^{3}.

The approach of metric-affine gravity initially distinguishes it from Einstein’s general relativity. Using an independent affine connection produces a theory with 10 + 64 unknowns. We write down the Yang-Mills action for the affine connection and produce the Yang-Mills equation and the so-called complementary Yang-Mills equation by independently varying with respect to the connection and the metric, respectively. We call this theory the Yang-Mielke theory of gravity. We construct explicit spacetimes with pp-metric and purely axial torsion and show that they represent a solution of Yang-Mills theory. Finally we compare these spacetimes to existing solutions of metric-affine gravity and present future research possibilities.

Einstein’s geometric theory of gravity can be summarised, paraphrasing Wheeler [

The motion of particles which are so small that their effect on the gravitational field they move in is negligible.

The nature of matter as a source for gravity.

Einstein’s equation, which shows how this matter source is related to the curvature of spacetime.

Einstein’s equation is at the centre of general relativity. It gives us a formulation of the relationship between spacetime geometry and the properties of matter. This equation is formulated using

Two problems with general relativity arose quite quickly after its initial introduction in 1915. Einstein considered that what are recognised locally as inertial properties of matter must be determined by the properties of the rest of the universe. Einstein’s efforts to discover to what extent general relativity manages to do this founded the modern study of cosmology. The second problem of general relativity was that electromagnetism is not included in the theory. As Einstein said in [

A number of recent developments in physics have evoked the possibility that the treatment of spacetime might involve more than just using the Riemannian spacetime of Einstein’s general relativity, like our failure to quantize gravity, the description of hadron or nuclear matter in terms of extended structures, the accelerating universe, the study of the early universe, and so forth (see Hehl et al. [

The smallest departure from a Riemannian spacetime of Einstein’s general relativity consists of admitting

As stated by Hehl et al. in [

This paper has the following structure. In Section

We consider spacetime to be a connected real 4-manifold

We use pp-waves, which are well known spacetimes in general relativity, in order to construct our solutions. In Section

We define a

A generalised pp-wave with purely axial torsion is a metric compatible spacetime with pp-metric and torsion

The real vector field

We list below the main properties of these generalised pp-waves. Note that here and further on we denote by

Our torsion completely corresponds to Singh’s axial torsion from [

The connection of a generalised pp-wave with purely axial torsion is clearly metric compatible. Since

The curvature of a generalised pp-wave is

The Ricci curvature is zero if Poisson’s equation

Ricci is parallel if

In this section we aim to use the spacetimes introduced in Section

Generalised pp-waves with purely axial torsion with parallel

Note that by

In the special case (

Since we know (see, e.g., [

Using the explicit formula for torsion (

Checking that all the terms in (

Vassiliev [

Constructing vacuum solutions of quadratic metric-affine gravity in terms of pp-waves is a recent development. Classical pp-waves of parallel Ricci curvature were first shown to be solutions of (

An interesting generalisation of the concept of a pp-wave was presented by Obukhov in [

In our previous work [

We aim to be able to do a similar feat using generalised pp-waves with purely axial torsion, that is, first showing that they are solutions of the field equations (

In [

It is interesting to note that in [

The two papers of Singh [

Our notation follows [

The authors declare that there is no conflict of interests regarding the publication of this paper.

Sincere thanks are due to Dmitri Vassiliev, Friedrich Hehl, and Milutin Blagojević for helpful advice. The authors also thank the anonymous reviewers for very useful comments and suggestions.