High-derivatives and massive electromagnetic models in the Lemaître–Tolman–Bondi spacetime

Eur. Phys. J. C (2020) 80:240 https://doi.org/10.1140/epjc/s10052-020-7787-z Regular Article - Theoretical Physics High-derivatives and massive electromagnetic models in the Lemaître–Tolman–Bondi spacetime Rafael L. Fernandes1,a , Everton M. C. Abreu2,3,4,b , Marcelo B. Ribeiro4,5,c 1 Instituto Federal de Educação, Ciência e Tecnologia da Bahia, Campus Juazeiro, Rodovia BA 210, S/N, Bairro Nova Juazeiro, Juazeiro, BA 48918-900, Brazil 2 Departamento de Física, Universidade Federal Rural do Rio de Janeiro, Seropédica, RJ 23890-971, Brazil 3 Departamento de Física, Universidade Federal de Juiz de Fora, Juiz de Fora, MG 36036-330, Brazil 4 Programa de Pós-Graduação Interdisciplinar em Física Aplicada, Instituto de Física, Universidade Federal do Rio de Janeiro, Rio de Janeiro, RJ 21941-972, Brazil 5 Instituto de Física, Universidade Federal do Rio de Janeiro, Rio de Janeiro, RJ 21941-972, Brazil Received: 19 January 2020 / Accepted: 21 February 2020 © The Author(s) 2020 Abstract The Maxwell electromagnetic theory embedded in an inhomogeneous Lemaître–Tolman–Bondi (LTB) spacetime background was described a few years back in the literature. However, terms concerning the mass or high-derivatives were no explored. In this work we studied the inhomogeneous spacetime effects on high-derivatives and massive electromagnetic models. We used the LTB metric and calculated the physical quantities of interest, namely the scale factor, density of the eletromagnetic field and Hubble constant, for the Proca and higher-derivative Podolsky models. We found a new singularity in both models, and that the magnetic field must be zero in the Proca model. 1 Introduction Homogeneity and isotropy, together with matter being treated as a specific gas, are the basic ingredients of the Friedmann– Lemaître–Robertson–Walker (FLRW) model. Since the pioneering work of E. Hubble, who suggested a correlation between the observed redshifts and distances of 24 galaxies, this is a very successfull model, being currently considered a very good measure to describe our cosmos. In the last decades a great effort was dedicated to understanding the local inhomogeneities that occur in the Universe. Such efforts suggest that these inhomogeneities may be related to the expansion of the Universe. Inhomogeneities, as an alternative to dark energy, were first discussed in Ref. [1] as a means of explaning the observational results of the a e-mail: b e-mail: (corresponding author) c e-mail: 0123456789().: V,-vol expansion of the Universe [2–8] without the need for postulating dark energy. Inhomogeneities inside astronomical objects can define different instability ranges, an effect that can describe distinct features of their evolution and structures formation. The current standard model of cosmology, the -Cold Dark Matter (CDM) model, is a homogeneous solutions of the FLRW Einstein’s field equations with only six free parameters that successfully accounts for most cosmological data, especially the characteristics of the Cosmic Microwave Background (CMB) and the structure formation on large scales considered through the theory of cosmological perturbations in homogeneous and isotropic background. However, in the last fifteen years “standard” inhomogeneous cosmological models that are generalizations of FLRW cosmologies have been the subject of growing interest in astrophysics community in order to investigate cosmological phenomena. Some authors have demonstrated that inhomogeneous models with spherical symmetry and dust source can be fitted to supernovae Ia (SNIa) data, as well as the position of the first peak of the CMB. These models show that the apparent accelerated expansion of the universe may not be a consequence of the repulsive gravity due to dark energy, but rather the result of inhomogeneities in the distribution of matter. In this context one needs to mention that an important cosmological model to describe the inhomogeneity universe is the Lemaître–Tolman–Bondi (LTB) spacetime [11–19], which is a spatially inhomogeneous description of a spherically symmetric distribution of dust matter in the Universe. In the last few years several studies analyzing the scenario of electrodynamics embedded in the isotropic and homogeneous FLRW gravitational background models [20] have been produced. Astrophysical effects were explored together 123 240 Page 2 of 11 with an anisotropic expansion of the Universe. The interesting result of polarized electromagnetic radiation occurs when it travels through local anisotropic regions. Concerning the inhomogeneities, in Ref. [21] the authors investigated that the inhomogeneity with electromagnetic field caused a new scale factor. The propagation of photons was also affected, which is important phenomenum since most information obtained about the Universe is by means of photons. Reference [22] studied the effects of Palatini f (R) gravity together with the so-called tilted observer on the dynamics of LTB spacetime embedded in an electromagnetic field. In this work we studied the effect of a high-derivative electromagnetic field embedded in the inhomogeneous LTB geometry. We analyzed the Proca and high-derivatives Podolsky electromagnetic models. The electrodynamic contribution was inserted separately through the energy-momentum tensor (EMT) of the respective models. For the Proca model, besides the matter density there also are the electromagnetic contributions obtained from the Lagrangian of the Proca model in curved space-time. An analysis of the Proca model in curved space-time was carried out for a particular case of interest by Bekenstein [9]. Here, we have analyzed the electrodynamics effects in LTB cosmological model and calculate the scale factor in LTB universe. We also computed the luminosity distance in the presence of electromagnetic field. Concerning the Podolsky electrodynamics effects in LTB cosmological model, the study of Podolsky model in curved space-time was made in Ref. [10] together with the analysis of the Bopp–Podolsky black holes. Therefore, we have obtained the line element and the equations that define the LTB model with Podolsky contributions. This paper is structured as follows. In Sect. 2 we review the main aspects of the LTB metric, where the Einstein tensor and the matter contribution for the EMT are obtained. In Sect. 3 we analyze the Proca model of electrodynamics in curved space-time, where the EMT and the Maxwell–Proca equations in curved space-time are derived. In Sect. 4 we solve the Einstein equations including the Proca contribution. We obtain the scale factor for this model and examine the inhomogeneities and luminosity distance. In Sect. 5 we analyze the Podolsky electrodynamics contributions for LTB model. Section 6 discusses the results and presents some final considerations. 2 The LTB cosmological model: a brief review The LTB model depicts a self-gravitating spherically symmetric distribution of inhomogeneous nondissip (...truncated)