We have found solutions of this kind in more general theories, for example, in
Lovelock gravity. In addition, if we admit matter that does not feel the connection,
the equivalence between formalisms can be extended from Einstein–Hilbert to any
theory with additional quadratic torsion terms in the action. Current work involves
the treatment with more general matter and with additional terms that introduce, for
example, dynamics for the torsion field.
Acknowledgements We are grateful to A. Bernal and M. Sánchez for previous collaborations. We
also wish to thank F. W. Hehl and J. Pons for useful comments and discussions. This work has been
funded by the Junta de Andalucía (FQM101), the Universidad de Granada (PP2015-03), the MEC
(FIS2016-78198-P) and the Unidad de Excelencia UCE-PP2016-02. In addition, J. A. O. was also
supported by a PhD contract of the Plan Propio of the Universidad de Granada, and A. J. C. by a
PhD contract of the FPU15 programme of the MEC.
References
Bernal A et al (2017) On the (non-) uniqueness of the Levi-Civita solution in the Einstein-HilbertPalatini formalism. Phys Lett B 768:280–287
Borunda M, Janssen B, Bastero-Gil M (2008) Palatini versus metric formulation in higher curvature
gravity. J Cosmol Astropart Phys 0811:008
Dadhich N, Pons JM (2012) On the equivalence of the Einstein-Hilbert and the Einstein-Palatini
formulations of general relativity for an arbitrary connection. Gen Relat Gravit 44
(9):2337–2352
Deser S (1996) Inequivalence of first and second-order formulations in d¼2 gravity models. Found
Phys 26(5):617–621
Hehl FW, Obukhov YN (2007) Elie Cartan’s torsion in geometry and field theory, an essay. Ann
Fond Broglie 32:157–194
Kleinert H, Pelster A (1999) Letter: autoparallels from a new action principle. Gen Relat Gravit 31
(9):1439
McCrea JD (1992) Irreducible decompositions of nonmetricity, torsion, curvature and Bianchi
identities in metric-affine spacetimes. Class Quant Gravit 9(2):553–568
5 (Non-)Uniqueness of Einstein–Palatini Gravity
59
Lovelock gravity. In addition, if we admit matter that does not feel the connection,
the equivalence between formalisms can be extended from Einstein–Hilbert to any
theory with additional quadratic torsion terms in the action. Current work involves
the treatment with more general matter and with additional terms that introduce, for
example, dynamics for the torsion field.
Acknowledgements We are grateful to A. Bernal and M. Sánchez for previous collaborations. We
also wish to thank F. W. Hehl and J. Pons for useful comments and discussions. This work has been
funded by the Junta de Andalucía (FQM101), the Universidad de Granada (PP2015-03), the MEC
(FIS2016-78198-P) and the Unidad de Excelencia UCE-PP2016-02. In addition, J. A. O. was also
supported by a PhD contract of the Plan Propio of the Universidad de Granada, and A. J. C. by a
PhD contract of the FPU15 programme of the MEC.
References
Bernal A et al (2017) On the (non-) uniqueness of the Levi-Civita solution in the Einstein-HilbertPalatini formalism. Phys Lett B 768:280–287
Borunda M, Janssen B, Bastero-Gil M (2008) Palatini versus metric formulation in higher curvature
gravity. J Cosmol Astropart Phys 0811:008
Dadhich N, Pons JM (2012) On the equivalence of the Einstein-Hilbert and the Einstein-Palatini
formulations of general relativity for an arbitrary connection. Gen Relat Gravit 44
(9):2337–2352
Deser S (1996) Inequivalence of first and second-order formulations in d¼2 gravity models. Found
Phys 26(5):617–621
Hehl FW, Obukhov YN (2007) Elie Cartan’s torsion in geometry and field theory, an essay. Ann
Fond Broglie 32:157–194
Kleinert H, Pelster A (1999) Letter: autoparallels from a new action principle. Gen Relat Gravit 31
(9):1439
McCrea JD (1992) Irreducible decompositions of nonmetricity, torsion, curvature and Bianchi
identities in metric-affine spacetimes. Class Quant Gravit 9(2):553–568
5 (Non-)Uniqueness of Einstein–Palatini Gravity
59