Optimal Extension Regularity at the McVittie Event Horizon

Abstract: We determine the optimal local extension regularity of the future black-hole event horizon in the exact spatially flat McVittie solutions sourced by a positive cosmological constant and a barotropic fluid with constant equation-of-state parameter $w&gt;-1$. Let $H_\infty$ be the asymptotic Hubble constant, $κ$ the surface gravity of the limiting black-hole root, and $p=3(1+w)H_\infty/κ$. Ingoing radial null geodesics reach the horizon in finite affine length. A parallelly propagated angular curvature component is asymptotic to $C s<sup>{p-2}$, with $C\ne0$ and $s$ the remaining affine distance, which excludes every anchored $C<sup>2$ extension for $0&lt;p\&lt;2$. For $p\ge2$ we construct a parameter-uniform Gaussian-null compactification and an explicit two-sided Lorentzian collar. If $p=N+\vartheta$ is nonintegral, with $N\ge2$ and $0&lt;\vartheta\&lt;1$, the optimal regularity is the standard big Hölder class $C^{N,\vartheta}$: extensions of this class exist, whereas no $C^{N,\vartheta&#39;}$ extension exists for $\vartheta&#39;&gt;\vartheta$. Every integer $p\ge2$ instead belongs to an analytic island and admits a real-analytic local extension. At the critical value $p=2$ the boundary Einstein endomorphism has a nonzero rank-one nilpotent part. The ratio of cosmological decay to horizon redshift therefore determines a sharp, arithmetic hierarchy of geometric regularity.

• The paper establishes a sharp regularity classification using p=3(1+w)H∞/κ: 0<p<2 forbids anchored C² extensions, noninteger p=N+θ gives exactly C^{N,θ} regularity, and integer p≥2 permits analytic extensions.
• The paper combines parallelly propagated curvature analysis with Gaussian-null coordinates, identifying a tidal term proportional to s^{p−2} and constructing two-sided collars when p≥2.
• The paper shows that scalar curvature invariants can remain finite while boosted null-frame curvature diverges, and that at p=2 a C² extension can exist despite a type-II nilpotent Einstein limit incompatible with a finite-velocity perfect-fluid boundary state.

Overview and main result

This paper determines the optimal local differentiability of the future black-hole event horizon in the exact spatially flat McVittie spacetime sourced by a positive cosmological constant and a barotropic perfect fluid with constant equation-of-state parameter $w>-1$. The central object is the dimensionless exponent

$p=\frac{3(1+w)H_\infty}{\kappa},$

the ratio of the exponential decay rate of the barotropic density to the surface gravity $\kappa$ of the limiting black-hole root. The main theorem establishes a sharp, arithmetic hierarchy: for $0$C^2$ Lorentzian extension exists; for nonintegral $p=N+\vartheta\ge 2$ the optimal class is exactly $C^{N,\vartheta}$ (extensions exist at this class, none exists at any higher H\"older exponent); and every integer $p\ge2$ lies on an "analytic island" admitting a real-analytic local extension. The result thus converts a decay-to-redshift ratio into a complete classification of horizon regularity.

Spacetime model and horizon identification

The background is the McVittie metric in areal-radius form,

$$ds^2/m^2=-f\,d\tau^2-\frac{2hx}{\sqrt S}\,d\tau\,dx+\frac{dx^2}{S}+x^2d\Omega^2,\qquad f=S-h^2x^2,$$

with $S(x)=1-2/x$, constant mass $p=\frac{3(1+w)H_\infty}{\kappa},$0 (no accretion), and an exact cosmology $p=\frac{3(1+w)H_\infty}{\kappa},$1 with $p=\frac{3(1+w)H_\infty}{\kappa},$2. The source decomposes into vacuum energy plus a barotropic component whose density decays as $p=\frac{3(1+w)H_\infty}{\kappa},$3 with $p=\frac{3(1+w)H_\infty}{\kappa},$4. Under the sub-Nariai condition $p=\frac{3(1+w)H_\infty}{\kappa},$5, the asymptotic marginal function $p=\frac{3(1+w)H_\infty}{\kappa},$6 has two simple positive roots; the smaller root $p=\frac{3(1+w)H_\infty}{\kappa},$7 carries positive surface gravity $p=\frac{3(1+w)H_\infty}{\kappa},$8. Invoking Nolan's radial-null completeness theorem, the ingoing radial null family reaches $p=\frac{3(1+w)H_\infty}{\kappa},$9 in finite affine parameter while outgoing rays escape to the cosmological end, so the limiting null tube is identified as the future event horizon $\kappa$0. The exact barotropic family matters because it fixes the expanding branch, monotonicity of $\kappa$1, and a stationary reference end under a single causal theorem.

Affine null geometry and the curvature obstruction

The paper constructs a complete parallelly propagated null frame $\kappa$2 along ingoing generators. Spherical symmetry leaves five independent tidal components; all remain finite except the boost-enhanced angular component

$\kappa$3

where $\kappa$4 grows like $\kappa$5 relative to remaining affine distance $\kappa$6. Linearizing at the simple root gives $\kappa$7 and $\kappa$8, whence

$\kappa$9

For $0anchored: the geodesic, its affine endpoint, and the transported frame are fixed by the original spacetime, so any hypothetical $0boost acts twice on the transverse tidal tensor.

Constructive side: Gaussian-null collar

On the constructive side, the paper builds a parameter-uniform Gaussian-null chart from a compact family of ingoing characteristics. The characteristic equation near $0ODE whose sole indicial root is removed by $0$C^2$0. A finite reflection operator matching three boundary jets extends both coefficients to negative $C^2$1, producing a two-sided $C^2$2 Lorentzian collar with continuous curvature across the endpoint hypersurface.

Optimal H\"older hierarchy and analytic islands

The full classification exploits the polyhomogeneous structure of the characteristic solution. The exponents form the locally finite index set $C^2$3; since the unique indicial root is occupied by the free mode $C^2$4 and every forced exponent exceeds one, no logarithms are generated. For nonintegral $C^2$5, all exponents below $C^2$6 are integers, so $C^2$7 on the one-sided collar; Lagrange-weight reflection operators of order $C^2$8 extend them jointly, using moment identities to match all mixed jets through order $C^2$9.

Sharpness again comes from the tidal scalar $p=N+\vartheta\ge 2$0, whose first nonstationary term is $p=N+\vartheta\ge 2$1 with explicitly positive coefficient. Differentiating $p=N+\vartheta\ge 2$2 times leaves a nonzero $p=N+\vartheta\ge 2$3 term, excluding any $p=N+\vartheta\ge 2$4 extension with $p=N+\vartheta\ge 2$5: a $p=N+\vartheta\ge 2$6 metric would have $p=N+\vartheta\ge 2$7 Riemann tensor, contradicting the anchored expansion. The argument is chart-independent because smooth endpoint changes only rescale the leading coefficient.

At integer $p=N+\vartheta\ge 2$8, the quotient defining $p=N+\vartheta\ge 2$9 is jointly analytic, so analytic ODE theory yields convergent two-sided series and a real-analytic collar. The corollary for integer classes states that a geometric $C^{N,\vartheta}$0 extension ($C^{N,\vartheta}$1) exists if and only if $C^{N,\vartheta}$2 or $C^{N,\vartheta}$3 is an integer $C^{N,\vartheta}$4 — a genuinely arithmetic distinction invisible to curvature-boundedness criteria alone.

Matter limits and physical consequences

At the critical value $C^{N,\vartheta}$5 the paper uncovers a tension between algebraic type and field equations. The boundary Einstein endomorphism equals its de Sitter value plus a nonzero rank-one nilpotent part, i.e., a Petrov-type-II limit with coefficient $C^{N,\vartheta}$6. Such an endomorphism admits no perfect-fluid decomposition with finite unit velocity. Nevertheless, two exact McVittie open regions can be joined via $C^{N,\vartheta}$7 into a $C^{N,\vartheta}$8 metric whose Einstein tensor is continuous, distributionally conserved (verified by a weak Bianchi argument), and free of delta-function curvature or null shells. The boundary algebraic type, the open-side equations, and the join's time orientation are therefore three independent pieces of information; notably, the natural expanding orientations of the two copies meet with opposite signs in this construction.

In terms of equation of state, the thresholds are $C^{N,\vartheta}$9. At the representative value $p\ge2$0 ($p\ge2$1), $p\ge2$2 and $p\ge2$3: dust and radiation fall below the $p\ge2$4 threshold (no $p\ge2$5 extension), while stiff matter admits analytic extension. For mixtures, the slowest nonvanishing component controls the exponent. Krolak and Tipler integrals show that in the range $p\ge2$6 accumulated tidal distortion is finite even though instantaneous curvature excludes a $p\ge2$7 completion, and angular Jacobi fields retain finite nonzero lengths throughout $p\ge2$8 — so the horizon is curvature-singular in the Tipler sense only for $p\ge2$9.

Limitations and open questions

The classification is explicitly local and anchored: it concerns compact families of ingoing generators and arbitrarily small collars, not global uniqueness or maximality of extensions. The paper concedes that $$ds^2/m^2=-f\,d\tau^2-\frac{2hx}{\sqrt S}\,d\tau\,dx+\frac{dx^2}{S}+x^2d\Omega^2,\qquad f=S-h^2x^2,$$0 and $$ds^2/m^2=-f\,d\tau^2-\frac{2hx}{\sqrt S}\,d\tau\,dx+\frac{dx^2}{S}+x^2d\Omega^2,\qquad f=S-h^2x^2,$$1 extensions, uniqueness of the attached side, maximality, and dynamically selected data beyond the endpoint remain open, as they require low-regularity causal theory and characteristic evolution. The analysis also assumes the sub-Nariai condition $$ds^2/m^2=-f\,d\tau^2-\frac{2hx}{\sqrt S}\,d\tau\,dx+\frac{dx^2}{S}+x^2d\Omega^2,\qquad f=S-h^2x^2,$$2 ensuring a simple root; the degenerate Nariai scaling at equality is not treated. Finally, the Hayward-profile comparison is presented as a local illustration of how the root slope shifts the threshold, with its global interpretation left specific to the realizing spacetime.

Conclusion

The paper resolves the regularity question for the McVittie event horizon completely within the classical curvature range $$ds^2/m^2=-f\,d\tau^2-\frac{2hx}{\sqrt S}\,d\tau\,dx+\frac{dx^2}{S}+x^2d\Omega^2,\qquad f=S-h^2x^2,$$3: the ratio of cosmological dilution rate to horizon redshift rate determines whether the horizon is $$ds^2/m^2=-f\,d\tau^2-\frac{2hx}{\sqrt S}\,d\tau\,dx+\frac{dx^2}{S}+x^2d\Omega^2,\qquad f=S-h^2x^2,$$4-inextendible, optimally $$ds^2/m^2=-f\,d\tau^2-\frac{2hx}{\sqrt S}\,d\tau\,dx+\frac{dx^2}{S}+x^2d\Omega^2,\qquad f=S-h^2x^2,$$5, or real-analytic, with integer exponents forming exceptional analytic islands. Methodologically it combines an invariant anchored curvature obstruction with parameter-uniform Gaussian-null estimates, and it demonstrates that a cosmological black-hole horizon can record matter information — here a type-II nilpotent Einstein limit at $$ds^2/m^2=-f\,d\tau^2-\frac{2hx}{\sqrt S}\,d\tau\,dx+\frac{dx^2}{S}+x^2d\Omega^2,\qquad f=S-h^2x^2,$$6 — even when a perfectly regular geometric completion exists.