<p>A well-known soliton (bubble) solution of five-dimensional Kaluza–Klein General Relativity is modified by imposing mass on the scalar field. By forcing the scalar field to be short-range, the failure of the original bubble solution to satisfy the equivalence principle is remedied, and the bubble acquires gravitational mass. Most importantly, the mass is quantized, even in this classical setting, and has a value <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10714_2025_3350_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="80" /> </InlineMediaObject> <EquationSource Format="TEX">$m_P / (4 \sqrt{\alpha })$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>m</mi> <mi>P</mi> </msub> <mo stretchy="false">/</mo> <mrow> <mo stretchy="false">(</mo> <mn>4</mn> <msqrt> <mi>α</mi> </msqrt> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10714_2025_3350_Article_IEq2.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="26" /> </InlineMediaObject> <EquationSource Format="TEX">$m_P$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>m</mi> <mi>P</mi> </msub> </math></EquationSource> </InlineEquation> is the Planck mass, and <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10714_2025_3350_Article_IEq3.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="TEX">$\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation> is the fine-structure constant. This result applies for any choice of scalar-field mass, as it is an attractor for the field equations.</p>

Kaluza–Klein bubble with massive scalar field

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2025年1月15日

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Darrell Jackson

摘要

A well-known soliton (bubble) solution of five-dimensional Kaluza–Klein General Relativity is modified by imposing mass on the scalar field. By forcing the scalar field to be short-range, the failure of the original bubble solution to satisfy the equivalence principle is remedied, and the bubble acquires gravitational mass. Most importantly, the mass is quantized, even in this classical setting, and has a value $m_P / (4 \sqrt{\alpha })$ $m_{P} / (4 \sqrt{α})$ , where $m_P$ $m_{P}$ is the Planck mass, and $\alpha $ $α$ is the fine-structure constant. This result applies for any choice of scalar-field mass, as it is an attractor for the field equations.