Abstract
We show that the path of any accelerated body in an arbitrary spacetime geometry \(g_{\mu \nu }\) can be described as a geodesic in a dragged metric \(\hat{q}_{\mu \nu }\) that depends only on the background metric and on the motion of the body. Such procedure allows the interpretation of all kinds of non-gravitational force as modifications of the spacetime metric. This method of effective elimination of the forces by changing the metric of the substratum can be understood as a generalization of the d’Alembert principle applied to all relativistic processes.
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Notes
Let us point out that the analysis we present in this section may be straightforwardly generalized to arbitrary curved Riemannian background.
Using the freedom in the definition of the four-vector \(v^{\mu }\), we set \(\eta _{\mu \nu }v^{\mu }v^{\nu } = 1.\) The acceleration is orthogonal to it, that is, \( a_{\mu } \, v^{\mu }=0\). We note that we are dealing with a collection of paths \(\Gamma \) that is usually called a congruence of curves. It is understood that each element of this collection concerns particles that have the same characteristics. For instance, if the acceleration is due to electromagnetic field, all particles of \(\Gamma \) must have the same charge-mass ratio, to wit a bunch of electrons.
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Acknowledgments
We would like to thank Dr Ivano Damião Soares for his comments in a previous version of this paper. We would like to thank FINEP, CNPq and FAPERJ for their partial financial support.
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M. Novello is Cesare Lattes ICRANet Professor.
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Novello, M., Bittencourt, E. Dragged metrics. Gen Relativ Gravit 45, 1005–1019 (2013). https://doi.org/10.1007/s10714-013-1507-z
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DOI: https://doi.org/10.1007/s10714-013-1507-z