<p>We investigate the smooth matching of a generalized Vaidya interior metric with an exterior Schwarzschild or Vaidya spacetime across a finite timelike boundary hypersurface, within the standard Israel-Darmois formalism. By explicitly deriving the extrinsic curvature components, we show that smooth matching generically requires that the mass function, evaluated at the boundary hypersurface, depend only on the ingoing Eddington-Finkelstein null coordinate, i.e., <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\partial m/\partial r|_{\Sigma } = 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mrow> <mi>∂</mi> <mi>m</mi> <mo stretchy="false">/</mo> <mi>∂</mi> <mi>r</mi> <mo stretchy="false">|</mo> </mrow> <mi mathvariant="normal">Σ</mi> </msub> <mo>=</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>. For <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\partial m/\partial r \ne 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>∂</mi> <mi>m</mi> <mo stretchy="false">/</mo> <mi>∂</mi> <mi>r</mi> <mo>≠</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>, the junction conditions lead to a discontinuity in the extrinsic curvature, inducing a thin shell with non-vanishing tangential surface stress. The discontinuity is also reflected in invariant quantities such as <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\mathcal {K}=K_{ab}K^{ab}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">K</mi> <mo>=</mo> <msub> <mi>K</mi> <mrow> <mi mathvariant="italic">ab</mi> </mrow> </msub> <msup> <mi>K</mi> <mrow> <mi mathvariant="italic">ab</mi> </mrow> </msup> </mrow> </math></EquationSource> </InlineEquation> and the Kodama current, indicating a mismatch in quasi-local energy flux across the boundary. To illustrate using examples, we consider models in which the interior Type-I matter is described by an ideal fluid with a linear equation of state. Within this class of models, enforcing <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\partial m/\partial r = 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>∂</mi> <mi>m</mi> <mo stretchy="false">/</mo> <mi>∂</mi> <mi>r</mi> <mo>=</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> at a finite boundary hypersurface either leads to singular behavior in physical quantities or drives the boundary towards an asymptotic limit. Our results therefore indicate that, for a broad class of generalized Vaidya models with non-trivial radial dependence, smooth matching to a Schwarzschild/Vaidya exterior either requires a surface stress or the configuration becomes effectively unbounded.</p>

On the limitations of a generalized Vaidya metric

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发表日期：

2026年6月14日

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Soumya Chakrabarti,
Naresh Dadhich,
Chiranjeeb Singha

摘要

We investigate the smooth matching of a generalized Vaidya interior metric with an exterior Schwarzschild or Vaidya spacetime across a finite timelike boundary hypersurface, within the standard Israel-Darmois formalism. By explicitly deriving the extrinsic curvature components, we show that smooth matching generically requires that the mass function, evaluated at the boundary hypersurface, depend only on the ingoing Eddington-Finkelstein null coordinate, i.e., $\partial m/\partial r|_{\Sigma } = 0$ ${\partial m / \partial r |}_{Σ} = 0$ . For $\partial m/\partial r \ne 0$ $\partial m / \partial r \neq 0$ , the junction conditions lead to a discontinuity in the extrinsic curvature, inducing a thin shell with non-vanishing tangential surface stress. The discontinuity is also reflected in invariant quantities such as $\mathcal {K}=K_{ab}K^{ab}$ $K = K_{ab} K^{ab}$ and the Kodama current, indicating a mismatch in quasi-local energy flux across the boundary. To illustrate using examples, we consider models in which the interior Type-I matter is described by an ideal fluid with a linear equation of state. Within this class of models, enforcing $\partial m/\partial r = 0$ $\partial m / \partial r = 0$ at a finite boundary hypersurface either leads to singular behavior in physical quantities or drives the boundary towards an asymptotic limit. Our results therefore indicate that, for a broad class of generalized Vaidya models with non-trivial radial dependence, smooth matching to a Schwarzschild/Vaidya exterior either requires a surface stress or the configuration becomes effectively unbounded.