Non-Vacuum Solutions in Cotton Theory

Maryam Ranjbar; Serkan Doruk Hazinedar; Yaghoub Heydarzade

Non-Vacuum Solutions in Cotton Theory

T0 review · 0 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Cotton theory, in its Codazzi formulation, is shown to admit exact non-vacuum solutions that generalize the Kiselev and Dymnikova black holes of general relativity, with extra linear and quadratic metric terms of geometric origin.

desk verdict Both exact-solution theorems are correct; the Dymnikova worry in the stress test comes from a sign slip, and the main soft spot is the thermodynamics treatment of effective charges. read the letter →

arxiv 2412.06953 v2 pith:2MKJXN74 submitted 2024-12-09 gr-qc

Serkan Doruk Hazinedar , Yaghoub Heydarzade , Maryam Ranjbar This is my paper · ORCID

classification gr-qc

keywords CottontheoryCodazziformulationKiselevsolutionDymnikovasphericallysymmetricexactsolutionsblackholethermodynamicsnullgeodesicsregularholes

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Cotton theory is a higher-derivative alternative to general relativity whose field equations are third-rank tensors. This paper shows that, within the theory's Codazzi formulation, there are two families of static spherically symmetric non-vacuum solutions: a Kiselev-type metric for an anisotropic quintessence-like fluid, with $\Phi(r)=1+\frac{c_0}{r}+c_1 r+c_2 r^2+c_3 r^{-(1+3\omega)}$, and a Dymnikova-type metric with an exponential density profile, $\Phi(r)=1+\frac{c_0}{r}\left(1-e^{-r^3/r_*^3}\right)+c_1 r+c_2 r^2$. In both families the linear $c_1 r$ and quadratic $c_2 r^2$ terms are of geometric origin: they already appear in the vacuum solutions of Cotton theory, and the fluid contributes only the remaining terms. If the paper is right, Cotton theory naturally contains these spacetimes and predicts horizon, thermodynamic, and geodesic behavior that differs from general relativity in ways the paper computes.

What carries the argument

The carrying object is the Codazzi formulation of Cotton gravity, in which the third-rank field equations are replaced by the pair $C_{ab}=S_{ab}-8\pi T_{ab}$ and $\nabla_a C_{bc}=\nabla_b C_{ac}$, where $S_{ab}$ is the Schouten tensor and $T_{ab}$ is the trace-adjusted energy-momentum tensor. For the static spherically symmetric line element $ds^2=-\Phi(r)dt^2+\Phi(r)^{-1}dr^2+r^2 d\Omega^2$, imposing these equations on the Kiselev anisotropic fluid reduces the system to a fourth-order Euler-Cauchy ODE for $\Phi$. The indicial polynomial $(n-1)(n+1)(n-2)(n+1+3\omega)=0$ produces exactly the powers $r^{-1}$, $r$, $r^2$, and $r^{-(1+3\omega)}$, which combine with the particular solution $\Phi_p=1$ to give the generalized Kiselev metric. For the Dymnikova source the same procedure gives a similar Euler-Cauchy equation whose solution combines the Schwarzschild-like term with $e^{-r^3/r_*^3}$ and the same geometric powers. This ODE reduction is the mechanism that lets the paper turn a fourth-order system into exact closed-form metrics.

What would settle it

Substitute the generalized Kiselev metric and its energy density $\rho(r)=3\omega c_3 r^{-3(1+\omega)}$ directly into the original third-rank Cotton field equations and check whether they hold identically; repeat for the Dymnikova-type metric with $\rho(r)=\rho_0 e^{-r^3/r_*^3}$. If the equations fail, or hold only after extra trace or boundary conditions are imposed, then the paper has produced solutions of the Codazzi system rather than of Cotton theory.

Watch

Extended reading notes

Core claim

The central claim is Theorem 1 and Theorem 2: Cotton theory in its Codazzi formulation admits the static spherically symmetric Kiselev spacetime as an exact non-vacuum solution, with metric function $\Phi(r)=1+\frac{c_0}{r}+c_1 r+c_2 r^2+c_3 r^{-(1+3\omega)}$ and energy density $\rho(r)=3\omega c_3 r^{-3(1+\omega)}$, and the Dymnikova spacetime as an exact non-vacuum solution, with metric function $\Phi(r)=1+\frac{c_0}{r}\left(1-e^{-r^3/r_*^3}\right)+c_1 r+c_2 r^2$ and energy density $\rho(r)=\rho_0 e^{-r^3/r_*^3}$. The paper reads the appearance of $c_1$ and $c_2$ as a signature of the theory's higher-derivative structure: these terms are present even when the source is absent, so the matter source is responsible only for the $c_3$ term, or for the exponential core. The paper then derives physical consequences: the generalized Kiselev solution stays singular at $r=0$; it may have one or several horizons; its first law acquires two effective charges associated with $c_1$ and $c_3$; and its null and timelike geodesics shift relative to general relativity. The generalized Dymnikova solution, unlike its general-relativistic counterpart, develops a curvature singularity at the center because of the linear $c_1 r$ term, and it contains two de Sitter regions.

Load-bearing premise

The argument adopts, from a cited earlier work rather than from a proof in this paper, the claim that the simplified second-order Codazzi equations used here are fully equivalent to the original third-order Cotton field equations for static spherically symmetric non-vacuum metrics; if that equivalence requires additional conditions, the new metrics may solve the simplified system without being solutions of Cotton theory itself.

Editorial extensions

If this is right

The GR Kiselev solutions for quintessence ($\omega=-2/3$) and a cosmological constant ($\omega=-1$) arise as vacuum solutions of Cotton theory, so effects that general relativity attributes to matter can be pure geometry in this theory.
The generalized Dymnikova black hole has two de Sitter regions—an inner core from the fluid and an outer asymptote from the geometric $c_2$ term—and a divergent Kretschmann scalar at $r=0$ unless $c_1=0$.
For the generalized Kiselev black hole the first law reads $\delta m=T\delta S+V\delta P+\Psi\delta Q_t$ with two effective charges carried by the $c_1$ and $c_3$ parameters, and the Hawking temperature differs from the Reissner-Nordstrom-de Sitter value in general relativity.
Null circular orbits coincide with the general-relativistic value $r_c=3m$ only under special parameter relations; otherwise the photon orbits are shifted, which gives an observational handle on the Cotton parameters.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The authors do not pursue it, but the presence of the linear $c_1 r$ term in non-vacuum solutions suggests that Cotton theory can mimic dark-energy-like or dark-matter-like radial effects in the presence of ordinary matter, not only in vacuum.
An implicit selection rule follows from the paper's singularity analysis: regular black holes in Cotton theory must satisfy $c_1=0$, since the linear term is what makes the generalized Dymnikova metric singular at the center.
Because the $c_2$ term is a cosmological-constant-like term of pure geometric origin, the paper's results imply that cosmic acceleration could in principle be geometric rather than sourced by a dark-energy field; this is an interpretation the paper states only for the solutions, not as a cosmological model.
The predicted shifts in circular photon orbits could be constrained by black-hole shadow measurements, though the paper does not compute shadow sizes or quasinormal modes.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Quick take: both theorems check out. The stress-test's objection to Theorem 2 is wrong. I re-derived Eq. (117) and the derivatives of the exponential piece Φ_exp = (ρ0 r*^3/3r) e^{-r^3/r*^3}; the r^2 and r^3 terms in the claimed residual never appear. The exponential piece cancels the source term exactly. So the Dymnikova half of the paper is fine.

What is actually new: the general Kiselev family with arbitrary ω, and the Dymnikova solution with the additional c1r and c2r^2 terms. The appendices are explicit and re-derivable; I verified the Kiselev metric satisfies the written field equation for generic ω. The c1,c2 terms genuinely arise from the higher-derivative structure of the theory rather than from the fluid, and the paper makes that point clearly.

Soft spots, in decreasing order. First, the thermodynamics section defines Q1=c1r0^3 and Q2=c3r0^{1-3ω} and then varies them as independent variables, even though they are functions of r0. That is not self-consistent; a referee should ask for a version in terms of c1 and c3, or a clear justification of the effective-charge treatment. Second, the paper assumes the Mantica-Molinari Codazzi formulation is fully equivalent to Harada's original third-rank equations, citing [20] rather than stating the conditions. This is standard in the literature, but a sentence would help. Third, novelty relative to Sussman and Najera [13] is asserted but not spelled out point-by-point. None of these are fatal; the central existence theorems are solid.

Who it is for: people working on Cotton gravity and exact black hole solutions in modified gravity. It is a competent, useful exact-solution paper. I would send it to a referee.

Referee Report

0 major / 5 minor

Summary. The paper studies non-vacuum static spherically symmetric solutions of Cotton theory in its Codazzi formulation. Theorem 1 presents a Kiselev-type metric Φ(r)=1+c0/r+c1r+c2r²+c3r^{-(1+3ω)} with ρ(r)=3ωc3r^{-3(1+ω)}, and Theorem 2 presents a Dymnikova-type metric Φ(r)=1+c0/r(1-e^{-r³/r*³})+c1r+c2r² with ρ(r)=ρ0e^{-r³/r*³}. The appendices give the algebraic derivations, and the body analyzes ADM mass, singularities, horizons, black-hole thermodynamics, and geodesics for both families.

Significance. These are useful, explicitly checkable extensions of known GR solutions into Cotton theory, and the main new point—that c1r and c2r² arise as integration constants of the higher-derivative structure rather than from the fluid source—is clearly illustrated. I independently verified the Dymnikova substitution: substituting Φ from Eq. (61) into Eq. (117) gives L[Φ]=ρ0e^{-r³/r*³}(12r^5/r*³-9r^8/r*⁶), which exactly cancels the source term, so the proposed metric does satisfy the field equation. The Kiselev reduction to the Euler-Cauchy equation (100) is also consistent. The main results therefore appear sound; the issues that remain are local and concern the interpretation of the thermodynamics and some asymptotic statements.

minor comments (5)

[3.2.3] Section 3.2.3, Eqs. (24)-(29): The definitions Q1=c1r0^3 and Q2=c3r0^{1-3ω} make Q1 and Q2 functions of the horizon radius, and the first law is obtained by treating them as independent variables in the identity Φ(r0)=0; this is a formal reparametrization rather than a physical first law with independent charges, and Ref. [2] itself reports vanishing conserved charges in CT. Please derive the relation in terms of the original parameters (m, Λ, c1, c3, ω) or explicitly label the potentials as formal.
[3.2.1] Section 3.2.1, Eqs. (10), (14), (15): The asymptotically flat range in Eq. (10) should exclude -1/3<ω<0, where Φ-1 decays slower than 1/r and the last term in Eq. (14) diverges. Also, Eq. (15) includes ω=0, for which Eq. (14) gives MADM=m-c3/2 rather than m unless c3 is absorbed into c0.
[3.1 and 3.3] The theorem statements specify only ρ(r); a complete statement of the energy-momentum tensor is needed. Theorem 1 should include the anisotropic equations of state Eqs. (87)-(88), and Theorem 2 should include the pressure (115), so that the solutions can be checked from the theorem statements themselves.
[Appendix I] In Eqs. (91)-(92), the Cotton tensor components are written as C=S−Tbar, omitting the 8π factor of Eq. (3). Please state the unit convention or keep the factor consistently throughout the appendices.
[Throughout] There are several presentation slips: a stray brace in Eq. (94); inconsistent notation between r*³ and r0²rg near Eq. (113); and the final line of Appendix II lists c3 among the arbitrary constants although the Dymnikova solution has no c3.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the Kiselev and Dymnikova solutions are obtained by direct integration of the stated Codazzi field equations, with all constants arising as integration constants rather than fitted parameters.

full rationale

The paper's central derivation is self-contained and non-circular. Theorem 1 (Appendix I) substitutes the Kiselev anisotropic fluid Ansatz (Eqs. 87-88) into the Codazzi field equations (Eqs. 93-94), reduces the system to the fourth-order Euler-Cauchy ODE (Eq. 100), and solves it with integration constants c0, c1, c2, c3; the energy density ρ(r)=3ωc3 r^{-3(1+ω)} follows from the field equations rather than being fitted. Theorem 2 (Appendix II) does the same for Dymnikova's exponentially decaying density (Eq. 111), obtaining the ODE (Eq. 117) and its general solution (Eq. 118); the final metric form (Eq. 119) is a reparameterization using r*^3=-r0^2 c0 and r0=sqrt(3/ρ0), not an assumption of the target metric. No fitted parameter is renamed as a prediction: the constants are integration constants, and the paper makes no benchmark-data comparison. The only external premise, that Harada's third-rank equations can be written as the Codazzi system (Eq. 3), is imported from Mantica and Molinari [20], which is not a self-citation of this paper's authors, and it is a stated input assumption rather than a derived conclusion. Self-citations such as [1] and [23]-[26] are contextual or concern prior applications and do not carry the derivation. A possible algebraic error in the Dymnikova substitution would be a correctness defect, not a circularity, because it does not consist in assuming the conclusion. Accordingly, the score is 0.

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

The central solutions rest on a standard metric ansatz, imposed fluid models from GR, the Codazzi equivalence, and free integration constants. No parameters are fitted to data. The invented entities are bookkeeping charges rather than new physical particles or forces.

free parameters (5)

c1
Integration constant of the fourth-order ODE; linear term of geometric origin in the metric. Not fitted to data.
c2
Integration constant interpreted as a cosmological-constant-like term; not fitted to data.
c3
Integration constant controlling the surrounding-field term in the Kiselev solution; not fitted to data.
omega
Equation-of-state parameter of the Kiselev anisotropic fluid; a free parameter of the source model.
rho0
Energy-density scale of the Dymnikova source; a free parameter of the source model.

assumptions (5)

domain assumption Codazzi formulation of Cotton gravity (Eq. 3) is equivalent to Harada's original third-rank field equations
Cited from [20]; all derivations in the paper assume this equivalence without independent proof.
standard math Static spherically symmetric metric ansatz (Eq. 83) is exhaustive for the solutions sought
The paper restricts to the standard Schwarzschild-like diagonal form, a standard ansatz for the considered symmetry class.
domain assumption Kiselev energy-momentum tensor components (Eqs. 87-88) with constant equation-of-state parameter omega
The source is imposed rather than derived; no energy conditions are checked.
domain assumption Dymnikova energy density (Eq. 111) is imposed
The exponential density profile is taken from the GR Dymnikova model and used as input.
standard math Euler-Cauchy ODE solution theory
Used to solve the fourth-order field equations in Appendices I and II.

invented entities (1)

Effective charges Q1=c1 r0^3 and Q2=c3 r0^{1-3ω}
purpose: Bookkeeping parameters to cast the first law of black hole thermodynamics in a Reissner-Nordstrom-de Sitter-like form
They are re-parametrizations of integration constants and the horizon radius, not independent physical charges; no falsifiable handle outside the paper.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Non-Vacuum Solutions in Cotton Theory." pith.science (2026). https://pith.science/paper/2MKJXN74

@misc{pith2026241206953,
  author       = {Pith},
  title        = {Pith review of: Non-Vacuum Solutions in Cotton Theory},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2MKJXN74}},
  note         = {Machine review of arXiv:2412.06953}
}

read the original abstract

Cotton theory (CT) introduces a higher derivative extension of General Relativity (GR) characterized by third-rank field equations. Recently, key distinctions between CT and GR concerning wave and vacuum solutions have been highlighted in [1, 2]. In this study, two particular non-vacuum solutions of CT are investigated within its Codazzi formulation. The motivation is to reveal how this theory might account for or adapt to the effects of non-vacuum sources, and whether it can provide new insights into the behavior of both singular and regular black holes in astrophysical contexts. It is shown that CT generalizes the Kiselev and Dymnikova solutions in GR. Some aspects of the generalized solutions, in particular concerning singularities, thermodynamics, and geodesics, are addressed in comparison to GR.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

60 extracted references · 55 canonical work pages

[20]

Codazzi tensors and th eir space-times and cotton gravity

C. A. Mantica and L. G. Molinari, “Codazzi tensors and th eir space-times and cotton gravity”, Gen. Relativ. Gravit, 55, 62, (2023)

work page 2023
[13]

Exact solutions of Cotton Gra vity in its Codazzi formulation

R. A. Sussman, S. Najera, “Exact solutions of Cotton Gra vity in its Codazzi formulation”, arXiv:2312.02115 [gr-qc]

arXiv
[1]

Wave Metrics in th e Cotton and Conformal Killing Gravity Theories

M. Gürses, Y. Heydarzade, Ç. Şentürk, “Wave Metrics in th e Cotton and Conformal Killing Gravity Theories”, Phys. Rev. D 110, 084082 (2024)

work page 2024
[2]

Vanishing of Conserved Charges in Cot ton Gravity

E. Altas, B. Tekin, “Vanishing of Conserved Charges in Cot ton Gravity”, Phys.Rev.D 111, L021503, (2025)

work page 2025
[3]

Sur les varietes a trois dimensions

E. Cotton, “Sur les varietes a trois dimensions”, Ann. Fac. Sci. Toulouse 2. 1, 385 (1899)

work page
[4]

Reine Inﬁnitesimalgeometrie

H. Weyl, “Reine Inﬁnitesimalgeometrie”, Math Z 2, 384-411 (1918)

work page 1918
[5]

Riemannian geometry

L. P. Eisenhart, “Riemannian geometry”, Princeton Univ ersity Press (1997), ISBN 0-691-02353-0

work page 1997
[6]

Cotton blend gravity pp wa ves

S. Deser, R. Jackiw, S.-Y. Pi, “Cotton blend gravity pp wa ves”, Acta Phys. Pol. B 36, 27 (2005)

work page 2005

Show all 60 references

[7]

Exact Vacuum Solution to C onformal Weyl Gravity and Galactic Rotation Curves

P. D. Mannheim and D. Kazanas, “Exact Vacuum Solution to C onformal Weyl Gravity and Galactic Rotation Curves”, Astrophys. J. 342, 635 (1989)

1989
[8]

General Structure of the G ravitational Equations of Motion in Conformal Weyl Gravity

P. D. Mannheim and D. Kazanas, “General Structure of the G ravitational Equations of Motion in Conformal Weyl Gravity”, Astrophys. J. Suppl. Ser. 76, 431 (1991)

1991
[9]

Emergence of the Cotton tensor for describin g gravity

H. Junpei, “Emergence of the Cotton tensor for describin g gravity”, Phys. Rev. D 103, L121502, (2021)

2021
[10]

Cotton gravity and 84 galaxy rotation curve s

H. Junpei, “Cotton gravity and 84 galaxy rotation curve s”, Phys. Rev. D 106, 064044, (2022)

2022
[11]

General spherica lly symmetric solution of Cotton gravity

M. Gogberashvili and A. Girgvliani, “General spherica lly symmetric solution of Cotton gravity”, Class. Quant. Gravity 41, 025010 (2024)

2024
[12]

Black bounces in Cotton gravity

Junior, E.L.B., Junior, J.T.S.S., Lobo, F.S.N. et al., “ Black bounces in Cotton gravity”, Eur. Phys. J. C 84, 1190 (2024)

2024
[14]

Comment on Emergence of the Cotton tensor f or describing gravity

P. Bargueno, “Comment on Emergence of the Cotton tensor f or describing gravity”, Phys. Rev. D 104, 088501 (2021)

2021
[15]

Reply to Comment on Emergence of the Cotton t ensor for describing gravity,

J. Harada, “Reply to Comment on Emergence of the Cotton t ensor for describing gravity,” Phys. Rev. D 104, 088502 (2021)

2021
[16]

Cotton gravity is not predic tive

G. Clement and K. Nouicer, “Cotton gravity is not predic tive”, Phys. Lett. B 856, 138947 (2024)

2024
[17]

Response to a critique of

R. A. Sussman, C. A. Mantica, L.G. Molinari, and S. Najer a, “Response to a critique of "Cotton Gravity"”, arXiv:2401.10479 [gr-qc]. 20

arXiv
[18]

Farewell to Cotton gravity

G. Clement and K. Nouicer, “Farewell to Cotton gravity” , arXiv:2401.16008 [gr-qc]

arXiv
[19]

Second Response to the critique of

R. A. Sussman, C. A. Mantica, L. G. Molinari, and S. Najer a, “Second Response to the critique of "Cotton Gravity"”, arXiv:2402.01992 [gr-qc]

arXiv
[21]

Quintessence and black holes

V. V. Kiselev, “Quintessence and black holes”, Class. Quant. Gravity 20, 1187 (2003)

2003
[22]

Vacuum nonsingular black hole

I. Dymnikova, “Vacuum nonsingular black hole”, Gen. Relativ. Gravit 24, 235242, (1992)

1992
[23]

Surrounded Vaidya solution by cosmological ﬁelds

Y. Heydarzade, F. Darabi, “Surrounded Vaidya solution by cosmological ﬁelds”, Eur. Phys. J. C 78, 582 (2018)

2018
[24]

Surrounded Vaidya black hol es: apparent horizon properties

Y. Heydarzade, F. Darabi, “Surrounded Vaidya black hol es: apparent horizon properties”, Eur. Phys. J. C 78, 342 (2018)

2018
[25]

D-b ound and the Bekenstein bound for the surrounded Vaidya black hole

H. Hadi, F. Darabi, K. Atazadeh, and Y. Heydarzade, “D-b ound and the Bekenstein bound for the surrounded Vaidya black hole”, Eur. Phys. J. C 80, 1126 (2020)

2020
[26]

D-b ound and Bekenstein bound for McVittie solution surrounded by dark energy cosmological ﬁelds

H. Hadi, Y. Heydarzade, F. Darabi, and K. Atazadeh, “D-b ound and Bekenstein bound for McVittie solution surrounded by dark energy cosmological ﬁelds”, Eur. Phys. J. Plus 135, 584 (2020)

2020
[27]

Distorted black holes

R. Geroch and J.B. Hartle, “Distorted black holes”, J. Math. Phys. 23, 680 (1982)

1982
[28]

Distorted Black Holes with Ch arge

S. Fairhurst, B. Krishnan, “Distorted Black Holes with Ch arge”, Int. J. Mod. Phys. D10, 691 (2001)

2001
[29]

The Evolution of Distorted Rota ting Black Holes III: Initial Data

S. R. Brandt, E. Seidel, “The Evolution of Distorted Rota ting Black Holes III: Initial Data”, Phys. Rev. D 54, 1403 (1996)

1996
[30]

Black holes surrounded by uniform ly rotating rings

M. Ansorg, D. Petroﬀ, “Black holes surrounded by uniform ly rotating rings”, Phys. Rev. D 72, 024019 (2005)

2005
[31]

Black holes in general relativity

S. W. Hawking, “Black holes in general relativity”, Commun. Math. Phys. 25, 152 (1972)

1972
[32]

Black holes with short hair

J. D. Brown and V. Husian, “Black holes with short hair”, Int. J. Mod. Phys. D 6, 563 (1997)

1997
[33]

New black hole solut ions with hair

S.Droz, M. Heusler, N. Straumann, “New black hole solut ions with hair”, Phys. Lett. B, 268, 371 (1991)

1991
[34]

Schwarzschild Black Holes can Wear Scalar Wigs

J. Barranco, A. Bernal, J.C. Degollado, A. Diez-Tejedor, M. Megevand, M. Alcubierre, D.Nunez, O. Sarbach, “Schwarzschild Black Holes can Wear Scalar Wigs”, Phys. Rev. Lett. 109, 081102 (2012)

2012
[35]

Regular magnetic black holes and monop oles from nonlinear electrodynamics

K. A. Bronnikov, “Regular magnetic black holes and monop oles from nonlinear electrodynamics”, Phys. Rev. D 63, 044005 (2001)

2001
[36]

Vacuum nonsingular black hole solutio ns in tetrad theory of gravitation

G. G. L. Nashed, “Vacuum nonsingular black hole solutio ns in tetrad theory of gravitation”, Gen. Rel. Grav. 34, 1047-1058 (2002)

2002
[37]

Nonsingular black holes on the brane

S. Shankaranarayanan, N. Dadhich, “Nonsingular black holes on the brane”, Int. J. Mod. Phys. D 13, 1095-1104 (2004)

2004
[38]

Regular electrically charged structur es in nonlinear electrodynamics coupled to general relativity

I. Dymnikova, “Regular electrically charged structur es in nonlinear electrodynamics coupled to general relativity”, Class. Quant. Grav 21, 4417 (2004). 21

2004
[39]

Formation and evaporation of regular bl ack holes

S. A. Hayward, “Formation and evaporation of regular bl ack holes”, Phys. Rev. Lett. 96, 031103 (2006)

2006
[40]

Regular phantom black holes

K.A. Bronnikov, “Regular phantom black holes”, Phys. Rev. Lett. 96, 251101 (2006)

2006
[41]

Re gular black holes in quadratic gravity

W. Berej, J. Matyjasek, D. Tryniecki, M. Woronowicz, “Re gular black holes in quadratic gravity”, Gen. Rel. Grav. 38, 885 (2006)

2006
[42]

Regular rotating de Sit ter-Kerr black holes and solitons

I. Dymnikova, E. Galakatinov, “Regular rotating de Sit ter-Kerr black holes and solitons”, Class. Quant. Grav. 33, 145010 (2016)

2016
[43]

Notes on nonsingular models of black hole s

V. P. Frolov, “Notes on nonsingular models of black hole s”, Phys. Rev. D 94, 104056 (2016)

2016
[44]

Kiselev and Schwarzschi ld-de Sitter black holes in higher derivative theories of gravitation

J. Matyjasek, and M. Telecka, “Kiselev and Schwarzschi ld-de Sitter black holes in higher derivative theories of gravitation”, Phys. Rev. D 107, 064058 (2023)

2023
[45]

Kis elev solution in power-Maxwell electrody- namics

Ma. Dariescu, C. Dariescu, V. Lungu, and C. Stelea, “Kis elev solution in power-Maxwell electrody- namics”, Phys. Rev. D 106, 064017 (2022)

2022
[46]

Regular black holes in f (R) gravity coupled to nonlinear electrodynamics

M. E. Rodrigues, E. L. B. Junior, G. T. Marques, and V. T. Za nchin, “Regular black holes in f (R) gravity coupled to nonlinear electrodynamics”, Phys. Rev. D 94, 024062 (2016)

2016
[47]

Regular multihorizon black holes in modiﬁed gravity with nonlinear elec- trodynamics

S. Nojiri, S. D. Odintsov, “Regular multihorizon black holes in modiﬁed gravity with nonlinear elec- trodynamics”, Phys. Rev. D 96, 104008 (2017)

2017
[48]

Regular blac k holes in Einstein-Gauss-Bonnet gravity

S. G. Ghosh, D. V. Singh, and S. D. Maharaj, “Regular blac k holes in Einstein-Gauss-Bonnet gravity”, Phys. Rev. D 97, 104050 (2018)

2018
[49]

Hairy black-hole solutions in generalized Proca theories

L. Heisenberg, R. Kase, M. Minamitsuji, and S. Tsujikaw a, “Hairy black-hole solutions in generalized Proca theories”, Phys. Rev. D 96, 084049 (2017)

2017
[50]

Regular multihori zon black holes in f (G) gravity with nonlinear electrodynamics

M. E. Rodrigues and M. V. de S. Silva, “Regular multihori zon black holes in f (G) gravity with nonlinear electrodynamics”, Phys. Rev. D 99, 124010 (2019)

2019
[51]

Friedmann equations in the Codazzi parametrization of Cotton and extended theories of gravity and the Dark Sector

C. A. Mantica, and L. G. Molinari, “Friedmann equations in the Codazzi parametrization of Cotton and extended theories of gravity and the Dark Sector”, Phys. Rev. D 109, 044059 (2024)

2024
[52]

Einstein spaces

A. Z. Petrov, “Einstein spaces”, Pergamon Press, Oxfor d, (1969), ISBN 10: 0080123155

1969
[53]

The covariant approa ch to the static spacetimes in Einstein and extended gravity theories

C. A. Mantica, and L. G. Molinari, “The covariant approa ch to the static spacetimes in Einstein and extended gravity theories”, Gen. Relativ. Gravit. 55, 100 (2023)

2023
[54]

Mechan ics of universal horizons

P. Berglund, J. Bhattacharyya, and D. Mattingly, “Mechan ics of universal horizons”, Phys. Rev. D 85, 124019 (2012)

2012
[55]

Static spherically symmetric solutions to Einstein-Maxwell dilaton ﬁeld equations in D dimensions

M. Gürses, E. Sermutlu, “Static spherically symmetric solutions to Einstein-Maxwell dilaton ﬁeld equations in D dimensions”, Class. Quant. Grav. 12, 2799 (1995)

1995
[56]

NAT black holes

M. Gürses, Y. Heydarzade, Ç. Şentürk, “NAT black holes” , Eur. Phys. J. C 79, 942 (2019)

2019
[57]

Positive-Deﬁnite Energy Density and Global Consequences for General Relativity

A. Komar, “Positive-Deﬁnite Energy Density and Global Consequences for General Relativity”, Phys. Rev. 129, 1873 (1963)

1963
[58]

The dynamics of gen eral relativity

R. Arnowitt, S. Deser, C. W. Misner, “The dynamics of gen eral relativity”, In L. Witten, editor, Gravitation: An Introduction to Current Research, Wiley, ( 1962). 22

1962
[59]

Black Holes and Entropy

J. D. Bekenstein, “Black Holes and Entropy”, Phys. Rev. D 7, 2333 (1973)

1973
[60]

Particle creation by black holes

S. W. Hawking, “Particle creation by black holes”, Commun. Math. Phys. 43, 199 (1975). 23

1975

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

[1] [20]

Codazzi tensors and th eir space-times and cotton gravity

C. A. Mantica and L. G. Molinari, “Codazzi tensors and th eir space-times and cotton gravity”, Gen. Relativ. Gravit, 55, 62, (2023)

work page 2023

[2] [13]

Exact solutions of Cotton Gra vity in its Codazzi formulation

R. A. Sussman, S. Najera, “Exact solutions of Cotton Gra vity in its Codazzi formulation”, arXiv:2312.02115 [gr-qc]

arXiv

[3] [1]

Wave Metrics in th e Cotton and Conformal Killing Gravity Theories

M. Gürses, Y. Heydarzade, Ç. Şentürk, “Wave Metrics in th e Cotton and Conformal Killing Gravity Theories”, Phys. Rev. D 110, 084082 (2024)

work page 2024

[4] [2]

Vanishing of Conserved Charges in Cot ton Gravity

E. Altas, B. Tekin, “Vanishing of Conserved Charges in Cot ton Gravity”, Phys.Rev.D 111, L021503, (2025)

work page 2025

[5] [3]

Sur les varietes a trois dimensions

E. Cotton, “Sur les varietes a trois dimensions”, Ann. Fac. Sci. Toulouse 2. 1, 385 (1899)

work page

[6] [4]

Reine Inﬁnitesimalgeometrie

H. Weyl, “Reine Inﬁnitesimalgeometrie”, Math Z 2, 384-411 (1918)

work page 1918

[7] [5]

Riemannian geometry

L. P. Eisenhart, “Riemannian geometry”, Princeton Univ ersity Press (1997), ISBN 0-691-02353-0

work page 1997

[8] [6]

Cotton blend gravity pp wa ves

S. Deser, R. Jackiw, S.-Y. Pi, “Cotton blend gravity pp wa ves”, Acta Phys. Pol. B 36, 27 (2005)

work page 2005