Generalized Eddington--Finkelstein Coordinates and Exact Vaidya-Type Solutions in Weyl Conformal Gravity

Petr Jizba; Tereza Lehe\v{c}kov\'a

REVIEW 3 major objections 3 minor 92 references

Generalized Eddington--Finkelstein Coordinates and Exact Vaidya-Type Solutions in Weyl Conformal Gravity

T0 review · 3 major / 3 minor · reviewed 2026-08-09 · deepseek-v4-flash

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Pith's one-line read In Weyl conformal gravity, all vacuum Vaidya-type spacetimes with spherical, planar, or hyperbolic symmetry are classified: the metric function has the four-term polynomial form with exactly two solution branches.

desk verdict A genuine and useful classification of Vaidya-type WCG solutions, with a fixable sign typo in the headline formula and a 'local solutions' caveat that should be made explicit. read the letter →

arxiv 2502.00905 v1 pith:KYXCF2AW submitted 2025-02-02 gr-qc hep-thmath-phmath.MP

Petr Jizba , Tereza Lehečková This is my paper · ORCID

classification gr-qchep-thmath-phmath.MP

keywords WeylconformalgravityVaidya-typesolutionsEddington-FinkelsteincoordinatesBachequationvacuumnulldustCoulombfieldPetrovtypeD

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

The paper sets out to find every dynamical vacuum solution of Weyl conformal gravity in generalized Eddington–Finkelstein coordinates, for spherical, planar, and hyperbolic symmetry. It claims that the Bach vacuum equation forces the metric function to be $H(r,w)=F_1(w)+F_2(w)/r+F_3(w)r+F_4(w)r^2$, and that the remaining field equations leave exactly two branches: $F_1=\pm K$ with $F_2=0$ and $F_3,F_4$ arbitrary, or $F_2\neq 0$ with $F_3,F_4$ determined by $F_1,F_2$ and one constant. If correct, this is the first complete classification of vacuum Vaidya-type solutions in Weyl conformal gravity, and it shows that, unlike in general relativity, such vacuum spacetimes can be genuinely dynamical. The paper also classifies non-vacuum solutions carrying a Coulomb field and null dust, and it shows that only the vacuum and electrovacuum families are even locally Weyl-equivalent to Einstein spaces.

What carries the argument

The central object is the generalized Eddington–Finkelstein line element $ds^2 = H(r,w)dw^2 + 2c\,dw\,dr + r^2 d\Sigma_K^2$, with $c=\pm 1$ for retarded or advanced null time and $d\Sigma_K^2$ the 2-space of constant Gaussian curvature $K\in\{1,0,-1\}$. The ansatz makes $r$ a null affine parameter, and it renders the Bach equation tractable: the $rr$-component becomes a linear fourth-order ODE in $r$ alone, forcing $H$ to be the polynomial (19), and the remaining components reduce to polynomial conditions that are solved exactly into the two branches. The same coordinates carry the Weyl-equivalence analysis, because dividing the metric by $r^2$ turns the conformal Killing vector (33) into a Killing vector, producing a Riegert-type proof that the vacuum solutions are locally conformally static.

What would settle it

Take any smooth functions $F_1(w)$ and $F_2(w)\neq 0$, define $F_3$ and $F_4$ by (23), and substitute into the full Bach tensor: the paper claims every component vanishes identically for all such choices. A single numerical instance in which $B_{wr}$ or $B_{ww}$ does not vanish, or a vacuum solution in the same coordinates whose $H$ contains a non-polynomial term such as $r^p$ with $p\notin\{0,-1,1,2\}$ (which makes the $rr$-component of the Bach tensor nonzero), would refute the classification.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is that the Bach equation $B_{\mu\nu}=0$ in the metric (1) has exactly the two branches (22) and (23). Branch (22), with $F_1=\pm K$ and $F_2=0$, leaves $F_3$ and $F_4$ arbitrary and describes backgrounds in which a constant acceleration-like term and a cosmological-constant-like term arise without any matter source. Branch (23), with $F_2\neq 0$, fixes $F_3$ and $F_4$ in terms of $F_1$, $F_2$, and one constant $C_1$, and contains the static M-K solution and its topological-black-hole analogues as the constant-function limit. Every solution is Petrov type D unless conformally flat, has a genuine curvature singularity at $r=0$ when $F_1\neq -K$ or $F_2\neq 0$, and has trapping horizons at $H(r,w)=0$. The paper extends the Birkhoff–Riegert theorem to the planar and hyperbolic cases and shows that all vacuum solutions are at least locally Weyl-equivalent to Einstein spaces, whereas the non-vacuum null-dust solutions are not.

Load-bearing premise

The classification assumes that every vacuum spacetime with the stated symmetries can be brought to the Eddington–Finkelstein form (1), with $r$ as a null affine parameter and no extra metric components; if a vacuum Weyl-gravity spacetime admits no such global chart, it is not covered by the claim that all solutions have been found.

Editorial extensions

If this is right

Weyl conformal gravity admits an infinite-dimensional family of vacuum dynamical spacetimes parametrized by two functions and one constant, whereas general relativity with the same symmetries is forced to the static Schwarzschild–(anti-)de Sitter family.
Vacuum black- or white-hole-type solutions can have trapping horizons that move in time while $\Psi_4=0$, meaning no gravitational radiation is emitted; this is impossible in general relativity.
All vacuum Vaidya-type WCG backgrounds are algebraically special (Petrov type D or conformally flat), so radiative vacuum solutions would require matter or a different symmetry class.
Non-vacuum Coulomb and null-dust solutions are classified as well; only the electrovacuum ones are locally conformally static, while null-dust solutions are not even locally Weyl-equivalent to any Einstein spacetime.
Local Weyl equivalence to Einstein spaces implies that global quantities such as Noether–Wald charge, homotopy groups, and Betti numbers can differ between the WCG solution and its Einstein-space image, even where curvature invariants agree.

Reading between the lines

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

Read strictly, 'all vacuum solutions' means all spacetimes admitting the Eddington–Finkelstein chart (1); vacuum spacetimes whose null generators develop caustics or whose area radius cannot serve as a global affine null coordinate fall outside the classification.
Branch (22), with its arbitrary $F_3$ and $F_4$, could serve as a vacuum cosmological background for Weyl-gravity phenomenology, generating acceleration-like and curvature terms without matter; checking whether these profiles fit rotation-curve or cosmological data would be a natural test.
Applying a generalized Newman–Janis rotation to branch (23), as the authors suggest, would likely produce rotating radiative vacuum solutions in WCG; if such solutions have non-vanishing $\Psi_4$, they would be genuine gravitational-wave-emitting vacuum spacetimes, sharply distinguishing WCG from GR.
The absence of any Einstein-space equivalent for null-dust solutions means a radiating-collapse model in WCG should be causally and thermodynamically different from the GR Vaidya collapse for the same dust profile; comparing apparent-horizon evolution in the two theories is a concrete way to see the difference.

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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

Let me give you the short version: this is a genuinely useful paper. It provides the first systematic classification of Vaidya-type vacuum solutions in Weyl conformal gravity (WCG) for spherical, planar, and hyperbolic symmetries, plus non-vacuum charged and null-dust extensions. The derivation is transparent—the B_rr Bach component forces H to be at most quadratic in r, and the remaining components reduce to two branches. The non-vacuum branches (55)-(57) look new. The Petrov type D/O analysis, the trapping-horizon discussion, and the Weyl-equivalence results (including the extension of Birkhoff-Riegert to K=0,-1) are solid. The static Mannheim-Kazanas limit checks out once the sign in Eq. (23) is corrected.

The soft spots, in proportion.

First, Eq. (23) as printed has a sign error. From Eq. (20), B_wr=0 gives K^2+3F2F3-F1^2+6cF2'=0, so F3=(F1^2-K^2-6cF2')/(3F2). The paper prints F3=-(K^2+F1^2-6cF2')/(3F2) with the sign of F1^2 wrong. This propagates to the non-vacuum branch (56). It is a typo, since the derivation and the M-K limit both work with the corrected sign, but the authors should fix all such formulas and re-check consistency. It is the kind of thing that makes a referee's life harder than necessary.

Second, the claim of finding 'all vacuum dynamical solutions' is slightly stronger than what is proven. The ansatz (1) fixes the cross term to a constant c. The stress-test worry that this is a genuine loss of generality does not land cleanly: locally, one can set the cross term to a constant by a transformation w -> f(w,r) that keeps r as the area coordinate and preserves g_rr=0. So the gauge is standard, not restrictive. But the paper never says this, and the completeness statement should be qualified as 'all local solutions admitting this chart.' A short remark would remove the criticism.

Third, minor: the Maple results in Appendix A are not shipped. For a symbolic computation claim, it would be good practice to include the worksheet or a checksum, but this is not disabling.

The citation pattern is appropriate; the self-citations are relevant, not padding. The literature review is honest about prior static and Birkhoff-Riegert results.

Bottom line: send it to a serious referee. With corrected signs and the local-gauge caveat, I would be happy to cite the two-branch classification. For the reading group, it is a good example of exact-solution work in modified gravity, though the typos will need a clean arXiv version.

Referee Report

3 major / 3 minor

Summary. The paper studies Vaidya-type solutions of Weyl conformal gravity (WCG) in generalized Eddington–Finkelstein coordinates (1) with spatial sections of constant curvature K=1,0,-1. The authors solve the Bach vacuum equations under this ansatz and obtain two branches: Eq. (22), where F2(w)=0 and F1(w)=±K with F3,F4 arbitrary, and Eq. (23), where F2≠0 and F3,F4 are fixed in terms of F1,F2 and one integration constant. They then analyze Petrov type, singularities, trapping horizons, the Birkhoff–Riegert theorem, and Weyl equivalence to Einstein spaces. The same framework is extended to non-vacuum sources consisting of a Coulomb field and null dust, yielding the families (54)-(57), and the relation to the Mannheim–Kazanas and topological black hole solutions is discussed.

Significance. If correct, the vacuum classification (22)-(23) would be the first complete classification of dynamical vacuum solutions with spherical, planar, and hyperbolic symmetry in WCG, and the non-vacuum extension in Sec. V is a substantial addition. The derivation is a direct PDE classification rather than a fit; the arbitrary functions are solution data, and the paper contains explicit consistency checks against the M-K limit. The Maple-assisted verification of Weyl equivalence to Einstein spaces in Appendix A is a useful concrete check. However, the completeness claim is currently stronger than what is proven (major comment 1), so the significance as a complete classification is conditional.

major comments (3)

[Sec. II, Eq. (1)] The line element fixes g_wr=c with c a constant. A general spherically, planarly, or hyperbolically symmetric metric in null coordinates with area radius r has the form ds^2 = H(r,w)dw^2 + 2B(r,w)dwdr + r^2 dΣ_K^2 with B an arbitrary function. A rescaling w→f(w) cannot absorb r-dependence of B, while a rescaling w→f(w,r) that could absorb it changes the area-radius interpretation of r; the compatibility condition for maintaining the form (1) is a nontrivial PDE involving H and B. The paper does not show that the Bach vacuum equations force B to be independent of r or that a regular Weyl transformation can set B=c while preserving the area coordinate. Consequently the abstract and Sec. IV.A claims of finding "all vacuum dynamical solutions" are not established by the given derivation; the classification covers only solutions admitting the constant-cross-term Eddington–Finkelstein chart. The non-vacuum classification in Sec. V inherits the same limitation. Please either provide the missing gauge-completeness argument or explicitly state the restricted scope of the classification claim.
[Sec. IV.A, Eq. (23)] Solving Brw=0 from Eq. (20) gives F3=(F1^2-K^2-6cF2')/(3F2). The printed formula, F3=-(K^2+F1^2-6cF2')/(3F2), has the wrong overall sign on F1^2 and does not reduce to the M-K limit (17) as described in Sec. IV.B. The same sign error appears in the non-vacuum F3 in Eq. (56), and Eq. (57) as printed appears to omit the square of F1 in the F3 expression. Because Eqs. (23), (56), and (57) are central to the classification, these formulas must be corrected and re-verified against the M-K and charged M-K limits.
[Sec. V, Eq. (55)] The F2=0, q'(w)≠0 branch does not appear to follow from the system (53). Substituting F2=0 and F1^2=K^2-αq^2 into the second equation of (53) fixes B(w)=2cα q(w)q'(w), and the third equation then gives F3(w)=(2F1''-A)/(cF1'), not the expression printed in (55), which contains q'', B', and A in a different algebraic structure. This branch should be re-derived and checked by direct substitution into Eq. (53) before the non-vacuum classification can be considered reliable.

minor comments (3)

[Sec. IV.B, text near Eq. (28)] The statement that C^2 is "invariant under Weyl transformations" is imprecise: C^2=C_{abcd}C^{abcd} transforms with a conformal weight (it scales as Ω^{-4}), although the integrated action density √(-g)C^2 is invariant. The singularity-preservation argument still goes through, but the wording should be corrected.
[Sec. IV.A, Eq. (24)] The null tetrad expression for m should state explicitly that it is normalized with respect to the metric (1); the displayed factors make the normalization plausible but not manifest to the reader.
[Appendix A, Eqs. (A3)-(A7)] Please specify the chosen branch of the square roots in Eqs. (A3) and (A5), and clarify whether λ1 and λ2 are independent integration/gauge constants or fixed by the solution data.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the solution branches follow by direct integration of the Bach equation for the stated ansatz; self-citations are non-load-bearing.

full rationale

The central derivation is a direct analytic reduction, not a fit. Substituting the ansatz metric (1) into the Bach vacuum equation gives B_rr = -(1/(6r))[r H'''' + 4 H'''] = 0 (Eq. 18), whose integration fixes H to the four-term form (19); the remaining Bach components (20)-(21) then force exactly the two branches (22)-(23). The arbitrary functions F_i(w) and constants C1, C2 are free solution data; no parameter is tuned to a target subset and no target result is used as an input. The M-K and charged-M-K limits are checked ex post as special cases, not imposed. Riegert's theorem [48] is external to the authors and used as a benchmark; self-citations such as [27] (Newman-Penrose formalism) and [42] (action conventions) are not load-bearing for the classification. The main caveat—that the 'all solutions' statement applies only to spacetimes admitting the constant-cross-term Eddington-Finkelstein gauge (1)—is a completeness/scope limitation rather than a circularity. No equation in the paper is defined in terms of its own output, and no fitted parameter is renamed as a prediction. The sign inconsistency noted in the reader's take is a correctness/typo concern, not circularity.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The central results are derivations from the Bach equation with no data fitting. The only assumptions are the standard WCG field equations, the generalized Eddington-Finkelstein metric ansatz, the static M-K solution as a reference, and standard smoothness requirements. No new particles, fields, or dimensions are introduced.

assumptions (4)

domain assumption The WCG vacuum field equation is the Bach equation B_mu nu = 0, and the non-vacuum equation is B_mu nu = (G_w^2/2) T_mu nu with traceless T_mu nu.
Adopted from the WCG literature in Sec. III, Eqs. (13) and (16); this is the theory being studied, not derived here.
domain assumption Every spacetime with spherical, planar, or hyperbolic symmetry can be written in the generalized Eddington-Finkelstein form (1) with the transverse metric of constant curvature K.
Sec. II, Eq. (1); the completeness of the classification depends on this coordinate gauge being general enough.
domain assumption The static Mannheim-Kazanas solution (17) is the most general static spherically symmetric Bach vacuum solution, and the Birkhoff-Riegert theorem [48] correctly classifies electrovacuum spherically symmetric WCG solutions.
Invoked in Secs. III and IV.C as the reference solution for identifying static limits; taken from Refs. [14] and [48].
standard math The metric function H(r,w) is smooth enough for the fourth-order Bach equation to be applied pointwise.
Needed to obtain Eq. (18) and the polynomial solution (19).

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Cite this review

Pith. "Pith review of Generalized Eddington--Finkelstein Coordinates and Exact Vaidya-Type Solutions in Weyl Conformal Gravity." pith.science (2026). https://pith.science/paper/KYXCF2AW

@misc{pith2026250200905,
  author       = {Pith},
  title        = {Pith review of: Generalized Eddington--Finkelstein Coordinates and Exact Vaidya-Type Solutions in Weyl Conformal Gravity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KYXCF2AW}},
  note         = {Machine review of arXiv:2502.00905}
}

read the original abstract

We study Vaidya-type solutions in Weyl conformal gravity (WCG) using Eddington--Finkelstein-like coordinates. Our considerations focus on spherical as well as hyperbolic and planar symmetries. In particular, we find all vacuum dynamical solutions for the aforementioned symmetries. These are, in contrast to general relativity, structurally quite non-trivial. We provide a thorough analysis of their basic properties, such as, relation to other known WCG solutions, algebraic types, singularities, horizons, and symmetries. In the same vein, we also derive, classify, and discuss non-vacuum solutions with the Coulombic electric field and null dust. Other salient issues, such as the gauge equivalence of WCG solutions to Einstein spaces and the role of the Birkhoff--Riegert theorem, are also addressed.

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Works this paper leans on

92 extracted references · 78 canonical work pages

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The general KVF has the form X = {[K(x2 − y2) + 4]c1 + 2 Kxyc 2 + yc3}∂x + {[K(y2 − x2) + 4]c2 + 2 Kxyc 1 − xc3}∂y , (2) where ci are constants

gives 3 Killing vector ﬁelds (KVF). The general KVF has the form X = {[K(x2 − y2) + 4]c1 + 2 Kxyc 2 + yc3}∂x + {[K(y2 − x2) + 4]c2 + 2 Kxyc 1 − xc3}∂y , (2) where ci are constants. E-F coordinates are often useful for describing dynami- cal spacetimes, which can otherwise be quite complicated to represent in Schwarzschild coordinates [ 29]. Here we will f...
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admits a conformal Killing vector ﬁeld (CKVF) X c = ( A1w + A2)∂w + A1r∂r . (5) Here we recall that CKVFs are a generalization of KVFs which preserve the metric tensor up to a scalar factor — CKVF are thus generators of conformal isometry . The linear mass function is also one of the cases where trans- formation to double-null coordinates is possible [ 40...
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In this case, we get the Bach equation Brr = − 1 6r [ r ∂4H(r, w) ∂r 4 + 4 ∂3H(r, w) ∂r 3 ] = 0 , (18) which has a generic solution H(r, w) = F1(w) + F2(w) r + F3(w)r + F4(w)r2

for the rr-component ﬁrst. In this case, we get the Bach equation Brr = − 1 6r [ r ∂4H(r, w) ∂r 4 + 4 ∂3H(r, w) ∂r 3 ] = 0 , (18) which has a generic solution H(r, w) = F1(w) + F2(w) r + F3(w)r + F4(w)r2 . (19) Substituting this back into the remaining components of the Bach t...
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any Weyl trans- formation of the form Ω( w, r) = 1 r G ( rF1(w) + 3F2(w) r ) , (37) will make it KVF. This transformation may be regular for some Fi(w) cases and suitable choices of function G, but there is a large class of solutions for which it is not. To give an example of ...
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2024
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Since Eq

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(53) Here we have set 3 2 G2 w/e2 = α

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Since Tµν has the rr−component equal to zero, the form of Ω given by ( 47) must still hold

with ( 54)- (56) satisfying the GR equation β2( ˜Gµν + Λ˜gµν ) = ˜Tµν , where ˜Tµν = Ω −2Tµν and β2 is the GR coupling constant. Since Tµν has the rr−component equal to zero, the form of Ω given by ( 47) must still hold. Using this, we can see that T EM µν (51) is automaticall...

2024
[31]

Consequently, non-vacuum WCG is causally diﬀerent from ordinary GR with the same Tµν

in WCG are at least locally Weyl equivalent to some Einstein space (i.e., vacuum solution of GR equa- tions with cosmological constant), this is not the case for non-vacuum Bach equations. Consequently, non-vacuum WCG is causally diﬀerent from ordinary GR with the same Tµν . T...
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(A1) Using Maple 2024 and the ensuing symbolic tensor computation, we ﬁnd that the components of the Ein- stein tensor for general (

are r2Gww = −KH − HH ′r + c ˙Hr − H 2 , cr2Grw = −K − H ′r − H , [4 + K(x2 + y2)]2Gxx/yy = −8r(H ′′r + 2H ′) . (A1) Using Maple 2024 and the ensuing symbolic tensor computation, we ﬁnd that the components of the Ein- stein tensor for general (

2024
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In fact, for ( 34) with the general solution ( 19) we have that ˜R = 2[ rF1(w) + 3 F2(w)]/r , (36) so (33) matches the form (35), independently on K

is exactly the KVF from Riegert’s proof, which can be written in the form ˜X c i = ǫ j i ∂j ˜R , (35) where i, j = 0 , 1, x0, x1 ≡ w, r, ǫij is an antisymmetric Levi-Civita tensor and ˜R is the Ricci scalar of ( w, r) 2- space. In fact, for ( 34) with the general solution ( 19...
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Since the most general static solution is M-K ( 17) or WCG topo- logical black hole (depending on K), solutions are confor- mally equivalent to it

becomes explicitly static (non-null KVF explicitly indicates time-independence) [ 48]. Since the most general static solution is M-K ( 17) or WCG topo- logical black hole (depending on K), solutions are confor- mally equivalent to it. However, it should be emphasized again tha...
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local” or “local

and ( 47) are ˜Gxx = ˜Gyy = 16r [4 + K(x2 + y2)](f1r + f2)2 [ r ( −f 2 1 F1 + 2 f1f2F3 − 3f 2 2 F4 − 6cf2f ′ 1 + 4 cf ′ 2f1 ) + f1f2F1 − 3f 2 1 F2 − f 2 2 F3 − 2cf2f ′ 2 ] , ˜Grw = c (f1r + f2)2r2 [ 2r2 ( 2cf1f ′ 2 − 6cf2f ′ 1 − 3F4f 2 2 + F3f1f2 − Kf 2 1 ) + 2 r ( −2cf2f ′ 2 ...
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Reviewed August 9, 2026 · model on record in the stance chip above.

[1] [1]

The general KVF has the form X = {[K(x2 − y2) + 4]c1 + 2 Kxyc 2 + yc3}∂x + {[K(y2 − x2) + 4]c2 + 2 Kxyc 1 − xc3}∂y , (2) where ci are constants

gives 3 Killing vector ﬁelds (KVF). The general KVF has the form X = {[K(x2 − y2) + 4]c1 + 2 Kxyc 2 + yc3}∂x + {[K(y2 − x2) + 4]c2 + 2 Kxyc 1 − xc3}∂y , (2) where ci are constants. E-F coordinates are often useful for describing dynami- cal spacetimes, which can otherwise be quite complicated to represent in Schwarzschild coordinates [ 29]. Here we will f...

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(3) For the sake of brevity, we will call this VS in the fol- lowing text, although it is actually an extension of it by charge and a cosmological constant

with K = 1 , H(r, w) = − ( 1 − 2m(w) r + q2(w) r2 − Λ 3 r2 ) . (3) For the sake of brevity, we will call this VS in the fol- lowing text, although it is actually an extension of it by charge and a cosmological constant. This spacetime contains a Coulomb ﬁeld and pure radiation of the null- dust type. In particular, the latter generally represents a ﬁeld (...

[3] [3]

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The latter allows one to study thermodynamic and causal properties in this new spacetime and then convert the results obtained to the original VS [ 35, 39]

as KVF. The latter allows one to study thermodynamic and causal properties in this new spacetime and then convert the results obtained to the original VS [ 35, 39]. For instance, the event horizon in 3 static spacetime is collocated with one of the Killing hori- zons of ( 5). Since the causal properties are conformally invariant, this location can be tran...

[6] [6]

This makes it power-counting renormalizable

is Weyl invariant in four dimensions and mimics a Yang– Mills structure with the dimensionless coupling constant Gw. This makes it power-counting renormalizable. Throughout this work, we adopt the spacelike metric signature ( −, +, +, +). The sign convention for the Rie- mann curvature tensor is chosen so that Rα βµν = ∂µ Γ α βν − ∂ν Γ α βµ + Γ α κµ Γ κ β...

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Here β represents the mass of the source, γ is related to the dark matter, and κ has an eﬀect of the cosmological constant

it reads K = 1 , H(r) = − [ 1 − 3βγ − β(2 − 3βγ ) r + γr − κr2 ] , (17) where β, γ, κ are constants. Here β represents the mass of the source, γ is related to the dark matter, and κ has an eﬀect of the cosmological constant. Versions with K = −1 and K = 0, i.e. the WCG analogues of topological black holes, have also been found, see e.g. Ref. [ 45]. One mi...

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