Spin Fluids in Homogeneous and Isotropic Space-times

We consider a Weyssenhoff fluid assuming that the space-time is homogeneous and isotropic, therefore being relevant for cosmological considerations of gravity theories with torsion. It is explicitly shown that theWeyssenhoff fluids obeying the Frenkel condition or the Papapetrou–Corinaldesi conditio...

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Дата:	2010
Автори:	Böhmer, C.G., Bronowski, P.
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Мова:	English
Опубліковано:	Відділення фізики і астрономії НАН України 2010
Назва видання:	Український фізичний журнал
Теми:	Астрофізика і космологія
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Цитувати:	Spin Fluids in Homogeneous and Isotropic Space-times / C.G. Böhmer, P. Bronowski // Український фізичний журнал. — 2010. — Т. 55, № 5. — С. 607-612. — Бібліогр.: 17 назв. — англ.

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spelling	nasplib_isofts_kiev_ua-123456789-562052025-02-09T22:32:47Z Spin Fluids in Homogeneous and Isotropic Space-times Спінові рідини в однорідному та ізотропному просторі-часі Böhmer, C.G. Bronowski, P. Астрофізика і космологія We consider a Weyssenhoff fluid assuming that the space-time is homogeneous and isotropic, therefore being relevant for cosmological considerations of gravity theories with torsion. It is explicitly shown that theWeyssenhoff fluids obeying the Frenkel condition or the Papapetrou–Corinaldesi condition are incompatible with the cosmological principle which restricts the torsion tensor to have only the vector and axial vector components. Moreover, it turns out that the Weyssenhoff fluid obeying the Tulczyjew condition is also incompatible with the cosmological principle. This condition has not been analyzed so far in this context. Based on this result, we propose to reconsider a number of previous works that analyzed cosmological solutions of the Einstein–Cartan theory, since their spin fluids did not obey usually the cosmological principle. Розглянуто рiдину Вiссенхофа у припущеннi, що простiр-час є однорiдним та iзотропним i тому придатний для космологiчного розгляду теорiй гравiтацiї з крученням. Показано, що рiдини Вiссенхофа, для яких виконується умова Френкеля або умова Папапетроу–Корiналдезi, є несумiсними з космологiчним принципом, згiдно з яким тензор кручення має тiльки векторну i аксiальну векторну компоненти. Бiльше того, виявилося, що рiдина Вiссенхофа, що задовольняє умову Тульчiєва, також несумiсна з космологiчним принципом. Але цю умову не було проаналiзовано повнiстю в цьому контекстi. Ґрунтуючись на цих результатах, запропоновано переглянути деякi попереднi роботи, в яких проаналiзовано космологiчнi розв’язки теорiї Ейнштейна–Картана, оскiльки їхнi спiновi рiдини звичайно не задовольняють космологiчний принцип. This invited submission is a part of the Project of Scientific Cooperation between the Austrian Academy of Sciences ( OAW) and the National Academy of Sciences of Ukraine (NASU) No. 01/04, Quantum Gravity, Cosmology, and Categorification. The work of CGB was supported by research grant BO 2530/1-1 of the German Research Foundation (DFG). Among the other forms of support, the author is especially grateful to the Austrian Academy of Sciences which in the framework of the collaboration with the National Academy of Sciences of the Ukraine co-financed the multilateral research project “Quantum Gravity, Cosmology and Categorification” and which also supported his travel expenses to Ukraine. The author also could have not succeeded in pursuing this program for many years without the collaborations with Prof. W. Kummer. 2010 Article Spin Fluids in Homogeneous and Isotropic Space-times / C.G. Böhmer, P. Bronowski // Український фізичний журнал. — 2010. — Т. 55, № 5. — С. 607-612. — Бібліогр.: 17 назв. — англ. 2071-0194 PACS 04.40.-b; 04.40.+h https://nasplib.isofts.kiev.ua/handle/123456789/56205 en Український фізичний журнал application/pdf Відділення фізики і астрономії НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
topic	Астрофізика і космологія Астрофізика і космологія
spellingShingle	Астрофізика і космологія Астрофізика і космологія Böhmer, C.G. Bronowski, P. Spin Fluids in Homogeneous and Isotropic Space-times Український фізичний журнал
description	We consider a Weyssenhoff fluid assuming that the space-time is homogeneous and isotropic, therefore being relevant for cosmological considerations of gravity theories with torsion. It is explicitly shown that theWeyssenhoff fluids obeying the Frenkel condition or the Papapetrou–Corinaldesi condition are incompatible with the cosmological principle which restricts the torsion tensor to have only the vector and axial vector components. Moreover, it turns out that the Weyssenhoff fluid obeying the Tulczyjew condition is also incompatible with the cosmological principle. This condition has not been analyzed so far in this context. Based on this result, we propose to reconsider a number of previous works that analyzed cosmological solutions of the Einstein–Cartan theory, since their spin fluids did not obey usually the cosmological principle.
format	Article
author	Böhmer, C.G. Bronowski, P.
author_facet	Böhmer, C.G. Bronowski, P.
author_sort	Böhmer, C.G.
title	Spin Fluids in Homogeneous and Isotropic Space-times
title_short	Spin Fluids in Homogeneous and Isotropic Space-times
title_full	Spin Fluids in Homogeneous and Isotropic Space-times
title_fullStr	Spin Fluids in Homogeneous and Isotropic Space-times
title_full_unstemmed	Spin Fluids in Homogeneous and Isotropic Space-times
title_sort	spin fluids in homogeneous and isotropic space-times
publisher	Відділення фізики і астрономії НАН України
publishDate	2010
topic_facet	Астрофізика і космологія
url	https://nasplib.isofts.kiev.ua/handle/123456789/56205
citation_txt	Spin Fluids in Homogeneous and Isotropic Space-times / C.G. Böhmer, P. Bronowski // Український фізичний журнал. — 2010. — Т. 55, № 5. — С. 607-612. — Бібліогр.: 17 назв. — англ.
series	Український фізичний журнал
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fulltext	ASTROPHYSICS AND COSMOLOGY ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 5 607 SPIN FLUIDS IN HOMOGENEOUS AND ISOTROPIC SPACE–TIMES∗ C.G. BÖHMER,1, 2 P. BRONOWSKI3 1ASGBG/CIU, Department of Mathematics, University of Zacatecas (UAZ) (Apartado Postal C-600, Zacatecas, Zac 98060, Mexico; e-mail: boehmer@ hep. itp. tuwien. ac. at ) 2Department of Mathematics and Institute of Origins, University College London (Gower Street, London, WC1E 6BT, United Kingdom; e-mail: c. boehmer@ ucl. ac. uk ) 3Institute for Theoretical Physics, University of Lodz (Pomorska 149/153, PL-90-236 Lodz, Poland; e-mail: pbronowski@ poczta. fm ) PACS 04.40.-b; 04.40.+h c©2010 We consider a Weyssenhoff fluid assuming that the space-time is homogeneous and isotropic, therefore being relevant for cosmolog- ical considerations of gravity theories with torsion. It is explicitly shown that the Weyssenhoff fluids obeying the Frenkel condition or the Papapetrou–Corinaldesi condition are incompatible with the cosmological principle which restricts the torsion tensor to have only the vector and axial vector components. Moreover, it turns out that the Weyssenhoff fluid obeying the Tulczyjew condition is also incompatible with the cosmological principle. This condition has not been analyzed so far in this context. Based on this result, we propose to reconsider a number of previous works that analyzed cosmological solutions of the Einstein–Cartan theory, since their spin fluids did not obey usually the cosmological principle. 1. Introduction Over the last years, cosmology has become a very active field of research containing many open questions that re- quire the further investigation. Hawking, Penrose, and others have shown that, under fairly general assump- tions, solutions of Einstein’s field equations evolve singu- larities. This, from a conceptual point of view, is rather unsatisfactory, we refer the reader to [1]. When cosmological models with torsion were first studied, it was hoped for that the inclusion of torsion would help to avoid these singularities. Unfortunately, ∗ This invited submission is a part of the Project of Scientific Cooperation between the Austrian Academy of Sciences (ÖAW) and the National Academy of Sciences of Ukraine (NASU) No. 01/04, Quantum Gravity, Cosmology, and Categorification. this could only be achieved assuming quite unrealistic matter models (see, e.g., [2]). It turns out, however, that most of these cosmological models with torsion did not satisfy the cosmological principle, sometimes also known as the Copernican principle, that strongly restricts the metric and the torsion tensor. The Copernican principle states that the Universe is spatially homogeneous and isotropic on very large scales. This principle takes the following mathematically pre- cise form. The four-dimensional 4d space-time manifold (M, g) is foliated by 3d constant time space-like hyper- surfaces which are the orbits of a Lie group G acting on M, with the isometry group SO(3). Following the Copernican principle [3], we assume all fields to be in- variant under the action of G Lξgµν = 0 , LξTλµν = 0 , (1) where ξ are the (six) Killing vectors generating the space-time isometries. The metric tensor is denoted by gµν , Tλµν denotes the torsion tensor, and Greek indices label the holonomic components. For the rest of the pa- per, only anholonomic components of tensors labelled by Latin indices are used. Kopczyński initiated the investigation of cosmologi- cal models with torsion in [4] and [5], who assumed a Weyssenhoff fluid to be the source of both curvature and torsion. In [4], a non-singular universe with torsion was constructed and an anisotropic model of the universe with torsion was analyzed in [5]. The cosmological prin- ciple in the above strict sense (1) was first developed in the Einstein–Cartan theory by Tsamparlis in [3], where it C.G. BÖHMER, P. BRONOWSKI was also suggested to reconsider the results in [4,5], since the Weyssenhoff fluid turns out to be incompatible with the cosmological principle (see also [6]). The spin tensor used by [5] in a cosmological context had just one non- vanishing component S23 = K, where K was assumed to be a function of the time variable K = K(t). Such a spin tensor, as we will show below, is not compatible with the cosmological principle, a fact that was noted by Kopczyński. It should also be pointed out that if we require only the metric of Friedman–Robertson–Walker (FRW) type and put no restrictions on the torsion ten- sor, then the Weyssenhoff fluid can consistently be used as a source for curvature and torsion (see the energy- momentum tensor Eq. (5.2) and (5.3) in [6]). However, by doing so, one must drop the second condition of (1), LξTλµν = 0, and use a weaker notion of the cosmologi- cal principle. For a recent example where the so-called cosmological model with macroscopic spin fluid was an- alyzed, see [7]. Applying the restrictions (1) to gµν yields the FRW- type metric ds2 = dt2 − ( a(t) 1 + k 4 r 2 )2 (dx2 + dy2 + dz2), (2) where r2 = x2 + y2 + z2, and the 3-space is spherical for k = 1, flat for k = 0, and hyperbolic for k = −1. If we impose the restrictions (1) on the torsion tensor [3], its allowed components are Txxt = Tyyt = Tzzt = h(t) , Txyz = Tzxy = Tyzx = f(t) , (3) where we follow the notation of [8]. Hence, the cos- mological principle allows a vector torsion component Tt = T aat = 3h, along the world lines, and an axial vec- tor component Txyz = fεxyz, within the hypersurfaces of constant time. Such a totally skew-symmetric tor- sion tensor in cosmology was considered earlier in [9], where h = 0 was assumed. The geometry parame- ter k was redefined to include the remaining torsion by k̄ = k− f2a2/2. Also with h = 0, the cosmological infla- tion could be explained by torsion in [10], using a rough model. 2. Cosmological Field Equations The spin-connection 1-form ω̃ij in theories with torsion can be split into a torsion-free part (the usual spin- connection 1-form ωij related to the Christoffel symbol Γkij) and a contortion 1-form part Ki j that takes the torsion of space-time into account: ω̃ij = ωij +Ki j . (4) Here, the torsion tensor (Cartan’s torsion) and the con- tortion tensor are related by the following algebraic re- lation T i = 1 2 T ijke i ∧ ej = Dei = dei + ω̃ije j = Ki j ∧ ej , (5) where we used that ωij is a torsion-free connection. The latter relation between torsion and contortion also im- plies that their vector and axial vector components are simply related by T[ijk] = K[ijk] , Tij j = 1 2 Kji j . (6) Since the metric and the contortion (or torsion) compo- nents that are compatible with the cosmological constant are fixed, one can compute any geometrical quantity of interest. Metric (2) gives rise to the basis 1-forms et = dt , ex,y,z = a(t) 1 + k 4 r 2 dx, y, z , (7) which together with the non-vanishing torsion compo- nents (4) yield the non-vanishing connection 1-forms ωtx = ȧ a ex + hex , ωty = ȧ a ey + hey , ωtz = ȧ a ez + hez , ωxy = ky 2a ex − kx 2a ey − f 2 ez , ωxz = kz 2a ex − kx 2a ez + f 2 ey , ωyz = kz 2a ey − ky 2a ez − f 2 ex (8) that are computed from (4); T i = dei + ω̃ije j , where the torsion two form can be obtained from (4) via T i = (1/2)T ijkej ∧ ek. The field equations of the Einstein– Cartan theory [2] are obtained by varying the usual Einstein–Hilbert action with respect to the vielbein and the spin-connection as independent variables Rij − 1 2 Rδij = 8πΣij , T ijk − δijT llk − δikT ljl = 8π sijk , (9) Σij is the canonical energy-momentum tensor, and sijk is the spin tensor. 608 ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 5 SPIN FLUIDS IN HOMOGENEOUS 3. Weyssenhoff Fluid The ideal Weyssenhoff fluid [11] is a generalization of the ideal fluid to take into account the properties of spin and torsion in space-time. Its canonical energy-momentum tensor is given by Σij = piuj + P (uiuj + gij) , pi = ρui − ul∇k(ukSli) , (10) sijk = uiSjk , (11) where pi is the momentum density of the fluid, and ui is the fluid’s velocity. By ρ and P, we denoted the en- ergy density and the pressure of the fluid, respectively. The intrinsic angular momentum tensor Sij satisfies the relation Sij = −Sji . (12) The spin tensor Sij can be decomposed into two 3- vectors µ := (S01, S02, S03) , (13) that, in case we assume the Frenkel condition [12], van- ishes in the rest-frame. The second vector σ := (S23, S31, S12) (14) in the rest-frame can be regarded as the spin density. Integrability of the particles’ equations of motion re- quires one more condition that the spin tensor has to satisfy, ζiSji = 0 , (15) where the vector ζi is usually taken to be the velocity vector of the fluid ui, following Frenkel [12]. It is also possible to choose the momentum density according to Tulczyjew [13]. Another frequently used condition was put forward by Papapetrou and Corinaldesi [14] who as- sumed the condition Sti = 0 , (16) where t stands for the time component of the spin ten- sor. In the following sections, we investigate whether a Weyssenhoff fluid obeying one of the three presented integrability conditions is compatible with the cosmolog- ical principle. 4. Frenkel Condition If we assume the Frenkel condition [12] ζi = ui, then the spin contribution of the energy-momentum tensor can be rewritten as ul∇k(ukSli) = ulSli∇kuk + uluk∇kSli = = uluk∇kSli = −alSli . (17) In the third and fourth steps, the Frenkel condition was necessary for the modifications, and we introduced the acceleration of the fluid aj defined by aj = (uk∇k)uj . Hence, Eqs. (10) and (11) with regard for the Frenkel condition yield Σij = ρuiuj − P (uiuj + gij) + alSliuj , sijk = uiSjk , uiSik = 0 . (18) This implies that, at the zero acceleration aj of a fluid, one is back at the Einstein gravity [15]. The interpreta- tion of the contribution of the spin angular momentum tensor in (10) in terms of the acceleration strongly de- pends on the Frenkel condition. The totally skew-symmetric part of the torsion ten- sor (3) is allowed by the cosmological principle. Since the four velocity ui enters the definition of the spin tensor (11), a Weyssenhoff-like fluid cannot be the source of the totally skew-symmetric torsion component (3). On the other hand, it is the Frenkel condition (15) with ζi = ui that does not allow the Weyssenhoff fluid to be a source of the trace components (3) of the torsion tensor. More explicitly, multiplying Eq. (9b) for the torsion field by δji leads to −2T llk = 8πsllk = 8πulSlk = 0 , (19) where (11) and the Frenkel condition were taken into account for the last steps. Therefore, we have explicitly shown that the Weyssenhoff fluid obeying the Frenkel condition is incompatible with the cosmological principle put forward in [3]. 5. Papapetrou–Corinaldesi Condition Assuming the Papapetrou–Corinaldesi [14] condition Stj has the following consequences for the torsion tensor im- plied by the spin fluid. As before, since the fluid’s four ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 5 609 C.G. BÖHMER, P. BRONOWSKI velocity enters the definition of a spin tensor, the to- tally skew-symmetric torsion has to vanish. Second, the traced torsion field equation (9b) yields −2T llk = 8πsllk = 8πulSlk = 8πδltSlk = 8πStk = 0 , (20) where the vanishing of the trace part of the torsion tensor is identically the condition of Papapetrou–Corinaldesi (16). Therefore, we again conclude that also the Weyssen- hoff fluid obeying the Papapetrou–Corinaldesi condition is incompatible with the cosmological principle (since we have ui = δi0, the Papapetrou–Corinaldesi condition and the Frenkel one in fact take the same form). 6. Tulczyjew Condition According to our information, it has not been analyzed so far if the Weyssenhoff fluid obeying the Tulczyjew condition [13] is compatible with the cosmological prin- ciple. As in the previous section, the fluid cannot be a source of the totally skew-symmetric component of the torsion tensor, because the four velocity ui of the fluid is present in Eq. (11). However, in this case, (15) does not vanish identically on general ground, and we arrive at −2T llk = 8πsllk = 8πulSlk , (21) where the last term on the right-hand side need not to vanish. The Tulczyjew condition, Eq. (15) with ζi = pi explicitely written out, leads to Sijp j = Sij ( ρuj − ul∇k(ukSlj) ) = 0 , (22) which can be used to express the last term on (21) by ρSiju j = Sijul∇k(ukSlj) . (23) The fluid’s four velocity simply reads uj = δj0, and Eq. (23) yields, by taking (8) into account, the following form of the Tulczyjew condition that the spin fluid has to satisfy: ρSi0 = Sij ( Ṡ0j + Γ0S0j ) , (24) where the dot means the differentiation with respect to time t. In contrast to the two previous cases, we find that this condition does not imply the vanishing of the resulting trace of the torsion tensor. For the trace of the Christoffel symbol, we find Γ0 = Γkk0 = 3(H+h), where, by H, we denoted the Hubble parameter defined by H = ȧ/a. Equations (24) provide us with for conditions (i = 0, 1, 2, 3) which we will analyze in more details. For i = 0, the left-hand side of (24) vanishes, and one is left with 0 = S0j ( Ṡ0j + Γ0S0j ) = −S0j ( Ṡ0j + Γ0S0j ) , (25) which can be written in an equivalent form by using the introduced vectors µ and σ in (13) and (14) and leads to 0 = ( µ̇ + Γ0 µ ) · µ , (26) where · means the usual inner product of vectors. From this, we conclude that the vectors µ and (µ̇ + Γ0µ) are orthogonal to each other. For the remaining values i = 1, 2, 3, condition (24) takes the following form in terms of the three-vector: µ = 1 ρ ( µ̇ + Γ0 µ ) × σ . (27) Here, we now see that (26) is not an independent equa- tion since we derive from the latter that µ ⊥ (µ̇ + Γ0µ) and, moreover, µ ⊥ σ. Therefore, we find the following non-vanishing components of the induced torsion tensor via the field equations (21): T ll0 = 0 , T lli = 4πµ , (28) where µ is given by Eq. (27). Note that the index i only takes the values 1, 2, 3, and we furthermore suppressed the explicit index for the vector µ. We can now try to continue the construction of a spin fluid that is compati- ble with cosmological principle, namely the Weyssenhoff fluid obeying the Tulczyjew condition. In principle, we have two possibilities: (a) we choose the vector µ of the spin tensor so that it satisfies condition (26), and we choose the spin density vector σ so that Eq. (27) is sat- isfied. On the other hand, (b) allows us to prescribe the spin density vector σ. Then, in order to get an allowed µ, one has to solve the vector differential equation (27), so that each solution satisfies (26). However, one must be careful with the above result. The cosmological prin- ciple allows a vector torsion component, but only along the world lines, T llt 6= 0. The other components are excluded. Therefore, also the Weyssenhoff fluid obeying the Tulczyjew condition is incompatible with the cosmo- logical principle, since (28a) vanishes identically. 610 ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 5 SPIN FLUIDS IN HOMOGENEOUS 7. Conclusions and Outlook The restrictions that follow from assuming the homo- geneity and the isotropy on the very large scales of the Universe (the cosmological principle) allow one met- ric component and two torsion components; the vector and axial vector components of the torsion tensor. We showed that the Weyssenhoff fluids obeying either the Frenkel or the Papapetrou–Corinaldesi condition are in- compatible with this principle. Furthermore, we ana- lyzed the Tulczyjew condition which, in principle, al- lows one to construct a non-trace-free torsion tensor. However, its time component, allowed by the cosmo- logical principle, vanishes identically. Therefore, it has been shown that no spin fluid obeying the common integrability conditions is compatible with the cosmo- logical principle. This rather surprising result shows the necessity to reconsider the previous works on cos- mology with torsion, since none of these results can be regarded as a truly cosmological model with tor- sion. Furthermore, it raises the question, whether an inte- grability condition exists that allows a spin fluid to have homogeneous and isotropic torsion components. The construction of such a spin fluid, if possible, could be a subject of the further research. If it will turn out that such a spin fluid does not exist, this would have quite significant consequences for the physical applicability of such models. The possible non-existence would indicate that the Weyssenhoff fluid is not a very good model for a macroscopic spin fluid. If, on the other hand, such a cosmological spin fluid can be constructed, it would be very interesting to study its properties. For example, the consequences of a truly cosmological spin fluid on the singularities, mentioned in the introduction, were worth a thorough investigation. Moreover, it would then be possible to reconsider some of the previously suggested models in a real cosmological fashion. Finally, we would like to mention the possibility of applying the cosmo- logical principle to the more general hyperfluid [16, 17]. However, a axial vector component for the torsion tensor cannot be obtained from the hyperfluid, since a gener- alized form of Eq. (11) essentially enters the spin ten- sor. The work of CGB was supported by research grant BO 2530/1-1 of the German Research Foundation (DFG). Among the other forms of support, the author is es- pecially grateful to the Austrian Academy of Sciences which in the framework of the collaboration with the Na- tional Academy of Sciences of the Ukraine co-financed the multilateral research project “Quantum Gravity, Cosmology and Categorification” and which also sup- ported his travel expenses to Ukraine. The author also could have not succeeded in pursuing this pro- gram for many years without the collaborations with Prof. W. Kummer. 1. S.W. Hawking and G.F.R. Ellis, The Large Scale Struc- ture of Space-Time (Cambridge University Press, Cam- bridge, 1973). 2. F.W. Hehl, P. von der Heyde, G.D. Kerlick, and J.M. Nester, Rev. Mod. Phys. 48, 393 (1976). 3. M. Tsamparlis, Phys. Lett. A 75, 27 (1979). 4. W. Kopczyński, Phys. Lett. A 39, 219 (1972). 5. W. Kopczyński, Phys. Lett. A 43, 63 (1973). 6. Y.N. Obukhov and V.A. Korotkii, Class. Quant. Grav. 4, 1633 (1987). 7. M. Szydlowski and A. Krawiec, Phys. Rev. D 70, 043510 (2004). 8. H.F.M. Goenner and F. Müller-Hoissen, Class. Quant. Grav. 1, 651 (1984). 9. P. Minkowski, Phys. Lett. B 173, 247 (1986). 10. C.G. Böhmer, Acta. Phys. Pol. B 36, 2841 (2005). 11. J. Weyssenhoff and A. Raabe, Acta. Phys. Pol. IX, 7 (1947). 12. J. Frenkel, Z. Phys. 37, 243 (1926). 13. W. Tulczyjew, Acta. Phys. Pol. XVIII, 393 (1959). 14. E. Corinaldesi and A. Papapetrou, Proc. Roy. Soc. Lond. A 209, 259 (1951). 15. J.B. Griffiths and S. Jogia, Gen. Rel. Grav. 14, 137 (1982). 16. Y.N. Obukhov and R. Tresguerres, Phys. Lett. A 184, 17 (1993). 17. Y.N. Obukhov, Phys. Lett. A 210, 163 (1996). Received 28.05.09 СПIНОВI РIДИНИ В ОДНОРIДНОМУ ТА IЗОТРОПНОМУ ПРОСТОРI-ЧАСI К.Г. Бомер, П. Броновскi Р е з ю м е Розглянуто рiдину Вiссенхофа у припущеннi, що простiр-час є однорiдним та iзотропним i тому придатний для космологiчно- го розгляду теорiй гравiтацiї з крученням. Показано, що рiдини Вiссенхофа, для яких виконується умова Френкеля або умова Папапетроу–Корiналдезi, є несумiсними з космологiчним прин- ципом, згiдно з яким тензор кручення має тiльки векторну i аксiальну векторну компоненти. Бiльше того, виявилося, що рiдина Вiссенхофа, що задовольняє умову Тульчiєва, також ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 5 611 C.G. BÖHMER, P. BRONOWSKI несумiсна з космологiчним принципом. Але цю умову не було проаналiзовано повнiстю в цьому контекстi. Ґрунтуючись на цих результатах, запропоновано переглянути деякi попереднi роботи, в яких проаналiзовано космологiчнi розв’язки теорiї Ейнштейна–Картана, оскiльки їхнi спiновi рiдини звичайно не задовольняють космологiчний принцип. 612 ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 5

Spin Fluids in Homogeneous and Isotropic Space-times

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