Five-Dimensional Primordial Rotating Black Holes

• Five-dimensional primordial rotating black holes are objects formed in the early universe with a compact extra spatial dimension, exhibiting discrete mass and spin quantization.
• Their metric solutions leverage harmonic function formalisms and gravitational spin–spin balance to sustain equilibrium multi-black-hole configurations beyond traditional 4D paradigms.
• Enhanced stability from quantum memory effects and slower Hawking evaporation make these black holes viable dark matter candidates with distinctive observational signatures.

A five-dimensional primordial rotating black hole is a gravitationally collapsed object formed in the early universe within a spacetime possessing one compactified extra spatial dimension. Such black holes generalize the four-dimensional paradigm by exhibiting novel quantization, stability, horizon topology, and evaporation properties unique to higher-dimensional and Kaluza-Klein frameworks. Their dynamics and potential cosmological roles, particularly as dark matter candidates, are determined by the interplay of higher-dimensional general relativity, compactification scale, extremal rotation, and quantum effects such as the memory burden.

1. Metric Constructions and Equilibrium Properties

The most comprehensive vacuum solutions describing stationary, rotating black holes in five dimensions with an asymptotic Kaluza–Klein circle leverage a harmonic function formalism. In the Matsuno–Ishihara–Kimura–Tatsuoka construction, spacetime is parametrized by coordinates $(t,x,y,z,v)$, where $v$ describes the compactified circle of circumference $L$. The family of multi-black hole solutions is characterized by the metric: $$ds^2 = -H^{-2} dt^2 + H^2 (dx^2 + dy^2 + dz^2) + 2\left[ (H^{-1} - 1)dt + \tfrac{L}{2\sqrt2} dv + w_i dx^i \right]^2$$ where

$$H(x, y, z) = 1 + \sum_{i=1}^N \frac{m_i}{|\mathbf{R} - \mathbf{R}_i|}$$

and $w$ is a one-form satisfying $\nabla \times w = \nabla H$. Away from the singularities $\mathbf{R} = \mathbf{R}_i$, these metrics are Ricci-flat and globally smooth (Matsuno et al., 2012).

Each center represents an extremal, maximally rotating black hole with mass-quantization set by horizon regularity and compactification size: $$m_i = \frac{n_i L}{2\sqrt2},\quad n_i \in \mathbb{N}$$ yielding discrete Komar mass and angular momentum: $$M_i = \frac{3L\,m_i\,A_{S^3}}{4\pi},\qquad J_i = \frac{L^2\,m_i\,A_{S^3}}{8\pi}$$ where $A_{S^3}=2\pi^2$ is the area of the unit 3-sphere. The extremality condition,

$J_i^2 = \frac{2\sqrt2}{27\pi n_i} M_i^3$

enforces maximal spin along the compact direction, with the discrete spectra set by the values of $n_i$.

Stationary multi-black hole configurations are sustained by a precise balance between gravitational monochromatic attraction and frame-dragging-induced spin–spin repulsion, generalizing the Majumdar–Papapetrou mechanism to pure gravity without electromagnetic charges. This enables equilibrium “crystals” of black holes in 5D Kaluza–Klein spacetimes (Matsuno et al., 2012).

2. Horizon Topology, Quantization, and Cosmological Implications

A key outcome in 5D Kaluza–Klein settings is the quantization of mass, angular momentum, and horizon area, all in integer steps proportional to the size $L$ of the extra dimension. The horizon cross section for each black hole is a lens space $L(n_i;1) = S^3/\mathbb{Z}_{n_i}$. When $n_i=1$, the horizon is a round 3-sphere; for $n_i>1$, it is a squashed lens space with nontrivial fundamental group.

This discrete topological labeling implies a spectrum of “black hole species” in a cosmological setting, potentially producing observable consequences or selection effects in early-universe black hole formation. The minimal possible 5D black hole mass is $\sim L^2/G_5$, and lighter black holes are forbidden, unlike the continuous spectrum in 4D. This quantization emerges fundamentally from the requirement of regular Killing horizons in higher dimensions (Matsuno et al., 2012).

In cosmological solutions constructed within Einstein–Maxwell–dilaton or “fake supergravity” frameworks, the presence of rotating Killing horizons persists within dynamic, FLRW-type backgrounds. The physical mass, spin, and horizon area scale with the cosmic expansion: for $n=1$ (the Kaluza–Klein regime), these evolve as

$$M_{\rm ADM} \sim Q\,a(\bar t)^{-1/3},\qquad J_{\rm ADM} \sim J\,a(\bar t)^{-3/2},\qquad A_H \sim a(\bar t)^{-1/2}$$

where $a(\bar t)$ is the scale factor (Nozawa et al., 2010). These backgrounds admit nondegenerate, rotating horizons, ergoregions (allowing for superradiant phenomena), and, generically near singularities, naked closed timelike curves unless parameters are tuned.

3. Hawking Evaporation and Black Hole Lifetime in 5D

The evaporation properties of 5D rotating black holes depart qualitatively from their 4D Kerr analogues. In the context of the Myers–Perry metric with a single nonvanishing spin parameter, the Hawking temperature for $n=1$ extra dimension is

$$T_H = \frac{1}{2\pi\,r_H\,(1+a_*^2)}\,,\qquad a_* = a/r_H$$

The semi-classical mass-loss rate, incorporating numerically determined greybody factors $C_M$, scales as

$$\dot M_{5D} \approx -\frac{3\pi\,C_M}{8\,G_5}\frac{1+a_*^2}{M}$$

in contrast to the four-dimensional case, where $\dot M_{4D}\propto-1/M^2$ (Leontaris et al., 11 Dec 2025). As a result, heavy 5D black holes evaporate much more slowly than their 4D counterparts. Integrating the rate yields a Hawking lifetime

$\tau_{5D} \propto M_0^2$

implying, for initial mass $M_0$, that $M_0 \gtrsim 10^{10} \;\text{g}$ allows survival to the present epoch, while in 4D, the lower bound is three to five orders of magnitude higher.

Table: Evaporation Law Comparison

Dimension	Mass-loss Rate $\dot M$	Lifetime Scaling $\tau$
4D	$-C/M^2$	$\tau \propto M_0^3$
5D	$-K/M$	$\tau \propto M_0^2$

Values from (Leontaris et al., 11 Dec 2025); C, K are model-dependent constants.

4. Quantum Memory Burden Effect and Enhanced Stability

Beyond the semiclassical evaporation paradigm, the “memory burden” effect, as formulated by Dvali, introduces quantum back-reaction by encoding information in black hole microstates. The 5D mass-loss rate is then further suppressed by a power of the black hole entropy,

$$\frac{dM}{dt} = -\frac{1}{S^p}\frac{C_M}{r_H^2},\qquad S = \frac{4\pi}{3} M r_H$$

With $S \propto M^{3/2}$, the evaporation rate for $p\geq1$ becomes highly suppressed as the PBH mass decreases, yielding lifetimes scaling as $M_0^{2+3p/2}$. For example, $p=1$ gives $\tau\propto M_0^{7/2}$; $p=2$ yields $\tau\propto M_0^{5}$, making even sub-$10^{10}$g PBHs cosmologically long-lived (Leontaris et al., 11 Dec 2025). This quantum effect thus dramatically enlarges the mass window for viable 5D primordial black-hole dark matter.

5. Dark Matter, Observational Prospects, and Cosmological Impact

In the “Dark Dimension” scenario, the survival and abundance of 5D primordial rotating black holes (PBHs) is a direct consequence of their slow Hawking evaporation and memory burden effects. The relic density is given by

$$\Omega_{\rm PBH} \simeq \beta(M_0) \frac{M_{\rm eq}}{M_0}$$

for PBHs of mass $M_0$, where $\beta(M_0)$ is the initial energy fraction at formation, and $M_{\rm eq} \sim 10^{17}$g is the PBH mass evaporating at matter–radiation equality (Leontaris et al., 11 Dec 2025). In contrast to 4D, the allowed mass range for all-dark-matter scenarios is expanded to $10^{10}$g $\lesssim M_0 \lesssim 10^{21}$g, with initial fraction $\beta$ in the range $10^{-7}$–$10^{-3}$.

Observational constraints are highly relaxed: extragalactic γ-ray, positron, and CMB bounds weaken due to the suppressed Hawking emission, with microlensing surveys only limiting heavier PBHs $M_0\gtrsim 10^{23}$g. Characteristic observational signatures include possible monochromatic particle bursts at energies set by the 5D Hawking temperature for a given surviving mass, in the 100 MeV regime for $M_0\sim 10^{12}$g.

A plausible implication is that “black hole crystals”—stationary multi-black-hole configurations—could have persisted through the early universe, and perturbations or mergers of such objects might generate unique gravitational-wave signals, particularly if compact lens-space horizon topologies can be encoded in the merger spectrum (Matsuno et al., 2012).

6. Embeddings in Dynamical Spacetimes and Higher-Dimensional Theories

Rotating 5D black holes have been embedded in dynamical, cosmologically relevant settings, such as solutions to Einstein–Maxwell–dilaton theory and five-dimensional “fake supergravity.” In these models, rotating PBHs evolve alongside a FLRW background, with their physical properties (mass, angular momentum, area) tracking cosmological expansion or contraction, enabling analysis of PBHs in time-dependent, inhomogeneous environments (Nozawa et al., 2010).

A distinctive feature is the occurrence of nondegenerate, rotating Killing horizons with constant angular velocity, and the emergence of ergoregions supporting superradiant modes. In the near-singularity regime, closed timelike curves (regions where the norm of azimuthal Killing vectors becomes negative) can appear unless the solution resides in an under-rotating regime.

For specific harmonic function choices, the 5D system admits uplift to eleven-dimensional supergravity, representing dynamically intersecting M2/M2/M2 brane systems in a rotating Kasner universe. This establishes a direct connection between 5D PBHs and higher-dimensional string/M-theory constructions, motivating further paper of their microphysics and possible origin mechanisms (Nozawa et al., 2010).

7. Summary and Outlook

Five-dimensional primordial rotating black holes substantially depart from four-dimensional black hole physics across quantization, rotation, topology, evaporation, quantum stability, and cosmological evolution. The presence of a compact extra dimension leads to quantized mass spectra, maximally aligned spin along the Kaluza-Klein circle, and the possibility of equilibrium configurations stabilized by gravitational and spin interactions. Their slow evaporation and quantum stabilization mechanisms enable plausible dark matter scenarios across a broad mass range [(Matsuno et al., 2012); (Leontaris et al., 11 Dec 2025)]. Observational probes include microlensing, gravitational waves, and future low-energy γ-ray or particle burst searches. Connecting 5D PBHs to higher-dimensional string theory frameworks remains an important direction for both theoretical consistency and phenomenological viability (Nozawa et al., 2010).