Wave operators, torsion, and Weitzenböck identities

Abstract

The current article offers a mathematical toolkit for the study of waves propagating on spacetimes with nonvanishing torsion. The toolkit comprises generalized versions of the Lichnerowicz–de Rham and the Beltrami wave operators, and the Weitzenböck identity relating them on Riemann–Cartan geometries. The construction applies to any field belonging to a matrix representation of a Lie (super) algebra containing an $\mathfrak {so} \left( \eta _{+}, \eta _{-} \right)$ subalgebra. These tools allow us to study the propagation of waves on an Einstein–Cartan background at different orders in the eikonal parameter. It stands in strong contrast with more traditional approaches that are restricted to studying only the leading order for waves on this kind of geometry (“plane waves”). The current article focuses only on the mathematical aspects and offers proofs and generalizations for some results already used in physical applications. In particular, the subleading analysis proves that torsion affects the propagation of amplitude and polarization for fields in some representations. These results suggest how one may use gravitational waves and multimessenger events as probes for torsion and the spin tensor of dark matter.

Appendices

A Open superalgebra of operators

In this appendix we deduce the open superalgebra Eqs. (70)–(75) satisfied by the operators $\mathrm {I}_{a}$, $\mathrm {D}$, and $\mathcal {D}_{b}$, with the commutator and anticonmmutator defined as

$$\begin{aligned} \left[ A,B\right]&:=AB-BA\,, \end{aligned}$$

(107)

$$\begin{aligned} \left\{ C,D\right\}&:=CD+DC\,. \end{aligned}$$

(108)

Let us start by consider a generic p-form

$$\begin{aligned} \Phi ^{m_{1}\cdots m_{m}}=\frac{1}{p!}\Phi ^{m_{1}\cdots m_{m}}{}_{c_{1}\cdots c_{p}}e^{c_{1}}\wedge \cdots \wedge e^{c_{p}}\,, \end{aligned}$$

(109)

and let us apply the commutator $\left[ \mathrm {I}_{a},\mathcal {D} _{b}\right]$ on it,

$$\begin{aligned} \left[ \mathrm {I}_{a},\mathcal {D}_{b}\right] \Phi ^{m_{1}\cdots m_{m} }=\mathrm {I}_{a}\mathcal {D}_{b}\Phi ^{m_{1}\cdots m_{m}}-\mathcal {D} _{b}\mathrm {I}_{a}\Phi ^{m_{1}\cdots m_{m}}\,. \end{aligned}$$

(110)

After some algebra, it is straightforward to prove that

$$\begin{aligned} \mathcal {D}_{b}\Phi ^{m_{1}\cdots m_{m}}=\frac{1}{p!}\mathcal {D}_{b}\Phi ^{m_{1}\cdots m_{m}}{}_{c_{1}\cdots c_{p}}e^{c_{1}}\wedge \cdots \wedge e^{c_{p}}+\mathrm {I}_{b}T^{c}\wedge \mathrm {I}_{c}\Phi ^{m_{1}\cdots m_{m}}\,, \end{aligned}$$

(111)

and therefore

$$\begin{aligned} \mathrm {I}_{a}\mathcal {D}_{b}\Phi ^{m_{1}\cdots m_{m}}= & {} \frac{1}{\left( p-1\right) !}\mathcal {D}_{b}\Phi ^{m_{1}\cdots m_{m}}{}_{ac_{2}\cdots c_{p} }e^{c_{2}}\wedge \cdots \wedge e^{c_{p}}-T^{c}{}_{ab}\mathrm {I}_{c}\Phi ^{m_{1}\cdots m_{m}}\nonumber \\&+\,\mathrm {I}_{b}T^{c_{1}}\wedge \mathrm {I}_{ac_{1}} \Phi ^{m_{1}\cdots m_{m}}\,. \end{aligned}$$

(112)

In a similar way, we have that

$$\begin{aligned} \mathcal {D}_{b}\mathrm {I}_{a}\Phi ^{m_{1}\cdots m_{m}}=\frac{1}{\left( p-1\right) !}\mathcal {D}_{b}\Phi ^{m_{1}\cdots m_{m}}{}_{ac_{2}\cdots c_{p} }e^{c_{1}}\wedge \cdots \wedge e^{c_{p}}+\mathrm {I}_{b}T^{c}\wedge \mathrm {I}_{ac}\Phi ^{m_{1}\cdots m_{m}},\nonumber \\ \end{aligned}$$

(113)

and replacing it in the commutator Eq. (110), we have that in fact

$$\begin{aligned} \left[ \mathrm {I}_{a},\mathcal {D}_{b}\right] =-T^{c}{}_{ab}\mathrm {I}_{c}\,. \end{aligned}$$

(114)

Regarding the commutator of Eq. (66), we have

$$\begin{aligned} \left[ \mathrm {D},\mathcal {D}_{b}\right]&=\mathrm {D}\mathcal {D} _{b}-\mathcal {D}_{b}\mathrm {D}\,, \end{aligned}$$

(115)

$$\begin{aligned}&=\mathrm {D}\left( \mathrm {I}_{b}\mathrm {D}+\mathrm {DI}_{b}\right) -\left( \mathrm {I}_{b}\mathrm {D}+\mathrm {DI}_{b}\right) \mathrm {D}\,, \end{aligned}$$

(116)

$$\begin{aligned}&=\mathrm {DI}_{b}\mathrm {D}+\mathrm {D}^{2}\mathrm {I}_{b}-\left( \mathrm {I}_{b}\mathrm {D}^{2}+\mathrm {DI}_{b}\mathrm {D}\right) \,, \end{aligned}$$

(117)

$$\begin{aligned}&=\mathrm {D}^{2}\mathrm {I}_{b}-\mathrm {I}_{b}\mathrm {D}^{2}\,. \end{aligned}$$

(118)

The definitions Eqs. (107)–(108) for the commutator and the anticommutator grant that the super Jacobi identity Eq. (67) closes. In fact,

$$\begin{aligned}&\left\{ \mathrm {D},\left[ \mathrm {I}_{b},\mathcal {D}_{a}\right] \right\} +\left[ \mathcal {D}_{a},\left\{ \mathrm {D},\mathrm {I}_{b}\right\} \right] -\left\{ \mathrm {I}_{b},\left[ \mathcal {D}_{a},\mathrm {D}\right] \right\} \nonumber \\&\quad =\mathrm {DI}_{b}\mathcal {D}_{a}-\mathrm {D}\mathcal {D}_{a}\mathrm {I} _{b}+\mathrm {I}_{b}\mathcal {D}_{a}\mathrm {D}-\mathcal {D}_{a}\mathrm {I} _{b}\mathrm {D} \end{aligned}$$

(119)

$$\begin{aligned}&\qquad +\,\mathcal {D}_{a}\mathrm {DI}_{b}+\mathcal {D}_{a}\mathrm {I}_{b}\mathrm {D} -\mathrm {DI}_{b}\mathcal {D}_{a}-\mathrm {I}_{b}\mathrm {D}\mathcal {D}_{a} \end{aligned}$$

(120)

$$\begin{aligned}&\qquad -\,\mathrm {I}_{b}\mathcal {D}_{a}\mathrm {D}+\mathrm {I}_{b}\mathrm {D} \mathcal {D}_{a}-\mathcal {D}_{a}\mathrm {DI}_{b}+\mathrm {D}\mathcal {D} _{a}\mathrm {I}_{b}\,, \end{aligned}$$

(121)

$$\begin{aligned}&\quad =0. \end{aligned}$$

(122)

From here, it is straightforward to obtain Eq. (69).

B Anomalous propagation of amplitude

In this Appendix we derive Eq. (101). Let us start by using Eq. (98) to write

$$\begin{aligned} k^{a}{\mathfrak {D}}_{a}\varphi ^{2}=\frac{1}{p!}C_{AB}\left( k^{a}{\mathfrak {D}}_{a}{\bar{\psi }}_{\left( 0\right) }^{Aa_{1}\cdots a_{p}}\psi _{\left( 0\right) a_{1}\cdots a_{p}}^{B}+{\bar{\psi }}_{\left( 0\right) }^{Aa_{1}\cdots a_{p}}k^{a}{\mathfrak {D}}_{a}\psi _{\left( 0\right) a_{1}\cdots a_{p}}^{B}\right) .\nonumber \\ \end{aligned}$$

(123)

Using Eq. (91), it is possible to prove that

$$\begin{aligned} \frac{1}{p!}e^{a_{1}}\wedge \cdots \wedge e^{a_{p}}k^{a}{\mathfrak {D}}_{a}\psi _{\left( 0\right) a_{1}\cdots a_{p}}^{A}+k^{a}\mathrm {I}_{a}{\mathbb {T}}^{b}\wedge \mathrm {I}_{b}\psi _{\left( 0\right) }^{A}+\frac{1}{2}\psi _{\left( 0\right) }^{A}{\mathfrak {D}}_{a}k^{a}=0. \end{aligned}$$

(124)

Combining both relations, we get

$$\begin{aligned} k^{a}{\mathfrak {D}}_{a}\varphi ^{2}&=-\left( -1\right) ^{\eta _{-}}C_{AB}*\left\{ k^{a}\left[ \mathrm {I}_{a}{\mathbb {T}}^{b}\wedge \mathrm {I}_{b}{\bar{\psi }}_{\left( 0\right) }^{A}\wedge *\psi _{\left( 0\right) }^{B}+{\bar{\psi }}_{\left( 0\right) }^{A}\wedge *\left( \mathrm {I}_{a}{\mathbb {T}}^{b}\wedge \mathrm {I}_{b}\psi _{\left( 0\right) }^{B}\right) \right] \right. \nonumber \\&\quad \left. +{\bar{\psi }}_{\left( 0\right) }^{A}\wedge *\psi _{\left( 0\right) }^{B}{\mathfrak {D}}_{a}k^{a}\right\} \end{aligned}$$

(125)

Replacing here Eq. (98), we get

$$\begin{aligned} {\mathfrak {D}}_{a}J^{a}={\mathbb {T}}_{abc}\Pi ^{ab}J^{c}, \end{aligned}$$

(126)

where ${\mathbb {T}}_{abc}$ are the orthonormal-basis components of the G-torsion, ${\mathbb {T}}_{a} = \frac{1}{2} {\mathbb {T}}_{abc} e^{b} \wedge e^{c}$,

$$\begin{aligned} J^{a}=\varphi ^{2}k^{a}, \end{aligned}$$

(127)

and

$$\begin{aligned} \Pi ^{ab}=\left( -1\right) ^{\eta _{-}}C_{AB}*\left[ \mathrm {I}^{a}{\bar{P}}^{A}\wedge *\mathrm {I}^{b}P^{B}+\mathrm {I}^{b}{\bar{P}}^{A}\wedge *\mathrm {I}^{a}P^{B}\right] . \end{aligned}$$

(128)

Splitting the covariant derivative as in Eq. (36), we find

$$\begin{aligned} {\mathfrak {D}}_{a}J^{a}&=-{\mathbb {D}}^{\ddag }J,\nonumber \\&=-\mathrm {\mathring{D}}^{\ddag }J+\mathrm {I}^{a}{\mathbb {K}}_{ab}J^{b},\nonumber \\&=-\mathrm {d}^{\dag }J+{\mathbb {T}}^{a}{}_{ab}J^{b}. \end{aligned}$$

(129)

This relation lead us to

$$\begin{aligned} \mathrm {d}^{\dag }J={\mathbb {T}}_{abc}\left( \eta ^{ab}-\Pi ^{ab}\right) J^{c}. \end{aligned}$$

(130)

for the anomalous propagation of amplitude when ${\mathbb {T}}^{a}\ne 0$ and Eq. (82).

C Anomalous propagation of polarization

In this Appendix we derive Eq. (103). Replacing Eq. (96) in Eq. (91) we get

$$\begin{aligned} k^{a}{\mathfrak {D}}_{a}P^{A}+\left( \frac{1}{\varphi }k^{a}{\mathfrak {D}}_{a}\varphi +\frac{1}{2}{\mathfrak {D}}_{a}k^{a}\right) P^{A}=0. \end{aligned}$$

(131)

Replacing here Eq. (126) with $J^{a}=\varphi ^{2}k^{a}$, we have

$$\begin{aligned} \frac{1}{\varphi }k^{a}{\mathfrak {D}}_{a}\varphi +\frac{1}{2}{\mathfrak {D}}_{a}k^{a}=\frac{1}{2\varphi ^{2}}{\mathbb {T}}_{abc}\Pi ^{ab}J^{c}, \end{aligned}$$

(132)

and therefore

$$\begin{aligned} k^{a}{\mathfrak {D}}_{a}P^{A}=-\frac{1}{2}{\mathbb {T}}_{abc}\Pi ^{ab}k^{c}P^{A}. \end{aligned}$$

(133)

From this equation, it is already evident that the orthonormal-basis components of torsion, ${\mathbb {T}}_{abc}$, modify the propagation of the polarization p-form $P^{A}$. To make this effect more explicit, let us split ${\mathbb {D}}$ (cf. Definition 3.5) in terms of the contorsion and G-contorsion one-forms as

$$\begin{aligned} {\mathfrak {D}}_{a} P^{A} = \mathring{\mathcal {D}}_{a} P^{A} + \mathrm {I}_{a} \left( \frac{1}{2} \kappa ^{mn} \left[ J_{mn} \right] ^{A}{}_{B} + a^{A}{}_{B} \right) \wedge P^{B} + {\mathbb {K}}_{ab} \wedge \mathrm {I}^{b} P^{A}. \end{aligned}$$

(134)

Writing the contorsion $\kappa ^{mn}$ and G-contorsion ${\mathbb {K}}_{ab}$ one-forms in terms of the torsion and G-torsion as in Eq. (37), and replacing this in Eq. (133), we finally find the general expression given in Eq. (103),

$$\begin{aligned} k^{a}\mathcal {\mathring{D}}_{a}P^{A}&=\frac{1}{2}\left\{ -{\mathbb {T}}_{abc}\Pi ^{ab}P^{A}+\left( {\mathbb {T}}_{bac}+{\mathbb {T}}_{abc}-{\mathbb {T}}_{cab}\right) e^{a}\wedge \mathrm {I}^{b}P^{A}\right. \nonumber \\&\quad \left. -\,\left[ \frac{1}{2}\left( T_{bac}-T_{abc}+T_{cab}\right) \left[ J^{ab}\right] ^{A}{}_{B}+2\mathrm {I}_{c}a^{A}{}_{B}\right] P^{B}\right\} k^{c}, \end{aligned}$$

(135)

from which it is evident that torsion modifies the propagation of polarization along the trajectory.