# Linear and second-order geometry perturbations on spacetimes with torsion

## Abstract

In order to study gravitational waves in any realistic astrophysical scenario, one must consider geometry perturbations up to second order. Here, we present a general technique for studying linear and quadratic perturbations on a spacetime with torsion. Besides the standard metric mode, a “torsionon” perturbation mode appears. This torsional mode will be able to propagate only in a certain kind of theories.

## 1 Introduction

Besides opening up the new field of gravitational-wave astronomy, gravitational waves have proven to be a valuable tool in order to constraint alternative theories of gravity. In particular, the detection of GW170817 [1] refuted whole families of theories [2, 3, 4]. Together with other experimental results [5], this amounts to a renewed challenge on alternative gravitational theories.

The simplest theory formulated using RC geometry is Einstein–Cartan–Sciama–Kibble (ECSK) gravity [6, 7, 8], but many other theories in four and higher dimensions are naturally expressed in this framework (see, e.g., Refs. [9, 10, 11]).

The presence of propagating torsion modes in this class of theories has been the subject of much investigation; for a recent assessment, see Refs. [12, 13, 14, 15] and references therein.

Traditionally, theoretical work on gravitational waves, whether including torsion or not, has used exclusively the tensor formalism, focusing on the metric and (if torsion is a concern) the affine connection perturbations. It is the purpose of this article to show that Cartan’s elegant and powerful exterior calculus, which, when applied to gravitational theory, treats the vielbein and spin connection one-forms as independent variables (see, e.g., Ref. [16]), is also up to the task. Crucial in accomplishing this goal is the introduction of certain operators whose action on differential *p*-forms mimics tensor operations without ever writing spacetime indices explicitly. In this paper, we use Cartan’s formalism to parametrize linear and second-order perturbations on the vielbein and the spin connection in a manner that clearly separates the metric from the affine degrees of freedom, while remaining well-suited for practical computations.

As part of an earlier work, in Ref. [17] we presented a quick introduction on how this can be accomplished to linear order. In this paper, we will show in detail how it is done up to second order in perturbations. Going to second order is essential in order to construct realistic astrophysical models. As explained in Chapter 1.4 of Ref. [18], we have to be able to split the field equations as a low-frequency background, \(g_{\mu \nu }\), plus a high-frequency term, \(h_{\mu \nu }\). Since the product of terms of similar high frequencies can give rise to low-frequency effects, we have to consider linear and quadratic terms in \(h_{\mu \nu }\) when constructing \(\bar{g}^{\mu \nu }\), \(\bar{\varGamma }_{\mu \nu }^{\lambda }\), \(\bar{R}^{\rho }_{\,\, \sigma \mu \nu }\), etc., in the context of Riemannian geometry. This means that second-order terms become essential to define an effective stress-energy tensor for the perturbations.

The current article deals only with *kinematics*. We will not construct any model, and we will not discuss any dynamics. The only purpose of this article is to provide a practical tool for anyone interested in studying geometry perturbations on RC geometry with nonvanishing torsion. Our results hold for any theory based on RC geometry.

Besides the graviton, there is a new mode, the “torsionon.”^{1} Whether or not it propagates depends on the choice of Lagrangian. For instance, in standard ECSK gravity and other similar theories [20], this new mode will not propagate in a vacuum. However, in nonminimally coupled theories (see, e.g., Refs. [17, 21]) or the MacDowell–Mansouri theory [22, 23], it will.

## 2 Geometry perturbations on a spacetime with torsion

### 2.1 First-order formalism

*M*be a spacetime manifold and let \(g_{\mu \nu }\) be the coordinate components of its metric tensor at some point

*P*. The orthonormal coframe \(e^{a}_{\,\, \mu }\) is implicitly defined through the relation

*vielbein*for the collection of one-forms given by \(e^{a} = e^{a}_{\,\, \mu } \mathrm {d} x^{\mu }\). Since knowledge of the vielbein at every spacetime point is equivalent to knowing the metric at every spacetime point, one can shift focus from \(g_{\mu \nu }\) to \(e^{a}\) when studying the spacetime geometry. The use of differential forms guarantees that the vielbein remains invariant under general coordinate transformations.

A crucial stage in the development of the first-order formalism comes with the realization that the orthonormal coframe at *P* is not unique. In terms of the vielbein, this means that the rotated vielbein \(e'^{a} = \varLambda ^{a}_{\,\, b} e^{b}\) is associated with the same spacetime metric \(g_{\mu \nu }\) as \(e^{a}\) [via Eq. (3)] provided that \(\varLambda ^{a}_{\,\, b}\) represents a Lorentz transformation, satisfying \(\eta _{ab} = \varLambda ^{c}_{\,\, a} \varLambda ^{d}_{\,\, b} \eta _{cd}\). Such a local Lorentz rotation amounts to a gauge transformation on \(e^{a}\), with the vielbein behaving as a Lorentz vector (of one-forms).

*spin connection*

^{2}\(\omega ^{ab}\) and define the Lorentz-covariant exterior derivative of the vielbein (and similarly for any Lorentz vector form) as

*torsion*two-form, \(T^{a} = \mathrm {D} e^{a}\).

*Lorentz curvature*. Despite being defined in terms of the non-tensorial spin connection, the Lorentz curvature transforms as a tensor under local Lorentz transformations.

No new objects appear by further application of the covariant exterior derivative, since (as can be easily shown) \(\mathrm {D} R^{ab} = 0\).

*affine*connection \(\varGamma ^{\lambda }_{\mu \nu }\). A direct link between the two can be established by means of the so-called “vielbein postulate,”

*contorsion*. The torsionless piece is uniquely determined [via Eq. (7)] in terms of the vielbein and its derivatives, meaning that the contorsion, defined as the difference \(\kappa ^{ab} = \omega ^{ab} - \mathring{\omega }^{ab}\), carries all affine degrees of freedom within itself. Torsion and contorsion are related by \(T^{a} = \kappa ^{a}_{\,\, b} \wedge e^{b}\).

*Riemann curvature*two-form \(\mathring{R}^{ab}\) as the purely metric concoction [cf. Eq. (5)]

With two connections, two curvatures, and two covariant exterior derivatives, things may get cumbersome. A large part of what follows is an attempt to keep the complications at bay by carefully parametrizing perturbations on the geometry in such a way as to easily track the metric and affine degrees of freedom involved, while keeping calculations as simple as possible.

### 2.2 Default parametrization of perturbations

Having established the relation between the metric and the vielbein perturbations, the following course of action presents itself to us. Let \(L = L \left( e^{a}, \omega ^{ab} \right) \) be the four-form Lagrangian density which defines the theory, so that its field equations can be written as \(\delta L / \delta e^{a} = 0\) and \(\delta L / \delta \omega ^{ab} = 0\). By replacing Eqs. (10)–(11) into the field equations, one might attempt to solve \(\delta L / \delta \omega ^{ab} = 0\) for \(u^{ab}\) and replace the result in \(\delta L / \delta e^{a} = 0\), thus eliminating \(u^{ab}\) from consideration. Experience shows this course of action to be an algebraic nightmare even in simple cases.

*d*-dimensional manifold with arbitrary signature.

This is certainly progress; we have managed to split the spin connection perturbation into a torsionless piece \(\mathring{u}^{ab}\) plus an independent contorsion perturbation \(q^{ab}\), and to cast \(\mathring{u}^{ab}\) as a series in \(H^{a}\), with the first two terms given by Eqs. (22)–(23). Unfortunately, this is not yet good enough. To see why, note that Eqs. (22)–(23) give us \(\mathring{u}^{ab}\) in terms of the torsionless covariant derivative \(\mathring{\mathrm {D}} = \mathrm {d} + \mathring{\omega }\). The field equations, however, use \(\mathrm {D} = \mathrm {d} + \omega \) and the Lorentz curvature \(R^{ab}\) instead of the Riemann curvature \(\mathring{R}^{ab}\). Mixing both operators and curvatures proves to be a recipe for algebraic disaster.

### 2.3 Useful parametrization of perturbations

Torsion is most naturally handled through the first-order formalism

Background | Perturbation | ||
---|---|---|---|

Default | Useful | ||

Independent | \(\kappa ^{ab}\) | \(q^{ab}\) | \(V^{ab}\) |

Derived | \(\mathring{\omega }^{ab}\) | \(\mathring{u}^{ab}\) | \(U^{ab}\) |

Mixed (sum) | \(\omega ^{ab}\) | \(u^{ab}\) | \(u^{ab}\) |

## 3 Final remarks

*H*-dependent contribution to the spin connection perturbation is given by [cf. Eqs. (32)–(34)]

*H*-independent torsional mode. See Appendix A for the definition of the \(\mathrm {I}\)-operators.

## Footnotes

## Notes

### Acknowledgements

We are grateful to José Barrientos, Jens Boos, Oscar Castillo-Felisola, Fabrizio Cordonier-Tello, Cristóbal Corral, Nicolás González, Perla Medina, Daniela Narbona, Julio Oliva, Francisca Ramírez, Patricio Salgado, Sebastián Salgado, Jorge Zanelli, and Alfonso Zerwekh for many enlightening conversations. FI acknowledges financial support from the Chilean government through Fondecyt grants 1150719 and 1180681. OV acknowledges VRIIP UNAP for financial support through Project VRIIP0258-18.

## References

- 1.B.P. Abbott et al., Phys. Rev. Lett.
**119**(16), 161101 (2017). https://doi.org/10.1103/PhysRevLett.119.161101 ADSCrossRefGoogle Scholar - 2.J.M. Ezquiaga, M. Zumalacárregui, Front. Astron. Space Sci.
**5**, 44 (2018). https://doi.org/10.3389/fspas.2018.00044 ADSCrossRefGoogle Scholar - 3.J.M. Ezquiaga, M. Zumalacárregui, Phys. Rev. Lett.
**119**(25), 251304 (2017). https://doi.org/10.1103/PhysRevLett.119.251304 ADSCrossRefGoogle Scholar - 4.J. Sakstein, B. Jain, Phys. Rev. Lett.
**119**(25), 251303 (2017). https://doi.org/10.1103/PhysRevLett.119.251303 ADSCrossRefGoogle Scholar - 5.S. Mukherjee, S. Chakraborty, Phys. Rev. D
**97**(12), 124007 (2018). https://doi.org/10.1103/PhysRevD.97.124007 ADSMathSciNetCrossRefGoogle Scholar - 6.T.W.B. Kibble, J. Math. Phys.
**2**, 212 (1961). https://doi.org/10.1063/1.1703702 ADSCrossRefGoogle Scholar - 7.D.W. Sciama, Rev. Mod. Phys.
**36**, 463 (1964). https://doi.org/10.1103/RevModPhys.36.1103. [Erratum: Rev. Mod. Phys. 36, 1103 (1964)]ADSCrossRefGoogle Scholar - 8.F.W. Hehl, P. von der Heyde, G.D. Kerlick, J.M. Nester, Rev. Mod. Phys.
**48**, 393 (1976). https://doi.org/10.1103/RevModPhys.48.393 ADSCrossRefGoogle Scholar - 9.F.W. Hehl, J.D. McCrea, E.W. Mielke, Y. Ne’eman, Phys. Rept.
**258**, 1 (1995). https://doi.org/10.1016/0370-1573(94)00111-F ADSCrossRefGoogle Scholar - 10.A. Mardones, J. Zanelli, Class. Quant. Grav.
**8**, 1545 (1991). https://doi.org/10.1088/0264-9381/8/8/018 ADSCrossRefGoogle Scholar - 11.M. Blagojević, F.W. Hehl (eds.) Gauge Theories of Gravitation: A Reader With Commentaries. Classification of Gauge Theories of Gravity (World Scientific, Singapore, 2013). https://doi.org/10.1142/p781 Google Scholar
- 12.J. Boos, F.W. Hehl, Int. J. Theor. Phys.
**56**(3), 751 (2017). https://doi.org/10.1007/s10773-016-3216-3 CrossRefGoogle Scholar - 13.M. Blagojević, B. Cvetković, Y.N. Obukhov, Phys. Rev. D
**96**(6), 064031 (2017). https://doi.org/10.1103/PhysRevD.96.064031 ADSMathSciNetCrossRefGoogle Scholar - 14.M. Blagojević, B. Cvetković, Phys. Rev. D
**98**, 024014 (2018). https://doi.org/10.1103/PhysRevD.98.024014 ADSCrossRefGoogle Scholar - 15.G.K. Karananas, Class. Quant. Grav.
**32**(5), 055012 (2015). https://doi.org/10.1088/0264-9381/32/5/055012. [Erratum: Class. Quant. Grav. 32(8), 089501 (2015). https://doi.org/10.1088/0264-9381/32/8/089501] - 16.J. Zanelli. In: Proceedings, 7th Mexican Workshop on Particles and Fields (MWPF 1999): Merida, Mexico, November 10–17, 1999 (2005)Google Scholar
- 17.J. Barrientos, F. Cordonier-Tello, F. Izaurieta, P. Medina, D. Narbona, E. Rodríguez, O. Valdivia, Phys. Rev. D
**96**(8), 084023 (2017). https://doi.org/10.1103/PhysRevD.96.084023 ADSMathSciNetCrossRefGoogle Scholar - 18.M. Maggiore,
*Gravitational Waves: Volume 1: Theory and Experiments*(Oxford University Press, Oxford, UK, 2007). https://doi.org/10.1093/acprof:oso/9780198570745.001.0001 - 19.F.W. Hehl, Four Lectures on Poincaré Gauge Field Theory, in
*Cosmology and Gravitation: Spin, Torsion, Rotation, and Supergravity*. NATO Advanced Study Institutes Series, ed. by P.G. Bergmann, V. De Sabbata (Springer, Boston, MA, USA, 1980), pp. 5–61. https://doi.org/10.1007/978-1-4613-3123-0 Google Scholar - 20.K. Shimada, K. Aoki, K.i. Maeda, (2018). Preprint arXiv:1812.03420 [gr-qc]
- 21.Y. Bonder, C. Corral, Phys. Rev. D
**97**(8), 084001 (2018). https://doi.org/10.1103/PhysRevD.97.084001 ADSMathSciNetCrossRefGoogle Scholar - 22.S.W. MacDowell, F. Mansouri, Phys. Rev. Lett.
**38**, 739 (1977). https://doi.org/10.1103/PhysRevLett.38.739 ADSMathSciNetCrossRefGoogle Scholar - 23.D.K. Wise, Class. Quant. Grav.
**27**, 155010 (2010). https://doi.org/10.1088/0264-9381/27/15/155010 ADSCrossRefGoogle Scholar

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