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REVIEW 2 major objections 6 minor 40 references

On the absence of conformally flat slicings of the Kerr spacetime

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Pith's one-line read No conformally flat slicing exists for the Kerr spacetime

desk verdict A careful perturbative no-go that genuinely extends Garat–Price, but the headline conclusion overreaches: the proof only rules out slicings analytic in the spin parameter. read the letter →

arxiv 1908.03456 v2 pith:ZSTGU62F submitted 2019-08-09 gr-qc

Antonio de Felice , François Larrouturou , Shinji Mukohyama , Michele Oliosi This is my paper · ORCID
classification gr-qc
keywords KerrspacetimeconformallyflatslicingCotton-YorktensorspinexpansionKerr-deSitterspatiallyfoliationnumericalrelativityinitialdataminimaltheoryofmassivegravity
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

This paper tries to settle whether the spinning Kerr black-hole spacetime can be sliced into three-dimensional spacelike hypersurfaces that are conformally flat, meaning their induced metric is a position-dependent rescaling of the flat Euclidean metric. The authors allow the slicing to reduce, when the spin vanishes, to any spherically symmetric time function, not just to constant-time Schwarzschild slices, so their search includes the Painlevé-Gullstrand family. They expand the slicing function in powers of the spin $a$ and demand that the Cotton-York tensor of each induced 3-metric vanish. The construction can be made to work order by order through fourth order in $a$, but at fifth order an obstruction $d^5 C^r{}_r/d(\cos\theta)^5$ is shown to be nonzero for every integration constant. They also prove that no analytic coordinate change makes the Kerr(-de Sitter) spatial metric exactly flat beyond linear order in $a$.

What carries the argument

The load-bearing object is the Cotton-York tensor $C^i{}_j=\varepsilon^{ikl}\nabla_k(R_{jl}-\frac14 R\gamma_{jl})$ of the induced 3-metric $\gamma_{ij}$; in three dimensions its vanishing is equivalent to conformal flatness. The argument expands the slicing function as $t=f_0(r)+\sum_{n\ge1}a^n f_n(r,\theta)$ and treats $C^i{}_j=0$ as a hierarchy of linear equations whose sources are built from lower-order $f_n$. The angular operator $B_n=(\partial_\theta^3+\cot\theta\,\partial_\theta^2+(1-\cot^2\theta)\partial_\theta)f_n/\sin\theta$ controls each order's homogeneous solution, and the free radial functions accumulate until the fifth-order derivative is explicitly computed and found non-vanishing. For the flat-slicing appendix, the analogous machinery is the linear operator $\mathcal{O}_{ij}$ acting on the first-order coordinate displacements around the Schwarzschild-de Sitter Painlevé-Gullstrand map, whose second-order source leaves the obstruction shown in Eq. (A22).

What would settle it

Construct a hypersurface $t=F(r,\theta,a)$ with $F(r,\theta,0)=f_0(r)$, or a similar coordinate map, for which the Cotton-York tensor of the induced metric vanishes for an open interval of $a$; the easiest loophole would be a non-analytic dependence on $a$ such as $\exp(-M/a)$, which the order-by-order calculation never samples. Alternatively, evaluate the fifth-order expression (25) at a value of $C_{0,1}$ not covered and find a zero.

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Extended reading notes

Core claim

The central claim is an absence result: within the axisymmetric ansatz $t=F(r,\theta,a)$ with $F(r,\theta,0)=f_0(r)$, there is no choice of the functions $f_n(r,\theta)$ that makes the Cotton-York tensor of the induced Kerr metric vanish. Solving $C^i{}_j=0$ order by order in the spin parameter $a$ splits at first order into two branches; one branch dies at second order, while the other can be pushed to fourth order. At fifth order the combination $d^5 C^r{}_r/d(\cos\theta)^5$ equals a nonzero rational expression, displayed as Eq. (25), built from the integration constant $C_{0,1}$, so the tensor cannot vanish identically. A separate argument in the appendix shows that a general coordinate change expanding around the Painlevé-Gullstrand slicing of Schwarzschild-de Sitter cannot produce a spatially flat induced metric for Kerr-de Sitter, because at second order in $a$ the derivative $\partial_\vartheta E^{(2)}_{\rho\rho}=9M^2\sin(2\vartheta)/(2\rho^3\mu)$ cannot vanish. The paper therefore strengthens the earlier no-go of [18] by removing its restriction to constant $F(r,\theta,0)$.

Load-bearing premise

The proof expands the slicing and the coordinate change as formal power series in the spin $a$ around $a=0$, so a candidate conformally flat or spatially flat slicing that depends on $a$ non-analytically would not be detected.

Editorial extensions

If this is right

  • Within the axisymmetric slicing class $t=F(r,\theta,a)$ with arbitrary $F(r,\theta,0)$, exact conformally flat Kerr foliations do not exist; earlier constructions that fixed the non-spinning slice to Schwarzschild time are not the only ones excluded.
  • Numerical initial data that assume conformal flatness for spinning black holes must be approximations; this proof shows the residual error cannot be removed by improving the slicing ansatz in this family, and it appears first at fifth order in the spin.
  • The Minimal Theory of Massive Gravity cannot inherit Kerr as a solution through the spatially flat slicing route, since no analytic coordinate change achieves spatial flatness even for Kerr-de Sitter; rotating black-hole solutions in that theory, if they exist, must differ from Kerr or enter through a different construction.
  • A spatially flat slicing of Kerr-de Sitter, if sought, must depend non-analytically on the spin, because the linearized construction can be extended only to first order in $a$.
  • An approximate conformally flat slicing accurate through fourth order in $a$ does exist and is parameterized by free functions, so high-order perturbative initial data are available before the fifth-order obstruction sets in.

Reading between the lines

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

  • The analyticity assumption is the true frontier: a slicing depending on $a$ non-analytically, for instance through $\exp(-M/a)$, would not be seen by this proof, and the no-go would need a separate argument to rule it out.
  • The same fifth-order obstruction likely indicates a rigidity of the Kerr geometry itself, tied to its multipole structure, rather than an artifact of this particular ansatz.
  • One could test the robustness by repeating the calculation for a metric with a slightly different quadrupole moment; if the obstruction shifts order or disappears, it would show exactly which Kerr feature forbids conformal flatness.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 6 minor

Summary. The paper investigates whether the Kerr spacetime admits conformally flat spacelike slicings of the form t = F(r,θ,a) in Boyer-Lindquist coordinates, with the non-spinning limit F(r,θ,0)=f0(r) left arbitrary. The authors expand F as a formal power series in the spin parameter a and solve the vanishing Cotton-York equations order by order. Two branches of solutions appear at first order; the first branch is shown to fail at second order, while the second branch can be solved through fourth order, but at fifth order the fifth θ-derivative of the rr-component of the Cotton-York tensor acquires a nonvanishing term (Eq. (25)) that cannot be removed by any choice of the integration constant C0,1. In Appendix A, an analogous expansion for a general coordinate change is used to argue that no spatially flat slicing of Kerr-de Sitter exists beyond linear order in a.

Significance. The result, if valid, meaningfully strengthens the earlier Garat-Price no-go by covering slicings that reduce to Painlevé-Gullstrand (spatially flat) slicings in the non-spinning limit, rather than only slicings that reduce to constant Schwarzschild time. The order-by-order strategy is transparent, and the explicit obstruction at fifth order is given in closed form. The paper is clearly motivated by the MTMG programme, and the no-go clarifies the limitations of constructing rotating black hole solutions via spatially flat slicings in that theory. Strengths include the careful separation of the two branches at first order, the explicit integration steps with free functions and integration constants, and the extension to the Kerr-de Sitter coordinate-change problem in Appendix A.

major comments (2)
  1. [Sec. IV; Eq. (3); App. A Eq. (A7)] The no-go result is proved only for slicings and coordinate changes that admit a formal power-series expansion in the spin parameter a, but the conclusion is stated unconditionally. The abstract and Sec. IV claim that 'it is not possible to find conformally flat hypersurfaces' and that 'no coordinate change can induce a spatially flat recasting' of Kerr(-de Sitter). A slicing or coordinate transformation that depends on a non-analytically (for instance through |a| or a^{1/2}) is outside the class covered by the order-by-order argument. Since the authors do not prove that any solution of the Cotton-York equations, or of the flat-slicing equations, must be analytic or smooth in a, the theorem should be explicitly restricted to this regularity class, or an argument establishing analyticity should be supplied. This is load-bearing because the entire proof is a perturbative obstruction.
  2. [Sec. III.B.e, Eq. (25), and App. B] The central fifth-order obstruction is presented through the functions N1, N2, D1, D2 in Appendix B, but the derivation of this coefficient, and in particular the claim that it is independent of the newly introduced function f5 and of the remaining lower-order free functions, is not shown. This is the key algebraic statement on which the no-go rests; please provide either a detailed derivation in Appendix B or a supplementary symbolic computation file so that the result can be independently verified.
minor comments (6)
  1. [Sec. III.A.b, Eq. (10)] The discussion of the λ→0 limit in the first branch is terse; please state explicitly that Γ5(r) is proportional to λ and hence the C^θ_φ expression diverges as 1/λ unless the numerator cancels it, which it does not.
  2. [Sec. III.B.d, around Eq. (22)] The text refers to 'source terms S4,2 and S4,4', but the following text and Appendix B define S4,1 and S4,3; please correct the inconsistency.
  3. [App. B, Eq. (B4)] The coefficient '26224M^5/r^5' in the denominator of the expression for f'_0 appears to be a typo for '26244M^5/r^5' used elsewhere (e.g., Eq. (B13)); please check.
  4. [Sec. III.B.e, near Eq. (25)] Please state explicitly that the displayed fifth-order derivative of C^r_r is independent of the free functions {f̄2, f̄3, f̄4, f4,c} and of f5, since this independence is essential for the obstruction to be robust.
  5. [Footnote 6] The claim that a φ-dependent ansatz also fails at fifth order for the second branch is not demonstrated; please provide the proof or clearly label it as a conjecture or outlook.
  6. [Abstract and Sec. IV] The abstract and conclusion should state the assumptions of the theorem explicitly: the slicing is axisymmetric (t = F(r,θ,a)) and depends smoothly or analytically on a. This will bring the preamble into line with the statement actually proved.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the no-go results are obtained by direct order-by-order solution of the Cotton-York and flat-slicing equations from the Kerr and Kerr-de Sitter metrics.

full rationale

The paper's central claims—failure of conformally flat slicings at fifth order and absence of spatially flat slicings beyond linear order—are derived by solving the Cotton-York equations and the flatness condition E_ij=0 order by order in the spin parameter, starting from the explicit Kerr and Kerr-de Sitter line elements. No parameter is fitted to enforce the desired outcome: the integration constants λ, C0,1, C1,1, κ1, and κ2 enter as free constants of the general solutions, and the final obstructions (Eq. (25) and Eq. (A22)) are shown to be nonzero independently of them. The authors' earlier works [26]-[28] appear only as motivation for the MTMG context and as the standard a=0 Painlevé-Gullstrand starting point; neither is used to supply the obstruction. The analyticity-in-a expansion is an explicit assumption, so the proof is limited to slicings and coordinate changes that are formal power series in a; this is a scope limitation, not a circular step. The derivation is self-contained within the stated ansatz, so no circularity is found.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The proof is self-contained once standard 3D conformal geometry and the Kerr metric are accepted. The main additional premises are analyticity in the spin parameter and spherical regularity, which are not proved and may exclude valid slicings. No empirical parameters or invented entities are used; the integration constants in the ODE solutions are free and the obstruction is independent of them.

assumptions (3)
  • standard math In three dimensions a metric is conformally flat if and only if its Cotton-York tensor vanishes.
    Invoked in Sec. II to turn conformal flatness into the equation C^i_j = 0; the paper does not prove this standard theorem.
  • domain assumption The candidate slicing and coordinate changes are analytic in the spin parameter a.
    Sec. II Eq. (3) and App. A Eq. (A7) expand t and the coordinates in powers of a; this excludes non-analytic slicings, which the paper does not discuss.
  • domain assumption Solutions are required to be regular on the sphere, so angular functions singular at the poles are discarded.
    Used in Sec. III A and III B when solving angular equations and fixing f1; polar-singular slicings are excluded without a separate argument.

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Pith. "Pith review of On the absence of conformally flat slicings of the Kerr spacetime." pith.science (2026). https://pith.science/paper/ZSTGU62F

@misc{pith2026190803456,
  author       = {Pith},
  title        = {Pith review of: On the absence of conformally flat slicings of the Kerr spacetime},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZSTGU62F}},
  note         = {Machine review of arXiv:1908.03456}
}
abstract

This work investigates the possibility of achieving a conformally flat slicing of the Kerr spacetime. We consider a hypersurface of the form $t = F(r,\theta,a)$, where $(t,r,\theta,\phi)$ are the Boyer-Lindquist coordinates, solve for a vanishing Cotton-York tensor of the induced metric order by order in the spin parameter $a$, and show that the procedure fails at the fifth order. We also prove that no coordinate change can induce a spatially flat recasting of the Kerr(-de Sitter) metric, beyond linear order in $a$, adopting a more general ansatz depending on $\phi$.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

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    (A2) This metric reduces to ( 1) in the Λ = 0 case

    Strategy Let’s recall the Kerr-de Sitter line element written in Boyer-Lindquis t-like coordinates ds2 = ¯gµν dxµ dxν = − ˜∆ − a2ζ sin2 θ Ξ dt2 + Σ ˜∆ dr2 + Σ ζ dθ2 + ( r2 + a2)2 ζ − a2 ˜∆ sin 2 θ Ξ sin2 θdφ2 − 2a [ 6M r − Λ( r2 + a2)Σ ] 3 Ξ dtdφ, (A1) where Λ is the cosmological constant and ˜∆ = ( r2 + a2)( 1 + Λ r2 3 ) − 2M r, Σ = r2 + a2 cos2 θ, Ξ = Σ...

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