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Geodesically completing regular black holes by the Simpson–Visser method

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Abstract

Regular black holes are often geodesically incomplete when their extensions to negative values of the radial coordinate are considered. Here, we propose to use the Simpson–Visser method of regularising a singular spacetime, and apply it to a regular solution that is geodesically incomplete, to construct a geodesically complete regular solution. Our method is generic, and can be used to cure geodesic incompleteness in any spherically symmetric static regular solution, so that the resulting solution is symmetric in the radial coordinate. As an example, we illustrate this procedure using a regular black hole solution with an asymptotic Minkowski core. We study the structure of the resulting metric, and show that it can represent a wormhole or a regular black hole with a single or double horizon per side of the throat. Further, we construct a source Lagrangian for which the geodesically complete spacetime is an exact solution of the Einstein equations, and show that this consists of a phantom scalar field and a nonlinear electromagnetic field. Finally, gravitational lensing properties of the geodesically complete spacetime are briefly studied.

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Notes

  1. The SV method has gained a lot of recent attention. For a non-exhaustive list of recent works, see [20,21,22,23,24,25,26,27,28,29,30,31,32,33].

  2. To the best of our knowledge, the phenomenological version of this metric was first proposed in [11], along with the rotating version in [12]. However, the full significance of this metric, from carefully chosen first principles was elucidated in [10, 34, 35].

  3. Note that, in the following we shall choose the parameter values in such a way that this equation has only two real roots.

  4. Since the metric extension is symmetric about \(r=0\) there are two horizons for negative values of r, at \(-r_{+}\) and \(-r_{-}\) respectively, as well.

  5. Notice that though sign of \(\rho +p_r\) depends on the metric function \({\mathcal {A}}(r)\), once we have made the above choice of the scalar field, the sign of the function h(r) is determined only by the areal radius \({\mathcal {B}}(r)\) and its second derivative.

  6. For the importance of the photon sphere and its role in gravitational lensing, see [38,39,40,41, 44].

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Acknowledgements

We thank Bidyut Dey and Rajibul Shaikh for discussions and help, and Tian Zhou for a useful correspondence. We also thank the anonymous referee for various constructive suggestions, which helped to improve the presentation of the paper.

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  1. Department of Physics, Indian Institute of Technology Kanpur, Kanpur, 208016, India

    Kunal Pal, Kuntal Pal & Tapobrata Sarkar

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  1. Kunal Pal
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Correspondence to Kuntal Pal.

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Pal, K., Pal, K. & Sarkar, T. Geodesically completing regular black holes by the Simpson–Visser method. Gen Relativ Gravit 55, 121 (2023). https://doi.org/10.1007/s10714-023-03168-7

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