arxiv: 2604.17279 · v2 · submitted 2026-04-19 · 🌀 gr-qc · hep-th

Recognition: unknown

Geometric deformations of symmetric spacetimes with a string cloud

Hiroshi Kozaki , Satsuki Matsuno , Tatsuhiko Koike , Yoshiyuki Morisawa , Hideki Ishihara

Authors on Pith no claims yet

Pith reviewed 2026-05-10 06:20 UTC · model grok-4.3

classification 🌀 gr-qc hep-th

keywords geometric deformationstring cloudEinstein equationssymmetric spacetimesFLRW cosmologyReissner-Nordström black holeseta-Einstein metricsKilling horizons

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The pith

A single-function deformation of eta-Einstein 3D metrics produces 4D spacetimes that satisfy Einstein's equations with string-cloud sources.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes a framework that deforms three-dimensional eta-Einstein metrics using one function to build four-dimensional metrics solving the Einstein equations sourced by a string cloud. This construction unifies the treatment of many highly symmetric spacetimes, from FLRW and Kantowski-Sachs cosmologies to Reissner-Nordström-(A)dS black holes with spherical, planar, or hyperbolic symmetry. In the cosmological cases the scale-factor evolution equations stay identical to the undeformed versions, so the expansion history is unchanged. For the black-hole cases the structure of the Killing horizons is also left insensitive to the deformation.

Core claim

Four-dimensional metrics are obtained from three-dimensional eta-Einstein metrics by a deformation controlled by a single function; the resulting spacetime solves the Einstein equations with a string-cloud stress-energy tensor that cancels the extra curvature terms generated by the deformation, thereby providing a unified description of FLRW, Kantowski-Sachs, LRS Bianchi, and symmetric Reissner-Nordström-(A)dS solutions in which expansion histories and horizon structures remain unchanged.

What carries the argument

The single-function deformation applied to an eta-Einstein three-dimensional metric, with the string-cloud source exactly canceling the curvature contributions introduced by the deformation.

If this is right

Cosmological models retain exactly the same scale-factor evolution equations and therefore the same expansion history as their undeformed counterparts.
The structure and location of Killing horizons in the deformed Reissner-Nordström-(A)dS black holes remain unchanged by the deformation.
FLRW, Kantowski-Sachs, LRS Bianchi, and Taub-NUT-(A)dS cosmologies as well as spherical, planar, and hyperbolic black holes are all treated inside one common construction.
The deformation can be applied uniformly to any three-dimensional eta-Einstein metric that admits a one-function deformation.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same deformation technique could be tested on other highly symmetric three-dimensional metrics that are not listed in the paper to generate new string-cloud solutions.
Because horizons and expansion are preserved, the framework offers a controlled way to study how string clouds affect thermodynamic quantities or perturbation spectra without changing the background geometry.
Time-dependent choices of the deformation function might introduce evolving string-cloud densities while still satisfying the Einstein equations.

Load-bearing premise

The starting three-dimensional metric must be eta-Einstein and must admit a deformation controlled by exactly one function, while the string-cloud stress-energy tensor must take the precise form needed to cancel all extra curvature terms.

What would settle it

Explicitly compute the Einstein tensor of one concrete deformed metric (for example the deformed flat FLRW solution) and check whether it equals the string-cloud energy-momentum tensor component by component.

read the original abstract

We establish a deformation framework for highly symmetric solutions to the Einstein equations. In this framework, four-dimensional metrics are constructed from three-dimensional {\eta}-Einstein metrics admitting a deformation determined by a single function. Under this deformation, the resulting spacetime solves the Einstein equations with a string-cloud source. Within this framework , a wide range of symmetric spacetimes can be treated in a unified manner. These include FLRW, Kantowski-Sachs, and LRS Bianchi cosmological models (including Taub-NUT-(A)dS solutions), as well as Reissner-Nordstr\"om-(A)dS black holes admitting spherical, planar, or hyperbolic symmetry. In the cosmological setting, the deformation leaves the evolution equations for the scale factors unchanged, and hence the expansion history coincides with that of the corresponding undeformed models. For the deformed Reissner-Nordstr\"om-(A)dS black holes, the structure of Killing horizons is insensitive to the deformation.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives a workable single-function deformation from 3D eta-Einstein metrics to 4D string-cloud spacetimes that preserves expansion history and horizons, but the curvature cancellation needs explicit checks for the less isotropic cases.

read the letter

The main takeaway is a geometric deformation that lets you add string clouds to a family of symmetric spacetimes without changing their scale-factor dynamics or horizon structure. They start with three-dimensional eta-Einstein metrics that allow a deformation controlled by a single function. The four-dimensional metric built from this solves the Einstein equations with the string-cloud source. This covers FLRW and LRS Bianchi cosmologies, including Taub-NUT-(A)dS, plus Reissner-Nordström-(A)dS black holes with spherical, planar, or hyperbolic symmetry. In cosmology the expansion history stays the same as the original models. In the black-hole cases the Killing horizons are unaffected. The unification across these cases is the real contribution. It turns what used to be separate constructions into instances of one procedure. The potential issue is the exact matching of the deformation-induced curvature to the string-cloud stress-energy. The eta-Einstein property handles the bulk of the terms, but the pieces coming from the deformation direction must lie in the span of the string-cloud projector. For the less isotropic cases, such as hyperbolic or Kantowski-Sachs, there could be residual components that do not fit unless the function is chosen specially. The abstract claims uniform validity, so the paper must contain the general proof or explicit checks that these residuals vanish. This paper is for specialists in exact solutions of Einstein's equations with matter sources like strings. A reader who works on symmetric cosmologies or black holes will see how to extend existing solutions. It is solid enough to go to peer review, though the referee should focus on verifying the curvature cancellation in the full component equations.

Referee Report

3 major / 2 minor

Summary. The manuscript establishes a deformation framework in which four-dimensional metrics are constructed from three-dimensional η-Einstein metrics via a single-function deformation. It claims that the resulting spacetimes satisfy the Einstein equations sourced by a string-cloud stress-energy tensor. The framework is applied uniformly to FLRW, LRS Bianchi (including Taub-NUT-(A)dS), and Reissner-Nordström-(A)dS solutions with spherical, planar, or hyperbolic symmetry. Cosmological expansion histories remain unchanged and Killing-horizon structures are preserved under the deformation.

Significance. If the central construction is verified, the work supplies a unified geometric method for generating exact string-cloud solutions across multiple symmetry classes. The invariance of scale-factor evolution and horizon properties under the deformation could simplify matching to cosmological observations or black-hole thermodynamics, and the single-function control may reduce the parameter space in modeling anisotropic sources.

major comments (3)

[§2] §2 (general deformation framework): The assertion that the 4D Einstein tensor lies exactly in the algebraic span of the string-cloud projector (u_a u_b − n_a n_b) after deformation is not supported by an explicit component-by-component calculation. The η-Einstein condition on the 3D base cancels many terms, but the residual contributions from Lie derivatives along the deformation direction and the extrinsic curvature of the foliation must be shown to vanish identically or to be absorbed without further restricting f; this step is load-bearing for the uniform claim.
[§4.2] §4.2 (hyperbolic RN-(A)dS case): The paper states that the construction works for all three horizon topologies without additional constraints on the deformation function, yet supplies no explicit verification that anisotropic-pressure or shear components orthogonal to the string-cloud ansatz are absent when the base 3D metric has negative curvature. A direct computation of the Einstein tensor components for this topology is required to confirm the cancellation holds identically.
[§3] §3 (cosmological models): While the evolution equations for the scale factors are stated to be unchanged, the consistency of the induced string-cloud density with the Friedmann and acceleration equations must be checked explicitly for at least the Kantowski-Sachs and LRS Bianchi cases, because the deformation may still source non-zero off-diagonal stress-energy terms that would alter the effective equation of state.

minor comments (2)

The introduction would benefit from a short paragraph outlining the precise algebraic form assumed for the string-cloud stress-energy tensor and the definition of the η-Einstein condition on the 3D base metric.
Notation for the deformation function f and the string-cloud parameters (energy density, tension) should be introduced once and used consistently; currently the abstract and later sections employ slightly different symbols.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful and constructive review of our manuscript. The comments highlight opportunities to strengthen the explicitness of our derivations, and we address each major point below with corresponding revisions.

read point-by-point responses

Referee: [§2] §2 (general deformation framework): The assertion that the 4D Einstein tensor lies exactly in the algebraic span of the string-cloud projector (u_a u_b − n_a n_b) after deformation is not supported by an explicit component-by-component calculation. The η-Einstein condition on the 3D base cancels many terms, but the residual contributions from Lie derivatives along the deformation direction and the extrinsic curvature of the foliation must be shown to vanish identically or to be absorbed without further restricting f; this step is load-bearing for the uniform claim.

Authors: We agree that greater explicitness would improve clarity. The general derivation in §2 already uses the η-Einstein condition together with the single-function deformation to show that the Einstein tensor takes the required algebraic form, but we will add a dedicated component-by-component expansion (as a new subsection or appendix) that isolates the Lie-derivative and extrinsic-curvature contributions and demonstrates their absorption into the string-cloud projector without imposing extra conditions on f. revision: yes
Referee: [§4.2] §4.2 (hyperbolic RN-(A)dS case): The paper states that the construction works for all three horizon topologies without additional constraints on the deformation function, yet supplies no explicit verification that anisotropic-pressure or shear components orthogonal to the string-cloud ansatz are absent when the base 3D metric has negative curvature. A direct computation of the Einstein tensor components for this topology is required to confirm the cancellation holds identically.

Authors: We will supply the requested explicit computation for the hyperbolic topology in the revised §4.2. Because the general framework of §2 is topology-independent once the base is η-Einstein, the same cancellations apply; the new calculation will confirm that no orthogonal anisotropic-pressure or shear terms survive for negative-curvature bases. revision: yes
Referee: [§3] §3 (cosmological models): While the evolution equations for the scale factors are stated to be unchanged, the consistency of the induced string-cloud density with the Friedmann and acceleration equations must be checked explicitly for at least the Kantowski-Sachs and LRS Bianchi cases, because the deformation may still source non-zero off-diagonal stress-energy terms that would alter the effective equation of state.

Authors: We will insert explicit verifications for the Kantowski-Sachs and LRS Bianchi models in the revised §3. These calculations will verify that symmetry together with the deformation ansatz forces all off-diagonal stress-energy components to vanish, so that the string-cloud density remains consistent with the unchanged Friedmann and acceleration equations. revision: yes

Circularity Check

0 steps flagged

No significant circularity: construction derives Einstein equations with string-cloud source via explicit curvature computation on deformed metrics.

full rationale

The framework starts from independent 3D η-Einstein metrics, applies a one-parameter geometric deformation to produce 4D spacetimes, and computes the resulting Einstein tensor to show it equals 8π times a string-cloud stress-energy tensor whose algebraic form is fixed by the deformation residuals. The η-Einstein condition and single-function ansatz are external inputs that enable term cancellation; they are not defined in terms of the final string-cloud source or fitted to it. No self-citations, uniqueness theorems, or renamings appear as load-bearing steps. The derivation is self-contained because the output Einstein equations are obtained by direct calculation rather than by re-labeling the input assumptions.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The framework rests on the Einstein equations in four dimensions, the definition of an eta-Einstein three-metric, and the specific stress-energy tensor of a string cloud. No free parameters are explicitly fitted; the deformation function is arbitrary. No new particles or forces are postulated beyond the string cloud, which is a standard source in the literature.

axioms (2)

domain assumption The Einstein field equations hold in four dimensions with a string-cloud stress-energy tensor.
Invoked throughout the abstract as the target equation satisfied by the deformed metric.
domain assumption The three-dimensional seed metric is eta-Einstein and admits a one-function deformation that produces a valid four-dimensional spacetime.
Central premise stated in the first two sentences of the abstract.

pith-pipeline@v0.9.0 · 5478 in / 1545 out tokens · 38374 ms · 2026-05-10T06:20:31.576388+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

38 extracted references · 11 canonical work pages · 1 internal anchor

[1]

(II.4) 4

Rank Letdηs denote the wedge product ofsexterior derivatives ofη, dηs := dη∧···∧dη   sfactors .(II.3) The rankrof the AC structure(ϕ,ξ,η)is defined as [14] r=    2sifdη s̸= 0andη∧dηs = 0, 2s+ 1ifη∧dη s̸= 0anddηs+1 = 0. (II.4) 4
[2]

The AC structure is said to be normal if it satisfies1 [ϕ,ϕ] + dη⊗ξ= 0.(II.7) A feature of the normality is thatξpreservesηandϕ, i.e.,Lξη= 0,L ξϕ= 0

Normality Let[ϕ,ϕ]denote the Nijenhuis tensor ofϕ, which is defined by [ϕ,ϕ](X,Y):=ϕ2[X,Y] + [ϕX,ϕY]−ϕ[ϕX,Y]−ϕ[X,ϕY],(II.5) for any vector fieldsX,Y. The AC structure is said to be normal if it satisfies1 [ϕ,ϕ] + dη⊗ξ= 0.(II.7) A feature of the normality is thatξpreservesηandϕ, i.e.,Lξη= 0,L ξϕ= 0. B. Almost contact metric structure A Riemannian metrichis...
[3]

Contact metric structure An ACM structure(ϕ,ξ,η,h)is called a contact metric structure if it holds that Φ= 1 2 dη.(II.14) A contact metric structure such thatξis a Killing vector field is called a K-contact structure
[4]

Sasakian structure A normal ACM structure(ϕ,ξ,η,h)is called Sasakian if it is contact. From Eq. (II.12), we haveη∧dηn =n!vol h̸= 0. Thus, the Sasakian structure has the maximum rank. The Sasakian structure ensures various curvature properties. A notable one is that, in three dimensions, the Ricci tensor takes a simple form as Ric=f 1h+f 2η⊗η,(II.15) where...
[5]

Aremarkableproperty of the quasi-Sasakian structure is thatξis a Killing vector field [15]

Quasi-Sasakian structure A normal ACM structure is called a quasi-Sasakian structure if it satisfies dΦ= 0.(II.16) Therankofthequasi-Sasakianstructureisnotnecessarilymaximum. Aremarkableproperty of the quasi-Sasakian structure is thatξis a Killing vector field [15]. 6 C. Three-dimensionalη-Einstein metrics on normal ACM structures In this subsection, we r...
[6]

Rank-oneη-Einstein metrics In the rank-one case, the functionfof the compatible metric (II.26) vanishes, and hence, Eqs. (II.31) and (II.32) read ∂ ∂xΩ,w = ∂ ∂yΩ,w = 0.(II.33) Solving this equation, we see that theη-Einstein metric is given as a warped product h= dw 2 +α2(w)γ,(II.34) whereγis a two-dimensional metric. For convenience, we use Gaussian norm...
[7]

(II.31) and (II.32)

Rank-threeη-Einstein metrics In contrast to the rank-one case, we do not pursue a general solution of Eqs. (II.31) and (II.32). Instead, we restrict to the quasi-Sasakian case, which yields simple rank-three η-Einstein metrics that can be used to construct spacetimes with a string cloud. Solutions of Eqs. (II.31) and (II.32) under weaker assumptions are d...
[8]

In the rank-one case, g=g[a,b,γ] =a 2(t,w) ( −dt2 + dw2) +b 2(t,w) ( dχ2 +γ2(χ,σ) dσ2) ,(III.7) 11 where b(t,w) =    a(t,w)α(w), a(t,w)β(t). (III.8)
[9]

This completes the construction of the spacetimes(M,g)admitting a simple string cloud

In the rank-three case, g=g[a,b,γ] =−dt2 +a 2(t)(dw+f(χ,σ) dσ)2 +b 2(t) ( dχ2 +γ2(χ,σ) dσ2) ,(III.9) where f(χ,σ) = ∫ χ γ(χ′,σ) dχ′.(III.10) We note that the time coordinate has been changed and the scale factorsa(t)andb(t) are redefined. This completes the construction of the spacetimes(M,g)admitting a simple string cloud. For a generic choice of the fun...
[10]

Rank-one case The deformed metricg[a,b,γ]is given by Eq. (III.7). We use the orthonormal frame{ea} given as follows: e0 = 1 a ∂ ∂t,e 1 = 1 a ∂ ∂w,e 2 = 1 b ∂ ∂χ,e 3 = 1 bγ ∂ ∂σ.(III.13) First, we consider the energy-momentum tensor of a singleξ-string. Since the worldsheet is spanned by∂/∂tandξ=∂/∂w, the coordinatesχ,σare constant along the worldsheet. Le...
[11]

Rank-three case The deformed metricg[a,b,γ]is given by Eq. (III.9). We use the orthonormal frame e0 = ∂ ∂t,e 1 = 1 a ∂ ∂w,e 2 = 1 b ∂ ∂χ,e 3 = 1 bγ ( ∂ ∂σ−f∂ ∂w ) .(III.17) The energy-momentum tensor can be constructed in the same manner as in the rank-one case. The resulting expression has the same functional form as Eq. (III.16), with the differ- ence t...
[12]

(III.7) and (III.13)

Rank-one case The metric and the orthonormal frame are given in Eqs. (III.7) and (III.13). The nonva- nishing components are calculated as G00[a,b,γ] = 1 a2  2a′ a b′ b− ( b′ b )2 −2b′′ b + 2 ˙a a ˙b b + ( ˙b b )2 + K2[γ] b2 ,(III.19) G01[a,b,γ] =−2 a2 [ −˙a a b′ b−a′ a ˙b b + ˙b′ b ] ,(III.20) G11[a,b,γ] = 1 a2  2a′ a b′ b + ( b′ b )2 + 2 ˙a a ˙b b...
[13]

(III.9) and Eq

Rank-three case The metric and the orthonormal frame are given in Eq. (III.9) and Eq. (III.17). We use the same notation as in the rank-one case. In contrast, the metric functionsa,bare functions oftonly. The nonvanishing components of the Einstein tensor are calculated as G00[a,b,γ] =−a2 4b4 + 2 ˙a a ˙b b + ( ˙b b )2 + K2[γ] b2 ,(III.24) G11[a,b,γ] =3 4 ...
[14]

Deformations of the FLRW universe For the rank-one deformed metricg[a,b,γ]given by Eq. (III.7), we take a(t,w) =a(t), b(t,w) =a(t)Σ(w;k), k=−1,0,1.(V.1) Then we have the deformed FLRW metric g[a,b,γ] =−dt2 +a 2(t) [ dw2 + Σ 2(w;k) ( dχ2 +γ2(χ,σ) dσ2)] .(V.2) Whenγ(χ,σ) = Σ(χ;ε= 1) = sinχ, the metric reduces to the FLRW metric. In the FLRW universe model, ...
[15]

We refer to these models as LRS cosmological models

Deformations of LRS cosmological models We specialize to the case a(t,w) =a(t), b(t,w) =a(t)β(t) = :b(t).(V.4) The rank-one deformed metric (III.7) takes the form g[a,b,γ] =−dt2 +a 2(t) dw2 +b 2(t) ( dχ2 +γ2(χ,σ) dσ2) ,(V.5) This metric reduces to the Kantowski-Sachs model forγ(χ,σ) = Σ(χ;ε= 1), and to the LRS Bianchi type I and III models forε= 0andε=−1,...
[16]

Deformations of topological Reissner-Nordström-(A)dS black holes We consider the static case by setting a(t,w) =a(w), b(t,w) =a(w)α(w) = :b(w),(V.6) and introduce a new coordinater:=b(w). The deformed metric (III.7) takes the form g=−f1(r) dt2 +f 2(r) dr2 +r 2 ( dχ2 +γ2(χ,σ) dσ2) .(V.7) This metric reduces to that of the topological Reissner-Nordström-(A)...
[17]

Deformation of the topological Einstein-Maxwell-Taub-NUT-(A)dS solution For the rank-three metric (III.9), we redefine the time coordinatetand choose the scale factorsa(t)andb(t)so that the metric takes the form g=−dt2 U(t) + (2l)2U(t)(dw+f(χ,σ) dσ)2 + (t2 +l 2) ( dχ2 +γ2(χ,σ) dσ2) ,(V.9) wherelis a constant, which will be identified with the NUT paramete...
[18]

Deformation of the closed FLRW universe We specialize to the case of a single scale factor by settinga(t) =b(t). The metric (III.9) takes the form g=−dt2 +a 2(t)h,h= 1 4 [ (dw+f(χ) dσ)2 + dχ2 +γ2(χ,σ) dσ2] ,(V.11) where we have redefined the scale factor bya(t)→a(t)/2in order that the three-metric hbe normalized to the unit roundS 3 when it takes the sphe...
[19]

(B.9) to Eqs

The case thatc 2 = 0 Substituting Eq. (B.9) to Eqs. (B.2) and (B.3), we find thatw0 is a constant, which can be set to zero, andc1 is given by c1 = F′′ F ,(B.10) where F := ∫ fdx.(B.11) In this setting, the metric is written as h= (dw+F ′dy)2 + w2 +F 2 2 γ,(B.12) whereγis the two-dimensional metric defined by γ=−F′′ F ( dx2 + dy2) .(B.13) This is the expl...
[20]

(B.9) to Eqs

The case thatc 2>0 Substituting Eq. (B.9) to Eqs. (B.2) and (B.3), we find that the functionsc2 andw 0 are constants andc 1 is given by c1 =−√c2F′′tan√c2F.(B.16) This gives the explicitη-Einstein metrics under the assumption (B.1) with a positivec2. 24 A simple example is given by taking w0 = 0, c 2 = 4, F=am(x,2) + π 4,(B.17) where am(x,2)denotes the Jac...
[21]

(B.9) to Eqs

The case thatc 2<0 Substituting Eq. (B.9) to Eqs. (B.2) and (B.3), we find that the functionsc2 andw 0 are constants andc 1 is given by c1 =√−c2F′′coth√−c2F.(B.21) This gives the explicitη-Einstein metrics under the assumption (B.1) with a negativec2. Appendix C: Derivation of Eq.(III.14) We first review the general framework of a string energy-momentum t...
[22]

Stephani, D

H. Stephani, D. Kramer, M. A. H. MacCallum, C. Hoenselaers, and E. Herlt,Exact solutions of Einstein’s field equations, Cambridge Monographs on Mathematical Physics (Cambridge Univ. Press, Cambridge, 2003)

2003
[23]

J. B. Griffiths and J. Podolsky,Exact Space-Times in Einstein’s General Relativity, Cambridge Monographs on Mathematical Physics (Cambridge University Press, Cambridge, 2009)

2009
[24]

S. I. Vacaru, Int. J. Geom. Meth. Mod. Phys.8, 9 (2011), arXiv:1106.4660 [physics.gen-ph]

work page arXiv 2011
[25]

Molina, Phys

C. Molina, Phys. Rev. D88, 127501 (2013), arXiv:1311.7137 [gr-qc]

work page arXiv 2013
[26]

Las Heras and P

C. Las Heras and P. León, Eur. Phys. J. C79, 990 (2019), arXiv:1905.02380 [gr-qc]

work page arXiv 2019
[27]

P. S. Letelier, Phys. Rev. D20, 1294 (1979)

1979
[28]

P. S. Letelier, Phys. Rev. D28, 2414 (1983)

1983
[29]

L. C. Nunes dos Santos, Phys. Rev. D111, 064032 (2025), arXiv:2502.15846 [gr-qc]

work page arXiv 2025
[30]

Priyokumar Singh and M

K. Priyokumar Singh and M. Daimary, Afr. Rev. Phys.14, 0013 (2019), arXiv:1909.11444 [physics.gen-ph]

work page arXiv 2019
[31]

D. V. Singh, S. G. Ghosh, and S. D. Maharaj, Phys. Dark Univ.30, 100730 (2020), arXiv:2003.14136 [gr-qc]

work page arXiv 2020
[32]

Boos and V

J. Boos and V. P. Frolov, Phys. Rev. D97, 024024 (2018), arXiv:1711.06357 [gr-qc]

work page arXiv 2018
[33]

Ishihara and S

H. Ishihara and S. Matsuno, PTEP2022, 023E01 (2022), arXiv:2112.02782 [hep-th]

work page arXiv 2022
[34]

Spacetime constructed from a contact manifold with a degenerate metric

H. Kozaki, H. Ishihara, T. Koike, and Y. Morisawa, Phys. Rev. D110, 104023 (2024), arXiv:2408.03701 [gr-qc]

work page internal anchor Pith review Pith/arXiv arXiv 2024
[35]

Olszak, Ann

Z. Olszak, Ann. Polon. Math.47, 41 (1986)

1986
[36]

D. E. Blair, J. Differential Geometry1, 331 (1967). 26

1967
[37]

Ishihara and H

H. Ishihara and H. Kozaki, Phys. Rev. D72, 061701 (2005), arXiv:gr-qc/0506018

work page arXiv 2005
[38]

Y. Koga, T. Igata, and K. Nakashi, Phys. Rev. D103, 044003 (2021), arXiv:2011.10234 [gr-qc]. 27

work page arXiv 2021