Hawking Radiation from the Dymnikova Regular Black Hole

Abstract: We study Hawking radiation of the Dymnikova regular black hole. This model replaces the central singularity by a smooth de Sitter core while remaining Schwarzschild-like far from the hole, and its black-hole branch ends in a cold extremal remnant. We compute the greybody factors of the Standard Model test fields and gravitons, compare the precise numerical scattering results with WKB estimates, and use the resulting spectra to estimate an adiabatic evaporation history. The main effect is not a dramatic change in the transmission probabilities: the greybody thresholds move only slightly as the geometry approaches the remnant. Instead, the rapid decrease of the Hawking temperature strongly suppresses the total luminosity. The photon, light-fermion and graviton channels all fade near the endpoint, with the gravitational contribution remaining subdominant. The residual massless flux becomes increasingly fermion dominated because the photon channel is suppressed more efficiently. The black hole approaches the cold remnant only asymptotically, so the quoted lifetime estimates should be interpreted as cutoff times to near-extremal configurations rather than as complete evaporation times.

• The paper computes electromagnetic, massless Dirac, and axial gravitational greybody factors through direct numerical integration, finding transmission thresholds shift by only a few percent toward the extremal limit.
• The paper shows that Hawking luminosity is suppressed primarily by the temperature collapse, with the photon-plus-fermion power falling to 1.06 × 10⁻⁵ of the Schwarzschild value near extremality.
• The paper finds that late-stage evaporation becomes increasingly fermion dominated and approaches a zero-temperature remnant, while estimated near-extremal lifetimes increase to roughly 265–2,830 times the Schwarzschild lifetime.

The Dymnikova regular black hole replaces the Schwarzschild central singularity with a smooth de Sitter core while remaining asymptotically flat, and its black-hole branch terminates in a zero-temperature extremal remnant. The paper under review computes greybody factors and Hawking energy-emission rates for electromagnetic, massless Dirac, and gravitational test fields on this background (2606.08631), quantifying how much of the late-stage evaporation suppression is a scattering effect versus a thermodynamic one. The central finding is that it is almost entirely thermodynamic: the transmission probabilities change only marginally along the branch, while the Hawking temperature collapses, quenching the luminosity by many orders of magnitude.

Geometry and extremal endpoint

In units $M=1$ the metric function is controlled by a single length parameter $h=(2r_0^2)^{1/3}$,

$f(r)=1-\frac{2}{r}\left[1-e^{-(r/h)^3}\right],$

with a de Sitter core $\Lambda_{\rm eff}=6/h^3$ near the origin. Solving the double-horizon conditions $f(r_{\rm ext})=f'(r_{\rm ext})=0$ via $u=(r/h)^3$, $e^u=1+3u$, yields the precise extremal values

Quantity	Value
$u_{\rm ext}$	1.903813694440
$r_{\rm ext}$	1.702001406975
$h_{\rm ext}$	1.373256824469
$h=(2r_0^2)^{1/3}$0	1.137922411780

For $h=(2r_0^2)^{1/3}$1 the hole has nonzero temperature; at $h=(2r_0^2)^{1/3}$2 the horizons merge and $h=(2r_0^2)^{1/3}$3. Notably, the temperature stays within a fraction of a percent of the Schwarzschild value $h=(2r_0^2)^{1/3}$4 until $h=(2r_0^2)^{1/3}$5, then falls steeply: from $h=(2r_0^2)^{1/3}$6 at $h=(2r_0^2)^{1/3}$7 to $h=(2r_0^2)^{1/3}$8 at $h=(2r_0^2)^{1/3}$9. This geometry also arises as the fixed point of an iterative renormalization-group improvement of Schwarzschild in asymptotically safe gravity, which motivates the model beyond pure phenomenology.

Field equations and numerical method

All three spin sectors reduce to one-dimensional wave equations in the tortoise coordinate with effective potentials: $f(r)=1-\frac{2}{r}\left[1-e^{-(r/h)^3}\right],$0 for electromagnetism; the supersymmetric-partner Dirac potential built from $f(r)=1-\frac{2}{r}\left[1-e^{-(r/h)^3}\right],$1; and the axial Regge–Wheeler-type gravitational potential of Dubinsky. An important caveat is stated explicitly: only the axial gravitational equation is solved, so the total graviton luminosity assumes approximate equality of polar and axial greybody factors, $f(r)=1-\frac{2}{r}\left[1-e^{-(r/h)^3}\right],$2. The derivation also treats the regular source as a fixed effective matter sector rather than perturbing it independently.

Greybody factors are obtained by direct outward integration with purely ingoing boundary conditions at the horizon; the maximum flux-balance residual over all tabulated curves is $f(r)=1-\frac{2}{r}\left[1-e^{-(r/h)^3}\right],$3. First- and third-order WKB estimates (Iyer–Will continuation) are computed for comparison but are not used in the spectra. Third order substantially reduces residuals for electromagnetic, gravitational, and higher Dirac modes, while the lowest Dirac mode retains the largest deviation — the expected limitation of a barrier-top approximation at low multipole number.

Greybody factors: mild deformation, no transparency enhancement

Half-transmission frequencies are remarkably stable along the branch:

Mode	$f(r)=1-\frac{2}{r}\left[1-e^{-(r/h)^3}\right],$4 ($f(r)=1-\frac{2}{r}\left[1-e^{-(r/h)^3}\right],$5)	$f(r)=1-\frac{2}{r}\left[1-e^{-(r/h)^3}\right],$6 ($f(r)=1-\frac{2}{r}\left[1-e^{-(r/h)^3}\right],$7)
EM $f(r)=1-\frac{2}{r}\left[1-e^{-(r/h)^3}\right],$8	0.253	0.252
Dirac $f(r)=1-\frac{2}{r}\left[1-e^{-(r/h)^3}\right],$9	0.189	0.183
Gravitational (axial) $\Lambda_{\rm eff}=6/h^3$0	0.378	0.375

The dominant thresholds shift by at most a few percent, and the Dirac threshold actually moves slightly downward toward extremality. The implication is that the Dymnikova deformation does not open any significant transparency window at fixed ADM mass; whatever happens to the luminosity must come from the thermal factor.

Energy-emission rates and selective quenching

Using the direct-integration greybody factors in Page-type sums truncated at $\Lambda_{\rm eff}=6/h^3$1, the integrated powers show a dramatic thermal collapse. The Schwarzschild row reproduces Page's benchmarks to $\Lambda_{\rm eff}=6/h^3$2 or better ($\Lambda_{\rm eff}=6/h^3$3 versus Page's $\Lambda_{\rm eff}=6/h^3$4; $\Lambda_{\rm eff}=6/h^3$5 versus $\Lambda_{\rm eff}=6/h^3$6), and the familiar $\Lambda_{\rm eff}=6/h^3$7/$\Lambda_{\rm eff}=6/h^3$8/$\Lambda_{\rm eff}=6/h^3$9 spin split is recovered. Relative to Schwarzschild, the Page proxy $f(r_{\rm ext})=f'(r_{\rm ext})=0$0 falls to:

$f(r_{\rm ext})=f'(r_{\rm ext})=0$1	Proxy ratio
1.25	$f(r_{\rm ext})=f'(r_{\rm ext})=0$2
1.32	$f(r_{\rm ext})=f'(r_{\rm ext})=0$3
1.36	$f(r_{\rm ext})=f'(r_{\rm ext})=0$4
1.37	$f(r_{\rm ext})=f'(r_{\rm ext})=0$5

Two compositional results follow. First, the photon channel is quenched more efficiently than the Dirac channel: $f(r_{\rm ext})=f'(r_{\rm ext})=0$6 drops from $f(r_{\rm ext})=f'(r_{\rm ext})=0$7 at $f(r_{\rm ext})=f'(r_{\rm ext})=0$8 to $f(r_{\rm ext})=f'(r_{\rm ext})=0$9 at $u=(r/h)^3$0, so the residual massless flux becomes progressively fermion dominated. Second, the graviton contribution remains subdominant throughout, with $u=(r/h)^3$1 falling from $u=(r/h)^3$2 to $u=(r/h)^3$3 (halved if the axial–polar equality assumption is dropped). A supplemental comparison with Konoplya's RG-improved analysis at mapped parameter values shows consistent qualitative behavior, with residual differences attributable to particle-counting prescriptions and WKB versus direct scattering.

Adiabatic evaporation time

Holding the core scale $u=(r/h)^3$4 fixed, evaporation drives $u=(r/h)^3$5 upward toward the remnant mass $u=(r/h)^3$6. Integrating the mass-loss equation with a monotone cubic spline in $u=(r/h)^3$7 plus a low-temperature power-law tail gives times to reach $u=(r/h)^3$8 of order $u=(r/h)^3$9 — roughly $e^u=1+3u$0 to $e^u=1+3u$1 times the Schwarzschild lifetime for the same initial mass, depending on the starting point. Most of this time is spent near the cold endpoint. Because the extremal configuration is reached only asymptotically, these numbers are cutoff times to near-extremal states, not complete evaporation times.

Limitations and open questions

The paper is candid about several restrictions. The calculation is semiclassical and valid only for $e^u=1+3u$2; if $e^u=1+3u$3 is Planckian, the final stage requires genuine quantum-gravity input, and the treatment is safest either for $e^u=1+3u$4 or for an effective scale $e^u=1+3u$5 above the Planck length. The massless Standard-Model count is a proxy: massive-species thresholds are not switched on dynamically, and any species with rest mass exceeding $e^u=1+3u$6 should be removed by a threshold treatment left for future work. The graviton channel rests on the unproven axial–polar degeneracy assumption, and the evaporation history is adiabatic rather than a back-reacting solution — the precise late-time cutoff depends on the low-temperature extrapolation. Open questions include massive thresholds, a full polar perturbation calculation, rotating Dymnikova geometries, and a dynamical model in which the core parameter and ADM mass evolve together.

A further significance claimed for the results is methodological: within the recently established correspondence between greybody factors and quasinormal modes, the frequency-dependent transmission data computed here can be used to infer ringdown spectral information from scattering alone, complementing direct quasinormal-mode calculations.

Conclusion

This work establishes that for the Dymnikova regular black hole, the approach to the cold extremal remnant is governed by thermodynamics rather than scattering: greybody thresholds move by at most a few percent while the vanishing surface gravity suppresses the luminosity by up to five orders of magnitude, preferentially quenching photons relative to fermions and keeping gravitons subdominant. Within the stated semiclassical regime and the axial–polar assumption, the result converts the Schwarzschild runaway into a long, remnant-dominated cooling phase whose quantitative lifetime awaits a fully dynamical, back-reacting treatment.