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The generalized Vaidya spacetime with polytropic equation of state

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Abstract

The process of the gravitational collapse might lead not only to a black hole but also to naked singularity formation. In this paper, we consider the generalized Vaidya spacetime with polytropic and generalized polytropic equations of state. We solve the Einstein and Einstein–Maxwell equations to obtain the explicit form of a mass function. We consider the limiting cases of solutions and find out, that generalized Vaidya spacetime might behave like Vaidya–de Sitter and Bonnor–Vaidya–de sitter solutions. Moreover, we explicitly show, that the part of solution, which depends on the polytropic index, is similar to cosmological fields surrounding both Vaidya and Bonnor–Vaidya black holes. The process of the gravitational collapse has been then considered. We have found out that the conditions of the naked singularity formation don’t depend on the polytropic index.

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Notes

  1. One should note that we can make this replacement only with the made assumption \(\gamma \ne 1\).

  2. One should note that there other possibilities of D(v) and the first two options of D(v) were chosen just to study the behavior of the mass function.

  3. Note, one can expand with respect to either \(\delta \) or \(\frac{D(v)}{2^{\gamma -1}\alpha r^{2-2\gamma }}\) and the result will be the same.

  4. Under notion ’eternal’ we mean that the singularity might form during the gravitational collapse and will never be covered with the apparent horizon.

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Acknowledgements

The work was performed as part of the SAO RAS government contract approved by the Ministry of Science and Higher Education of the Russian Federation.

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  1. Physics Department, Herzen State Pedagogical University of Russia, 48 Moika Emb., Saint Petersburg, Russia, 191186

    Vitalii Vertogradov

  2. SPb Branch of SAO RAS, 65 Pulkovskoe Rd, Saint Petersburg, Russia, 196140

    Vitalii Vertogradov

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Vertogradov, V. The generalized Vaidya spacetime with polytropic equation of state. Gen Relativ Gravit 56, 59 (2024). https://doi.org/10.1007/s10714-024-03244-6

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