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An exploration of the curvature and inheritance properties of the Interior black hole spacetime

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Pith's one-line read The interior black hole spacetime satisfies a family of pseudosymmetry-type curvature identities and admits generalized curvature inheritance for a specific non-Killing vector field.

arxiv 2504.20060 v1 pith:72AD3HLE submitted 2025-04-16 physics.gen-ph

Absos Ali Shaikh , Kamiruzzaman This is my paper · ORCID
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keywords spacetimecdotgeneralizedinheritanceblackcurvatureholeinterior
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The reading

The interior black hole (IBH) spacetime is the metric inside the Schwarzschild horizon written in coordinates where the radial coordinate behaves like time. This paper computes several standard curvature objects for that metric and asks whether they satisfy special linear relations. In differential geometry, when one curvature-like tensor is a multiple or linear combination of others, the spacetime is called pseudosymmetric. The authors show that, for this metric, operators built from the Riemann tensor and the conformal tensor satisfy such linear relations, for example the commutator built from conformal and Riemann curvature is a combination of two simpler tensors. They also establish that the metric is 2-quasi-Einstein and Einstein of level 2, that certain curvature 2-forms are recurrent, and that the energy-momentum tensor derived from Einstein's equations has pseudosymmetric type. Finally they exhibit a non-Killing vector field along which the metric obeys generalized curvature inheritance, a symmetry condition weaker than having a Killing field. A caution: most computations treat the function in the metric as arbitrary and carry along its time derivatives. The physical interior Schwarzschild solution has that function constant, and in that case the Ricci and energy-momentum tensors vanish, so several advertised statements about energy-momentum pseudosymmetry become trivial. The algebraic identities are the real content; their physical relevance depends on whether the generalized family is intended.
Extended reading notes

Core claim

In an IBH spacetime, the commutator C.R minus R.C is linearly dependent with Q(g,R) and Q(S,R) as well as Q(g,C) and Q(S,C); and R.R is linearly dependent with Q(S,R) and Q(g,C) (Theorem 3.3(i)-(ii)). If correct, these are explicit linear algebraic relations among six-index curvature operators, not merely asymptotic or approximate statements.

Load-bearing premise

The paper treats the function in the IBH metric (3.1) as an arbitrary smooth function of time and carries all results with its first, second, and third derivatives. The physical interior black hole solution cited in Section 1 has this function fixed by matching to Schwarzschild, i.e., constant, so the derived non-trivial energy-momentum and inheritance results apply to a generalized family rather than necessarily to the IBH spacetime as defined. If the function is constant, the Ricci and energy-momentum tensors vanish and several advertised theorems become vacuous.

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Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

The central claims rest on treating xi(t) as a free function, invertibility of several denominators, and standard curvature-tensor definitions. No constants are fitted to data and no new physical entities are introduced. The most delicate input is the xi(t) generalization, which the cited physical solution does not justify.

free parameters (1)
  • xi(t) = arbitrary smooth function
    Section 3 defines V_i and all results in Theorems 3.3, 4.1, and 5.1 contain xi-dot, xi-ddot, and xi-third-derivative. No equation fixes xi beyond smoothness and denominator conditions; the physical IBH solution has constant xi.
assumptions (5)
  • ad hoc to paper The IBH metric (3.1) is studied with xi an arbitrary smooth function of t.
    Section 3 defines V_i and all results contain derivatives of xi; the cited physical solution has constant xi, so this is a paper-specific generalization.
  • domain assumption Non-vanishing denominators: V5 = 6 xi - 4 t xi-dot + t^2 xi-ddot, 3 xi - t^2 xi-dot + 2 t xi (t xi-ddot - xi-dot), and 6 xi - 2 t^2 xi-dot + 4 t xi (t xi-ddot - xi-dot) are nonzero.
    Stated before Theorem 3.3; these conditions are needed to define the coefficients L1, L2, and L3.
  • domain assumption The energy-momentum tensor is T = S - (kappa/2) g with units 8 pi G / c^4 set to 1.
    Section 4 uses this Einstein field equation form to derive the energy-momentum pseudosymmetry results.
  • domain assumption The coordinate domain excludes t = 0 and V2 = t - 2 xi = 0.
    The Christoffel symbols in (3.2) contain denominators t and V2, so the stated formulas are valid only away from these loci.
  • standard math Standard definitions of R, C, W, K, P, the Tachibana tensors, and Lie derivatives are accepted as background.
    Section 2 collects these definitions and the paper relies on them without proof.

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Pith. "Pith review of An exploration of the curvature and inheritance properties of the Interior black hole spacetime." pith.science (2026). https://pith.science/paper/72AD3HLE

@misc{pith2026250420060,
  author       = {Pith},
  title        = {Pith review of: An exploration of the curvature and inheritance properties of the Interior black hole spacetime},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/72AD3HLE}},
  note         = {Machine review of arXiv:2504.20060}
}
abstract

In continuation of the study in \cite{SDHK_interior_2020}, the present article explores the geometric and curvature properties of the interior black hole (briefly, IBH) spacetime. It is shown that in an IBH spacetime the operator $R\cdot C$ and $C\cdot R$ does not commute with each other and infact the commutator $C\cdot R-R\cdot C$ is linearly dependent with $Q(g,R)$ and $Q(S,R)$ as well as $Q(g,C)$ and $Q(S,C)$. Also in IBH spacetime $R \cdot R$ is linearly dependent with $Q(S,R)$ and $Q(g,C)$. It is exhibited that IBH spacetime is $2$-quasi Einstein, Ein$(2)$ and generalized quasi Einstein spacetime in the sense of Chaki, and its conformal $2$-forms are recurrent. We have derived the universal form of the compatible tensors in such a spacetime. We have also demonstrated that the nature of energy momentum tensor of IBH spacetime is pseudosymmetric (see, Theorem $4.1$). Again it is exposed that with respect to the non-Killing vector field $\frac{\partial}{\partial t},$ the IBH spacetime obeys the generalized curvature inheritance, generalized Ricci inheritance, special type of generalized conformal, concircular, conharmonic and generalized Weyl projective inheritance. Finally a comparison between IBH spacetime and KIselev Black Hole (KBH) is displayed.

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