# Quantum Phenomena Inside a Black Hole: Quantization of the Scalar Field Iniside Horizon in Schwarzschild Spacetime

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## Abstract

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## 1. Introduction

## 2. “T-Sphere” Model—An Anisotropic Cosmological Model

## 3. Scalar Free Field in a T-Model

#### Quantization

- (a)
- ${\widehat{A}}_{\epsilon lm}$, ${\widehat{A}}_{\epsilon lm}^{\u2020},$ are the annihilation and creation operators, respectively, i.e., the only nonvanishing commutator is$$\left[{\widehat{A}}_{\epsilon lm},{\widehat{A}}_{{\epsilon}^{\prime}{l}^{\prime}{m}^{\prime}}^{\u2020}\right]={\delta}_{\epsilon {\epsilon}^{\prime}}{\delta}_{l{l}^{\prime}}{\delta}_{m{m}^{\prime}}$$
- (b)
- the Wronskian (21) must hold.

## 4. Hamiltonian of the Scalar Field in a T-Model

## 5. Discussion

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A

## References

- Frolov, V.P.; Novikov, I.D. Black Hole Physics: Basic Concepts and New Developments; Kluwer Academic: Dordrecht, The Netherlands, 1998. [Google Scholar]
- Hartle, J.B. Gravitation; Addison Wesley: Reading, MA, USA, 2003. [Google Scholar]
- Bańados, M.; Silk, J.; West, S.M. Kerr Black Holes as Particle Accelerators to Arbitrarily High Energy. Phys. Rev. Lett.
**2009**, 103, 111102. [Google Scholar] [CrossRef] [Green Version] - Harada, T.; Kimura, M. Black holes as particle accelerators: A brief review. Class. Quantum Gravity
**2014**, 31, 243001. [Google Scholar] [CrossRef] - Zaslavskii, O.B. The Bańados-Silk-West effect with immovable particles near static black holes and its rotational counterpart. Gravit. Cosmol.
**2023**, 29, 74–78. [Google Scholar] [CrossRef] - Gusin, P.; Augousti, A.T.; Formalik, F.; Radosz, A. The (A)symmetry between the Exterior and Interior of a Schwarzschild Black Hole. Symmetry
**2018**, 10, 366. [Google Scholar] [CrossRef] [Green Version] - Doran, R.; Lobo, F.S.; Crawford, P. Interior of a Schwarzschild black hole revisited. Found. Phys.
**2008**, 38, 160. [Google Scholar] [CrossRef] [Green Version] - Ruban, V.A. Spherically Symmetric T-Models in the General Theory of Relativity. Gen. Relativ. Gravit.
**2001**, 33, 375–394. [Google Scholar] [CrossRef] - Radosz, A.; Gusin, P.; Augousti, A.; Formalik, F. Inside spherically symmetric black holes or how a uniformly accelerated particle may slow down. Eur. Phys. J. C
**2019**, 79, 876. [Google Scholar] [CrossRef] [Green Version] - Toporensky, A.V.; Zaslavskii, O.B. Zero-momentum trajectories inside a black hole and high energy particle collisions. J. Cosmol. Astropart. Phys.
**2019**, 2019, 063. [Google Scholar] [CrossRef] [Green Version] - Augousti, A.; Gusin, P.; Kuśmierz, B.; Masajada, J.; Radosz, A. On the speed of a test particle inside the Schwarzschild event horizon and other kinds of black holes. Gen. Relativ. Gravit.
**2018**, 50, 131. [Google Scholar] [CrossRef] [Green Version] - Toporensky, A.V.; Zaslavskii, O.B. Redshift of a photon emitted along the black hole horizon. Eur. Phys. J.
**2017**, 77, 179. [Google Scholar] [CrossRef] [Green Version] - Gal’tsov, D.V.; Donets, E.E. Power—law mass inflation in Einstein—Yang—Mills—Higgs black holes. Comptes Rendus L’académie Sci. Ser. IIB Mech. Phys. Chem. Astron.
**1997**, 325, 649–657. [Google Scholar] [CrossRef] [Green Version] - Donets, E.E.; Gal’tsov, D.V.; Zotov, M.Y. Internal Structure of Einstein–Yang–Mills Black Holes. Phys. Rev. D
**1997**, 56, 3459. [Google Scholar] [CrossRef] [Green Version] - Yajnik, U.A.; Narayan, K. CPT invariance and canonical quantization inside the Schwarzschild black hole. Class. Quantum Gravity
**1998**, 15, 1315. [Google Scholar] [CrossRef] [Green Version] - Parker, E.; Toms, D.J. Quantum Field Theory in Curved Spacetime, Quantized Fields and Gravity; Cambridge University Press: Cambridge, UK, 2009. [Google Scholar]
- Habib, S.; Molina-Paris, C.; Mottola, E. Energy-Momentum Tensor of Particles Created in an Expanding Universe. Phys. Rev. D
**2000**, 61, 024010. [Google Scholar] [CrossRef] [Green Version] - Tsoupros, G. Conformal Scalar Propagation inside the Schwarzschild Black Hole. Gen. Relativ. Gravit.
**2012**, 44, 309–351. [Google Scholar] [CrossRef] [Green Version] - Oshita, N. Resolution to the firewall paradox: The black hole information paradox and highly squeezed interior quantum fluctuations. Class. Quantum Gravity
**2017**, 34, 195002. [Google Scholar] [CrossRef] [Green Version] - Almeida, C.S.; Rodrigues, D.C. Quantization of a black-hole gravity: Geometrodynamics and the quantum. Class. Quantum Gravity
**2023**, 40, 035004. [Google Scholar] [CrossRef] - Giddings, S.B.; Perkins, J. Quantum evolution of the Hawking state for black holes. Phys. Rev. D
**2022**, 106, 065011. [Google Scholar] [CrossRef] - Hamilton, A.J.S.; Polhemus, G. Stereoscopic visualization in curved spacetime: Seeing deep inside a black hole. New J. Phys.
**2010**, 12, 123027–123052. [Google Scholar] [CrossRef] - Christodoulou, M.; Rovelli, C. How big is a black hole? Phys. Rev. D
**2015**, 91, 064046. [Google Scholar] [CrossRef] [Green Version] - Gusin, P.; Radosz, A. The volume of the black holes—The constant curvature slicing of the spherically symmetric spacetime. Mod. Phys. Lett. A
**2017**, 32, 1750115. [Google Scholar] [CrossRef] - Zaslavskii, O.B. Schwarzschild Black Hole as Accelerator of Accelerated Particles. JETP Lett.
**2020**, 111, 260–263. [Google Scholar] [CrossRef] [Green Version] - Birrel, N.D.; Davies, P.C.W. Quantum Fields in Curved Space; Cambridge University Press: Cambridge, UK, 1984. [Google Scholar]
- Rajeev, K.; Chakraborty, S.; Padmanabhan, T. Inverting a normal harmonic oscillator: Physical interpretation and applications. Gen. Relativ. Gravit.
**2018**, 50, 116. [Google Scholar] [CrossRef] [Green Version] - Zel’dovich, Y.B.; Starobinsky, A.A. Particle production and vacuum polarization in an anisotropic gravitational field. Sov. J. Exp. Theor. Phys.
**1972**, 34, 1159. [Google Scholar]

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**MDPI and ACS Style**

Gusin, P.; Radosz, A.; Augousti, A.T.; Polonyi, J.; Zaslavskii, O.B.; Ściborski, R.J.
Quantum Phenomena Inside a Black Hole: Quantization of the Scalar Field Iniside Horizon in Schwarzschild Spacetime. *Universe* **2023**, *9*, 299.
https://doi.org/10.3390/universe9070299

**AMA Style**

Gusin P, Radosz A, Augousti AT, Polonyi J, Zaslavskii OB, Ściborski RJ.
Quantum Phenomena Inside a Black Hole: Quantization of the Scalar Field Iniside Horizon in Schwarzschild Spacetime. *Universe*. 2023; 9(7):299.
https://doi.org/10.3390/universe9070299

**Chicago/Turabian Style**

Gusin, Pawel, Andrzej Radosz, Andy T. Augousti, Janos Polonyi, Oleg B. Zaslavskii, and Romuald J. Ściborski.
2023. "Quantum Phenomena Inside a Black Hole: Quantization of the Scalar Field Iniside Horizon in Schwarzschild Spacetime" *Universe* 9, no. 7: 299.
https://doi.org/10.3390/universe9070299