Symmetry evolution for the imperfect fluid under perturbations

Symmetry evolution fo r the imp erfect fluid under p erturbations Alcides Ga rat 1 1. F ormer Pr ofessor at Universidad de la R ep´ ublic a, A v. 18 de Julio 1824-1850, 11200 Montevide o, Uruguay. a) (Dated: 24 Marc h 2026) 1 Ev er since a new symmetry was found for the imp erfect fluid with v orticit y the ques- tion of the effect of p erturbations on the symmetry itself has b een raised. This new symmetry arose when realizing that lo cal four-v elo cit y gauge-lik e transformations w ould render the left hand side of the Einstein equations inv ariant. Because the metric tensor w ould be inv ariant under this new kind of local transformations. Then the point was raised ab out the inv ariance of such a kind of transformations of the stress-energy tensor on the right hand side of the Einstein equations in curved four- dimensional Loren tz spacetimes. It w as verified that these in v ariances do not work with plain p erfect fluid but they do w ork for imp erfect fluids. The imperfect fluid stress-energy tensor will b e in v ariant under local four-v elo cit y gauge-like transforma- tions when additional transformations are in troduced for sev eral v ariables included in the stress-energy tensor itself. This lo cal in v ariance w as also the criteria introduced in order to present a new stress-energy tensor for v orticit y as w ell. New tetrads are at the core of the realization of the existence of this new symmetry b ecause it is through these new tetrads that this new symmetry is realized. It is through the lo cal transformation of these new tetrad v ectors that w e can pro v e that the metric tensor is in v ariant. This new kind of symmetry has its origins in a similar tetrad form ulation as to the Einstein-Maxw ell spacetimes formalism presented in previous manuscripts. In this man uscript we will in tro duce lo cal p erturbations b y external agen ts to the relev an t ob jects in the imp erfect fluid geometry . W e will demonstrate a theorem that pro ves that the symmetries under four-v elo cit y gauge-lik e transformations will b e instan taneously broken but at the same time transformed into new symmetries. Because the lo cal orthogonal planes determined by these new tetrads, which happen to b e the lo cal planes of symmetry will tilt under lo cal p erturbations. There will b e a symmetry ev olution under p erturbations. P A CS n um b ers: 04.40.Nr; 04.20.Gz; 04.80.Cc; 04.20.Cv; 11.15.-q; 02.40.ky MSC2020: 85A30 , 85A04, 83C05 , 83C20, 76Y05, 22E70 , 22E05 , 51F25 Keyw ords: imperfect fluid; vorticit y; four-velocity gauge-lik e transformations; new tetrads; symmetry ev olution a) garat.alcides@gmail.com 2 I. INTR ODUCTION In manuscripts 1,2 it was pro ved that the existence of new tetrads in curv ed four- dimensional Loren tz spacetimes with signature (-+++) enabled the pro of about a new symmetry of spacetime in the presence of imp erfect fluids with vorticit y . These tetrads ha v e remark able prop erties that create this new mathematical scenario where new relev ant geometrical features are visualized with simplicit y . These new tetrads are built with t wo structure comp onents, the skeletons and the gauge vectors. The timelike-spacelik e lo cal plane, w e call plane one. The orthogonal lo cal spacelik e-spacelike plane is plane t wo. Ho w ever in our present case w e will start our analysis with an imp erfect fluid where the stress-energy tensor of a source can b e describ ed by the following equation 3–6 , T imp µν = ( ρ + p ) u µ u ν + p g µν + ( q µ u ν + q ν u µ ) + τ µν , (1) where the viscous stress-energy tensor τ µν is giv en b y 6 , τ µν = − η ( u µ ; ν + u ν ; µ + u µ u α u ν ; α + u ν u α u µ ; α ) − ( ζ − 2 3 η ) u α ; α ( g µν + u µ u ν ) , (2) where ρ is the energy-density of the fluid, p the isotropic pressure and where q µ is the heat flux relativ e to u µ its four-v elo city field, g µν is the metric tensor. The parameter η is the coefficient of shear viscosity and the parameter ζ is the co efficien t of bulk viscosity . If in addition this fluid has v orticit y ω µν , then w e can build the new tetrads for this particular case followin g the method developed in reference 7–11 and explained from different angles in references 12–17 . These new tetrads introduced and describ ed in detail in man uscript 1 will manifestly and co v ariantly diagonalize the stress-energy tensor for the p erfect fluid terms T p µν = ( ρ + p ) u µ u ν + p g µν at every spacetime ev ent. How ever in this man uscript we are in terested in the study of the lo cal symmetries when an external agen t p erturbs the source of relev ant fields. Source of gra vitational field, a source with a stress-energy tensor given by equations (1) and (2). F or p erturbations to imperfect fluids see references 18–20 and references therein. In order to find the new tetrads for the new p erturbation problem w e pro ceed then to in tro duce the fluid p erturb ed extremal field or the p erturb ed v elo cit y curl-extremal field through the lo cal duality transformation given by , 3 ξ µν = cos α u [ µ ; ν ] − sin α ∗ u [ µ ; ν ] , (3) where ∗ u [ µ ; ν ] = 1 2 ϵ µν σ τ g σ ρ g τ λ u [ ρ ; λ ] is the dual tensor of u [ µ ; ν ] and the lo cal complexion α is defined through the condition ξ µν ∗ ξ µν = 0 . (4) The sym b ol ; stands for co v arian t deriv ativ e with resp ect to the metric tensor g µν and the ob ject ϵ µν σ τ is the alternating tensor, see reference 7 . The ob jects with tildes represen t the p erturb ed ob jects. F or example u µ = ( u µ + ε δ u µ ) / q | ( u ρ + ε δ u ρ ) g ρλ ( u λ + ε δ u λ ) | where δ u µ is a lo cal p erturbation to the lo cal unit four-v elo city u λ with a suitable p erturbation parameter ε and u µ g µν u ν = − 1. F or the gra vitational field as another example g µν = g µν + ε δ g µν where δ g µν is a lo cal p erturbation to the lo cal metric tensor with the same suitable p erturbation parameter ε . All these p erturbations are of a physical nature and not the result of a co ordinate transformation. Similar for all the other v ariables in the ph ysical setup. It must b e stressed that ev en if the notation and the pap er structure are similar to reference 1 the con tent is different b ecause unlik e man uscript 1 in this man uscript w e are studying imp erfect fluids under p erturbations. W e are trying to prov e the dynamic nature of lo cal symmetries and this pro of requires a similar pap er structure and notation with resp ect to reference 1 but the conten t is substan tially different. F ollo wing exactly the same construction steps as in sections I and I I in man uscript 1 w e w ould find the follo wing results, this time for the p erturb ed quan tities. The complexion, which is a lo cal scalar, can then b e found b y plugging equation (3) in to equation (4) as, tan(2 α ) = −  u [ µ ; ν ] g σ µ g τ ν ∗ u [ σ ; τ ]  /  u [ λ ; ρ ] g λα g ρβ u [ α ; β ]  . (5) After introducing the new p erturb ed velocity curl-extremal field we proceed to write the four p erturb ed orthogonal vectors that will b ecome an in termediate step in constructing the tetrad that diagonalizes the p erturb ed p erfect fluid stress-energy tensor terms as T p µν = ( ρ + p ) u µ u ν + p g µν (see further in to the man uscript in equation (23) and a detailed analysis in reference 1 ), 4 V α (1) = ξ αλ ξ ρλ X ρ (6) V α (2) = ξ αλ X λ (7) V α (3) = ∗ ξ αλ Y λ (8) V α (4) = ∗ ξ αλ ∗ ξ ρλ Y ρ , (9) In order to pro v e the orthogonality of the tetrad (6-9) it is necessary to use the identit y A µα B ν α − ∗ B µα ∗ A ν α = 1 2 δ ν µ A αβ B αβ . (10) whic h is v alid for every pair of an tisymmetric tensors in a four-dimensional Lorentzian spacetime 17 for the case A µα = ξ µα and B ν α = ξ ν α w e w ould obtain, ξ µα ξ ν α − ∗ ξ µα ∗ ξ ν α = 1 2 δ ν µ Q , (11) where Q = ξ µν ξ µν is assumed not to b e zero. When (10) is applied to the case A µα = ξ µα and B ν α = ∗ ξ ν α w e obtain an equiv alen t condition to (4), ξ µρ ∗ ξ µλ = 0 . (12) The v ector fields X α and Y α in equations (6-9) are the gauge v ectors and we are free to c ho ose them as long as the four v ector fields (6-9) do not b ecome trivial. The tetrad vectors ha v e t wo essential components. F or instance in v ector V α (1) there are t wo main structures. First, the skeleton, in this case ξ αλ ξ ρλ , and second, the gauge vector X ρ . In v ector V α (3) the sk eleton is ∗ ξ αλ , and the gauge v ector Y λ . It is clear that if our c hoice for these gauge v ector fields is X α = Y α = u α , then the follo wing orthogonalit y relations will hold, g ρµ u ρ V µ (2) = g ρµ u ρ ξ µλ u λ = 0 (13) g ρµ u ρ V µ (3) = g ρµ u ρ ∗ ξ µλ u λ = 0 , (14) b ecause of the an tisymmetry of the p erturb ed v elo city curl-extremal field ξ µν . Then, at the p oin ts in spacetime where the set of four v ectors (6-9) is not trivial, we can normalize, 5 ˆ U α = ξ αλ ξ ρλ u ρ / ( q − Q/ 2 q u µ ξ µσ ξ ν σ u ν ) (15) ˆ V α = ξ αλ u λ / ( q u µ ξ µσ ξ ν σ u ν ) (16) ˆ Z α = ∗ ξ αλ u λ / ( q u µ ∗ ξ µσ ∗ ξ ν σ u ν ) (17) ˆ W α = ∗ ξ αλ ∗ ξ ρλ u ρ / ( q − Q/ 2 q u µ ∗ ξ µσ ∗ ξ ν σ u ν ) . (18) In analogy with the electromagnetic case and without tampering with an ything funda- men tal w e assume for simplicit y that u µ ξ µσ ξ ν σ u ν > 0, u µ ∗ ξ µσ ∗ ξ ν σ u ν > 0 and − Q > 0. W e also assume that ˆ U α ˆ U α = − 1. Using the tetrad vectors (15-18) and the metho d developed in man uscript 7 for the antisymmetric electromagnetic field, we can express the four-velocity curl in its maximal simplest form, u [ µ ; ν ] = − 2 q − Q/ 2 cos α ˆ U [ α ˆ V β ] + 2 q − Q/ 2 sin α ˆ Z [ α ˆ W β ] . (19) The metric tensor will b e written as, g αβ = − ˆ U α ˆ U β + ˆ V α ˆ V β + ˆ Z α ˆ Z β + ˆ W α ˆ W β . (20) The pair of v ectors ( ˆ U α , ˆ V α ) span the lo cal plane one. The vectors ( ˆ Z α , ˆ W α ) span the lo cal orthogonal plane t w o. The gauge-lik e Ab elian group of transformations of the four- v elo cit y X α = Y α = u α → X α = Y α = u α + Λ α induces sev eral kind of local tetrad transformations either in the lo cal plane one or the lo cal orthogonal plane t w o. W e use the notation Λ α = Λ ,β g β α for Λ a lo cal scalar. In the lo cal plane one the vectors that span this plane undergo local tetrad transformations under the lo cal group LB1 lea ving these v ectors inside the original lo cal plane. The group LB1 is comp osed by S O (1 , 1) plus tw o discrete transformations t w o by tw o. One of them is the full inv ersion or minus the identit y t w o b y t wo. The other discrete transformation is a reflection given by Λ o o = 0, Λ o 1 = 1, Λ 1 o = 1, Λ 1 1 = 0. LB2 for the lo cal tetrad transformations in plane tw o is just S O (2). All these p ossible cases hav e b een discussed in detail in manuscript 7 . The metric tensor will remain inv ariant under LB1 and LB2 as prov en in manuscript 7 for the electromagnetic field. Under all these transformations the four-velocity curl u [ µ ; ν ] and the extremal field ξ µν will also remain inv ariant. W e w ould lik e to know ab out the Einstein equations under this “four-v elo cit y gauge transformation”. The Einstein equations are given b y , 6 R µν − 1 2 g µν R = T µν = ( ρ + p ) u µ u ν + p g µν , (21) as we notice from equation T µν = ( ρ + p ) u µ u ν + p g µν . Under the transformation u α → u α + Λ α w e see that the right hand side will change b ecause u µ will change. The metric tensor remains inv ariant as already pro v ed, therefore the left hand side of equation (21) and the metric tensor on the second term of T µν will remain in v ariant. W e will b e able to write the transformed new stress-energy tensor as, T new µν = ( ρ + p ) ( u µ + Λ µ ) ( u ν + Λ ν ) + p g µν . (22) Then the perfect fluid stress-energy tensor will not b e in v ariant under gauge-like four- v elo cit y transformations on its o wn for the p erturb ed formulation either. In order to mak e it in v arian t we w ould hav e to add to the right hand side of the Einstein equations in (21) differen t kinds of terms including heat flow curren ts ( ρ + p ) ( u µ q ν + u ν q µ ) where u µ q µ = 0 is satisfied and terms with viscous stresses in the fluid τ µν where u µ τ µν = 0 is satisfied as w ell, see references 21–23 and section I I. This kind of fluid will b e ideal no longer, ho wev er w e are in terested in the in v ariance geometrical prop erties of the p erturb ed Einstein equations under the gauge-lik e four-v elo cit y transformations. I I. PER TURBED IMPERFECT FLUID STRESS-ENER GY TENSOR SYMMETR Y Let us consider a p erturb ed imp erfect fluid stress-energy tensor of the kind 6 , T imp µν = ( ρ + p ) u µ u ν + p g µν + ( q µ u ν + q ν u µ ) + τ µν , (23) where q µ is the perturb ed heat flux relativ e to the unit v elo city u µ and the perturb ed viscous stress-energy tensor τ µν is giv en b y 6 , τ µν = − η ( u µ ; ν + u ν ; µ + u µ u α u ν ; α + u ν u α u µ ; α ) − ( ζ − 2 3 η ) u α ; α ( g µν + u µ u ν ) . (24) The parameter η is the co efficien t of shear viscosit y and the parameter ζ is the co efficien t of bulk viscosit y . The p erturb ed velocity will ha ve to be normalized as in section I. Under 7 the four-v elo cit y gauge-like transformation u α → u α + Λ α the stress-energy tensor (23) will c hange and will not b e in v arian t if this is the only transformation that w e carry out. Let us remind ourselv es that the p erturb ed Einstein equations are giv en no w by , R µν − 1 2 g µν R = T imp µν . (25) The left hand side of equations (25) will remain in v ariant since it has been pro ven already in section I and reference 7 that the metric tensor is explicitly inv arian t under u α → u α + Λ α . The cov arian t deriv atives, meaning the Christoffel sym b ols will also b e inv arian t for the same reason. Therefore b y using this symmetry argumen t we realize that the righ t hand side of equations (25) will also hav e to be in v arian t. The transformations u α → u α + Λ α on the righ t hand side of equations (25) will not lea ve the tensor T imp µν in v arian t if we apply only this kind of four-v elo cit y gauge-lik e transformation. In order to mak e it in v arian t w e need to in tro duce the follo wing additional lo cal transformations, ρ → ρ + e ρ (26) p → p + e p (27) q µ → q µ + e q µ (28) τ µν ( u ) → τ µν ( u + Λ) + e τ µν . (29) By τ µν ( u ) we mean the equation (24) while b y τ µν ( u + Λ) w e mean equation (24) under the transformation u α → u α + Λ α . This notation will shorten the detailed writing of long expressions with man y terms. There is also a state equation p ( ρ ) and there is also another one e p ( e ρ ). There is then only one local scalar indep enden t v ariable that w e consider to be e ρ . After imposing in v ariance of the p erturb ed stress-energy tensor T imp µν w e will obtain ten equations for the fifteen v ariables e ρ , e q µ and e τ µν . Additionally we also impose that q α u α = 0 and τ µν u ν = 0 whic h are five more equations. The extension of these last t w o equations to the case under the transformation u α → u α + Λ α w e find ( q µ + e q µ ) ( u µ + Λ µ ) = 0 and ( τ µν ( u + Λ) + e τ µν ) ( u µ + Λ µ ) = 0. F rom these last t wo “transv erse” equations plus the original ones q α u α = 0 and τ µν u ν = 0 w e obtain the following equations, 8 q µ Λ µ + e q µ ( u µ + Λ µ ) = 0 (30) τ µν ( u ) Λ ν + τ µν (Λ) ( u ν + Λ ν ) + e τ µν ( u ν + Λ ν ) = 0 . (31) By τ µν (Λ) w e mean all the terms of the style u µ Λ ν ; α u α + u ν Λ µ ; α u α or Λ µ u ν ; α Λ α + Λ ν u µ ; α Λ α or Λ µ Λ ν ; α Λ α + Λ ν Λ µ ; α Λ α just to sho w a few examples. The tensor τ µν ( u ) is just (24) and w e can write τ µν ( u + Λ) = τ µν ( u ) + τ µν (Λ). Next w e apply in expression (23) the simultaneous transformations u α → u α + Λ α plus transformations (26-29). W e find, T imp µν → T imp µν + ( ρ + p ) ( u µ Λ ν + u ν Λ µ + Λ µ Λ ν ) + ( e ρ + e p ) ( u µ + Λ µ ) ( u ν + Λ ν ) + e p g µν + Λ µ q ν + Λ ν q µ + + e q µ ( u ν + Λ ν ) + e q ν ( u µ + Λ µ ) + τ µν (Λ) + e τ µν . (32) The next step is imp osing inv ariance, ( ρ + p ) ( u µ Λ ν + u ν Λ µ + Λ µ Λ ν ) + ( e ρ + e p ) ( u µ + Λ µ ) ( u ν + Λ ν ) + e p g µν + Λ µ q ν + Λ ν q µ + e q µ ( u ν + Λ ν ) + e q ν ( u µ + Λ µ ) + τ µν (Λ) + e τ µν = 0 . (33) W e contract equation (33) with ( u ν + Λ ν ) and use equations (30-31) in order to find the ob ject e q µ , e q µ = { ( q α Λ α ) ( u µ + Λ µ ) + τ µα ( u ) Λ α − [( ρ + p ) ( u µ Λ α + u α Λ µ + Λ µ Λ α ) + Λ µ q α + Λ α q µ + ( e ρ + e p ) ( u µ + Λ µ ) ( u α + Λ α ) + e p g µα ] ( u α + Λ α ) } / ( u ν + Λ ν ) ( u ν + Λ ν ) . (34) Con tracting again with ( u µ + Λ µ ) and using one more time the condition (30) w e can obtain after some straigh tforw ard algebraic work the equation corresp onding to e ρ and e p , ( e ρ + e p ) [( u µ + Λ µ ) ( u µ + Λ µ )] 2 + e p g µα ( u α + Λ α ) ( u µ + Λ µ ) = { ( q α Λ α ) ( u µ + Λ µ ) ( u µ + Λ µ ) + τ µα ( u ) Λ α ( u µ + Λ µ ) − [( ρ + p ) ( u µ Λ α + u α Λ µ + Λ µ Λ α ) + Λ µ q α + Λ α q µ ] . . ( u α + Λ α ) ( u µ + Λ µ ) } − ( u ν + Λ ν ) ( u ν + Λ ν ) ( − q α Λ α ) . (35) 9 W e know this equation of state e p ( e ρ ) from the outset indep enden tly from the ab ov e anal- ysis, and w e also get to find e ρ just b y equating these t w o indep enden t expressions of e p ( e ρ ) (the one w e know from the outset plus the one from equation (35)) and then replacing b oth e p ( e ρ ) and e ρ back in equation (34). Once e q µ is found in equation (34) we replace it along with e p ( e ρ ) and e ρ in equation (33) and find e τ µν . W e start with the p erturb ed imp erfect stress- energy tensor (23) and then we implemen t simultaneous transformations u α → u α + Λ α plus corresp onding transformations (26-29). The original (23) plus the four-v elo cit y gauge-lik e transformation scalar Λ or the gradient Λ µ are all known from the outset as are also kno wn from the start the equations of state p ( ρ ), e p ( e ρ ) and e p ( p ( ρ ) , ρ, e ρ ). In fact from equation (35) w e observ e that e p is e p ( p ( ρ ) , ρ, e ρ ). These are the t wo expressions for e p that we equate in order to obtain e ρ in terms of the ob jects given at the outset. Next we imp ose in v ariance through equations (33) under all of these four-velocity gauge-lik e lo cal transformations. Finally and using the conditions (30-31) w e obtain by algebraic work b oth e q µ and e τ µν plus e p ( e ρ ) and e ρ . W e ha ve av ailable ten equations or in v ariance conditions through equations (33) plus the five conditions (30-31) and found fifteen lo cal ob jects e q µ , e τ µν and e p ( e ρ ) or e ρ . This system of ideas ensures the gauge-lik e inv ariance in the righ t hand side of the system (25) knowing from the outset that the left hand side of the system (25) is explicitly and manifestly inv arian t under this lo cal group of transformations since this latter claim has b een pro ven in sections I-I I-I I I of reference 1 and separately also in reference 7 . It is also clear that the conserv ation equations ∇ ν T imp µν = 0 are also in v arian t under the new lo cal symmetry b ecause on one hand the stress-energy tensor T imp µν is in v arian t and on the other hand the p erturb ed metric tensor g µν is inv arian t under this lo cal symmetry and therefore the Christoffel sym b ols are also in v arian t. I I I. COMMENTS ON THE STRESS-ENER GY PER TURBED V OR TICITY TENSOR In section I I a thorough discussion is provided on ho w a p erturb ed imp erfect fluid stress- energy tensor can b e in v ariant under gauge-lik e four-velocity lo cal transformations u α → u α + Λ α . It is concluded that sp ecific transformations ha v e to be sim ultaneously satisfied for the p erturbed heat flow currents, the viscous stresses, the densit y and pressure for the right hand side of the Einstein equations for the p erturb ed imp erfect fluid to be in v ariant under 10 this kind of lo cal transformation. W e w ould also lik e to pro ceed in complete analogy to the electromagnetic case presented in reference 7 and also reference 1 for the unp erturb ed case in order to present a proposal for this p erturbed v orticit y stress-energy tensor and to the study of its prop erties, see references 24–28 . Let us remember that the unp erturb ed v orticity stress-energy tensor w as presented in pap er 1 . Let us pro ceed to introduce the follo wing symmetric tensor, T v ort µν = ξ µλ ξ λ ν + ∗ ξ µλ ∗ ξ λ ν . (36) When w e consider equations (11) and (12) it b ecomes simple to pro ve that the tetrad sets (6-9) and (15-18) diagonalize lo cally and cov arian tly the stress-energy tensor (36). W e consider this tensor to b e a stress-energy tensor b ecause it is built in the same w ay as in Einstein-Maxw ell spacetimes just replacing the electromagnetic four-potential b y the four- v elo cit y . Using the inv erse of the local dualit y rotation in tro duced in equation (3) and giv en b y u [ µ ; ν ] = cos α ξ µν + sin α ∗ ξ µν w e get, T v ort µν = u [ µ ; λ ] u [ ν ; ρ ] g ρλ + ∗ u [ µ ; λ ] ∗ u [ ν ; ρ ] g ρλ = ξ µλ ξ λ ν + ∗ ξ µλ ∗ ξ λ ν . (37) Lea ving for the moment p ossible constant units factors aside it is also clear that this tensor is not the whole stress-energy tensor on the right hand side of the p erturb ed Einstein fluid equations. The complete tensor fulfilling the conserv ation equations ∇ ν T µν = 0 would include the perturb ed p erfect fluid terms plus perturb ed heat flo w plus p erturb ed viscous stress plus the p erturb ed v orticit y stress-energy , see section IV. V ectors (6-7) or the normal- ized (15-16) generate the lo cal plane one where all v ectors are eigen vectors of the tensor (36) with eigen v alue Q/ 2 = ξ µλ ξ λ µ / 2 whic h w e assume to b e Q  = 0. V ectors (8-9) or the normal- ized (17-18) generate the lo cal orthogonal plane t w o where all vectors are eigen vectors of the tensor (36) with eigen v alue − Q/ 2 = − ξ µλ ξ λ µ / 2. Under the transformation u α → u α + Λ α the v ectors that span the lo cal plane one w ould undergo LB1 transformations inside this plane while the v ectors that span the lo cal plane tw o would undergo S O (2) transformations inside this second plane. Since the p erturb ed curl field u [ µ ; ν ] is locally in v ariant under this transformation, the perturb ed v orticity stress-energy (37) is in v arian t as well. The electro- magnetic case analyzed in reference 7 is identical in mathematical structure to our present v orticit y case. The principle of symmetry is the hidden principle that has b een guiding u s 11 in our searc h for the v orticity stress-energy symmetric tensor. IV. CONCLUSIONS In a previous man uscript 1 it was found a new symmetry for an imp erfect fluid in curved four-dimensional Loren tz spacetimes. The k ey element in this disco v ery was that when there is vorticit y we can use the curl of the four-velocity as an antisymmetric field in order to build new tetrads with remark able prop erties. Through a lo cal duality transformation b y a lo cal angle that we called the complexion in equation (3) it was found an extremal field that enabled the construction of tetrad skeletons. These new tetrads that diagonalize the p erfect fluid stress-energy tensor and the v orticity stress-energy tensor 1 as w ell hav e t w o construction comp onen ts, the skeletons and the gauge v ectors. The gauge v ectors are a c hoice that we can mak e and w e c hose the gauge v ectors to be the four-v elo cit y . There arose the idea of four-v elo city gauge-like transformations, b ecause w e can also gauge the tetrads with the four-v elo city plus a gradien t and see ho w the tetrads transform. These new tetrads define at ev ery point in spacetime t w o orthogonal planes. The timelike-spacelik e plane or plane one and the spacelike-spacelik e plane or plane t wo. The tetrad vectors that span these lo cal planes transform under four-v elo cit y gauge-like transformations in such a w a y that they do not leav e these planes after the transformation. In fact it w as previously pro v ed in manuscript 7 that the local group of Abelian four-v elo cit y gauge-like transformations in plane one is isomorphic to the group LB1 of tetrad transformations and in plane t w o to the group LB2 of tetrad transformations. The group LB1 is giv en b y S O (1 , 1) × Z 2 × Z 2 where S O (1 , 1) is prop er ortho chronous. The first Z 2 is given by { I 2 × 2 , − I 2 × 2 } and the second Z 2 is giv en by { I 2 × 2 , the swap (01 | 10) } . W e w ould hav e to add in order to complete the image of the map S O (1 , 1) × Z 2 × Z 2 L { l ig ht cone g aug e } where the light cone gauge includes the four solutions to the differential equations in the lo cal future and past light cones established in manuscripts 7–10 . One of these discrete transformations is the full inv ersion or min us the identit y t wo by tw o. It is a Loren tz transformation and w e designated this discrete transformation ab o ve by the notation, − I 2 × 2 . The other discrete transformation is giv en by Λ o o = 0, Λ o 1 = 1, Λ 1 o = 1, Λ 1 1 = 0, whic h is not a Lorentz transformation because it is a reflection, see reference 7 for the whole analysis. W e designated this discrete transformation ab o v e by the notation, the swap (01 | 10). The lo cal group of Abelian four-v elo cit y gauge-like 12 transformations is pro ven indep endently to be isomorphic to the lo cal group LB2 of tetrad transformations on the lo cal plane t wo and it is S O (2). W e can mention applications in relativistic astroph ysics suc h as 18 : 1. Regarding p erturbations to imp erfect fluids in cosmology w e can cite reference 18 and w e quote “W e present a new prescription for analysing cosmological p erturbations in a more general class of scalar-field dark-energy mo dels where the energy-momen tum tensor has an imp erfect-fluid form. This class includes Brans-Dick e mo dels, f ( R ) gra vit y , theories with kinetic gravit y braiding and generalised galileons. W e employ the intuitiv e language of fluids, allo wing us to explicitly maintain a dep endence on ph ysical and p oten tially measurable prop erties. W e demonstrate that hydrodynamics is not alw ays a v alid description for describing cosmological p erturbations in general scalar-field theories and present a consisten t alternative that nonetheless utilises the fluid language. W e apply this approac h explicitly to a w orked example: k-essence non- minimally coupled to gra vity . This is the simplest case which captures the essen tial new features of these imperfect-fluid mo dels. W e demonstrate the generic existence of a new scale separating regimes where the fluid is p erfect and imp erfect. W e obtain the equations for the ev olution of dark-energy densit y p erturbations in b oth these regimes. The mo del also features tw o other known scales: the Compton scale related to the breaking of shift symmetry and the Jeans scale whic h w e show is determined b y the sp eed of propagation of small scalar-field perturbations, i.e. causalit y , as op- p osed to the frequently used definition of the ratio of the pressure and energy-density p erturbations.” 2. Let us see for example a simple result from neutron star perfect fluid approximations. W e quote from reference 29 “The macroscopic neutron vorticit y ϖ n (w e use Greek letters for spacetime indices) ϖ n = q ϖ µν ϖ µν 2 where the vorticit y 2-form ϖ n is defined b y ϖ µν = ∇ µ p n ν − ∇ ν p n µ , p n µ denoting the conjugate superfluid momen tum. W e note here that, on length scales smaller than the interv ortex separation d v , typically of the order of d v ∼ n − 1 / 2 v ∼ 10 − 3 cm (see Eq. (1)), ϖ n strictly v anishes b ecause p n µ should b e lo cally proportional to the gradien t of a quan tum scalar phase. Nevertheless, on the large scales w e are in terested in here, the neutron v orticity 2-form is non-v anishing, as w ell as its corresponding scalar amplitude ϖ n ”, see references 29–41 . Since a neutron 13 star migh t accrete matter from a companion star for man y y ears the neutron-star crust migh t b e set out of its thermal equilibrium with the core. After the accretion ends the heated crust relaxes tow ards a state of equilibrium. The time of thermal relaxation dep ends sp ecially on the crust heat capacity . W ere the neutrons not superfluid they w ould be able to store so m uch heat that the thermal relaxation w ould last longer than the observed times. How ever, the thermal relaxation mechanism of these systems is not completely understo o d. F or instance, additional heat sources unkno wn so far need to b e considered in order to repro duce accurately the observ ations 42,43 . W e wonder if this discrepancy migh t b e related to the non-consideration of a vorticit y stress-energy tensor. In what follows we will consider the comp onen ts of the Ricci tensor when w e can neglect the imp erfect fluid terms in the p erturb ed equation (23) and also in equation (24). W e will only k eep the p erturb ed p erfect fluid terms plus the p erturbed v orticit y terms. W e are not sa ying that the lo cal four-v elo city gauge-lik e symmetry that w e ha ve b een studying in this manuscript is no longer v alid, rather w e consider that ev en though the symmetry still exists, for practical purp oses the p erturb ed stress- energy tensor imp erfect fluid terms can b e dismissed or neglected with resp ect to the p erturb ed p erfect fluid stress-energy terms for example. In neutron star ph ysics there is ample literature, w e just cite some pap ers and reviews where more references can b e found therein 44–53 . W e apply this new tec hnique using only lo cal algebraic co v arian t analysis whic h will not add any more substan tial computational time and how ever simplify further applications like in spacetime dynamical ev olution. The stress-energy tensor under all of the ab o v e simplifications can b e expressed b y , T µν = ( ρ + p ) u µ u ν + p g µν + T v ort µν , (38) where the v orticity stress-energy tensor is presen ted in equation (36). If w e call Q v ort = ξ µλ ξ µλ and we keep in mind that the metric tensor is given b y equation (20) w e can write the tensor (38) as, T µν = ( ρ + p ) u µ u ν + p g µν + Q v ort 2 [ − ˆ U µ ˆ U ν + ˆ V µ ˆ V ν − ˆ Z µ ˆ Z ν − ˆ W µ ˆ W ν ] . (39) W e can also chec k using the iden tit y equations (11) and (12) that, 14 ˆ U α T β α = ( ρ + p ) ( ˆ U α u α ) u β + [ p + Q v ort 2 ] ˆ U β (40) ˆ V α T β α = [ p + Q v ort 2 ] ˆ V β (41) ˆ Z α T β α = [ p + Q v ort 2 ] ˆ Z β (42) ˆ W α T β α = ( ρ + p ) ( ˆ W α u α ) u β + [ p − Q v ort 2 ] ˆ W β . (43) W e can also introduce the expression u β = − ( ˆ U α u α ) ˆ U β + ( ˆ W α u α ) ˆ W β and from equations (16-17) we kno w that ˆ V α u α = 0 and ˆ Z α u α = 0. Using these results and the equations (40-43) w e can immediately conclude that only fiv e components of the stress-energy tensor (38) or the equiv alen t (39) are not zero and this is a relev an t set of simplifications. The tetrad that w e hav e found (15-18) will b e used in order to find if there is a p ossible simplification on the left hand side of the Einstein equations indep enden tly of our stress-energy tensor pro cedure of diagonalization on the righ t hand side of the Einstein equations for p erturb ed fields. W e consider the follo wing equation that can b e found in the bibliography 22,54 , v µ ; ν ; ρ − v µ ; ρ ; ν = − R σ µν ρ v σ . (44) Equation (44) is v alid in general for an y vector field v σ . In order to v erify a particular example of simplification let us use the previous algorithm starting with equation (44) applied to the v ector (18) after con tracting this equation on b oth sides with g µρ , ˆ W µ ; ν ; µ − ˆ W µ ; µ ; ν = − R σ ν ˆ W σ . (45) Finally , we contract with vector (17), ˆ Z ν ˆ W µ ; ν ; µ − ˆ Z ν ˆ W µ ; µ ; ν = − ˆ Z ν R σ ν ˆ W σ = − ˆ Z ν ( T σ ν − 1 2 δ σ ν T µ µ ) ˆ W σ = 0 , (46) 15 since ˆ Z α T β α = [ p + Q vor t 2 ] ˆ Z β according to equation (42) and also using the orthogonality ˆ Z σ ˆ W σ = 0. Equation (46) is another source of simplification. This idea can b e rep eated with other comp onen ts as w ell for this p erturb ed formulation. W e notice that all the simplifications brough t ab out by the new tetrads in the p erturb ed case as compared to the unp erturb ed case, remain. Resuming our original analysis, it w as pro v ed that the lo cal group of Ab elian four-v elo cit y gauge-like transformations w as a symmetry of the metric tensor. Therefore, a symmetry of the Einstein equations left hand side. It was concluded that it should also b e a symmetry of the Einstein equations right hand side and a symmetry of the imp erfect fluid stress-energy tensor. It is not eviden tly a symmetry of the perfect fluid stress-energy tensor but it w as found to b e a symmetry of the imp erfect fluid stress-energy tensor. The p erfect fluid stress- energy tensor will not b e in v ariant under gauge-like four-velocity transformations by itself. In order to mak e it in v arian t w e w ould ha ve to add to the righ t hand side of the Einstein equations in (21) terms including heat flow curren ts ( ρ + p ) ( u µ q ν + u ν q µ ) where u µ q µ = 0 is satisfied and terms with viscous stresses in the fluid τ µν where u µ τ µν = 0 is also satisfied, see references 21–23 and section I I. The whole symmetry pro of of the imp erfect fluid was presen ted in man uscript 1 . In this present manuscript w e hav e dev elop ed the pro of for a source and gravitational field of imp erfect fluid under p erturbations. Once again w e m ust stress that ev en if the notation and the paper structure are similar to reference 1 the con tent is different b ecause unlik e man uscript 1 in this man uscript w e are studying imp erfect fluids under p erturbations. W e ha v e prov en the dynamic nature of lo cal symmetries and this pro of requires a similar pap er structure and notation with resp ect to reference 1 but the con tent is substantially different. The instan taneous symmetry is the ob ject of our study . W e pro ved that the lo cal orthogonal planes one and t w o tilt under p erturbations con tin uous or discrete and that the symmetries b ecome instantaneous. The whole system of ideas for the unp erturb ed imp erfect fluids, see reference 1 , w as pro v ed in this man uscript to b e reproduced for the perturb ed scenario. The key to understand the concept that w e are presenting in this manuscript is that there is for fluids with vorticit y a sector of analogous ideas as presen ted in the paper 7 for the electromagnetic and gra vitational fields in Einstein-Maxwell spacetimes. In Einstein-Maxw ell spacetimes there is a non-trivial 16 curl of the electromagnetic p oten tial four-v ector. There is manifest electromagnetic gauge in v ariance of the metric tensor as expressed in terms of tetrads of an analogous nature as to (6-9) or (15-18). The main difference b etw een the Einstein-Maxw ell spacetimes and the p erfect fluid spacetimes is the stress-energy tensor. In Einstein-Maxwell spacetimes the stress-energy tensor is in v ariant under electromagnetic gauge transformations while in p erfect fluid spacetimes the stress-energy tensor is not necessarily inv arian t under lo cal gauge-lik e transformations of the four-velocity v ectors. It is necessary to in tro duce heat flo ws and viscous stresses to make it in v ariant. W e pro ceeded in this direction in manuscript 1 where we in tro duced a new kind of lo cal transformation in the heat flux vector, the viscous stress-energy tensor, the density and pressure in order to make the whole imp erfect fluid stress-energy tensor inv ariant. In this present manuscript w e pro v ed that for p erturbed form ulations the basic constructions, prop erties and symmetries, remain instan taneously , see also 55–59 . The summary of all these findings can b e stated in the follo wing theorem, Theorem 1 The lo c al four-velo city gauge-like symmetry alr e ady found for imp erfe ct fluids with vorticity b e c omes instantane ous under p erturb ations. The lo c al planes of symmetry tilt under p erturb ations and even though the symmetries ar e c ontinuously or discr etely br oken at the p oints in sp ac etime, new symmetries arise. Ther e is a lo c al symmetry evolution. V. DECLARA TION OF INTEREST ST A TEMENT The authors declare that they ha v e no known competing financial interests or personal relationships that could ha v e app eared to influence the work rep orted in this pap er. VI. D A T A A V AILABILITY ST A TEMENT There is no data to b e rep orted in this pap er. REFERENCES 1 A. 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