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Hyperboli heat equation in Kaluza's magnetohydrodynami s A. Sandoval-Villalbazo and A. L. Gar ía-Per iante Departamento de Físi a y Matemáti as, Universidad Iberoameri ana, Prolonga ión Paseo de la Reforma 880, Méxi o D. F. 01210, Méxi o. 6 L. S. Gar ía-Colín 0 0 Departamento de Físi a, Universidad Autónoma Metropolitana-Iztapalapa, 2 Av. Purísima y Mi hoa án S/N, Méxi o D. F. 09340, n a J Méxi o. Also at El Colegio Na ional, Luis González Obregón 23, 1 1 Centro Históri o, Méxi o D. F. 06020, Méxi o. 1 (Dated: 4th February 2008) v 2 4 Abstra t 0 1 Thispaper showsthatahyperboli equation forheat ondu tion anbeobtaineddire tly 0 6 using thetenets of linear irreversible thermodynami s inthe ontext of the (cid:28)vedimensional 0 / spa e-time metri originally proposed by T. Kaluza ba k in 1922. The asso iated speed c q - of propagation is slightly lower than the speed of light by a fa tor inversely proportional r g : to the spe i(cid:28) harge of the (cid:29)uid element. Moreover, onsisten y with the se ond law of v i X thermodynami s is a hieved. Possible impli ations in the ontext of physi s of lusters of r a galaxies of this result are brie(cid:29)y dis ussed. 1 I. INTRODUCTION It is today a well established fa t that knowledge of magnetohydrodynami s (MHD) is essential in our understanding of a vast number situations whi h o ur in astrophysi s [1℄. Less known is the relationship whi h exists between the basi equations of MHD with non-equilibrium thermodynami s, spe ially in the ase in whi h one wishes to in lude dissipative e(cid:27)e ts into the s heme. Four years ago, two of us [2℄ wrote a paper showing learly that when this is the ase, the lassi al non-relativisti stru ture of linear irreversible thermodynami s (LIT) restri ts the onstitutive equations relating the ele tri al urrent to an ele tri (cid:28)eld to the well- known Ohm's equation when the se ond law is satis(cid:28)ed. Non-ohmi e(cid:27)e ts an thus be in luded by attempting an extension of LIT to in lude su h ases, but a theory with these hara teristi s whi h is also at grips with the se ond law of thermodynami s does not yet exists [3℄. The other alternative is to approa h the problem by in orporating the e(cid:27)e ts of the ele tromagneti (cid:28)eld using a relativisti multidimensional theory, a step that also brings the theory to the framework of general relativity. Many multidimensional theories have been formulated sin e Einstein's idea of unifying nature's fundamental for es in a single theory was put forward. One of the pioneers in this topi was the mathemati ian T. Kaluza [4℄. Although he was not the (cid:28)rst one to propose an extended metri , his idea of adding extra elements to 4 × 4 the general relativity spa e-time an be onsidered as the origin of a variety of multidimensional theories that has been proposed in the last (cid:28)ve de ades. In this work we formulate the thermo-hydrodynami s for an ionized (cid:29)uid in the ontext of Kaluza's original (cid:28)ve-dimensional theory. This framework, whi h does not a ount for weak and strong intera tions, has su(cid:30) ient information to in lude gravitation and ele tromagnetism. In Ref.[2℄ it was shown how the urvature arising from the extra dimension a - 2 ounts for the ele tromagneti e(cid:27)e ts in the hydrodynami equations. This was a hieved by using Meixner's theory of non-equilibrium thermodynami s [5, 6, 7℄ and 5 × 5 the metri proposed by Kaluza. In that work the entropy produ tion was al ulated and it was shown how, using this formalism, the onstitutive equation for the ele tri urrent an be other than Ohm's law, whi h as we know is not always valid. However, the ausality issue in the heat equation was not addressed. In this work we omplete the al ulation in Ref.[2℄ and add a new fundamental ξ element in Kaluza's magnetohydrodynami s. A parameter is introdu ed in the metri whi h ultimately adds non-negligible orre tions in the transport equations. The ausality issue in the transport equations is also addressed by al ulating the heat equation, whi h is shown to be of the hyperboli type, and thus does not allow heat waves propagating with a speed larger than the speed of light. Se tion II is a brief summary of Kaluza's formalism. In Se t.III we establish the entropy produ tionandtheheat equationshowinghow ausalityisnotviolatedifthe (cid:28)ve dimensionalformalismis onsidered. This is a omplishedby using the standard hµ ν proje tor [8℄ whi h eliminates the somewhat deli ate on ept of dissipation in time [9℄. A summary and a brief dis ussion of the results here obtained is in luded in Se t.IV. In Appendix A some relevant details of Kaluza's theory are shown ξ in luding the relevan e of the parameter . II. KALUZA'S FORMALISM The theory formulated by T. Kaluza [4℄, uni(cid:28)es general relativityand ele tromagnetism by extending the 4-dimensional manifold into a 5-dimensional spa e-time. In this metri , the (cid:28)fth olumn ontains the omponents of the ele tromagneti 3 potentials: A ξ 1 A ξ  2 g = A ξ , µ5  3  (1)    φξ   c     1      A φ µ where are the omponents of the ve tor potential and is the ele trostati s alar ξ potential. The onstant is a parameter whose value is set equal to 16πGǫ 0 ξ = , c2 (2) r φ by mat hing Einstein's (cid:28)eld equation to a Poisson equation for , in the non- relativisti limit, as is shown in Appendix A. For simpli ity we will negle t the magneti (cid:28)eld and assume that the 4-manifoldis given by a quasi-Minkowski metri . ψ That is, if the gravitational potential is , the metri tensor an be written as 1 0 0 0 0 0 1 0 0 0  g = 0 0 1 0 0 , µν   (3)   0 0 0 −1− 2ψ −φξ   c2 c    0 0 0 −φξ 1   c    and 1 0 0 0 0 0 1 0 0 0  gµν = 0 0 1 0 0 ,   (4)   0 0 0 −1+ 2ψ −φξ   c2 c    0 0 0 −φξ 1   c    4 1/c3 negle ting terms of order and lower. In this framework, the Christo(cid:27)el symbols are proportional to the omponents of the for es and dx5 q = , dt mξ (5) so that the equation of motion for a harged parti le in a gravitational and ele tri (cid:28)eld orresponds to a geodesi in this spa e-time as shown in Appendix A. It is also required that quantities have no variation in the (cid:28)fth dimension by setting ∂/∂x5 = 0 . This is alled the (cid:16) ylindri al ondition(cid:17) [4℄ and follows the idea that the extra dimension is losed on itself as is now assumed in some multi-dimensional theories. III. HEAT CONDUCTION INKALUZA'S MAGNETOHYDRODYNAMICS As shown in Ref.[2℄, the MHD equations an be obtained within the (cid:28)ve- dimensional theory. The e(cid:27)e ts of the ele tri al for e arise in a natural way from the urvature of spa e-time. No external for es are introdu ed sin e the omponents of the ele tromagneti (cid:28)eld are in luded in the equations through the Christo(cid:27)el symbols in the ovariant derivatives (see Appendix A). n The onservation of parti les is una(cid:27)e ted by the (cid:28)fth dimension. That is, if is the lo al parti le density n˙ +nθ = 0, (6) θ = uµ u5 where ;µ is the divergen e of the velo ity for whi h the last term, ,5, vanishes due to the ylindri al ondition. For simpli ity, we will assume that the al ulations θ = 0 are performed in the omoving frame. Thus, and the parti le onservation law n˙ = 0 is given by . 5 The stress tensor within this framework is given by [2℄ Tµ = ρuµu +phµ +Ξµ, ν ν ν ν (7) ρ p Ξµ ν where is the mass-energy density, the hydrostati pressure and is the vis ous hµ ν stress tensor. The spatial proje tor is de(cid:28)ned in the standard way so that uνhµ = 0, ν (8) and, as will be seen, u Ξµ = 0. µ ν (9) This is a hieved by having 1 hµ = δµ + u uµ, ν ν α2 ν (10) α2 = c2−u u5 5 where . Some authorsin lude a heat terminthe de(cid:28)nitionof thestress tensor followingE kart's formalism [8℄. In Meixner's theory, heat is viewed as a non- me hani al form of energy and is in luded in a separate equation. A dis ussion on the di(cid:27)eren e between both formalisms and a possible way of dis erning the orre t way to treat heat in general relativity via a light s attering experiment an be found in Refs.[9, 10℄. In Meixner's thermodynami s, one onsiders the total energy (cid:29)ux: Jµ = uνTµ +ne uµ +Jµ , [T] ν int [Q] (11) uνTµ e uµ ν int where and are the me hani al and internal energy (cid:29)uxes respe tively Jµ Jµ = 0 and [Q] is the heat (cid:29)ux. Total energy onservation requires [T];µ whi h, using the form of the stress tensor given by Eq.(7), yields (uνTµ) = uν Ξµ. ν ;µ ;µ ν (12) Tµ = 0 To obtain Eq.(12) we used that, a ording to Einstein's (cid:28)eld equations, ν;µ and uµu = α2 (uµu ) = 0 that µ is a onstant. Thus, in the omoving frame µ ;ν . Hen e, the 6 balan e equation for the internal energy is ne˙ = −uν Ξµ −Jµ . int ;µ ν [Q];µ (13) The ele tri ontribution to the internal energy is, as mentioned above, ontained in the ovariant derivatives of Eq.(13). It is pertinent to point out that Eq.(13) would di(cid:27)er from its E kart(cid:1)s type ounterpart by a term proportional to the hydrodynam- i al a eleration. In both ases, in the omoving frame, both formalisms would lead to Eq.(13). A. Entropy produ tion In Meixner's theory, the entropy produ tion is obtained by using the lo al equilibrium assumption [5, 6℄ namely s = s(n, e ), int (14) n˙ = 0 for whi h, sin e , one an write n ns˙ = e˙ , int T (15) or, introdu ing Eq.(13), Jµ 1 Jµ ns˙ + [Q] = − uν Ξµ − [Q]T . T ! T ;µ ν T2 ,µ (16) ;µ One an identify the right side of Eq.(16) with Clausius' un ompensated heat or the entropy produ tion whi h, a ording to the se ond law of thermodynami s is required to be non-negative: Jµ − [Q]T −uν Ξµ ≥ 0. T2 ,µ ;µ ν (17) 7 Thisrequirementsuggests onstitutiveequationsasexpressionsofea hsour etermin Eq.(16) as a produ t of a thermodynami for e andits orresponding (cid:29)ux. Following this reasoning we write a Fourier law for the heat (cid:29)ux: Jµ = −κhµT,ν. [Q] ν (18) In E kart's theory a se ond term, proportional to the a eleration, is in luded in ui = 0 Eq.(18) whi h is not present here. This is not only be ause in the omoving frame but primarily sin e, as mentioned above, the heat (cid:29)ux is treated separately in Meixner's formalism [10℄. Anyway, both formalisms yield the same expressions for the entropy produ tion, heat (cid:29)ux and transport equations in the omoving frame. In order to guarantee that the se ond term in Eq.(17) is positive we write Ξµ = −ησµ, ν ν (19) η where isa(cid:16)vis osity(cid:17) oe(cid:30) ient. Equation(19)relatesthevis ousstresstensorwith σµ ν the symmetri alpart of the (tra eless) velo ity gradient . We want to remark that Eq.(19) in the omoving frame is equivalent to the standard onstitutive equation Ξµ = −ηhµhβσα σµ ν α ν β ν . Bulk vis osity has been negle ted. A al ulation for in the omoving frame leads to (see Appendix A): 1 q 1 σi = ui = u4 = ψ + φ σi = ui = −u5 = ξφ , 4 ;4 ;i c ,i 2m ,i and 5 ;5 ;i 2 ,i (20) (cid:16) (cid:17) i = 1, 2, 3 for and the rest of the oe(cid:30) ients vanish. Sin e in Einstein's equation, G µν is symmetri al, the vis ous stress tensor does not have antisymmetri al omponents and thus we an assume that the oupling onstant whi h would orrespond to a rotational vis osity vanishes. Then, Ξi = 0 i, j = 1, 2, 3, 5, j for (21) and η q Ξ4 = − ψ + φ . i c ,i 2m ,i (22) (cid:16) (cid:17) 8 When these onstitutive equations are substituted in Eq.(16), the entropy balan e equation reads Jµ T,νT 1 q q ns˙ + [Q] = κhµ ,µ +η ψ + φ ψ,i + φ,i . T ! ν T2 c2 ,i 2m ,i 2m (23) ;µ (cid:16) (cid:17)(cid:16) (cid:17) These ondtermintherightsideofEq.(23)isduetothe(cid:28)eldspresent intheproblem and is obtained here as a onsequen e of the urvature of spa e-time. B. Heat equation The heat equation is obtained by using the lo al equilibrium assumption for the internal energy ∂T ne˙ = nC , int v ∂t (24) together with Eq.(13) and the orresponding onstitutive equations given by Eqs.(19) and (22). The (cid:28)rst three terms in the divergen e of the heat (cid:29)ux give the standard result Ji = −κ∇2T i = 1, 2, 3. [Q] ;i for (25) (cid:0) (cid:1) Note that, in the standard theory, the heat (cid:29)ux only ontains the three terms in Eq.(25) whi h, when introdu ed in Eq.(24), yields a paraboli heat equation. How- ever, in the (cid:28)ve-dimensional theory, the fourth omponent of the heat (cid:29)ux is nonzero retaining its onsisten y with the orthogonality ondition in Eq.(8). That is, sin e h4 = 0 i = 1, 2, 3 i for (26) c2 h4 = 1− , 4 α2 (27) T,5 = 0 and by means of the ylindri al ondition , the fourth omponent of the heat (cid:29)ux is 1 J4 = −κ 1− T,4, [Q] 1−δ2 (28) (cid:18) (cid:19) 9 δ2 = 1 q2 where 16πǫ0Gm2. Taking the ovariant derivative in Eq.(28) leads to 1 κ 1 ∂2T J4 = −κ 1− T,4 = 1− −ψ T,i . [Q] ;4 1−δ2 ;4 c2 1−δ2 ∂t2 ,i (29) (cid:18) (cid:19) (cid:18) (cid:19)(cid:18) (cid:19) (cid:0) (cid:1) (cid:0) (cid:1) ∇ψ ·∇T The nonlinear term ouples the gravitational (cid:28)eld to the temperature gra- 1/c4 dient. It is worth mentioning that there is also a oupling of order between the ele trostati (cid:28)eld and the temperature gradient whi h onsistently has been negle ted. An analysis of the onsequen es of these ouplings is an interesting topi whi hwillbeaddressed inthefuture. Withtheseextraterms,Eq.(24) anbewritten as ∂T 1 1 ∂2T η 1 q q D = ∇2T− 1− −ψ T,i + ψ + φ ψ,i + φ,i , T ,i ,i ,i ∂t c2 1−δ2 ∂t2 κc2 2m 2m (cid:18) (cid:19)(cid:18) (cid:19) (cid:16) (cid:17)(cid:16) (cid:17) (30) D = nC /κ T v where is the thermal di(cid:27)usivity. Equation (30) is a hyperboli , and η thus ausal, heat equation. Sin e the oe(cid:30) ient plays the role of a vis osity, the fa tor that multiplies the dissipative sour e on the third term on the right side an be interpreted as a Prandtl number that quanti(cid:28)es the ratio between vis ous and thermal dissipations. Di(cid:27)usion dominates for small values of this parameter and, assuming the (cid:29)uid is neutral, the standard heat equation is re overed. Otherwise, ǫ G 0 sin e is a very small quantity, even a very small net spe i(cid:28) harge will yield δ2 > 1 1/ξ . That is, be ause of the large value of the orre tions due to the ele tri q/m harge of matter, and thus to the (cid:28)fth dimension, are always signi(cid:28) ant unless is identi ally zero. The speed of propagation for the wave-like solution of Eq.(30) is lower than the c˜ δ2 ≫ 1 speed of light. If is the modi(cid:28)ed speed and sin e we obtain 1 c˜2 = c2 1− . δ2 (31) (cid:18) (cid:19) As an example, onsider an element of (cid:29)uid purely onstituted by ele trons. In that 10