Spatiotemporal jumps as particular solutions in geodesic trajectories with the Gödel metric on an extended manifold

Mauricio Bellini

doi:10.1140/epjc/s10052-022-10786-z

2023 Impact factor 4.2

Particles and Fields

Open Access

Eur. Phys. J. C (2022) 82: 817
https://doi.org/10.1140/epjc/s10052-022-10786-z

Regular Article - Theoretical Physics

Spatiotemporal jumps as particular solutions in geodesic trajectories with the Gödel metric on an extended manifold

Mauricio Bellini¹^,2^a

¹ Departamento de Física, Facultad de Ciencias Exactas y Naturales, Universidad Nacional de Mar del Plata, Funes 3350, C.P. 7600, Mar del Plata, Argentina
² Instituto de Investigaciones Físicas de Mar del Plata (IFIMAR), Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET), Mar del Plata, Argentina

^a mbellini@mdp.edu.ar

Received: 17 May 2022
Accepted: 7 September 2022
Published online: 15 September 2022

Abstract

Using the Gödel metric, we obtain some relevant solutions compatible with spatiotemporal jumps for the geodesic equations, by using an extension of General Relativity with nonzero boundary terms, which are described on an extended manifold generated by the connections $δ Γ_{α β}^{μ} = b U^{μ} g_{α β}$ $\delta \Gamma ^{\mu }_{\alpha \beta } = b\,U^{\mu }\,g_{\alpha \beta }$ . These terms are given by a flow of velocities with components $U^{ν}$ $U^{\nu }$ : $3 b^{2} \nabla_{ν} U^{ν} = g^{α β} δ R_{α β} = λ [s (x^{α})] g^{α β} δ g_{α β}$ $3\,b^2\,\nabla _{\nu }U^{\nu }=g^{\alpha \beta }\, \delta R_{\alpha \beta } = \lambda \left[ s\left( x^{\alpha }\right) \right] \,g^{\alpha \beta }\, \delta g_{\alpha \beta }$ in the varied Einstein–Hilbert action. The solutions are valid for an arbitrary equation of state with ordinary matter: $Ω = P / (c^{2} ρ) = \frac{{(\frac{ω}{c})}^{2} - λ (s)}{{(\frac{ω}{c})}^{2} + λ (s)}$ $\Omega =P/(c^2\,\rho ) = \frac{\left( \frac{\omega }{c}\right) ^2-\lambda (s)}{\left( \frac{\omega }{c}\right) ^{2}+\lambda (s)}$ .

© The Author(s) 2022

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