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Two-oscillator Kantowski–Sachs model of the Schwarzschild black hole interior

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Abstract

In this paper the interior of the Schwarzschild black hole, which is presented as a vacuum, homogeneous and anisotropic Kantowski–Sachs minisuperspace cosmological model, is considered. Lagrangian of the model is reduced by a suitable coordinate transformation to Lagrangian of two decoupled oscillators with the same frequencies and with zero energy in total (an oscillator-ghost-oscillator system). The model is presented in a classical, a p-adic and a noncommutative case. Then, within the standard quantum approach Wheeler–DeWitt equation and its general solutions, i.e. a wave function of the model is written, and then an adelic wave function is constructed. Finally, thermodynamics of the model is studied by using the Feynman–Hibbs procedure.

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Acknowledgments

The work of G. Djordjevic and Lj. Nesic is partially supported by Ministry of Education, Science and Technological Development of the Republic of Serbia under Grant Nos. 174020 and 176021, and ICTP-SEENET-MTP Project PRJ09 Cosmology and Strings in the frame of the Southeastern European Network in Theoretical and Mathematical Physics, the work of D. Radovancevic is partially supported within this frame, as well. G. Djordjevic is thankful to the CERN-TH group for financial support and hospitality during his stay, where a part of this paper was finalized.

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Authors and Affiliations

  1. Faculty of Science and Mathematics, University of Nis, P.O. Box 224, Nis, 18000, Serbia

    Goran S. Djordjevic, Ljubisa Nesic & Darko Radovancevic

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  1. Goran S. Djordjevic
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  2. Ljubisa Nesic
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  3. Darko Radovancevic
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Correspondence to Darko Radovancevic.

Appendix: Coefficients \(\alpha _{ij}\) and \(K_i^\pm \)

Appendix: Coefficients \(\alpha _{ij}\) and \(K_i^\pm \)

If \(A=\frac{2\theta \omega ^2{\varOmega }_1}{\omega _\theta ^2-{\varOmega }_1^2}\), \(B=\frac{2\theta \omega ^2{\varOmega }_2}{\omega _\theta ^2-{\varOmega }_2^2}\) and

\({\varDelta }=-2AB[1-\cos ({\varOmega }_1T)\cos ({\varOmega }_2T)]+(A^2+B^2)\sin ({\varOmega }_1T)\sin ({\varOmega }_2T)\not =0\), then:

$$\begin{aligned} \alpha _{11}= & {} \frac{1}{{\varDelta }}\left[ -AB+B^2\sin \left( {\varOmega }_1T\right) \sin \left( {\varOmega }_2T\right) +AB\cos \left( {\varOmega }_1T\right) \cos \left( {\varOmega }_2T\right) \right] , \nonumber \\ \alpha _{12}= & {} \frac{1}{{\varDelta }}\left[ AB\left( \cos \left( {\varOmega }_2T\right) -\cos \left( {\varOmega }_1T\right) \right) \right] , \nonumber \\ \alpha _{13}= & {} \frac{1}{{\varDelta }}\left[ B\sin \left( {\varOmega }_1T\right) \cos \left( {\varOmega }_2T\right) -A\sin \left( {\varOmega }_2T\right) \cos \left( {\varOmega }_1T\right) \right] , \nonumber \\ \alpha _{14}= & {} \frac{1}{{\varDelta }}\left[ A\sin \left( {\varOmega }_2T\right) -B\sin \left( {\varOmega }_1T\right) \right] , \nonumber \\ \alpha _{21}= & {} \frac{1}{{\varDelta }}\left[ AB\sin \left( {\varOmega }_1T\right) \cos \left( {\varOmega }_2T\right) -B^2\cos \left( {\varOmega }_1T\right) \sin \left( {\varOmega }_2T\right) \right] , \nonumber \\ \alpha _{22}= & {} \left. \frac{1}{{\varDelta }}\left[ B^2\sin \left( {\varOmega }_2T\right) -AB\sin \left( {\varOmega }_1T\right) \right) \right] , \nonumber \\ \alpha _{23}= & {} \frac{1}{{\varDelta }}\left[ B-B\cos \left( {\varOmega }_1T\right) \cos \left( {\varOmega }_2T\right) -A\sin \left( {\varOmega }_1T\right) \sin \left( {\varOmega }_2T\right) \right] , \nonumber \\ \alpha _{24}= & {} \frac{1}{{\varDelta }}\left[ B\left( \cos \left( {\varOmega }_1T\right) -\cos \left( {\varOmega }_2T\right) \right) \right] , \nonumber \\ \alpha _{31}= & {} \frac{1}{{\varDelta }}\left[ -AB+AB\cos \left( {\varOmega }_1T\right) \cos \left( {\varOmega }_2T\right) +A^2\sin \left( {\varOmega }_1T\right) \sin \left( {\varOmega }_2T\right) \right] , \nonumber \\ \alpha _{32}= & {} \frac{1}{{\varDelta }}\left[ AB\left( \cos \left( {\varOmega }_1T\right) -\cos \left( {\varOmega }_2T\right) \right) \right] , \nonumber \\ \alpha _{33}= & {} \frac{1}{{\varDelta }}\left[ A\sin \left( {\varOmega }_2T\right) \cos \left( {\varOmega }_1T\right) -B\sin \left( {\varOmega }_1T\right) \cos \left( {\varOmega }_2T\right) \right] , \nonumber \\ \alpha _{34}= & {} \frac{1}{{\varDelta }}\left[ A\sin \left( {\varOmega }_2T\right) -B\sin \left( {\varOmega }_1T\right) \right] , \nonumber \\ \alpha _{41}= & {} \frac{1}{{\varDelta }}\left[ AB\cos \left( {\varOmega }_1T\right) \sin \left( {\varOmega }_2T\right) -A^2\sin \left( {\varOmega }_1T\right) \cos \left( {\varOmega }_2T\right) \right] , \nonumber \\ \alpha _{42}= & {} \frac{1}{{\varDelta }}\left[ A^2\sin \left( {\varOmega }_1T\right) -AB\sin \left( {\varOmega }_2T\right) \right] , \nonumber \\ \alpha _{43}= & {} \frac{1}{{\varDelta }}\left[ A-B\sin \left( {\varOmega }_1T\right) \sin \left( {\varOmega }_2T\right) -A\cos \left( {\varOmega }_1T\right) \cos \left( {\varOmega }_2T\right) \right] , \nonumber \\ \alpha _{44}= & {} \frac{1}{{\varDelta }}\left[ A\left( \cos \left( {\varOmega }_2T\right) -\cos \left( {\varOmega }_1T\right) \right) \right] . \end{aligned}$$
(75)
$$\begin{aligned} K_1^{\pm }= & {} K_1^{\pm }\left( {\varOmega }_1\right) =\left( 1{\mp }A^2\right) \left[ \left( 1+\theta ^2\omega ^2\right) {\varOmega }_1^2{\mp }\omega ^2\right] -2\theta {\varOmega }_1A\left( \omega ^2\pm \omega ^2\right) , \nonumber \\ K_2^{\pm }= & {} K_2^{\pm }\left( {\varOmega }_2\right) =\left( 1{\mp }B^2\right) \left[ \left( 1+\theta ^2\omega ^2\right) {\varOmega }_2^2{\mp }\omega ^2\right] -2\theta {\varOmega }_2B\left( \omega ^2\pm \omega ^2\right) , \nonumber \\ K_3^{\pm }= & {} K_3^{\pm }\left( {\varOmega }_1,{\varOmega }_2\right) =\left[ \left( 1+\theta ^2\omega ^2\right) {\varOmega }_1{\varOmega }_2{\mp }\omega ^2\right] \left( 1{\mp }AB\right) {\mp }\left( A{\pm }B\right) \theta \omega ^2\left( {\varOmega }_1{\pm }{\varOmega }_2\right) .\nonumber \\ \end{aligned}$$
(76)

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Djordjevic, G.S., Nesic, L. & Radovancevic, D. Two-oscillator Kantowski–Sachs model of the Schwarzschild black hole interior. Gen Relativ Gravit 48, 106 (2016). https://doi.org/10.1007/s10714-016-2102-x

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