Quantum Black Hole as a Harmonic Oscillator from the Perspective of the Minimum Uncertainty Approach
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| Format: | Preprint |
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2024
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| author | Carpio, Wilfredo Yupanqui Obregón, Octavio |
| author_facet | Carpio, Wilfredo Yupanqui Obregón, Octavio |
| contents | Starting from the eigenvalue equation for the mass of a black hole derived by Mäkelä and Repo, we show that, by reparametrizing the radial coordinate and the wave function, it can be rewritten as the eigenvalue equation of a quantum harmonic oscillator. We then study the interior of a Schwarzschild black hole using two quantization approaches. In the standard quantization, the area and mass spectra are discrete, characterized by a quantum number $n$, but the wave function is not square-integrable, limiting its physical interpretation. In contrast, a minimal-uncertainty quantization approach yields an area spectrum that grows as $n^2$, and consequently the mass $M$ also increases. In this framework, the wave function is finite and square-integrable, with convergence requiring that the deformation parameter $β$ be regulated by a discrete quantum number $m$. The wave function exhibits quantum tunneling connecting the black hole interior with both its exterior and a white hole region, effects that disappear in the limit $β\to 0$. These results demonstrate how minimal-length effects both regularize the wave function and modify the semiclassical structure of the black hole. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2409_09181 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Quantum Black Hole as a Harmonic Oscillator from the Perspective of the Minimum Uncertainty Approach Carpio, Wilfredo Yupanqui Obregón, Octavio General Relativity and Quantum Cosmology High Energy Physics - Theory Starting from the eigenvalue equation for the mass of a black hole derived by Mäkelä and Repo, we show that, by reparametrizing the radial coordinate and the wave function, it can be rewritten as the eigenvalue equation of a quantum harmonic oscillator. We then study the interior of a Schwarzschild black hole using two quantization approaches. In the standard quantization, the area and mass spectra are discrete, characterized by a quantum number $n$, but the wave function is not square-integrable, limiting its physical interpretation. In contrast, a minimal-uncertainty quantization approach yields an area spectrum that grows as $n^2$, and consequently the mass $M$ also increases. In this framework, the wave function is finite and square-integrable, with convergence requiring that the deformation parameter $β$ be regulated by a discrete quantum number $m$. The wave function exhibits quantum tunneling connecting the black hole interior with both its exterior and a white hole region, effects that disappear in the limit $β\to 0$. These results demonstrate how minimal-length effects both regularize the wave function and modify the semiclassical structure of the black hole. |
| title | Quantum Black Hole as a Harmonic Oscillator from the Perspective of the Minimum Uncertainty Approach |
| topic | General Relativity and Quantum Cosmology High Energy Physics - Theory |
| url | https://arxiv.org/abs/2409.09181 |