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<p>The quantum tunneling processes related to the black hole determine the black hole thermodynamics. The Hawking temperature is determined by the quantum tunneling processes of emission of particles from the black hole. On the other hand, the Bekenstein–Hawking entropy of the black hole is obtained by consideration of the macroscopic quantum tunneling processes of splitting of black hole to the smaller black holes. These tunneling processes also determine the composition rule for the black hole entropy, which coincides with the composition rule for the non-extensive Tsallis–Cirto <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11448_2025_4111_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="40" /> </InlineMediaObject> <EquationSource Format="TEX">\(\delta = 2\)</EquationSource> <!--JETPLet2460346Volovik-m1--> </InlineEquation> entropy. This composition rule suggests that the mass spectrum of the black hole is equidistant, <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11448_2025_4111_Article_IEq2.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="79" /> </InlineMediaObject> <EquationSource Format="TEX">\(M = N{{M}_{0}}\)</EquationSource> <!--JETPLet2460346Volovik-m2--> </InlineEquation>. Here, <i>N</i> is an integer number and <i>M</i><sub>0</sub>&#xa0;=&#xa0;<InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11448_2025_4111_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="51" /> </InlineMediaObject> <EquationSource Format="TEX">\(\sqrt 2 {{m}_{{\text{P}}}}\)</EquationSource> <!--JETPLet2460346Volovik-m3--> </InlineEquation> is the mass quantum expressed via the reduced Planck mass <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11448_2025_4111_Article_IEq4.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="24" /> </InlineMediaObject> <EquationSource Format="TEX">\({{m}_{{\text{P}}}}\)</EquationSource> <!--JETPLet2460346Volovik-m4--> </InlineEquation>. The Bekenstein–Hawking entropy of the black hole with mass <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11448_2025_4111_Article_IEq5.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="79" /> </InlineMediaObject> <EquationSource Format="TEX">\(M = N{{M}_{0}}\)</EquationSource> <!--JETPLet2460346Volovik-m5--> </InlineEquation> is <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11448_2025_4111_Article_IEq6.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="100" /> </InlineMediaObject> <EquationSource Format="TEX">\({{S}_{{{\text{BH}}}}}(N) = {{N}^{2}}\)</EquationSource> <!--JETPLet2460346Volovik-m6--> </InlineEquation>.</p>

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Tsallis–Cirto Entropy of a Black Hole and a Black Hole Atom

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  • 发表日期：

    2025年2月10日

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  • G. E. Volovik

摘要

The quantum tunneling processes related to the black hole determine the black hole thermodynamics. The Hawking temperature is determined by the quantum tunneling processes of emission of particles from the black hole. On the other hand, the Bekenstein–Hawking entropy of the black hole is obtained by consideration of the macroscopic quantum tunneling processes of splitting of black hole to the smaller black holes. These tunneling processes also determine the composition rule for the black hole entropy, which coincides with the composition rule for the non-extensive Tsallis–Cirto \(\delta = 2\) entropy. This composition rule suggests that the mass spectrum of the black hole is equidistant, \(M = N{{M}_{0}}\) . Here, N is an integer number and M0 =  \(\sqrt 2 {{m}_{{\text{P}}}}\) is the mass quantum expressed via the reduced Planck mass \({{m}_{{\text{P}}}}\) . The Bekenstein–Hawking entropy of the black hole with mass \(M = N{{M}_{0}}\) is \({{S}_{{{\text{BH}}}}}(N) = {{N}^{2}}\) .