On the (non-)uniqueness of the Levi-Civita solution in the Einstein–Hilbert–Palatini formalism
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AuthorBernal, Antonio N.; Janssen, Bert; Jiménez-Cano, Alejandro; Orejuela, José Alberto; Sánchez, Miguel; Sánchez Moreno, Pablo
Dynamics of a particleDifferentiable dynamical systemsMathematicsPhysics
Bernal, A.N.; et al. On the (non-)uniqueness of the Levi-Civita solution in the Einstein–Hilbert–Palatini formalism. Physics Letters B, 768: 280-287 (2017). [http://hdl.handle.net/10481/46542]
SponsorshipOpen Access funded by SCOAP³ - Sponsoring Consortium for Open Access Publishing in Particle Physics; The work of B.J. and J.A.O. was partially supported by the Junta de Andalucía (FQM101) and the Universidad de Granada (PP2015-03). J.A.O. is also supported by a PhD contract of the Plan Propio de la Universidad de Granada. M.S. has been partially financed by the Spanish Ministry of Economy and Competitiveness and European Regional Development Fund (ERDF) through the project MTM2016-78807-C2-1-P.
We study the most general solution for affine connections that are compatible with the variational principle in the Palatini formalism for the Einstein–Hilbert action (with possible minimally coupled matter terms). We find that there is a family of solutions generalising the Levi-Civita connection, characterised by an arbitrary, non-dynamical vector field AμAμ. We discuss the mathematical properties and the physical implications of this family and argue that, although there is a clear mathematical difference between these new Palatini connections and the Levi-Civita one, both unparametrised geodesics and the Einstein equation are shared by all of them. Moreover, the Palatini connections are characterised precisely by these two properties, as well as by other properties of its parallel transport. Based on this, we conclude that physical effects associated to the choice of one or the other will not be distinguishable, at least not at the level of solutions or test particle dynamics. We propose a geometrical interpretation for the existence and unobservability of the new solutions.