# PP-waves with torsion: a metric-affine model for the massless neutrino

## Abstract

In this paper we deal with quadratic metric-affine gravity, which we briefly introduce, explain and give historical and physical reasons for using this particular theory of gravity. We then introduce a generalisation of well known spacetimes, namely pp-waves. A classical pp-wave is a 4-dimensional Lorentzian spacetime which admits a nonvanishing parallel spinor field; here the connection is assumed to be Levi-Civita. This definition was generalised in our previous work to metric compatible spacetimes with torsion and used to construct new explicit vacuum solutions of quadratic metric-affine gravity, namely *generalised* pp-waves of parallel Ricci curvature. The physical interpretation of these solutions we propose in this article is that they represent a *conformally invariant metric-affine model for a massless elementary particle*. We give a comparison with the classical model describing the interaction of gravitational and massless neutrino fields, namely *Einstein–Weyl theory* and construct pp-wave type solutions of this theory. We point out that generalised pp-waves of parallel Ricci curvature are very similar to pp-wave type solutions of the Einstein–Weyl model and therefore propose that our generalised pp-waves of parallel Ricci curvature represent a metric-affine model for the massless neutrino.

### Keywords

Quadratic metric-affine gravity pp-Waves Torsion Massless neutrino Einstein–Weyl theory## 1 Introduction

The smallest departure from a Riemannian spacetime of Einstein’s general relativity would consist of admitting *torsion* (8), arriving thereby at a Riemann–Cartan spacetime, and, furthermore, possible nonmetricity (11), resulting in a ‘*metric-affine*’ spacetime. Metric-affine gravity is a natural generalisation of Einstein’s general relativity, which is based on a spacetime with a Riemannian metric \(g\) of Lorentzian signature.

We consider spacetime to be a connected real 4-manifold \(M\) equipped with a Lorentzian metric \(g\) and an affine connection \({\varGamma }\). The 10 independent components of the (symmetric) metric tensor \(g_{\mu \nu }\) and the 64 connection coefficients \({{\varGamma }^\lambda }_{\mu \nu }\) are the unknowns of our theory. Note that the characterisation of the spacetime manifold by an *independent* linear connection \({\varGamma }\) initially distinguishes metric-affine gravity from general relativity. The connection incorporates the inertial properties of spacetime and it can be viewed, according to Weyl [72], as the guidance field of spacetime. The metric describes the structure of spacetime with respect to its spacio-temporal distance relations.

According to Hehl et al. [33], in Einstein’s general relativity the linear connection of its Riemannian spacetime is metric-compatible and symmetric. The symmetry of the Levi–Civita connection translates into the closure of infinitesimally small parallelograms. Already the transition from the flat gravity-free Minkowski spacetime to the Riemannian spacetime in Einstein’s theory can locally be understood as a deformation process. The lifting of the constraints of metric-compatibility and symmetry yields nonmetricity and torsion, respectively. The continuum under consideration, here classical spacetime, is thereby assumed to have a non-trivial microstructure, similar to that of a liquid crystal or a dislocated metal or feromagnetic material etc. It is gratifying to have the geometrical concepts of nonmetricity and torsion already arising in the (three-dimensional) continuum theory of lattice defects, see [39, 40].

*after*the variation in \({\varGamma }\) is carried out. It is known [68] that in this case for a generic 11-parameter action Eq. (5) reduces to

Let us here mention the contributions of Yang [74] and Mielke [46] who showed, respectively, that Einstein spaces satisfy Eqs. (2) and (3). There is a substantial bibliography devoted to the study of the system (2), (3) in the special case (4) and one can get an idea of the historical development of the Yang–Mielke theory of gravity from [13, 19, 20, 50, 55, 64, 66, 67, 73].

The motivation for choosing a model of gravity which is purely quadratic in curvature is explained in Section 1 of [71]. The study of Eqs. (2), (3) for specific purely quadratic curvature Lagrangians has a long history. Quadratic curvature Lagrangians were first discussed by Weyl [72], Pauli [54], Eddington [17] and Lanczos [42, 43, 44] in an attempt to include the electromagnetic field in Riemannian geometry.

The idea of using a purely quadratic action in General Relativity goes back to Hermann Weyl, as given at the end of his paper [72], where he argued that the most natural gravitational action should be quadratic in curvature and involve all possible invariant quadratic combinations of curvature, like the square of Ricci curvature, the square of scalar curvature, etc. Unfortunately, Weyl himself never afterwards pursued this analysis. Stephenson [64] looked at three different quadratic invariants: scalar curvature squared, Ricci curvature squared and the Yang–Mills Lagrangian (4) and varied with respect to the metric and the affine connection. He concluded that every equation arising from the above mentioned quadratic Lagrangians has the Schwarzschild solution and that the equations give the same results for the three ‘crucial tests’ of general relativity, i.e. the bending of light, the advance of the perihelion of Mercury and the red-shift.

Higgs [36] continued in a similar fashion to show that in scalar squared and Ricci squared cases, one set of equations may be transformed into field equations of the Einstein type with an arbitrary ‘cosmological constant’ in terms of the ‘new gauge-invariant metric’.

One can get more information and form an idea on the historical development of the quadratic metric-affine theory of gravity from [13, 19, 20, 36, 46, 50, 55, 64, 66, 67, 73, 74].

It should be noted that the action (1) contains only purely quadratic curvature terms, so it excludes the Einstein–Hilbert term (linear in curvature) and any terms quadratic in torsion (8) and nonmetricity (11). By choosing a purely quadratic curvature Lagrangian we are hoping to describe phenomena whose characteristic wavelength is sufficiently small and curvature sufficiently large.

We should also point out that the action (1) is conformally invariant, i.e. it does not change if we perform a Weyl rescaling of the metric \(g\rightarrow e^{2f}g\), \(f:M\rightarrow \mathbb {R}\), without changing the connection \({\varGamma }\).

In our previous work [53], we presented new non-Riemannian solutions of the field equations (2), (3). These new solutions were to be constructed explicitly and the construction turned out to be very similar to the classical construction of a pp-wave, only with torsion. This paper aims to give additional information on these spacetimes, provide their physical interpretation, additional calculations and future possible applications.

The paper has the following structure. In Sect. 3 we recall basic facts about pp-waves with torsion, in Sect. 3.1 we provide information about classical pp-waves, in Sect. 3.2 we recall the way pp-waves were generalised in [53] and list the properties of these spacetimes with torsion and in Sect. 3.3 we present the pp-waves spinor formalism. In Sect. 4 we present our attempt at giving a physical interpretation to the solutions of the field equations from [53]. In Sect. 4.1 we provide a reminder on the classical model describing the interaction of gravitational and massless neutrino fields (Einstein–Weyl theory), while in Sect. 4.1.1 we present a brief review of known solutions of this theory. In Sect. 4.2 we present our pp-wave type solutions of this theory. In Sect. 4.3 we compare the Einstein–Weyl solutions to our conformally invariant solutions. Finally, “Appendix A” provides the spinor formalism used throughout our work, “Appendix B” gives detailed calculations involved in comparing our solutions to Einstein–Weyl theory and “Appendix C” provides a correction of a mistake found in our previous work [53] and gives the motivation for future work.

## 2 Notation

*tensor torsion*,

*trace torsion*, and

*axial torsion*respectively. We say that our connection \({\varGamma }\) is metric compatible if \(\nabla g\equiv 0\). The interval is \(\mathrm{d}s^2:=g_{\mu \nu }\,d x^\mu \,d x^\nu \). Given a scalar function \(f:M\rightarrow \mathbb {R}\) we write for brevity \(\displaystyle \int f:=\int _Mf\,\sqrt{|\det g|}\,\mathrm{d}x^0\mathrm{d}x^1\mathrm{d}x^2\mathrm{d}x^3\,, \det g:=\det (g_{\mu \nu }). \) We define

*nonmetricity*by

## 3 PP-waves with torsion

In this section, where we mostly follow the exposition from [53], we provide background information about *pp-waves*, starting with the notion of a classical pp-wave, then introducing a generalisation with the addition of torsion and lastly presenting the particular spinor formalism of pp-waves.

### 3.1 Classical pp-waves

PP-waves are well known spacetimes in general relativity, first discovered by Brinkmann [10], and subsequently rediscovered by several authors, for example Peres [57]. There are differing views on what the ‘pp’ stands for. According to Griffiths [23] and Kramer et al. [38] ‘pp’ is an abbreviation for ‘plane-fronted gravitational waves with parallel rays’. See e.g. [1, 5, 8, 10, 21, 23, 38, 48, 49, 51, 53, 57, 58, 70, 71] for more information on pp-waves and pp-wave type solutions of metric-affine gravity.

**Definition 1**

A *pp-wave* is a Riemannian spacetime which admits a nonvanishing parallel spinor field.

*fixed*. Put

^{1}Then \(l\) is a nonvanishing parallel real null vector field. Now we define the real scalar function

*phase*. It is defined uniquely up to the addition of a constant and possible multi-valuedness resulting from a nontrivial topology of the manifold. Put

**Definition 2**

*pp-wave*is a Riemannian spacetime whose metric can be written locally in the form

*Remark 1*

Note that the Cotton tensor of classical pp-waves with parallel Ricci curvature vanishes, as classical pp-waves are metric compatible spacetimes with zero scalar curvature. In the theory of conformal spaces the main geometrical objects to be analysed are the Weyl and the Cotton tensors, see [22]. It is well known that for conformally flat spaces the Weyl tensor has to vanish and consequently the Cotton tensor has to vanish too. Note that the Cotton tensor is only conformally invariant in three dimensions.

### 3.2 Generalised pp-waves

One natural way of generalising the concept of a classical pp-wave is simply to extend Definition 1 to general metric compatible spacetimes, i.e. spacetimes whose connection is not necessarily Levi-Civita. However, this gives a class of spacetimes which is too wide and difficult to work with. We choose to extend the classical definition in a more special way better suited to the study of the system of field equations (2), (3).

^{2}

**Definition 3**

*generalised pp-wave*is a metric compatible spacetime with pp-metric and torsion

We list below the main properties of generalised pp-waves. Note that here and further on we denote by \(\{\!\nabla \!\}\) the covariant derivative with respect to the Levi-Civita connection which should not be confused with the full covariant derivative \(\nabla \) incorporating torsion.

*Remark 2*

From Eq. (27) we can clearly see that torsion has 4 independent non-zero components.

In the beginning of Sect. 3.1 we introduced the spinor field \(\chi \) satisfying \(\{\!\nabla \!\}\chi =0\). It becomes clear that this spinor field also satisfies \(\nabla \chi =0\).

**Lemma 1**

The generalised pp-wave and the underlying classical pp-wave admit the same nonvanishing parallel spinor field.

*Proof*

*Remark 3*

In view of Lemma 1, it is clear that both the generalised pp-wave and the underlying classical pp-wave admit the same nonvanishing parallel real null vector field \(l\) and the same nonvanishing parallel complex 2-form (14), (15).

**Lemma 2**

^{3}i.e.

*Proof*

The curvatures generated by the Levi-Civita connection and torsion simply add up [compare formulae (19) and (25)].

The second term in the RHS of (25) is purely Weyl. Consequently, the Ricci curvature of a generalised pp-wave is completely determined by the pp-metric.

- Clearly, generalised pp-waves have the same non-zero irreducible pieces of curvature as classical pp-waves, namely symmetric trace-free Ricci and Weyl. Using special local coordinates (16), (18), these can be expressed explicitly aswhere \(m_1=\mathrm{Re } (m), m_2=\mathrm{Im }(m)\), \(f_{\alpha \beta } := \partial _{\alpha }\partial _{\beta } f\) and \(w_{jk}\) are real scalars given by$$\begin{aligned} Ric&= \frac{1}{2} (f_{11} + f_{22}) \, l\otimes l,\\ \fancyscript{W}&= \sum _{j,k=1}^2 w_{jk} (l\wedge m_j)\otimes (l\wedge m_k), \end{aligned}$$Compare these to the corresponding Eqs. (20) and (21) for classical pp-waves.$$\begin{aligned} w_{11} = \frac{1}{4} [-f_{11} +f_{22} + \mathrm{Re } ((h^2)'')],&\quad w_{22} = -w_{11},\\ w_{12} = \pm \frac{1}{2} f_{12}-\frac{1}{4} \mathrm{Im } ((h^2)''),&\quad w_{21} = w_{12}. \end{aligned}$$
- The curvature of a generalised pp-wave has all the usual symmetries of curvature in the Riemannian case, that is,$$\begin{aligned} R_{\kappa \lambda \mu \nu }&= R_{\mu \nu \kappa \lambda },\end{aligned}$$(28)$$\begin{aligned} \varepsilon ^{\kappa \lambda \mu \nu }R_{\kappa \lambda \mu \nu }&= 0,\end{aligned}$$(29)$$\begin{aligned} R_{\kappa \lambda \mu \nu }&= -R_{\lambda \kappa \mu \nu },\end{aligned}$$(30)Of course, (31) is true for any curvature whereas (30) is a consequence of metric compatibility. Also, (30) follows from (28) and (31).$$\begin{aligned} R_{\kappa \lambda \mu \nu }&= -R_{\kappa \lambda \nu \mu }. \end{aligned}$$(31)
The second term in the RHS of (23) is pure gauge in the sense that it does not affect curvature (25). It does, however, affect torsion (24).

- The Ricci curvature of a generalised pp-wave is zero if and only ifand the Weyl curvature is zero if and only if$$\begin{aligned} f_{11}+f_{22}=0 \end{aligned}$$(32)Here we use special local coordinates (16), (18) and denote \(f_{\alpha \beta }:=\partial _\alpha \partial _\beta f\).$$\begin{aligned} f_{11}-f_{22}=\mathrm{Re}\left( (h^2)''\right) , \qquad f_{12}=\pm \frac{1}{2}\mathrm{Im}\left( (h^2)''\right) . \end{aligned}$$(33)
The curvature of a generalised pp-wave is zero if and only if we have both (32) and (33). Clearly, for any given function \(h\) we can choose a function \(f\) such that \(R=0\): this \(f\) is a quadratic polynomial in \(x^1\), \(x^2\) with coefficients depending on \(x^3\). Thus, as a spin-off, we get a class of examples of Weitzenböck spaces (\(T\ne 0\), \(R=0\)).

### 3.3 Spinor formalism for generalised pp-waves

*Remark 4*

In the case \(f=0\), formulae (34) do not turn into the Minkowski space Pauli matrices, since we write the metric in the form (16). This is a matter of convenience in calculations.

*Remark 5*

Note that we could have chosen a different set of Pauli matrices \(\sigma ^\alpha {}_{a\dot{b}}\) in (34), namely with the opposite sign in every Pauli matrix, as they are a basis in the real vector space of Hermitian \(2\times 2\) matrices \(\sigma _{a\dot{b}}\) satisfying \(\sigma ^\alpha {}_{a\dot{b}}\sigma ^{\beta c\dot{b}} +\sigma ^\beta {}_{a\dot{b}}\sigma ^{\alpha c\dot{b}} =2g^{\alpha \beta }\delta _a{}^c\), defined uniquely up to a Lorentz transformation. See “Appendix A” for more on our chosen spinor formalism.

*Remark 6*

In view of Eqs. (35) and (36), it is easy to see that \(\chi F(\varphi )\) is a solution of the massless Dirac equation. Here \(F\) is an arbitrary function of the phase (13) and \(\chi \) is the parallel spinor introduced in Sect. 3.1.

Note that as the second order Pauli matrices \(\sigma ^{\alpha \beta }{}_{ab}\) are antisymmetric over the tensor indices, i.e. \(\sigma ^{\alpha \beta }{}_{ab}=-\sigma ^{\beta \alpha }{}_{ab}\), we only give the independent non-zero terms.

## 4 Physical interpretation of generalised pp-waves

It was shown in our previous work [53] that using the generalised pp-waves described in Sect. 3.2 we can construct new vacuum solutions of quadratic metric-affine gravity. The main result of [53] is the following

**Theorem 1**

Generalised pp-waves of parallel Ricci curvature are solutions of the system of Eqs. (2), (3).

The observation that one can construct vacuum solutions of quadratic metric-affine gravity in terms of pp-waves is a recent development. The fact that classical pp-spaces of parallel Ricci curvature are solutions was first pointed out in [69, 70, 71].

*Remark 7*

There is a slight error in our calculations of the explicit form of the field equations from [53], which in no way influences the main result. The error was noticed in producing [52], where the generalised version of the explicit field equations can be found, see “Appendix C” and for more details.

Einstein spaces (\(Ric=\Lambda g\)), and

classical pp-spaces of parallel Ricci curvature.

^{4}and Einstein–Weyl theory.

^{5}The difference with our model is that Einstein–Maxwell and Einstein–Weyl theories contain the gravitational constant which dictates a particular relationship between the strengths of the fields in question, whereas our model is conformally invariant and the amplitudes of the two curvatures (38) and (39) are totally independent.

In the remainder of this subsection we outline an argument in favour of interpreting our torsion wave (24), (23) as a mathematical model for some massless particle.

Formula (43) shows that our rank 4 spinor \(\omega \) has additional algebraic structure: it is the 4th tensor power of a rank 1 spinor \(\xi \). Consequently, the complexified curvature generated by our torsion wave is completely determined by the rank 1 spinor field \(\xi \).

We claim that the spinor field (44) satisfies the massless Dirac equation, see (35) or (36). Indeed, as \(\chi \) is parallel checking that \(\xi \) satisfies the massless Dirac equation reduces to checking that \(\,(r^{1/4})'\,\sigma ^\mu {}_{a\dot{b}}\,l_\mu \,\chi ^a=0\,\). The latter is established by direct substitution of the explicit formula (12) for \(l\).

### 4.1 Einstein–Weyl field equations

In this section we aim to provide a reminder of Einstein–Weyl theory and the field equations arising from this classical model describing the interaction of gravitational and massless neutrino fields, then to provide pp-wave type solutions within this model, provide the previously known solutions of this type and, finally, to compare them to the pp-wave type solutions of our conformally invariant metric-affine model of gravity from [53].

*Remark 8*

Note that in Einstein–Weyl theory the connection is assumed to be Levi-Civita, so we only vary the action (45) with respect to the metric and the spinor.

*Remark 9*

When the Eq. (50) is satisfied, we have that the energy-momentum tensor (48) is trace free and the second line of (49) vanishes, see e.g. end of section 2 of Griffiths and Newing [25].

#### 4.1.1 Known solutions of Einstein–Weyl theory

In one of the early works on this subject, Griffiths and Newing [24] show how the solutions of Einstein–Weyl equations can be constructed and present five examples of solutions and a later work by the same authors [25] presents a more general solution of Kundt’s class. Audretsch and Graf [4] derive a differential equation representing radiation solutions of the general relativistic Weyl’s equation and study the corresponding energy-momentum tensor and they present an exact solution of Einstein–Weyl equations in the form of pp-waves. Audretsch [3] continues to study the asymptotic behaviour of the neutrino energy-momentum tensor in curved space-time with the sole aid of generally covariant assumptions about the nature of the Weyl field and the author shows that these Weyl fields behave asymptotically like neutrino radiation.

Griffiths [26] expanded on his previous work and this paper is of particular interest to us as in section 5 of [26] the author presents solutions whose metric is the pp-wave metric (16) and the author presents a condition on the function \(f\) from the pp-metric (16). Griffiths [27] identifies a class of neutrino fields with zero energy momentum tensor and stipulates that these spacetimes may also be interpreted as describing gravitational waves. Collinson and Morris [14] showed that these could be either pp-waves or Robinson-Trautman type N solutions presented in [24]. Subsequently these were called ‘*ghost neutrinos*’ by Davis and Ray in [16].

Kuchowicz and Żebrowski [41] expand on the work on ghost neutrinos trying to resolve this anomaly by considering non-zero torsion in the framework of Einstein–Cartan theory. Griffiths [28] also considers the possibility of non-zero torsion and in a more general work [29] he showed that neutrino fields in Einstein-Cartan theory must have metrics that belong to the family of solutions of Kundt’s class, which include the pp-waves. Singh and Griffiths [62] corrected several mistakes from [29] and showed that neutrino fields in Einstein–Cartan theory also include the Robinson–Trautman type N solutions and that any solution of the Einstein–Weyl equations in general relativity has a corresponding solution in Einstein–Cartan theory. Thus pp-wave type solutions of Einstein–Weyl equations have corresponding solutions in Einstein–Cartan theory. This paper was one of the main inspirations behind the result in Sect. 4.2.

### 4.2 PP-wave type solutions of Einstein–Weyl theory

### 4.3 Comparison of metric-affine and Einstein–Weyl solutions

To make our comparison clearer, let us compare these models in the case of mono-chromatic solutions of both models using local coordinates (16), (18) and Pauli matrices (34).

#### 4.3.1 Monochromatic metric-affine solutions

#### 4.3.2 Monochromatic Einstein–Weyl solutions

#### 4.3.3 Comparison of monochromatic metric-affine and Einstein–Weyl solutions

The main difference between the two models is that in the metric-affine model our generalised pp-wave solutions have parallel Ricci curvature, whereas in the Einstein–Weyl model the pp-wave type solutions do not necessarily have parallel Ricci curvature. However, when we look at monochromatic pp-wave type solutions in the Einstein–Weyl model their Ricci curvature also becomes parallel. The only remaining difference is in the right-hand sides of Eqs. (54) and (56): in (54) the constant is arbitrary whereas in (56) the constant is expressed via the characteristics of the spinor wave and the gravitational constant.

In other words, comparing Eqs. (54) and (56) we see that while in the metric-affine case the Laplacian of \(f\) can be *any* constant, in the Einstein–Weyl case it is required to be a *particular* constant. This should not be surprising as our metric-affine model is conformally invariant, while the Einstein–Weyl model is not.

We also want to clarify that \(f\) and the quantities \(a,b,c\) appearing in this Sect. 4.3 are generally arbitrary functions of the null coordinate \(x^3\). As such, if these quantities are non-zero only for a short finite interval of \(x^3\), the solutions represent spinors, curvature and torsion components which propagate at the speed of light.

Hence we can conclude that generalised pp-waves of parallel Ricci curvature are very similar to pp-type solutions of the Einstein–Weyl model, which is a classical model describing the interaction of massless neutrino and gravitational fields. Therefore we suggest that

*Generalised pp-waves of parallel Ricci curvature represent a metric-affine model for the massless neutrino.*

## Footnotes

## Notes

### Acknowledgments

The authors are very grateful to D Vassiliev, J B Griffiths and F W Hehl for helpful advice and to the Ministry of Education and Science of the Federation of Bosnia and Herzegovina, which supported our research.

### References

- 1.Adamowicz, W.: Plane waves in gauge theories of gravitation. Gen. Relativ. Gravit.
**12**, 677–691 (1980)MathSciNetCrossRefMATHADSGoogle Scholar - 2.Alekseevsky, D.V.: Holonomy groups and recurrent tensor fields in Lorentzian spaces. In: Stanjukovich, K.P. (ed.) Problems of the Theory of Gravitation and Elementary Particles issue 5, pp. 5–17. Atomizdat, Moscow (in Russian) (1974)Google Scholar
- 3.Audretsch, J.: Asymptotic behaviour of neutrino fields in curved space-time. Commun. Math. Phys.
**21**, 303–313 (1971)MathSciNetCrossRefADSGoogle Scholar - 4.Audretsch, J., Graf, W.: Neutrino radiation in gravitational fields. Commun. Math. Phys.
**19**, 315–326 (1970)MathSciNetCrossRefMATHADSGoogle Scholar - 5.Babourova, O.V., Frolov, B.N., Klimova, E.A.: Plane torsion waves in quadratic gravitational theories in RiemannCartan space. Class. Quantum Grav.
**16**, 1149–1162 (1999). gr-qc/9805005 - 6.Berestetskii, V.B., Lifshitz, E.M., Pitaevskii, L.P.: Quantum Electrodynamics (Course of Theoretical Physics vol. 4) 2nd edn. Pergamon Press, Oxford (1982)Google Scholar
- 7.Blagojevic, M.: Gravitation and Gauge Symmetries. Institute of Physics Publishing, Bristol (2002)CrossRefMATHGoogle Scholar
- 8.Blagojević, M., Hehl, F.W.: Gauge Theories of Gravitation. A Reader with Commentaries. Imperial College Press, London (2013)CrossRefMATHGoogle Scholar
- 9.Brill, D.R., Wheeler, J.A.: Interaction of neutrinos and gravitational fields. Rev. Mod. Phys.
**29**, 465–479 (1957)MathSciNetCrossRefMATHADSGoogle Scholar - 10.Brinkmann, M.W.: On Riemann spaces conformal to Euclidean space. Proc. Natl. Acad. Sci. USA
**9**, 1–3 (1923)CrossRefADSGoogle Scholar - 11.Bryant, R.L.: Pseudo-Riemannian metrics with parallel spinor fields and vanishing Ricci tensor. Global Analysis and Harmonic Analysis (Marseille-Luminy, 1999) Sémin. Congr.
**4**(Paris: Soc. Math. France), 53–94 (2000). math/0004073 - 12.Buchbinder, I.L., Kuzenko, S.M.: Ideas and Methods of Supersymmetry and Supergravity. Institute of Physics Publishing, Bristol (1998)MATHGoogle Scholar
- 13.Buchdahl, H.A.: Math. Rev.
**20**, 1238 (1959)Google Scholar - 14.Collinson, C.D., Morris, P.B.: Space-time admitting neutrino fields with zero energy and momentum. J. Phys.
**A6**, 915–916 (1972)ADSGoogle Scholar - 15.Cotton, É.: Sur les varits a trois dimensions. Annales de la Facult des Sciences de Toulouse II
**14**(14), 385–438 (1899)MathSciNetCrossRefGoogle Scholar - 16.Davis, T.M., Ray, J.R.: Ghost neutrinos in general relativity. Phys. Rev. D
**9**, 331–333 (1974)CrossRefADSGoogle Scholar - 17.Eddington, A.S.: The Mathematical Theory of Relativity, 2nd edn. The University Press, Cambridge (1952)Google Scholar
- 18.Esser, W.: Exact Solutions of the Metric-Affine Gauge Theory of Gravity. Diploma Thesis, University of Cologne, Cologne (1996)Google Scholar
- 19.Fairchild Jr, E.E.: Gauge theory of gravitation. Phys. Rev. D
**14**, 384–391 (1976)MathSciNetCrossRefADSGoogle Scholar - 20.Fairchild Jr, E.E.: Erratum: gauge theory of gravitation. Phys. Rev. D
**14**, 2833 (1976)MathSciNetCrossRefADSGoogle Scholar - 21.García, A., Macías, A., Puetzfeld, D., Socorro, J.: Plane fronted waves in metric affine gravity. Phys. Rev. D
**62**, 044021 (2000). gr-qc/0005038 - 22.García, A., Hehl, F.W., Heinicke, C., Macías, A.: The cotton tensor in Riemannian spacetimes. Class. Quantum Grav.
**21**, 10991118 (2004). gr-qc/0309008 - 23.Griffiths, J.B.: Colliding Plane Waves in General Relativity. Oxford University Press, Oxford (1991)MATHGoogle Scholar
- 24.Griffiths, J.B., Newing, R.A.: The two-component neutrino field in general relativity. J. Phys. A
**3**, 136–149 (1970)MathSciNetCrossRefADSGoogle Scholar - 25.Griffiths, J.B., Newing, R.A.: Tetrad equations for the two-component neutrino field in general relativity. J. Phys. A
**3**, 269–273 (1970)MathSciNetCrossRefADSGoogle Scholar - 26.Griffiths, J.B.: Some physical properties of neutrino-gravitational fields. Int. J. Theor. Phys.
**5**, 141–150 (1972)CrossRefGoogle Scholar - 27.Griffiths, J.B.: Gravitational radiation and neutrinos. Commun. Math. Phys.
**28**, 295–299 (1972)CrossRefADSGoogle Scholar - 28.Griffiths, J.B.: Ghost neutrinos in Einstein–Cartan theory. Phys. Lett. A
**75**, 441–442 (1980)CrossRefADSGoogle Scholar - 29.Griffiths, J.B.: Neutrino fields in Einstein–Cartan theory. Gen. Relativ. Gravit.
**13**, 227–237 (1981)CrossRefMATHADSGoogle Scholar - 30.Hehl, F.W.: Spin und Torsion in der Allgemeinen Relativitätstheorie oder die Riemann-Cartansche Geometrie der Welt. Technischen Universität Clausthal, Habilitationsschrift (1970)Google Scholar
- 31.Hehl, F.W.: Spin and torsion in general relativity I: foundations. Gen. Relativ. Gravit.
**4**, 333–349 (1973)MathSciNetCrossRefADSGoogle Scholar - 32.Hehl, F.W.: Spin and torsion in general relativity II: geometry and field equations. Gen. Relativ. Gravit.
**5**, 491–516 (1974)MathSciNetCrossRefADSGoogle Scholar - 33.Hehl, F.W., McCrea, J.D., Mielke, E.W., Ne’eman, Y.: Metric-affine gauge theory of gravity: field equations, Noether identities, world spinors, and breaking of dilation invariance. Phys. Rep.
**258**, 1–171 (1995). gr-qc/9402012 - 34.Hehl, F.W., Macías, A.: Metric-affine gauge theory of gravity II. Exact solutions. Int. J. Mod. Phys. D
**8**, 399–416 (1999). gr-qc/9902076 - 35.Hehl, F.W., von der Heyde, P., Kerlick, G.D., Nester, J.M.: General relativity with spin and torsion: foundations and prospects. Rev. Mod. Phys.
**48**, 393–416 (1976)CrossRefADSGoogle Scholar - 36.Higgs, P.W.: Quadratic Lagrangians and general relativity. Nuovo Cimento
**11**, 816–820 (1959)MathSciNetCrossRefMATHGoogle Scholar - 37.King, A.D., Vassiliev, D.: Torsion waves in metric-affine field theory. Class. Quantum Grav.
**18**, 2317–2329 (2001). gr-qc/0012046 - 38.Kramer, D., Stephani, H., Herlt, E., MacCallum, M.: Exact Solutions of Einstein’s Field Equations. Cambridge University Press, Cambridge (1980)MATHGoogle Scholar
- 39.Kröner, E.: Continuum theory of defects. In: Balian R., et al. (eds.) Physics of Defects, Les Houches, Session XXXV, 1980. North-Holland, Amsterdam (1980)Google Scholar
- 40.Kröner, E.: The continuized crystal—a bridge between micro- and macromechanics. Gesellschaft angewandte Mathematik und Mechanik Jahrestagung Goettingen West Germany Zeitschrift Flugwissenschaften, vol. 66 (1986)Google Scholar
- 41.Kuchowicz, C., Żebrowski, J.: The presence of torsion enables a metric to allow a gravitational field. Phys. Lett. A
**67**, 16–18 (1978)MathSciNetCrossRefADSGoogle Scholar - 42.Lanczos, C.: A remarkable property of the Riemann–Christoffel tensor in four dimensions. Ann. Math.
**39**, 842–850 (1938)MathSciNetCrossRefGoogle Scholar - 43.Lanczos, C.: Lagrangian multiplier and Riemannian spaces. Rev. Mod. Phys.
**21**, 497–502 (1949)MathSciNetCrossRefMATHADSGoogle Scholar - 44.Lanczos, C.: Electricity and general relativity. Rev. Mod. Phys.
**29**, 337–350 (1957)MathSciNetCrossRefMATHADSGoogle Scholar - 45.Landau, L.D., Lifshitz, E.M.: The Classical Theory of Fields (Course of Theoretical Physics vol. 2) 4nd edn. Pergamon Press, Oxford (1975)Google Scholar
- 46.Mielke, E.W.: On pseudoparticle solutions in Yang’s theory of gravity. Gen. Relativ. Gravit.
**13**, 175–187 (1981)MathSciNetCrossRefMATHADSGoogle Scholar - 47.Nakahara, M.: Geometry, Topology and Physics. Institute of Physics Publishing, Bristol (1998)Google Scholar
- 48.Obukhov, Y.N.: Generalized plane fronted gravitational waves in any dimension. Phys. Rev. D
**69**, 024013 (2004). gr-qc/0310121 - 49.Obukhov, Y.N.: Plane waves in metric-affine gravity. Phys. Rev. D
**73**, 024025 (2006). gr-qc/0601074 - 50.Olesen, P.: A relation between the Einstein and the Yang–Mills field equations. Phys. Lett. B
**71**, 189–190 (1977)MathSciNetCrossRefADSGoogle Scholar - 51.Pasic, V.: New vacuum solutions for quadratic metric-affine gravity—a metric affine model for the massless neutrino? Math. Balk. New Ser.
**24**, Fasc 3–4, 329 (2010)Google Scholar - 52.Pasic, V., Barakovic, E., Okicic, N.: A new representation of the field equations of quadratic metric-affine gravity. Adv. Math. Sci. J.
**3**(1), 33–46 (2014)Google Scholar - 53.Pasic, V., Vassiliev, D.: PP-waves with torsion and metric-affine gravity. Class. Quantum Grav.
**22**, 3961–3975 (2005). gr-qc/0505157 - 54.Pauli, W.: Zur Theorie der Gravitation und der Elektrizität von Hermann Weyl. Physik. Zaitschr.
**20**, 457–467 (1919)MATHGoogle Scholar - 55.Pavelle, R.: Unphysical solutions of Yang’s gravitational-field equations. Phys. Rev. Lett.
**34**, 1114 (1975)CrossRefADSGoogle Scholar - 56.Penrose, O., Rindler, W.: Propagating modes in gauge field theories of gravity, vol. 2. Cambridge University Press, Oxford (1984, 1986)Google Scholar
- 57.Peres, A.: Some gravitational waves. Phys. Rev. Lett.
**3**, 571–572 (1959)CrossRefMATHADSGoogle Scholar - 58.Peres, A.: PP—WAVES preprint (reprinting of [57]) (2002). hep-th/0205040
- 59.Pirani, F.A.E.: Introduction to Gravitational Radiation Theory. Lectures on General Relativity. Prentice-Hall, Inc. Englewood Cliffs, New Jersey (1964)Google Scholar
- 60.Singh, P.: On axial vector torsion in vacuum quadratic Poincaré gauge field theory. Phys. Lett. A
**145**, 7–10 (1990)MathSciNetCrossRefADSGoogle Scholar - 61.Singh, P.: On null tratorial torsion in vacuum quadratic Poincaré gauge field theory. Class. Quantum Grav.
**7**, 2125–2130 (1990)CrossRefMATHADSGoogle Scholar - 62.Singh, P., Griffiths, J.B.: On neutrino fields in Einstein–Cartan theory. Phys. Lett. A
**132**, 88–90 (1988)MathSciNetCrossRefADSGoogle Scholar - 63.Singh, P., Griffiths, J.B.: A new class of exact solutions of the vacuum quadratic Poincaré gauge field theory. Gen. Relativ. Gravit.
**22**, 947–956 (1990)MathSciNetCrossRefMATHADSGoogle Scholar - 64.Stephenson, G.: Quadratic Lagrangians and general relativity. Nuovo Cimento
**9**, 263–269 (1958)MathSciNetCrossRefMATHGoogle Scholar - 65.Streater, R.F., Wightman, A.S.: PCT, spin and statistics, and all that. In: Princeton Landmarks in Physics, Princeton University Press, Princeton. ISBN:0-691-07062-8, corrected third printing of the 1978 edition (2000)Google Scholar
- 66.Thompson, A.H.: Yang’s gravitational field equations. Phys. Rev. Lett.
**34**, 507–508 (1975)CrossRefADSGoogle Scholar - 67.Thompson, A.H.: Geometrically degenerate solutions of the Kilmister–Yang equations. Phys. Rev. Lett.
**35**, 320–322 (1975)MathSciNetCrossRefADSGoogle Scholar - 68.Vassiliev, D.: Pseudoinstantons in metric-affine field theory. Gen. Relativ. Gravit.
**34**, 1239–1265 (2002). gr-qc/0108028 - 69.Vassiliev, D.: Pseudoinstantons in metric-affine field theory. In: Brambilla, N., Prosperi, G.M. (eds.) Quark Confinement and the Hadron Spectrum V, pp. 273–275. World Scientific, Singapore (2003)CrossRefGoogle Scholar
- 70.Vassiliev, D.: Quadratic non-Riemannian gravity. J. Nonlinear Math. Phys.
**11**(Supplement), 204–216 (2004)MathSciNetCrossRefADSGoogle Scholar - 71.Vassiliev, D.: Quadratic metric-affine gravity. Ann. Phys. (Lpz.)
**14**, 231–252 (2005). gr-qc/0304028 - 72.Weyl, H.: Eine neue Erweiterung der Relativitätstheorie. Ann. Phys. (Lpz.)
**59**, 101–133 (1919)CrossRefMATHADSGoogle Scholar - 73.Wilczek, F.: Geometry and interaction of instantons. In: Stump, D.R., Weingarten, D.H. (eds.) Quark Confinement and Field theory, pp. 211–219. Wiley-Interscience, New York (1977)Google Scholar
- 74.Yang, C.N.: Integral formalism for gauge fields. Phys. Rev. Lett.
**33**, 445–447 (1974)MathSciNetCrossRefADSGoogle Scholar