# The homogeneous and isotropic Weyssenhoff fluid

###### Abstract

We consider a Weyssenhoff fluid assuming that the spacetime is homogeneous and isotropic, therefore being relevant for cosmological considerations of gravity theories with torsion. In this paper, it is explicitely 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 a vector and an axial vector component. Moreover it turns out that the Weyssenhoff fluid obeying the Tulczyjew condition is also incompatible with the cosmological principle. Based on this result we propose to reconsider a number of previous works that analysed cosmological solutions of Einstein-Cartan theory, since their spin fluids usually did not obey the cosmological principle.

Keywords: Spin fluids, Einstein-Cartan theory, Cosmology

## 1 Introduction

Cosmology, over the last years, has become a very active field of research containing many open questions that require further investigation. Hawking, Penrose and others have shown that under fairly general assumptions solutions of Einstein’s field equations evolve singularities. 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 that the inclusion of torsion would help to avoid these singularities. Unfortunately 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 precise form. The four-dimensional spacetime manifold is foliated by constant time spacelike hypersurfaces, which are the orbits of a Lie group acting on , with isometry group . Following the Copernican principle [3], we assume all fields to be invariant under the action of

(1) |

where are the (six) Killing vectors generating the spacetime isometries. The metric tensor is denoted by , denotes the torsion tensor and Greek indices label the holonomic components. For the rest of the paper only anholonomic components of tensors are used, labelled by Latin indices.

Kopczyński initiated the investigation of cosmological models with torsion in Ref. [4] and Ref. [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 in [5] an anisotropic model of the universe with torsion was analysed. The cosmological principle in the above strict sense (1) was first developed in Einstein-Cartan theory by Tsamparlis in [3], where it 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 , where was assumed to be a function of the time variable . 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 tensor, 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 Ref. [6]). However, by doing so, one must drop the second condition of (1), and use a weaker notion of the cosmological principle. For a recent example where a so-called cosmological model with macroscopic spin fluid was analysed see Ref. [7].

Applying the restrictions (1) to yields the FRW type metric

(2) |

where and where the 3-space is spherical for , flat for and hyperbolic for . If we impose the restrictions (1) on the torsion tensor [3], its allowed components are

(3) |

where we follow the notation of [8]. Hence, the cosmological principle allows a vector torsion component , along the world lines, and an axial vector component , within the hypersurfaces of constant time. Such a totally skew-symmetric torsion tensor in cosmology was considered earlier in [9], where was assumed. The geometry parameter was redefined to include the remaining torsion by . Also with , the cosmological inflation could be explained by torsion in Ref. [10], using a rough model.

## 2 The cosmological field equations

The spin-connection 1-form in theories with torsion can be split into a torsion free part (the usual spin-connection 1-form related to the Christoffel symbol ) and a contortion 1-form part , that takes the torsion of spacetime into account

(4) |

where the torsion tensor and the contortion tensor are related by the following algebraic relation

(5) |

where we used that is a torsion-free connection. The latter relation between torsion and contortion also implies that their vector and axial vector component are simply related by

(6) |

Since the metric and the contortion (or torsion) components that are compatible with the cosmological constant are fixed, one can compute any geometrical quantity of interest.

Metric (2) gives rise to the following basis 1-forms

(7) |

which together with the non-vanishing torsion components (4) yields the following non-vanishing connection 1-forms

(8) |

that are computed from (4) , where the torsion two form can be obtained from (4) via . The field equations of 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

(9) |

is the canonical energy-momentum tensor and is the tensor of spin.

## 3 The Weyssenhoff fluid

The ideal Weyssenhoff fluid [11] is a generalisation of the ideal fluid to take into account the properties of spin and torsion in spacetime. Its canonical energy-momentum tensor is given by

(10) | |||

(11) |

where is the momentum density of the fluid and is the fluid’s velocity. By and we denoted the energy density and the pressure of the fluid, respectively. The intrinsic angular momentum tensor satisfies

(12) |

The spin tensor can be decomposed into two 3-vectors

(13) |

that in case we assume the Frenkel condition [12], vanishes in the rest-frame. The second vector

(14) |

in the rest-frame can be regarded as the spin density.

Integrability of the particles’ equations of motion requires one more condition that the spin tensor has to satisfy

(15) |

where the vector is usually taken to be the velocity vector of the fluid , 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 assumed the condition

(16) |

where with stands for the time component of the spin tensor. In the following sections we investigate whether a Weyssenhoff fluid obeying one of the three presented integrability conditions is compatible with the cosmological principle.

## 4 The Frenkel condition

If we assume the Frenkel condition [12] then the spin contribution of the energy-momentum tensor can be rewritten to given

(17) |

In the third and fourth step the Frenkel was necessary for the modifications, and we introduced the acceleration of the fluid , defined by . Hence equations (10) and (11) taking the Frenkel condition into account yield

(18) |

This implies that for vanishing acceleration of the fluid one is back at Einstein gravity [15]. The interpretation of the contribution of the spin angular momentum tensor in (10) in terms of the acceleration strongly depends on the Frenkel condition.

The totally skew-symmetric part of the torsion tensor (3) is allowed by the cosmological principle. Since the four velocity enters the definition of the tensor of spin (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 which does not allow the Weyssenhoff fluid to be the source of the trace components (3) of the torsion tensor. More explicitely, multiplying the torsion field equation (9b) by leads to

(19) |

where for the last steps (11) and the Frenkel condition were taken into account. Therefore we have explicitely shown that the Weyssenhoff fluid obeying the Frenkel condition is incompatible with the cosmological principle put forward in Ref. [3].

## 5 The Papapetrou-Corinaldesi condition

Assuming the Papapetrou-Corinaldesi [14] condition has the following consequences for the torsion tensor implied by the spin fluid. As before, since the fluid’s four velocity enters the definition of the spin tensor, the totally skew-symmetric torsion has to vanish. Secondly, the traced torsion field equation (9b) yields

(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 Weyssenhoff fluid obeying the Papapetrou-Corinaldesi condition is incompatible with the cosmological principle (since we have the Papapetrou-Corinaldesi and the Frenkel condition in fact take the same form).

## 6 The Tulczyjew condition

According to our information, it has not been analysed so far if the Weyssenhoff fluid obeying the Tulczyjew condition [13] is compatible with the cosmological principle. 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 of the fluid is present in Eq. (11). However, in this case (15) does not vanish identically on general ground and we arrive at

(21) |

where the last term on the right hand side need not to vanish. The Tulczyjew condition, Eq. (15) with explicitely written out leads to

(22) |

which can be used to express the last term on (21) by

(23) |

The fluid’s four velocity simply reads , and equation (23) by taking (8) into account yields the following form of the Tulczyjew condition that the spin fluid has to satisfy

(24) |

where the dot means differentiation with respect to time . 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 where by we denoted the Hubble parameter defined by . Equations (24) provide us with for conditions () which we will analyse in more detail. For the left-hand side of (24) vanishes and one is left with

(25) |

which can be written in an equivalent form by using the introduced vectors and in (13) and (14) and leads to

(26) |

where by we mean the usual inner product of vectors. From this we conclude that the vectors and are orthogonal to each other. For the remaining values the condition (24) takes the following form in terms of the three vector

(27) |

Here we now see that (26) is not an independent equation since from the latter we derive that and moreover that . Therefore we find the following non-vanishing components of the induced torsion tensor via the field equations (21)

(28) |

where is given by equation (27). Note that the index only takes the values and that we furthermore suppressed the explicit index for the vector . We can now try to continue the construction of a spin fluid that is compatible with cosmological principle, namely the Weyssenhoff fluid obeying the Tulczyjew condition. In principle we have two possibilities: (a) we choose the vector of the spin tensor so that it satisfies the condition (26) and we choose the spin density vector so that equation (27) is satisfied. On the other hand, (b) let us prescribe the spin density vector . Then, in order to get an allowed , 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 principle allows a vector torsion component, but only along the world lines, . The other components are excluded. Therefore, also the Weyssenhoff fluid obeying the Tulczyjew condition is incompatible with the cosmological principle, since (28a) vanishes identically.

## 7 Conclusions and outlook

The restrictions that follow from assuming homogeneity and isotropy on the very large scales of the universe (the cosmological principle) allow one metric component and two torsion components; a vector and an axial vector component of the torsion tensor. We showed that the Weyssenhoff fluids obeying either the Frenkel or the Papapetrou-Corinaldesi condition are incompatible with this principle. Furthermore we analysed the Tulczyjew condition which, in principle, allows one to construct a non-trace-free torsion tensor. However, its time component, allowed by the cosmological principle, vanishes identically. Therefore it has been shown that no spin fluid obeying the common integrability conditions is compatible with the cosmological principle. This rather surprising result shows the necessity to reconsider the previous works on cosmology with torsion, since none of these results can be regarded as a truly cosmological model with torsion.

Furthermore it raises the question, whether an integrability 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 the subject of further research. If it turned 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 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 cosmological 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 generalised form of equation (11) essentially enters the tensor of spin.

## Acknowledgements

This work is partially supported by the Polish Ministry of Scientific Research
and Information Technology under the grant No. PBZ/MIN/008/P03/2003
and by the University of Lodz.
The work of CGB was supported by research grant BO 2530/1-1 of the
German Research Foundation (DFG).

This work is part of the research project Nr. 01/04
Quantum Gravity, Cosmology and Categorification
of the Austrian Academy of Sciences (ÖAW) and the National
Academy of Sciences of Ukraine (NASU).

## References

- [1] S. W. Hawking and G. F .R. Ellis. The Large Scale Structure of Space-Time, Cambridge University Press (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. A75 27 (1979).
- [4] W. Kopczyński. Phys. Lett. A39 219 (1972).
- [5] W. Kopczyński. Phys. Lett. A43 63 (1973).
- [6] Y. N. Obukhov and V. A. Korotkii. Class. Quant. Grav. 4 1633 (1987).
- [7] M. Szydlowski and A. Krawiec. Phys. Rev. D70 043510 (2004).
- [8] H. F. M. Goenner and F. Müller-Hoissen. Class. Quant. Grav. 1 651 (1984).
- [9] P. Minkowski. Phys. Lett. B173 247 (1986).
- [10] C. G. Böhmer. Acta. Phys. Pol. B36 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. A209 259 (1951).
- [15] J. B. Griffiths and S. Jogia. Gen. Rel. Grav. 14 137 (1982).
- [16] Y. N. Obukhov and R. Tresguerres, Phys. Lett. A184 17 (1993).
- [17] Y. N. Obukhov, Phys. Lett. A210 163 (1996).