<p>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 <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10714_2025_3481_Article_IEq1.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="TEX">$n$</EquationSource> <EquationSource Format="MATHML"><math> <mi>n</mi> </math></EquationSource> </InlineEquation>, 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 <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10714_2025_3481_Article_IEq2.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="TEX">$n^2$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>n</mi> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation>, and consequently the mass <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10714_2025_3481_Article_IEq3.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="21" /> </InlineMediaObject> <EquationSource Format="TEX">$M$</EquationSource> <EquationSource Format="MATHML"><math> <mi>M</mi> </math></EquationSource> </InlineEquation> also increases. In this framework, the wave function is finite and square-integrable, with convergence requiring that the deformation parameter <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10714_2025_3481_Article_IEq4.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="15" /> </InlineMediaObject> <EquationSource Format="TEX">$\beta $</EquationSource> <EquationSource Format="MATHML"><math> <mi>β</mi> </math></EquationSource> </InlineEquation> be regulated by a discrete quantum number <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10714_2025_3481_Article_IEq5.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="TEX">$m$</EquationSource> <EquationSource Format="MATHML"><math> <mi>m</mi> </math></EquationSource> </InlineEquation>. The wave function exhibits quantum tunneling connecting the black hole interior with both its exterior and a white hole region, effects that disappear in the limit <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10714_2025_3481_Article_IEq6.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="48" /> </InlineMediaObject> <EquationSource Format="TEX">$\beta \rightarrow 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>β</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>. These results demonstrate how minimal-length effects both regularize the wave function and modify the semiclassical structure of the black hole.</p>

Quantum black hole as a harmonic oscillator from the perspective of the minimum uncertainty approach

期刊论文
发表日期：

2025年10月9日

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Wilfredo Yupanqui Carpio,
Octavio Obregón

摘要

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 $n$ $n$ , 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 $n^2$ $n^{2}$ , and consequently the mass $M$ $M$ also increases. In this framework, the wave function is finite and square-integrable, with convergence requiring that the deformation parameter $\beta $ $β$ be regulated by a discrete quantum number $m$ $m$ . 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 $\beta \rightarrow 0$ $β \to 0$ . These results demonstrate how minimal-length effects both regularize the wave function and modify the semiclassical structure of the black hole.