The Palatini formalism of the theory of gravity

J. G. de Lima; P. H. R. S. Moraes; E. Brito; J. A. S. Fortunato

doi:10.1140/epjc/s10052-024-13731-4

2023 Impact factor 4.2

Particles and Fields

Open Access

Eur. Phys. J. C (2025) 85: 38
https://doi.org/10.1140/epjc/s10052-024-13731-4

Regular Article - Theoretical Physics

The Palatini formalism of the $f (R, L_{m}, T)$ $f(R,{\mathcal {L}}_{m},T)$ theory of gravity

J. G. de Lima Jr.¹^a, P. H. R. S. Moraes², E. Brito³ and J. A. S. Fortunato⁴

¹ Instituto de Física, Universidade Federal do Alagoas (UFAL), Avenida Lourival Melo Mota S/N, 57072-970, Maceió, AL, Brazil
² Laboratório de Física Teórica e Computacional (LFTC), Universidade Cidade de São Paulo (UNICID), Rua Galvão Bueno 868, 01506-000, São Paulo, Brazil
³ Centro de Ciências Exatas e das Tecnologias, Universidade Federal do Oeste da Bahia, Rua Bertioga 892, 47810-059, Barreiras, BA, Brazil
⁴ Centro de Ciências Exatas, Departamento de Física, PPGCosmo, Universidade Federal do Espírito Santo (UFES), Avenida Fernando Ferrari 514, 29075-910, Vitória, ES, Brazil

^a grimario.lima@fis.ufal.br

Received: 25 November 2024
Accepted: 19 December 2024
Published online: 21 January 2025

Abstract

We present the first formulation of the recently proposed $f (R, L_{m}, T)$ $f(R,{\mathcal {L}}_m,T)$ theory of gravity within the Palatini formalism, a well-known alternative variational approach where the metric and connection are treated as independent variables. By applying this formalism, we derive a new set of field equations that exhibit, as expected, distinct properties compared to their metric formalism counterparts. We particularly present the Newtonian limit of this formalism, as well as the resulting Friedmann-like equations. We highlight that potential observational signatures may distinguish between the metric and Palatini frameworks. Our results open new pathways for exploring the phenomenology of modified gravity theories and their testability with observational data.

© The Author(s) 2025

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