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.

Appendices

Appendix A: Spinor formalism

This appendix provides the spinor formalism used throughout our work. Unless otherwise stated, we work in a general metric compatible spacetime with torsion. When introducing our spinor formalism, we were faced with the problem that there doesn’t seem to exist a uniform convention in the existing literature on how to treat spinors. Optimally, we would have wanted to achieve the following:

• (i) consecutive raising and lowering of a spinor index does not change the sign of a rank 1 spinor;

• (ii) the metric spinor $\epsilon ^{ab}$ is the raised version of $\epsilon _{ab}$ and vice versa;

• (iii) the spinor inner product is invariant under raising and lowering of indices, i.e. $\xi _a \eta ^a = \xi ^a \eta _a$.

Unfortunately, it becomes clear that it is not possible to satisfy all three desired properties, as shown in [59]. This inconsistency is related to the well known fact (see for example Section 19 in [6] or Section 3–5 in [65]), that a spinor does not have a particular sign—for example, a spatial rotation of the coordinate system by $2\pi$ leads to a change of sign. Also see [56] for more helpful insight about the problem of choice of the spinor formalism, as well as e.g. [6, 7, 12, 29, 59, 63] for insight to various approaches to spinor formalism.

We decided to define our spinor formalism in the following way. We define the ‘metric spinor’ as

$$\begin{aligned} \epsilon _{ab}=\epsilon _{\dot{a}\dot{b}}= \epsilon ^{ab}=\epsilon ^{\dot{a}\dot{b}}= \left( \begin{array}{cc} 0&{}\quad 1\\ -1&{}\quad 0 \end{array} \right) \end{aligned}$$

(57)

with the first index enumerating rows and the second enumerating columns. We raise and lower spinor indices according to the formulae

$$\begin{aligned} \xi ^a=\epsilon ^{ab}\xi _b, \qquad \xi _a=\epsilon _{ab}\xi ^b, \qquad \eta ^{\dot{a}}=\epsilon ^{\dot{a}\dot{b}}\eta _{\dot{b}}, \qquad \eta _{\dot{a}}=\epsilon _{\dot{a}\dot{b}}\eta ^{\dot{b}}. \end{aligned}$$

(58)

Our definition (57), (58) has the following advantages:

• The spinor inner product is invariant under the operation of raising and lowering of indices, i.e. $(\epsilon _{ac}\xi ^c)(\epsilon ^{ad}\eta _d)=\xi ^a\eta _a$.

• The ‘contravariant’ and ‘covariant’ metric spinors are ‘raised’ and ‘lowered’ versions of each other, i.e. $\epsilon ^{ab}=\epsilon ^{ac}\epsilon _{cd}\epsilon ^{bd}$ and $\epsilon _{ab}=\epsilon _{ac}\epsilon ^{cd}\epsilon _{bd}$.

The disadvantage of our definition (57), (58) is that the consecutive raising and lowering of a single spinor index leads to a change of sign, i.e. $\epsilon _{ab}\epsilon ^{bc}\xi _c=-\xi _a$. In formulae where the sign is important we will be careful in specifying our choice of sign; see, for example, (59), (63). We in a sense intentionally ‘sacrificed’ this property in order to guarantee that the other two properties, which in our view have greater physical significance, are satisfied.

Let $\mathfrak {v}$ be the real vector space of Hermitian $2\times 2$ matrices $\sigma _{a\dot{b}}$. Pauli matrices $\sigma ^\alpha {}_{a\dot{b}}$, $\alpha =0,1,2,3$, are a basis in $\mathfrak {v}$ satisfying $\sigma ^\alpha {}_{a\dot{b}}\sigma ^{\beta c\dot{b}} +\sigma ^\beta {}_{a\dot{b}}\sigma ^{\alpha c\dot{b}} =2g^{\alpha \beta }\delta _a{}^c$ where

$$\begin{aligned} \sigma ^{\alpha a\dot{b}}:= \epsilon ^{ac}\sigma ^\alpha {}_{c\dot{d}}\epsilon ^{\dot{b}\dot{d}}. \end{aligned}$$

(59)

At every point of the manifold $M$ Pauli matrices are defined uniquely up to a Lorentz transformation. Define

$$\begin{aligned} \sigma _{\alpha \beta ac}:=\frac{1}{2} \bigl ( \sigma _{\alpha a\dot{b}}\epsilon ^{\dot{b}\dot{d}}\sigma _{\beta c\dot{d}} - \sigma _{\beta a\dot{b}}\epsilon ^{\dot{b}\dot{d}}\sigma _{\alpha c\dot{d}} \bigr ). \end{aligned}$$

(60)

These ‘second order Pauli matrices’ are polarized, i.e.

$$\begin{aligned} *\sigma =\pm i\sigma \end{aligned}$$

(61)

depending on the orientation of ‘basic’ Pauli matrices $\sigma ^\alpha {}_{a\dot{b}}$, $\alpha =0,1,2,3$.

We define the covariant derivatives of spinor fields as

$$\begin{aligned} \nabla _\mu \xi ^a&= \partial _\mu \xi ^a+{\varGamma }^a{}_{\mu b}\xi ^b, \qquad \nabla _\mu \xi _a=\partial _\mu \xi _a-{\varGamma }^b{}_{\mu a}\xi _b,\\ \nabla _\mu \eta ^{\dot{a}}&= \partial _\mu \eta ^{\dot{a}} +\bar{\varGamma }^{\dot{a}}{}_{\mu \dot{b}}\eta ^{\dot{b}}, \qquad \nabla _\mu \eta _{\dot{a}}=\partial _\mu \eta _{\dot{a}} -\bar{\varGamma }^{\dot{b}}{}_{\mu \dot{a}}\eta _{\dot{b}}, \end{aligned}$$

where $\bar{\varGamma }^{\dot{a}}{}_{\mu \dot{b}}=\overline{{\varGamma }^a{}_{\mu b}}$. The explicit formula for the spinor connection coefficients ${\varGamma }^a{}_{\mu b}$ can be derived from the following two conditions:

$$\begin{aligned} \nabla _\mu \epsilon ^{ab}=0, \quad \nabla _\mu j^\alpha =\sigma ^\alpha {}_{a\dot{b}}\nabla _\mu \zeta ^{a\dot{b}}, \end{aligned}$$

(62)

where $\zeta$ is an arbitrary rank 2 mixed spinor field and $j^\alpha :=\sigma ^\alpha {}_{a\dot{b}}\zeta ^{a\dot{b}}$ is the corresponding vector field (current). Conditions (62) give a system of linear algebraic equations for $\mathrm{Re}\,{\varGamma }^a{}_{\mu b}$, $\mathrm{Im}\,{\varGamma }^a{}_{\mu b}$ the unique solution of which is

$$\begin{aligned} {\varGamma }^a{}_{\mu b}=\frac{1}{4} \sigma _\alpha {}^{a\dot{c}} \left( \partial _\mu \sigma ^\alpha {}_{b\dot{c}} +{\varGamma }^\alpha {}_{\mu \beta }\sigma ^{\beta }{}_{b\dot{c}} \right) . \end{aligned}$$

(63)

See section 3 of [24] for more background on covariant differentiation of spinors.

Appendix B: Massless Dirac equation

The generally accepted point of view [29–32, 35] is that a massless neutrino field is a metric compatible spacetime with or without torsion, described by the action

$$\begin{aligned} S_\mathrm{neutrino}:=2i\int \Bigl ( \xi ^a\,\sigma ^\mu {}_{a\dot{b}}\,(\nabla _\mu \bar{\xi }^{\dot{b}}) \ -\ (\nabla _\mu \xi ^a)\,\sigma ^\mu {}_{a\dot{b}}\,\bar{\xi }^{\dot{b}} \Bigr ), \end{aligned}$$

(64)

see formula (11) of [29]. We first vary the action (64) with respect to the spinor $\xi$, while keeping torsion and the metric fixed. A straightforward calculation produces the massless Dirac (or Weyl’s) equation

$$\begin{aligned} \sigma ^\mu {}_{a\dot{b}}\nabla _\mu \,\xi ^a -\frac{1}{2}T^\eta {}_{\eta \mu }\sigma ^\mu {}_{a\dot{b}}\,\xi ^a=0, \end{aligned}$$

(65)

which can be equivalently rewritten as

$$\begin{aligned} \sigma ^\mu {}_{a\dot{b}}\{\!\nabla \!\}_\mu \,\xi ^a \pm \frac{\mathrm{i}}{4} \varepsilon _{\alpha \beta \gamma \delta }T^{\alpha \beta \gamma } \sigma ^\delta {}_{a\dot{b}}\,\xi ^a=0. \end{aligned}$$

(66)

1.1 Energy momentum tensor

In this subsection we give the derivation of the energy momentum tensor of the action $S_\mathrm{neutrino}$, where we vary the metric keeping the spinor fixed. The covariant and contravariant metric change in the following way

$$\begin{aligned} g_{\alpha \beta }\mapsto g_{\alpha \beta } + \delta g_{\alpha \beta }, \qquad g^{\alpha \beta }\mapsto g^{\alpha \beta } -g^{\alpha \alpha '}(\delta g_{\alpha '\beta '})g^{\beta \beta '}, \end{aligned}$$

(67)

while the Pauli matrices transform in the following way

$$\begin{aligned} \sigma _{\alpha }\mapsto \sigma _{\alpha } + \frac{1}{2}\delta g_{\alpha \beta } g^{\beta \gamma }\sigma _{\gamma }, \ \quad \ \sigma ^\alpha \mapsto \sigma ^\alpha - \frac{1}{2}g^{\alpha \beta }(\delta g_{\beta \gamma }) \sigma ^{\gamma }. \end{aligned}$$

(68)

Formulae describe a ‘symmetric’ variation of the Pauli matrices caused by the (symmetric) variation of the (symmetric) metric.

Remark 10

We do most of the following calculations under the assumption that the metric is the Minkowski metric $g_{\mu \nu }=\mathrm diag (1,-1,-1,-1)$ and that the connection is Levi-Civita.

Now we need to look at the $\delta {\varGamma }^{\alpha }{}_{\beta \gamma }$. Using the definition of the Levi-Civita connection, Eq. (67) and metric compatibility ($\nabla g \equiv 0$), we get that the connection transforms as

$$\begin{aligned} \delta {\varGamma }^{\kappa }{}_{\mu \nu } = \frac{1}{2}g^{\kappa \lambda }(\nabla _\mu \delta g_{\lambda \nu } +\nabla _\nu \delta g_{\lambda \mu }-\nabla _\lambda \delta g_{\mu \nu }). \end{aligned}$$

(69)

Lemma 3

The variation of the covariant derivative of $\xi$ with respect to the metric is

$$\begin{aligned} \delta \nabla _\mu \xi ^a =\frac{1}{8}\left( \sigma _\alpha {}^{a\dot{d}}\sigma ^{\beta }{}_{c\dot{d}} -\sigma ^\beta {}^{a\dot{d}}\sigma _{\alpha }{}_{c\dot{d}}\right) \xi ^c\delta {\varGamma }^\alpha {}_{\mu \beta }. \end{aligned}$$

(70)

Proof

Using Eq. (63), the fact that $\xi$ does not contribute to the variation and the assumptions in Remark 10, we obtain

$$\begin{aligned} 4\delta \nabla _\mu \xi ^a = \sigma _\alpha {}^{a\dot{d}}\left( \partial _\mu (\delta \sigma ^\alpha {}_{c\dot{d}}) +(\delta {\varGamma }^\alpha {}_{\mu \beta })\sigma ^{\beta }{}_{c\dot{d}} \right) \xi ^c. \end{aligned}$$

Using Eq. (67) and metric compatibility we get that

$$\begin{aligned} \partial _\mu (\delta \sigma ^\alpha {}_{c\dot{d}})= -\frac{1}{2} g^{\alpha \eta }\sigma _{\zeta }{}_{c\dot{d}}\ \delta {\varGamma }^{\zeta }{}_{\mu \eta } -\frac{1}{2} \delta ^\alpha {}_{\zeta }\ \sigma ^{\xi }{}_{c\dot{d}}\ \delta {\varGamma }^{\zeta }{}_{\mu \xi }. \end{aligned}$$

Combining this with the formula for the variation of $\nabla \xi$, we get the equivalent to Eq. (70)

$$\begin{aligned} 4\delta \nabla _\mu \xi ^a = -\frac{1}{2}\sigma ^\beta {}^{a\dot{d}} \sigma _{\alpha }{}_{c\dot{d}}\ \xi ^c \delta {\varGamma }^{\alpha }{}_{\mu \beta } -\frac{1}{2}\sigma _\alpha {}^{a\dot{d}} \ \sigma ^{\beta }{}_{c\dot{d}}\ \xi ^c \delta {\varGamma }^{\alpha }{}_{\mu \beta } +\sigma ^{\beta }{}_{c\dot{d}}\sigma _\alpha {}^{a\dot{d}}\xi ^c \delta {\varGamma }^\alpha {}_{\mu \beta }. \end{aligned}$$

$\square$

We now combine Eqs. (69) and (70) to get

$$\begin{aligned} \delta \nabla _\mu \xi ^a =\frac{1}{16}\left( \sigma ^\lambda {}^{a\dot{d}}\sigma ^{\beta }{}_{c\dot{d}} -\sigma ^\beta {}^{a\dot{d}}\sigma ^{\lambda }{}_{c\dot{d}}\right) \xi ^c\ \left( \partial _\mu \delta g_{\lambda \beta } +\partial _\beta \delta g_{\lambda \mu } -\partial _\lambda \delta g_{\mu \beta }\right) . \end{aligned}$$

As the first derivative is symmetric over $\lambda ,\beta$ and the Pauli matrices are antisymmetric over these indices, we get

$$\begin{aligned} \left( \sigma ^\lambda {}^{a\dot{d}}\sigma ^{\beta }{}_{c\dot{d}} -\sigma ^\beta {}^{a\dot{d}}\sigma ^{\lambda }{}_{c\dot{d}}\right) \partial _\mu \delta g_{\lambda \beta }= -\left( \sigma ^\lambda {}^{a\dot{d}}\sigma ^{\beta }{}_{c\dot{d}} -\sigma ^\beta {}^{a\dot{d}}\sigma ^{\lambda }{}_{c\dot{d}}\right) \partial _\mu \delta g_{\beta \lambda } =0. \end{aligned}$$

Hence,

$$\begin{aligned} \delta \nabla _\mu \xi ^a&= \frac{1}{16}\left( \sigma ^\lambda {}^{a\dot{d}}\sigma ^{\beta }{}_{c\dot{d}} -\sigma ^\beta {}^{a\dot{d}}\sigma ^{\lambda }{}_{c\dot{d}}\right) \xi ^c \partial _\beta \delta g_{\lambda \mu }\\&-\frac{1}{16}\left( \sigma ^\beta {}^{a\dot{d}}\sigma ^{\lambda }{}_{c\dot{d}} -\sigma ^\lambda {}^{a\dot{d}}\sigma ^{\beta }{}_{c\dot{d}}\right) \xi ^c\partial _\beta \delta g_{\mu \lambda }. \end{aligned}$$

So finally, we get the formula for the variation of the covariant derivative of $\xi$:

$$\begin{aligned} \delta \{\! \nabla \! \}_\mu \xi ^a =\frac{1}{8}\xi ^c \left( \sigma ^\alpha {}^{a\dot{d}}\sigma ^{\beta }{}_{c\dot{d}} -\sigma ^\beta {}^{a\dot{d}}\sigma ^{\alpha }{}_{c\dot{d}}\right) \partial _\beta \delta g_{\mu \alpha }. \end{aligned}$$

(71)

Lemma 4

The energy momentum tensor of the action (64) is Eq. (48).

Proof

Varying the action (64) with respect to the metric, we get

$$\begin{aligned} \delta S&= 2i\delta \int \left( \xi ^a\,\sigma ^\eta {}_{a\dot{b}}\,(\{\! \nabla \! \}_\eta \overline{\xi }^{\dot{b}}) \ -\ (\{\! \nabla \! \}_\eta \xi ^a)\,\sigma ^\eta {}_{a\dot{b}}\,\overline{\xi }^{\dot{b}}\right) \sqrt{|\det g|} \\&= 2i\int \xi ^a\,(\delta \sigma ^\eta {}_{a\dot{b}})\,(\{\! \nabla \! \}_\eta \overline{\xi }^{\dot{b}}) +\xi ^a\,\sigma ^\eta {}_{a\dot{b}}\,(\delta \{\! \nabla \! \}_\eta \overline{\xi }^{\dot{b}}) - (\delta \{\! \nabla \! \}_\eta \xi ^a)\,\sigma ^\eta {}_{a\dot{b}}\,\overline{\xi }^{\dot{b}} \\&\quad -\,(\{\! \nabla \! \}_\eta \xi ^a)\,(\delta \sigma ^\eta {}_{a\dot{b}})\,\overline{\xi }^{\dot{b}} \\&\quad +\,\frac{1}{2}\left( \xi ^a\,\sigma ^\eta {}_{a\dot{b}}\,(\{\! \nabla \! \}_\eta \overline{\xi }^{\dot{b}}) \ -\ (\{\! \nabla \! \}_\eta \xi ^a)\,\sigma ^\eta {}_{a\dot{b}}\,\overline{\xi }^{\dot{b}}\right) g^{\mu \nu }\delta g_{\mu \nu } \end{aligned}$$

and using Eq. (68) we get

$$\begin{aligned} \delta S&= 2i\int \left( -\frac{1}{4}\xi ^a\,g^{\eta \mu }\sigma ^{\nu }{}_{a\dot{b}}\, \left( \{ \nabla \}_\eta \overline{\xi }^{\dot{b}}\right) -\frac{1}{4}\xi ^a\,g^{\eta \nu }\sigma ^{\mu }{}_{a\dot{b}}\, \left( \{ \nabla \}_\eta \overline{\xi }^{\dot{b}}\right) \right. \\&\quad +\,\left. \frac{1}{4}\left( \{\! \nabla \! \}_\eta \xi ^a\right) \,g^{\eta \mu }\sigma ^{\nu }{}_{a\dot{b}}\,\overline{\xi }^{\dot{b}}+\frac{1}{4}\left( \{\! \nabla \! \}_\eta \xi ^a\right) \,g^{\eta \nu }\sigma ^{\mu }{}_{a\dot{b}}\,\overline{\xi }^{\dot{b}} +\frac{1}{2}\xi ^a\,\sigma ^\eta {}_{a\dot{b}}\,(\{\! \nabla \! \}_\eta \overline{\xi }^{\dot{b}})g^{\mu \nu } \right. \\&\quad -\,\left. \frac{1}{2}\left( \{\! \nabla \! \}_\eta \xi ^a\right) \,\sigma ^\eta {}_{a\dot{b}}\,\overline{\xi }^{\dot{b}}g^{\mu \nu }\right) \delta g_{\mu \nu } \\&\quad +\,\xi ^a\,\sigma ^\eta {}_{a\dot{b}}\,(\delta \{\! \nabla \! \}_\eta \overline{\xi }^{\dot{b}}) -(\delta \{\! \nabla \! \}_\eta \xi ^a)\,\sigma ^\eta {}_{a\dot{b}}\,\overline{\xi }^{\dot{b}}. \end{aligned}$$

Now we look at the terms involving the variation of $\{\! \nabla \! \} \xi$ on their own. Using Eq. (71) we get

$$\begin{aligned} I_1&= \frac{i}{4} \int \xi ^a\,\sigma ^\mu {}_{a\dot{b}}\, \overline{\xi }^{\dot{d}}(\sigma ^\nu {}^{c\dot{b}}\sigma ^{\eta }{}_{c\dot{d}} -\sigma ^\eta {}^{c\dot{b}}\sigma ^{\nu }{}_{c\dot{d}})\partial _\eta \delta g_{\mu \nu }\\&- \xi ^c (\sigma ^\nu {}^{a\dot{d}}\sigma ^{\eta }{}_{c\dot{d}} -\sigma ^\eta {}^{a\dot{d}}\sigma ^{\nu }{}_{c\dot{d}}) \partial _\eta \delta g_{\mu \nu } \,\sigma ^\mu {}_{a\dot{b}}\,\overline{\xi }^{\dot{b}}. \end{aligned}$$

Integrating by parts and using the simplifications from Remark 10, we get

$$\begin{aligned} I_1&= \frac{i}{4} \int \overline{\xi }^{\dot{b}}\{\! \nabla \! \}_\eta \xi ^a \, \left( -\sigma ^\mu {}_{a\dot{d}}\sigma ^\nu {}^{c\dot{d}}\sigma ^{\eta }{}_{c\dot{b}} +\sigma ^\mu {}_{a\dot{d}}\sigma ^\eta {}^{c\dot{d}}\sigma ^{\nu }{}_{c\dot{b}}+ \sigma ^{\eta }{}_{a\dot{d}}\sigma ^\nu {}^{c\dot{d}}\sigma ^\mu {}_{c\dot{b}} \right. \\&\quad \left. -\sigma ^{\nu }{}_{a\dot{d}}\sigma ^\eta {}^{c\dot{d}}\sigma ^\mu {}_{c\dot{b}}-\sigma ^\mu {}_{a\dot{d}}\sigma ^\nu {}^{c\dot{d}}\sigma ^{\eta }{}_{c\dot{b}} +\sigma ^\mu {}_{a\dot{d}}\sigma ^\eta {}^{c\dot{d}}\sigma ^{\nu }{}_{c\dot{b}} +\sigma ^{\eta }{}_{a\dot{d}}\sigma ^\nu {}^{c\dot{d}}\sigma ^\mu {}_{c\dot{b}} \right. \\&\quad \left. -\sigma ^{\nu }{}_{a\dot{d}}\sigma ^\eta {}^{c\dot{d}}\sigma ^\mu {}_{c\dot{b}} \right) \delta g_{\mu \nu }. \end{aligned}$$

Since we have $\displaystyle (\sigma ^\mu {}_{a\dot{d}}\sigma ^\eta {}^{c\dot{d}}\sigma ^{\nu }{}_{c\dot{b}} -\sigma ^{\nu }{}_{a\dot{d}}\sigma ^\eta {}^{c\dot{d}}\sigma ^\mu {}_{c\dot{b}}) \delta g_{\mu \nu } =0,$ as it is a product of symmetric and antisymmetric tensors, as well as (after a lengthy but straightforward calculation)

$$\begin{aligned} \sigma ^{\eta }{}_{a\dot{d}}\sigma ^\nu {}^{c\dot{d}}\sigma ^\mu {}_{c\dot{b}}+ \sigma ^{\eta }{}_{a\dot{d}}\sigma ^\mu {}^{c\dot{d}}\sigma ^\nu {}_{c\dot{b}} -\sigma ^\mu {}_{a\dot{d}}\sigma ^\nu {}^{c\dot{d}}\sigma ^{\eta }{}_{c\dot{b}} -\sigma ^\nu {}_{a\dot{d}}\sigma ^\mu {}^{c\dot{d}}\sigma ^{\eta }{}_{c\dot{b}} = 0, \end{aligned}$$

we have shown that the terms involving $\delta \{\! \nabla \! \} \xi$ do not contribute to the variation, i.e. $I_1 = 0.$ We now return to the variation of the whole action, which after some simplification becomes

$$\begin{aligned} \frac{\delta S}{\delta g}&= \frac{i}{2}\int \left( \sigma ^{\nu }{}_{a\dot{b}} ((\{\! \nabla \! \}^\mu \xi ^a)\overline{\xi }^{\dot{b}}-\xi ^a\{\! \nabla \! \}^\mu \overline{\xi }^{\dot{b}}) +\sigma ^{\mu }{}_{a\dot{b}} ((\{\! \nabla \! \}^\nu \xi ^a)\overline{\xi }^{\dot{b}}-\xi ^a\{\! \nabla \! \}^\nu \overline{\xi }^{\dot{b}})\right) \delta g_{\mu \nu }\\&\quad +\,i\int \left( \xi ^a\,\sigma ^\eta {}_{a\dot{b}}\,(\{\! \nabla \! \}_\eta \overline{\xi }^{\dot{b}})g^{\mu \nu } - (\{\! \nabla \! \}_\eta \xi ^a)\,\sigma ^\eta {}_{a\dot{b}}\,\overline{\xi }^{\dot{b}}g^{\mu \nu }\right) \delta g_{\mu \nu }. \end{aligned}$$

Finally we can conclude that the energy momentum tensor of the action (64) is exactly Eq. (48). $\square$

Appendix C: Correction of explicit form of the second field equation and future work

In calculating the Bianchi identity for curvature in “Appendix B” from [53], there was an arithmetic error, hence equation (B.11) from that work should read

$$\begin{aligned} \nabla _\eta \textit{Ric}^\eta {}_\lambda = -\frac{1}{2} \textit{Ric}^\eta {}_\xi T_\eta {}^\xi {}_{\lambda } -\frac{1}{2} \fancyscript{W}^{\eta \zeta }{}_{\lambda \xi }(T_\eta {}^\xi {}_{\zeta }-T_\zeta {}^\xi {}_{\eta }), \end{aligned}$$

and from here equation (B.12) should read

$$\begin{aligned} \nabla _\eta \fancyscript{W}^\eta {}_{\mu \lambda \kappa }&= \fancyscript{W}^\eta {}_{\mu \kappa \xi }(T_\eta {}^\xi {}_{\lambda }-T_\lambda {}^\xi {}_{\eta } )+\fancyscript{W}^\eta {}_{\mu \lambda \xi }(T_\kappa {}^\xi {}_{\eta }-T_\eta {}^\xi {}_{\kappa })\\&\quad +\,\frac{1}{4}(T_\zeta {}^\xi {}_{\eta }-T_\eta {}^\xi {}_{\zeta })(g_{\mu \lambda }\fancyscript{W}^{\eta \zeta }{}_{\kappa \xi }-g_{\mu \kappa }\fancyscript{W}^{\eta \zeta }{}_{\lambda \xi })\\&+\,\frac{1}{4} \textit{Ric}^\eta {}_\xi (g_{\mu \lambda }T_\eta {}^\xi {}_{\kappa }-g_{\mu \kappa }T_\eta {}^\xi {}_{\lambda })\\&\quad +\,\frac{1}{2}\left[ \nabla _\lambda \textit{Ric}_{\mu \kappa }- \nabla _\kappa \textit{Ric}_{\mu \lambda } + \textit{Ric}^\eta {}_\kappa (T_{\lambda \eta \mu }-T_{\eta \lambda \mu }) \right. \\&\left. +\textit{Ric}^\eta {}_\lambda (T_{\eta \kappa \mu }-T_{\kappa \eta \mu })\right] \end{aligned}$$

Consequently, when these two are used in calculating the explicit form of the field equation (3) this produces a different result to the one presented in [53]. Namely, Eq. (3) in its explicit form (i.e. equation (3) from [53]) should read

$$\begin{aligned}&d_6 \nabla _\lambda \textit{Ric}_{\kappa \mu } - d_7 \nabla _\kappa \textit{Ric}_{\lambda \mu }\\&\quad +\,d_6\left( Ric^\eta {}_\kappa \left( T_{\eta \mu \lambda }-T_{\lambda \mu \eta }\right) +\frac{1}{2} g_{\mu \lambda } \fancyscript{W}^{\eta \zeta }{}_{\kappa \xi }(T_\eta {}^\xi {}_{\zeta }-T_\zeta {}^\xi {}_{\eta }) +\frac{1}{2} g_{\mu \lambda } \textit{Ric}^\eta {}_\xi T_\eta {}^\xi {}_{\kappa } \right) \\&\quad +\,d_7 \left( \textit{Ric}^\eta {}_\lambda \left( T_{\eta \mu \kappa } - T_{\kappa \mu \eta } \right) +\frac{1}{2} g_{\kappa \mu } \fancyscript{W}^{\eta \zeta }{}_{\lambda \xi }(T_\eta {}^\xi {}_{\zeta }-T_\zeta {}^\xi {}_{\eta }) +\frac{1}{2} g_{\kappa \mu }\textit{Ric}^\eta {}_\xi T_\eta {}^\xi {}_{\lambda } ) \right) \\&\quad +\,b_{10} \left( T_\eta {}^\xi {}_{\zeta }-T_\zeta {}^\xi {}_{\eta })(g_{\mu \kappa }\fancyscript{W}^{\eta \zeta }{}_{\lambda \xi }-g_{\mu \lambda }\fancyscript{W}^{\eta \zeta }{}_{\kappa \xi }\right) \\&\quad +\,2b_{10}\left( \fancyscript{W}^\eta {}_{\mu \kappa \xi }(T_\eta {}^\xi {}_{\lambda }-T_\lambda {}^\xi {}_{\eta } ) + \fancyscript{W}^\eta {}_{\mu \lambda \xi }(T_\kappa {}^\xi {}_{\eta }-T_\eta {}^\xi {}_{\kappa }) - \fancyscript{W}^{\xi \eta }{}_{\kappa \lambda } T_{\eta \mu \xi } \right) =0. \end{aligned}$$

This change in no way affects the proof of the main theorem, as this form of the field equation is even simpler and does not contain any additional terms, but we believed it was important to point out for purposes of future work. This mistake was noticed in the process of generalising the explicit form of the field equations (2), (3), i.e. the equations equation (26) and (27) from [53], see [52].

The two papers of Singh [60, 61] are of particular interest to us, where the author constructs solutions for the Yang–Mills case (4) with purely axial and purely trace torsion respectively, see (9), and unlike the solution of [63], $\{Ric\}$ is not assumed to be zero. It is obvious that these solutions differ from the ones presented in our work, as the torsion of generalised pp-waves is assumed to be purely tensor. It would however be of interest to us to see whether this construction of Singh’s can be expanded to our most general $\mathrm{O}(1,3)$-invariant quadratic form $q$ with 16 coupling constants.

We also hope to see whether it is possible to produce torsion waves which are purely axial or trace and combine them with the pp-wave metric, in a similar fashion as was done with purely tensor torsion waves, in order to produce new solutions of quadratic metric-affine gravity and also give their physical interpretation in the near future.