Horizonless Black-Hole Mimickers

• Horizonless black-hole mimickers are ultra-compact objects that lack a true event horizon while closely replicating the exterior Schwarzschild field with Planck-scale corrections.
• They emerge in modified and quadratic gravity models, offering solutions like the 2-2-hole that address the information paradox with unique interior metric behavior.
• Observational strategies such as gravitational wave echoes and high-resolution imaging can distinguish these mimickers, thereby probing quantum gravity and dark matter implications.

Horizonless black-hole mimickers are ultra-compact, horizonless astrophysical objects whose exterior gravitational field closely resembles that of classical black holes, yet whose internal structure or boundary conditions depart fundamentally from the predictions of General Relativity’s black hole solutions. These objects arise in modified gravity theories, quantum gravity-inspired scenarios, or as phenomenological constructs designed to address the information paradox and test the limits of strong gravity. A prominent example, the “2-2-hole” in asymptotically free quadratic gravity, exemplifies how such constructs can reproduce black hole phenomenology (including the exterior Schwarzschild solution down to Planck-scale corrections), while introducing new physics in the near-horizon and interior regions. Horizonless mimickers constitute a key target for current and future observational tests of gravity via gravitational waves, electromagnetic signatures, and high-resolution imaging.

1. Theoretical Motivation and Defining Properties

Horizonless black-hole mimickers are motivated by longstanding theoretical challenges including the black hole information paradox and the search for a UV-complete theory of gravity. In classical General Relativity, compact objects (with vanishing pressure at the surface and satisfying energy conditions) have a lower bound on their radius (the Buchdahl limit), preventing horizon formation except for black holes. Many modified gravity models—such as quadratic (higher-derivative) gravity—permit static, spherically symmetric solutions that evade the formation of true event horizons while achieving extreme compactness.

Key defining features, as illustrated by the 2-2-hole solution in quadratic gravity (Holdom et al., 2016), include:

• Absence of an event horizon: The exterior metric matches Schwarzschild (or Kerr for rotating cases) for $r \gtrsim r_H = 2M$, but the horizon is replaced by a sharply squeezed transition region near $r_H$.
• Distinct interior: The metric functions $A(r)$ and $B(r)$ both vanish as $r \rightarrow 0$, indicating a highly squeezed (shrinking four-volume) interior with a timelike singularity at the origin.
• Universal scaling: In solutions like the 2-2-hole, curvature invariants and interior metric components become universal functions of the scaled variable $r/M$, largely independent of the mass $M$.
• Extreme gravitational potential: The interior potential well is so deep that particles and radiation are efficiently trapped.
• No pathologies in finite-energy propagation: Despite the central singularity, the dynamics of finite-energy fields remain uniquely defined; the wave operator (e.g., for the Klein-Gordon field) is essentially self-adjoint.

2. Solution Structure in Quadratic Gravity: The 2-2-Hole

Classical quadratic gravity (CQG), as a UV extension of General Relativity, allows a broader space of static, spherically symmetric solutions. In CQG, the near-origin (small $r$) expansions divide into families classified by the leading powers of $r$ in the metric functions:

• The (2,2)-family, and specifically the even-power subclass $(2,2)_e$, gives rise to the 2-2-hole. Here:

$$A(r) = a_2 r^2 + a_4 r^4 + \ldots,\quad B(r) = b_2 r^2 + b_4 r^4 + \ldots$$

with parameters tuned so the full nonlinear solution is connected to an appropriate matter source.

• Under gravitational collapse of sufficiently dense matter, traditional star-like (0,0) solutions cease to exist once the star radius drops below $r_H$. Only the (2,2)-family survives, leading naturally to the formation of a horizonless ultra-compact object—interpreted as the endpoint of collapse in CQG.
• The exterior ($r \gtrsim r_H$) is essentially indistinguishable from Schwarzschild, differing only at separations of order the Planck length. However, inside $r_H$, the metric quickly transitions to the 2-2-hole profile with rapidly vanishing $A(r)$ and $B(r)$, producing an extreme gravitational redshift and a highly suppressed four-volume.
• The interior timelike singularity at $r=0$ has finite action and does not obstruct the unique regular evolution of classical or quantum fields.

3. Particle Dynamics and Gravitational Trapping

The interior structure of 2-2-hole mimickers has important dynamical consequences:

• The geodesic radial equation for particles in the equatorial plane,

$$A(r)B(r) \left(\dfrac{dr}{d\zeta}\right)^2 + B(r) \left( \dfrac{L^2}{r^2} + \vartheta \right) = E^2,$$

with $\vartheta = 1$ (massive) or $0$ (massless), becomes dominated by the rapid drop in $B(r)$ for small $r$.

• The phase-space for escaping radiation is suppressed by a factor $\sim 1/M^2$ in the large-mass limit, making the interior a near-perfect trap for matter and photons. The escape threshold for massless particles is highly restrictive.
• Collisions in the interior reach Planck-scale center-of-mass energies due to the $1/B(r)$ enhancement:

$E_{\text{cm}}^2 \approx \dfrac{4E_1E_2}{B(r)}$

for colliding particles with energies $E_1, E_2$.

This efficient trapping, combined with the absence of a horizon, means that the object appears nearly as “dark” as a classical black hole for practical purposes, while still being fundamentally different in its causal structure.

4. Gravitational Collapse, Exterior Mimicry, and Quantum Gravity Implications

Horizonless black-hole mimickers in quadratic gravity represent a generic collapse endpoint:

• Formation proceeds via gravitational collapse past $r_H$, at which point star-like configurations are no longer viable. The system selects a $(2,2)$ solution, resulting in a 2-2-hole.
• The exterior geometry remains indistinguishable from Schwarzschild down to Planck-scale corrections, satisfying all current astrophysical constraints on compactness and redshift.
• No event horizon forms; therefore, information is not fundamentally lost—the classical or quantum field evolution is well-posed outside the origin. The propagation is governed by a unique self-adjoint extension of the wave operator.
• The interior universality (scaling with $r/M$) suggests that quantum gravity effects are confined to the deepest interior, and higher-derivative corrections remain hidden from the exterior—supporting a resolution to the information paradox.
• Statistical entropy computed in a brick-wall model for the 2-2-hole shows an area-law scaling $S \propto M^2$, matching black hole thermodynamics despite the absence of a horizon.

5. Gravitational Wave Echoes and Observational Distinctions

A distinguishing observational signature of horizonless mimickers such as 2-2-holes is the occurrence of gravitational wave echoes in the post-merger phase:

• Gravitational waves entering the deep interior potential are reflected at the singular origin (or the effective “inner boundary”) and repeatedly scatter off the outer light ring, producing a sequence of delayed echoes in the waveform.
• The echo delay time $\Delta t$ is approximately twice the light travel time from the origin to the photon sphere, and for astrophysical masses (e.g., $30M_\odot$) is estimated as:

$$\Delta t \approx \left[ 700 + 7 \ln\left(\frac{M}{30M_\odot}\right) \right] M$$

corresponding to $100-125$ ms—two orders of magnitude longer than black hole ringdown damping for comparable masses.

• Observing such echoes would constitute strong evidence for the absence of a true event horizon and would provide constraints on quantum gravity modifications of black hole interiors.

6. Thermodynamics, Remnants, and Phenomenological Consequences

Horizonless black-hole mimickers show intriguing thermodynamic behavior:

• For large masses, they mimic black hole thermodynamics, with temperature at infinity $T_\infty \propto 1/M$ and entropy $S \propto M^2$ arising from ordinary matter (e.g., a trapped thermal gas) redshifted by the deep gravitational potential.
• As mass decreases (through Hawking-like evaporation or other processes), the heat capacity transitions from negative to positive, and the object settles into a stable remnant phase with mass determined by the fundamental spin-2 mode ($M_{\text{min}} \sim M_\text{Pl}^2/m_2$ in quadratic gravity) (Aydemir, 2020).
• Stable Planck-scale remnants may serve as natural dark matter candidates and avoid issues of information loss.
• The absence of an event horizon ensures unitarity of quantum evolution for all field modes except perhaps in the immediate vicinity of the central singularity, which is dynamically protected.
• 2-2-hole mimickers predict gravitational wave and electromagnetic signatures (e.g., echoes, subtle differences in tidal deformability) that may become testable with next-generation detectors and imaging arrays.

7. Relation to Other Ultra-Compact Mimickers and Open Questions

The 2-2-hole is one member of a broader class of horizonless mimickers, ranging from gravastars to black shells to nonlocal stars:

• Gravastars and related models typically rely on anisotropic matter or exotic equations of state and may feature similar or even larger compactness, but with different internal stress profiles.
• The existence and stability of stable photon spheres in horizonless compact objects raises concerns about nonlinear instabilities, as these surfaces may trap perturbations for extremely long times (Franzin et al., 2023). Evidence points to possible late-time energy accumulation and the development of turbulence or nonperturbative phenomena.
• Observationally, distinguishing horizonless mimickers from true black holes remains challenging, given their near-perfect exterior mimicry; gravitational wave echoes and precision timing of orbital precession or ringdown modes provide the most promising avenues.
• The broader theoretical landscape includes connections to string theory, nonlocal gravity, and alternative quantizations of the gravitational field, with each framework predicting distinct internal structure and phenomenology.

Horizonless black-hole mimickers thus constitute a central class of solutions in the paper of strong gravity, quantum gravity, and astrophysical compact objects, providing both a playground for theoretical exploration and concrete targets for multi-messenger observations.