Extreme Black Points in Gravitational Physics

Updated 4 September 2025

Extreme black points are degenerate black hole solutions characterized by singular horizons and vanishing interior proper times in the extremal limit.
They exhibit distinct thermodynamic behavior with zero temperature and entropy, contrasting sharply with conventional black hole models.
Observational signatures, including altered photon shadows and gravitational wave patterns, make extreme black points key probes in quantum gravity research.

Extreme black points are a class of black hole and compact object solutions characterized by their degeneracy, singular horizon structure, or limited-access interior regions, often emerging in the limiting cases of extremal charge and/or spin, nonlinear electrodynamics, or gravitational collapse. This concept spans multiple physical regimes, including classical and quantum gravity, nonlinear field theories, observational astrophysics, and early-universe cosmology. In many cases, extreme black points mark the boundary beyond which standard distinctions between interior/exterior, regular/singular, and area/volume scaling break down. The following sections delineate the dominant theoretical, mathematical, thermodynamic, and observational properties derived from primary research literature.

1. Horizon Structure and Singularity Formation

In classical general relativity, extreme black holes are solutions in which the event horizon coincides with the inner Cauchy horizon, as in the extreme Reissner–Nordström (RN) metric. However, physical considerations show that the horizon in actual extreme black holes is effectively singular (Marolf, 2010). Any test particle, defined by negligible back-reaction, encounters a curvature singularity immediately upon crossing the horizon. The mathematical expression for the proper time between horizons in a non-extreme RN black hole is

$\Delta \tau = \frac{r_+ - r_-}{\gamma} + O\left(\frac{(r_+ - r_-)^3}{r_+^2 \gamma^3}\right),$

where $r_+$ and $r_-$ are the outer and inner horizons and $\gamma$ the energy per unit mass of the infalling particle. In the extremal limit ( $r_- \rightarrow r_+$ ), $\Delta \tau$ vanishes: the infalling observer contacts a null singularity at the horizon, with no "interior" experience. This conclusion is robust in gravitational collapse models, e.g., charged thin shells leading to nearly extremal black holes (Garfinkle, 2011). There, the time between horizon crossing and singularity is governed by a small scale $m \epsilon$ ( $m$ is mass, $\epsilon = \sqrt{1-q^2/m^2}$ ), which becomes negligible as extremality is approached.

2. Thermodynamics and Phase Structure

An extreme black point, especially when realized via nonlinear electrodynamics (NED) such as the Born–Infeld theory, is a charged black hole solution whose horizon degenerates to a point coinciding with the singularity (Sokolov, 30 Aug 2025). In this state, distinct thermodynamic behavior arises:

The temperature $T$ follows

$T = \frac{1}{4\pi r_h}[1 + \Phi'(r_h)],$

with $r_h \rightarrow 0$ for a black point. For generic configurations, $T$ diverges as $r_h \to 0$ ; for the extreme black point with fine-tuned parameters, $T \to 0$ . The entropy $S = \pi r_h^2$ also vanishes.

The third law of thermodynamics in Planck's formulation is satisfied: as $T \to 0$ , $S \to 0$ . This sharply contrasts with the extremal Reissner–Nordström solution ( $Q = M$ ), where $S$ remains finite as $T \to 0$ , mildly violating the third law.
The heat capacity $C$ is continuous and vanishes in the black point state, with no latent heat or discontinuity characteristic of first-order transitions. Thus, formation of a black point is not a phase transition of first order.

This thermodynamic regime can be achieved for mass and charge values satisfying $M_{\text{cr}} \approx 0.437 a$ and $|Q_{\text{cr}}| = a/2$ , where $a$ is the Born–Infeld parameter.

3. Spacetime Geometry and Effective Metrics

The spacetime structure of extreme black points is distinct from that of traditional black holes. The metric component $g_{00}$ admits a double root at $r=0$ (the horizon). For massive particles, the pseudo-Riemannian background possesses a timelike singularity at the origin. However, in NED, the null geodesics (photon trajectories) are governed by an effective metric

$G_{kn} = \frac{g_{kn} - a^2 F_{kn}^{(2)}}{1 - \frac{a^2}{2} J_2 - \frac{a^4}{4} J_4 + \frac{a^4}{8} J_2^2},$

where $F_{kn}^{(2)}$ , $J_2$ , and $J_4$ are constructed from the field strength tensor (Sokolov, 30 Aug 2025). At the extreme black point, the scalar curvature of the effective metric and all Carminati–McLenaghan invariants remain regular everywhere, so photons propagate through a fully regular geometry—even though massive particles "see" a singularity.

This duality places extreme black points as intermediates between traditional (singular) and regular black holes.

4. Observational and Phenomenological Signatures

Extreme black points can influence observable phenomena:

Black hole shadows and photon spheres: For extremal black holes, the photon orbit may lie exactly on the event horizon and be stable (static spherically symmetric), unstable (Kerr), or split (Kerr–Newman, depending on parameters) (Khoo et al., 2016). The size and regularity of the shadow can be modified for extreme black points; for instance, its angular radius can increase linearly with the Born–Infeld parameter $a$ ( $R\sin\varphi_\infty \simeq 1.827 a$ for extreme black points).
Gravitational wave signatures: Collisions and inspirals involving extreme mass-ratio black hole binaries bridge the gap between non-linear and point-particle approximations (Sperhake et al., 2011), providing calibration for extreme mass-ratio inspiral models.
Accretion and X-ray spectral mapping: In high-spin active galactic nuclei and black hole binaries, innermost regions (within $\sim$ 1 gravitational radius) can be mapped, revealing strong relativistic blurring and reflection-dominant spectra conditioned by extreme spacetime properties (Fabian, 2015).
Dark matter environments: Spikes in DM density near black holes can cause substantial deformations of event horizons and ergospheres, altering shadow structures and gravitational wave signals by orders of magnitude compared to diffuse halos (Xu et al., 2021).
Black hole imaging artifacts: Extreme ring or black point features observed by EHT may be PSF artifacts, as the ring diameters align with dirty beam sidelobe spacings (Miyoshi et al., 26 Sep 2024). Rigorous assessment of deconvolution and calibration is required to disentangle physical features from imaging artifacts.

5. Quantum and Microphysical Considerations

The singular horizon in the extremal limit of classical black holes (Reissner–Nordström, Kerr) invalidates the physical relevance of analytic interior extensions: region B in the conformal diagram is inaccessible to any infalling observer (Marolf, 2010). Quantum effects may instead become significant outside the horizon, possibly leading to nonperturbative corrections in entropy and microphysics. Models predicated on smooth classical interiors (D-brane, fuzzball) must reconcile with the near-horizon singularity for realistic black points.

In quantum field theory considerations:

Scalar hair in extremal black holes: Aretakis and collaborators show that minimally coupled scalar fields can generate permanent conserved charges (Aretakis hair) on extremal black holes, distinguishing them from nearly/extreme and non-extreme cases, where hair decays quadratically or vanishes (Burko et al., 2019).
Stability: Linearized perturbation analyses reveal that apparent divergences in curvature scalars (e.g., Weyl scalar $\psi_4$ in Kerr) are gauge artifacts; curvature polynomials remain finite. A suitable choice of tetrad (dynamical Hartle–Hawking) regularizes all observables at late times (Burko et al., 2017).

6. Cosmological Formation and Statistical Properties

Extreme black points manifest in cosmological settings as the most massive primordial black holes (PBHs) formed by rare, large overdensities. Applying extreme-value theory (Chongchitnan et al., 2019, Kuhnel et al., 2021), block-maxima statistics yield narrower, high-mass-peaked PBH mass functions, and the predicted PBH abundance is boosted by many orders of magnitude versus Gaussian extrapolation. Observational bounds (e.g., from gamma-ray constraints and tidal deformation searches) limit the possible existence of such objects near Earth (Namigata et al., 2022).

These statistical approaches offer strong tests for PBH formation theories and may provide connections between rare, observable black points and models for dark matter and galaxy evolution.

7. Generalizations and Conceptual Frameworks

The concept of "maximal black room" (MBR) provides a geometric and causal generalization of the shadow region associated with a black hole. The MBR's boundary is a globally unstable, non-spacelike hypersurface formed from tangential null geodesics (the "rays' surface") (Siino, 2023). In static, spherically symmetric spacetime, this reduces to the photon sphere, but in more general settings accommodates shadows of exotic compact objects and transitions to area-scaling entropy regimes during collapse (see nearly black stars, where depth information evaporates for external observers and the entropy transitions from volume to area law) (Lin, 2019).

These conceptual and mathematical generalizations guide the understanding of extreme black points as end-states of collapse, limiting thermodynamic systems, and ultimate probes for spacetime singularity regularization and quantum gravitational effects.

The extreme black point paradigm thus subsumes singular horizon structures, vanishing interior proper time, distinctive thermodynamic limits, regular effective photon geometry, altered observational shadows, enhanced quantum and statistical phenomena, and generalized causal boundaries—all unified as manifestations of highly degenerate, strong-field gravitational states across classical, quantum, and astrophysical domains.