A non-trivial connection for the metric-affine Gauss-Bonnet theory in \(D = 4\). (English) Zbl 1421.83084
Summary: We study non-trivial (i.e., non-Levi-Civita) connections in metric-affine Lovelock theories. First, we study the projective invariance of general Lovelock actions and show that all connections constructed by acting with a projective transformation of the Levi-Civita connection are allowed solutions, albeit physically equivalent to Levi-Civita. We then show that the (non-integrable) Weyl connection is also a solution for the specific case of the four-dimensional metric-affine Gauss-Bonnet action, for arbitrary vector fields. The existence of this solution is related to a two-vector family of transformations, that leaves the Gauss-Bonnet action invariant when acting on metric-compatible connections. We argue that this solution is physically inequivalent to the Levi-Civita connection, giving thus a counterexample to the statement that the metric and the Palatini formalisms are equivalent for Lovelock gravities. We discuss the mathematical structure of the set of solutions within the space of connections.
MSC:
| 83D05 | Relativistic gravitational theories other than Einstein’s, including asymmetric field theories |
| 53Z05 | Applications of differential geometry to physics |
| 83C10 | Equations of motion in general relativity and gravitational theory |
| 53B15 | Other connections |
Software:
xActReferences:
| [1] | Dadhich, N.; Pons, J. M., Gen. Relativ. Gravit., 44, 2337 (2012) · Zbl 1250.83047 |
| [2] | Bernal, A. N.; Janssen, B.; Jiménez-Cano, A.; Orejuela, J. A.; Sánchez, M.; Sánchez-Moreno, P., Phys. Lett. B, 768, 280-287 (2017) · Zbl 1370.53062 |
| [3] | Janssen, B.; Jiménez-Cano, A.; Orejuela, J. A.; Sánchez-Moreno, P., (Non-)uniqueness of Einstein-Palatini gravity |
| [4] | Eisenhart, L. P., Non-Riemannian Geometry (1927), American Mathematical Society: American Mathematical Society New York · JFM 53.0681.02 |
| [5] | Julia, B.; Silva, S., Class. Quantum Gravity, 15, 2173 (1998) · Zbl 0935.83001 |
| [6] | Capozziello, S.; De Laurentis, M., Phys. Rep., 509, 167 (2011) |
| [7] | Cotsakis, S.; Miritzis, J.; Querella, L., J. Math. Phys., 40, 3063 (1999) · Zbl 0948.83050 |
| [8] | Querella, L., Variational principles and cosmological models in higher-order gravity · Zbl 1161.83310 |
| [9] | Allemandi, G.; Borowiec, A.; Francaviglia, M.; Odintsov, S. D., Phys. Rev. D, 72, Article 063505 pp. (2005) |
| [10] | Sotiriou, T. P.; Liberati, S., Ann. Phys., 322, 935 (2007) · Zbl 1112.83049 |
| [11] | Li, B.; Barrow, J. D.; Mota, D. F., Phys. Rev. D, 76, Article 104047 pp. (2007) |
| [12] | Bauer, F.; Demir, D. A., Phys. Lett. B, 665, 222-226 (2008) |
| [13] | Capozziello, S.; Darabi, F.; Vernieri, D., Mod. Phys. Lett. A, 26, 65-72 (2011) · Zbl 1208.83087 |
| [14] | Bauer, F., Class. Quantum Gravity, 28, Article 225019 pp. (2011) · Zbl 1230.83096 |
| [15] | Olmo, G., Introduction to Palatini theories of gravity and nonsingular cosmologies |
| [16] | Olmo, G. J.; Rubiera-García, D.; Sánchez-Puente, A., Eur. Phys. J. C, 76, 3, 14 (2016) |
| [17] | Olmo, G. J.; Rubiera-García, D., Universe, 1, 2, 173 (2015) |
| [18] | Bambi, C.; Cárdenas-Avendano, A.; Olmo, G. J.; Rubiera-García, D., Phys. Rev. D, 93, 6, Article 064016 pp. (2016) |
| [19] | Borowiec, A.; Stachowski, A.; Szydłowski, M.; Wojnar, A., J. Cosmol. Astropart. Phys., 01, Article 040 pp. (2016) |
| [20] | Bejarano, C.; Olmo, G. J.; Rubiera-García, D., Phys. Rev. D, 95, 6, Article 064043 pp. (2017) |
| [21] | Szydłowski, M.; Stachowski, A., Phys. Rev. D, 97, Article 103524 pp. (2018) |
| [22] | Exirifard, Q.; Sheikh-Jabbari, M. M., Phys. Lett. B, 661, 158-161 (2008) · Zbl 1246.83167 |
| [23] | Borunda, M.; Janssen, B.; Bastero-Gil, M., J. Cosmol. Astropart. Phys., 0811, Article 008 pp. (2008) |
| [24] | Dadhich, N.; Pons, J. M., Phys. Lett. B, 705, 139-142 (2011) |
| [25] | Zumino, B., Phys. Rep., 137, 109 (1986) |
| [26] | Capozziello, S.; Darabi, F.; Vernieri, D., Mod. Phys. Lett. A, 25, 3279-3289 (2010) · Zbl 1208.83086 |
| [27] | Iosifidis, D.; Koivisto, T., Scale transformations in metric-affine geometry |
| [28] | Borowiec, A.; Ferraris, M.; Francaviglia, M.; Volovich, I., Class. Quantum Gravity, 15, 43-55 (1998) · Zbl 0908.53049 |
| [29] | Beltrán Jimenez, J.; Koivisto, T. S., Class. Quantum Gravity, 31, Article 135002 pp. (2014) · Zbl 1297.83037 |
| [30] | Beltrán Jimenez, J.; Koivisto, T. S., Phys. Lett. B, 756, 400-404 (2016) · Zbl 1400.83042 |
| [31] | Beltrán Jimenez, J.; Heisenberg, L.; Koivisto, T. S., J. Cosmol. Astropart. Phys., 1604, Article 04 pp. (2016), 046 |
| [32] | Iosifidis, D.; Petkou, A. C.; Tsagas, C. G., Torsion/non-metricity duality in f(R) gravity · Zbl 1419.83050 |
| [33] | Martín-García, J. M., xAct: efficient tensor computer algebra for the Wolfram Language |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
