Abstract
We present the Jacobi metric formalism for dynamical wormholes. We show that in isotropic dynamical spacetimes , a first integral of the geodesic equations can be found using the Jacobi metric, and without any use of geodesic equation. This enables us to reduce the geodesic motion in dynamical wormholes to a dynamics defined in a Riemannian manifold. Then, making use of the Jacobi formalism, we study the circular stable orbits in the Jacobi metric framework for the dynamical wormhole background. Finally, we also show that the Gaussian curvature of the family of Jacobi metrics is directly related, as in the static case, to the flare-out condition of the dynamical wormhole, giving a way to characterize a wormhole spacetime by the sign of the Gaussian curvature of its Jacobi metric only.
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Notes
Sometimes they are called cosmological wormholes.
We impose the condition \(a(t) >0\) in (2.1) to avoid cosmological singularities.
\(\delta =0\) for null geodesics, although we know that for timelike geodesics the equation of motion is going to be the same.
For static wormholes this condition is reduced to \(u_{c}^n=\frac{1}{b_{o}^n}\) which implies that the only circular orbit is located at the throat of the wormhole [4].
The form factor a(t) comes from the Friedman equation. We have set all constants equal to one.
Note that the Jacobi metric is a Riemannian metric.
References
Pin, O.C.: Curvature and mechanics. Adv. Math. 15(3), 269–311 (1975)
Gibbons, G.W.: The Jacobi-metric for timelike geodesics in static spacetimes. Class. Quant. Grav. 33(2), 025004 (2016). https://doi.org/10.1088/0264-9381/33/2/025004. arXiv:1508.06755 [gr-qc]
Das, P., Sk, R., Ghosh, S.: Motion of charged particle in Reissner–Nordström spacetime: a Jacobi-metric approach. Eur. Phys. J. C 77(11), 735 (2017). https://doi.org/10.1140/epjc/s10052-017-5295-6. arXiv:1609.04577 [gr-qc]
Argañaraz, M., Lasso Andino, O.: Dynamics in wormhole spacetimes: a Jacobi metric approach. Class. Quant. Grav. 38(4), 045004 (2021). https://doi.org/10.1088/1361-6382/abcf86. arXiv:1906.11779 [gr-qc]
Argañaraz, M.A., Andino, O.L.: The generalized Jacobi metric. arXiv:2112.10910 [gr-qc]
Chanda, S., Gibbons, G.W., Guha, P.: Jacobi–Maupertuis metric and Kepler equation. Int. J. Geom. Methods Mod. Phys. 14(07), 1730002 (2017). arXiv:1612.07395 [math-ph]
Chanda, S., Gibbons, G.W., Guha, P.: Jacobi–Maupertuis–Eisenhart metric and geodesic flows. J. Math. Phys. 58(3), 032503 (2017). arXiv:1612.00375
Bera, A., Ghosh, S., Majhi, B.R.: Hawking radiation in a non-covariant frame: the Jacobi metric approach. Eur. Phys. J. Plus 135(8), 670 (2020). https://doi.org/10.1140/epjp/s13360-020-00693-1. arXiv:1909.12607 [gr-qc]
Cataldo, M., del Campo, S., Minning, P., Salgado, P.: Evolving Lorentzian wormholes supported by phantom matter and cosmological constant. Phys. Rev. D 79, 024005 (2009). https://doi.org/10.1103/PhysRevD.79.024005. arXiv:0812.4436 [gr-qc]
Hayward, S.A.: Dynamic wormholes. Int. J. Mod. Phys. D 8, 373–382 (1999). https://doi.org/10.1142/S0218271899000286. arXiv:gr-qc/9805019 [gr-qc]
Rehman, M., Saifullah, K.: Thermodynamics of dynamical wormholes. JCAP 06, 020 (2021). https://doi.org/10.1088/1475-7516/2021/06/020. arXiv:2001.08457 [gr-qc]
Mishra, A., Chakraborty, S.: On the trajectories of null and timelike geodesics in different wormhole geometries. Eur. Phys. J. C 78(5), 374 (2018). https://doi.org/10.1140/epjc/s10052-018-5854-5. arXiv:1710.06791 [gr-qc]
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Duenas-Vidal, Á., Lasso Andino, O. The Jacobi metric approach for dynamical wormholes. Gen Relativ Gravit 55, 9 (2023). https://doi.org/10.1007/s10714-022-03060-w
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DOI: https://doi.org/10.1007/s10714-022-03060-w