:16) e (2) β ; µ e β (1) e µ (1) + a (2) (cid:17) − (cid:0) σ (2)(2) + σ (1)(1) (cid:1) (cid:16) e µ (2); µ − a (2) (cid:17) = 8 πq (2) . (A6)Finally, from the Bianchi identities, the following two equations describing the evolution of the magnetic part ofthe Weyl tensor, are obtained (Eqs. B17, B18 in [2]). − a (2) E (1)(1) + 2 a (1) E (2)(1) − E δ δ e − AY I,θ √ A B r + G + Y III,r B − (cid:20)
13 (2 Y I + Y II ) e (1) β ; δ + 13 (2 Y II + Y I ) e ν (1) e (2) ν ; δ e (2) β + Y III ( e (2) ν ; δ e ν (1) e (1) β + e (2) β ; δ ) (cid:21) ǫ γδβ e (3) γ + H (3)(1) ,δ V δ + H (3)(1) (cid:0) Θ + σ (2)(2) − σ (1)(1) (cid:1) + Ω H (2)(3) = − π µ ,θ e + 12 π Ω q (1) + 4 πq (2) (cid:0) σ (1)(1) + Θ (cid:1) , (A7)2 a (1) E (2)(2) − a (2) E (2)(1) + E δβ ; δ e β (1) + Y II,r B − AY III,θ √ A B r + G − (cid:20) −
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