Regular Black-Hole Evaporation Models

A regular effective metric is a self-consistent semiclassical solution only if its source is a conserved renormalized expectation value in a specified admissible quantum state, the metric and state solve one boundary-value problem, and curvature, frequencies, stress fluctuations, and omitted terms remain below cutoff. Reverse-engineering a finite tensor from a nonsingular metric satisfies none of the state-existence or approximation requirements by itself.

Helpful background. Quantum Fields in Curved Spacetime supplies Hadamard states and stress renormalization; Einstein Equations supplies conservation; Black-Hole Geometry and Mechanics supplies inner and outer horizons; Semiclassical Gravity supplies self-consistency tests; Black-Hole Information treats fine-grained recovery; and Quantum Gravity studies microscopic regularization of a singular core.

From a regular metric to a source

For spherical symmetry in ingoing coordinates, write

$$ds^2=-e^{2\psi(v,r)}F(v,r)\,dv^2 +2e^{\psi(v,r)}\,dv\,dr+r^2d\Omega^2, \qquad F=1-\frac{2Gm(v,r)}{r}.$$

Regularity at the center requires more than finite metric functions. Scalar curvature, freely falling tidal components, and the stress measured in regular frames must remain finite. A necessary mass scaling in many smooth spherical models is $m(v,r)=O(r^3)$ as $r\to0$, together with suitable smoothness and radial behavior of $\psi$. For example, $m=m_3r^3+O(r^5)$ and a radially constant leading lapse give a locally de Sitter-like core if $m_3\gt0$. The big-O statement alone neither fixes the lapse nor proves a smooth, geodesically complete extension.

One can always define

$$T_{ab}^{\rm eff} =\frac{1}{8\pi G}\bigl(G_{ab}+\Lambda g_{ab}\bigr).$$

The Bianchi identity makes this tensor conserved. But a semiclassical source must have the stronger form

$$T_{ab}^{\rm eff} =\langle T_{ab}\rangle_{\omega}^{\rm ren} +T_{ab}^{\rm local}(\alpha_i),$$

for a specified field theory, Hadamard state $\omega$, renormalized couplings $\alpha_i$, and boundary data. It must reproduce the anomaly and state-dependent flux, and its fluctuations must permit replacement by a mean. Conservation is necessary, not sufficient.

A finite-curvature metric can still reach the cutoff

As an illustrative static mass profile, take $\psi=0$ and

$$m(r)=\frac{Mr^3}{r^3+2GM\ell^2},\qquad M\gt0,\quad\ell\gt0.$$

At large radius $m\to M$. At small radius $F=1-r^2/\ell^2+O(r^5)$, so the center has de Sitter curvature $R\to12/\ell^2$ and Kretschmann scalar $K\to24/\ell^4$. These finite limits demonstrate the local smoothing mechanism used in Hayward-type models, discussed in Frolov 2016, § II, PDF. Yet $\ell_{\rm P}^2\sqrt K\to\sqrt{24}(\ell_{\rm P}/\ell)^2$: choosing $\ell\sim\ell_{\rm P}$ makes the curvature finite but uncontrolled in the low-energy expansion. This profile is a geometric example, not a constructed quantum source or a claim of global regularity.

Horizons, matching, and instability

A regular core generally creates an outer and an inner trapping horizon. The outgoing Hawking flux fixes a negative-energy ingoing balance at the outer horizon, but that does not determine the state at the inner horizon. Counterstreaming perturbations can be exponentially blueshifted there, producing mass inflation or large stress even when scalar invariants of the background ansatz were designed to remain finite.

Matching an exterior evaporating solution to a core requires continuity of the induced metric and, absent a shell, the appropriate extrinsic curvature. If the jump is nonzero, the model contains a surface stress tensor whose energy, state, and stability must be included. A time-dependent transition radius also carries flux; assigning it by hand can violate the Bondi balance.

Bardeen constructs a detailed nonsingular-core scenario and explicitly relies on assumptions about the core and its growth Bardeen 2014, §§ II–V, PDF. It is a useful kinematic model, not a derivation of $\langle T_{ab}\rangle_\omega^{\rm ren}$ from a four-dimensional state.

Self-consistency and cutoff tests

The following conditions should be checked together:

1. State: a globally or patchwise well-defined Hadamard state produces the claimed flux and regularity.
2. Renormalization: finite counterterms are fixed once; they are not retuned as the geometry evolves.
3. Equation: every independent component of the semiclassical Einstein equation and the constraints is satisfied to the same order.
4. Fluctuations: the smeared noise kernel is small enough on the response scale, or stochastic corrections are included.
5. Cutoff: $\ell_{\rm P}^2\sqrt{K}\ll1$, local frequencies remain below the effective cutoff, and higher-order terms are smaller.
6. Stability: perturbations of both horizons and the core remain within the assumed branch over the claimed time.

A model can be geometrically regular and still fail items 1, 4, or 6. It can solve an effective equation yet fail item 5. Regularity is therefore one column in the evidence, not a synonym for self-consistency.

Results and open questions through 2026-08-19

The evidence through 2026-08-19 contains informative constructions and partial self-consistent solutions, but no established regular four-dimensional evaporation model satisfying all six tests through the endpoint.

Evidence or construction	Result	Scope and limitation
Bardeen 2014, §§ II–V, PDF	Gives a nonsingular evaporating-core scenario with explicit causal and matching structure	Core physics and stress state are assumed; inner-horizon stability is not derived from a complete renormalized source
Two-dimensional comparisons in Callan et al. 1992, Eqs. (1)–(23), PDF and Russo, Susskind, and Thorlacius 1992, §§ III–IV, PDF	Provide controlled backreaction laboratories and distinct late-time behavior	Action, counterterm, and dimensional differences prevent a universal four-dimensional inference
del Río and Marañón-González 2026, Eqs. (6)–(12) and p. 4, PDF	Constructs a conserved stress-tensor family satisfying selected asymptotic flux conditions and derives a quasi-stationary geometry at retained perturbative order	The local coincidence data are not accompanied by a reconstructed positive Hadamard two-point bisolution; the calculation does not demonstrate a regular microscopic endpoint
Reverse-engineered regular metrics, exemplified by Bardeen 2014, §§ II–V, PDF	Guarantee a finite designed geometry and conserved effective tensor	Do not establish a Hadamard state, noise control, cutoff hierarchy, or stability

The conservative position requires an explicit state-to-stress calculation and perturbative control before calling a model self-consistent. The competing construction-first position treats a regular conserved effective tensor as a phenomenological parameterization to be tested. These approaches differ along state, renormalization, inner-boundary, fluctuation, stability, and cutoff axes; comparing metrics alone hides the disagreement.

The 2026 arXiv record was checked on 2026-09-26 and still lists v1 without a journal reference. The substantive issue is state realizability, whether or not the work appears in a journal: coincidence data satisfying conservation and the trace identity are necessary consequences of a Hadamard state, not sufficient proof that a positive two-point bisolution with the required global boundary conditions exists.

Counterexample and open questions

Conservation counterexample. A smooth tensor chosen to support the metric may obey $\nabla^aT_{ab}=0$ but have a trace incompatible with the field theory’s anomaly. It cannot equal that theory’s renormalized expectation value even though the Einstein equation is algebraically satisfied.

Regular-background counterexample. A finite background curvature does not control a perturbation that grows exponentially at an inner horizon. The perturbed stress can reach cutoff while the unperturbed metric remains regular.

Classical-limit check. When state-dependent quantum terms and the regular-core scale are removed, the exterior should recover Schwarzschild or the appropriate charged/rotating family. A model retaining a finite core in that limit has inserted new classical matter rather than isolated a quantum correction.

The central question remains unresolved for complete evaporation. The conclusions could change after an independent mode-sum or point-splitting calculation reproduces a proposed source, a nonlinear perturbation analysis establishes or destroys inner-horizon control, or a corrected peer-reviewed four-dimensional solution satisfies the cutoff and noise tests; a theorem ruling out a relevant Hadamard state could also change the conclusion.

Exercise

A regular metric gives a conserved $T_{ab}^{\rm eff}$ with finite components, but the trace differs from the known conformal anomaly by a nonlocal state-independent term. Is the model a self-consistent solution for that conformal field?

Solution

No. Conservation and regularity are necessary, but the renormalized stress tensor must also satisfy the field theory’s trace identity up to allowed local finite counterterms. A nonlocal state-independent discrepancy cannot be repaired by changing those couplings. The metric remains a phenomenological effective geometry until a different matter theory or a consistent source construction is supplied.

References

• Bardeen, J. M. (2014). “Black Hole Evaporation without an Event Horizon.” arXiv:1406.4098. Open PDF.
• Callan, C. G., Jr., Giddings, S. B., Harvey, J. A., and Strominger, A. (1992). “Evanescent Black Holes.” Physical Review D 45, R1005–R1009. doi:10.1103/PhysRevD.45.R1005. Open PDF.
• del Río, A., and Marañón-González, F. J. (2026). “Black-Hole Evaporation from the Semiclassical Einstein Equations in 3+1 Dimensions.” arXiv:2607.11737v1. Open PDF.
• Frolov, V. P. (2016). “Notes on Nonsingular Models of Black Holes.” Physical Review D 94, 104056. DOI. Open PDF.
• Russo, J. G., Susskind, L., and Thorlacius, L. (1992). “Black Hole Evaporation in 1+1 Dimensions.” Physics Letters B 292, 13–18. doi:10.1016/0370-2693(92)90601-Y. Open PDF.