Singularity-Free Black Holes in Temporal Geometry

The Singularity Problem

Black hole singularities represent one of the most profound failures of classical general relativity. At the center of every black hole the theory predicts infinite curvature, infinite density, and the breakdown of spacetime itself.

This is mathematically ugly and it violates core physical principles:

• Energy conservation: Infinite density means infinite energy density in finite volume
• Unitarity: Information appears to be destroyed rather than processed
• Predictability: Physics breaks down entirely at the singularity

The temporal geometry framework offers a surprisingly elegant resolution: when we properly account for how time flows in extreme gravity, singularities simply cannot form.

How Our Approach Differs from Previous "Regular Black Hole" Models

The idea of replacing a black hole singularity with something finite is not new. In fact, since the 1960s physicists have explored whether the extreme densities inside a collapsing star might create a smooth "vacuum-like" state instead of an infinite blow-up.

• Sakharov and Gliner (1966): proposed that ultra-dense matter could act like a vacuum with negative pressure (p = -ρ), producing a de Sitter–like interior.
• Bardeen (1968): introduced the first "regular black hole" solution, in which the singularity was replaced with a finite, nonsingular core.
• Hayward, Dymnikova, and others (1990s–2000s): developed variations where the Schwarzschild exterior transitions smoothly into a de Sitter-like interior, often with free parameters chosen for mathematical convenience.

These pioneering ideas showed that singularity-free black holes were mathematically possible but they never became mainstream, largely because:

1. Ad hoc transitions: The "switch" from Schwarzschild to de Sitter was usually imposed by hand, without a guiding principle.
2. Free parameters: The density scale of the core (ρ_c) was left arbitrary, making the models feel like curve-fitting rather than predictions.
3. Quantum gravity detour: Many believed singularities could only be cured by a future quantum gravity theory, so classical de Sitter cores were set aside.

Our Unique Angle

In the time-first formulation of gravity, the replacement of singularities by finite cores is not just possible it is natural and mathematically enforced.

• Horizon regularity is automatic: In lapse-first variables, horizons are pure gauge effects. The only real issue to fix is the central singularity.
• Selector equation (no tuning): Matching Schwarzschild to a de Sitter interior using Israel junction conditions yields
  r_c³ = 3Mc² / (4πρ_c)
  The core size is determined, not chosen.
• Cosmology anchor: Unlike earlier models, we tie ρ_c to the inflationary plateau energy density V_*. This creates a direct link between black holes and the early universe a bridge across the smallest and largest scales.
• Absorbing boundary condition: Instead of speculative bounces or wormholes, we adopt the most conservative boundary rule (Φ'(r_c)=0): geodesics end smoothly at a finite, absorbing surface, with curvature everywhere finite.

Why This Matters

Earlier regular black hole models were mathematical possibilities.
Our construction is a predictive framework: the exterior remains pure GR, the interior is tied to cosmology, and the matching leaves no free knobs to adjust.

This makes singularity-free black holes not just a thought experiment, but a minimal, testable extension of classical GR built deliberately on the most conservative assumptions to provide a solid foundation for future, more speculative explorations (such as quantum bounces or wormhole continuations).

The Temporal Solution

In the temporal geometry framework, gravity is fundamentally about the flow of time. This is characterized by the lapse function N = e^Φ. The key insights are:

1. Time stops at horizons: The lapse function goes to zero, meaning clocks effectively stop. This creates a natural boundary
2. Energy conservation constrains geometry: The total energy-momentum must remain finite
3. Absorbing cores emerge naturally: Near r = 0, the solution transitions to a finite-density absorbing state

The Mathematical Structure

The solution takes a remarkably simple form. Outside the quantum core (r > r_c):

This is exactly the Schwarzschild solution. But near r = 0, instead of diverging, we find:

Where r_c ~ l_P(M/M_P)^1/3 is the absorbing core radius. The metric remains smooth and finite everywhere.

Physical Interpretation

The absorbing core that replaces the singularity has fascinating properties:

• Maximal time dilation: Time flows arbitrarily slowly but never stops completely
• Finite energy density: ρ ~ ρ_inf anchored to the inflationary energy scale
• Information processing: The core can absorb and potentially process information
• Thermodynamic consistency: Maintains black hole thermodynamics and Hawking radiation

Observational Consequences

Crucially, this resolution preserves all confirmed observations of black holes:

1. External geometry: Identical to Schwarzschild outside the quantum core
2. Event horizons: Form exactly as in classical GR
3. Gravitational waves: LIGO/Virgo signals unchanged
4. Accretion dynamics: All astrophysical processes preserved

The differences only appear in regimes we cannot yet observe:

• Very late-time Hawking radiation might carry information
• Core dynamics at the inflationary energy scale
• Modified dynamics for primordial black holes

Connection to Quantum Gravity

This solution bridges classical and quantum gravity without requiring a full theory of quantum gravity. The key is recognizing that:

1. Energy conservation prevents infinite densities
2. Temporal geometry naturally provides cutoffs at finite scales
3. The absorbing core anchors to known physics (inflationary energy density)

The absorbing core represents a conservative baseline that avoids singularities while remaining within well-understood physics. You cannot have both zero radius (r = 0) and infinite energy density. Something must give way first.

Implications for Information Paradox

The absence of a true singularity has profound implications for the black hole information paradox:

• No information destruction: Information is absorbed by the finite-density core, not destroyed at a singularity
• Unitary evolution possible: The complete spacetime avoids the breakdown that would prevent unitary evolution
• Page curve: Information can potentially return after the Page time
• No firewalls needed: Smooth horizons without paradox

Mathematical Rigor

The solution satisfies several crucial mathematical requirements:

1. Einstein equations: Solved exactly with a well-defined stress-energy tensor
2. Energy conditions: Weak energy condition satisfied everywhere
3. Coordinate independence: Results hold in all coordinate systems
4. Asymptotic flatness: Proper falloff at infinity

A Conservative First Step

This paper represents our first attempt to understand black holes within the temporal geometry framework. We deliberately chose a conservative approach. We use a simple de Sitter core at the inflationary energy density to establish a solid mathematical foundation that we can build upon.

While more exotic possibilities exist (quantum bounces, temporal tunneling, connections to the origin-of-time framework), we focus here on the minimal modification needed to resolve the singularity problem while preserving all observational consequences. This conservative path ensures our solution is robust and provides a reliable starting point for future explorations.

Future Directions

Having established this solid foundation, the framework opens several exciting research directions:

Immediate Extensions

• Rotating black holes: Extension to Kerr geometry with frame-dragging and more complex matching conditions
• Binary mergers: Quantum core dynamics during coalescence and their signatures in gravitational waves
• Primordial black holes: Formation and evaporation in the early universe with modified dynamics
• Alternative core densities: Exploring other physical scales beyond the inflationary energy density

Bold New Possibilities

• Euclidean bounce models: Instead of ending at a finite core, spacetime might undergo a quantum "bounce" to another region, echoing the origin-of-time framework where the universe nucleated via Euclidean-to-Lorentzian transition
• Temporal tunneling: Black hole cores as windows into the substrate that gave rise to time itself
• Full quantum completions: Moving beyond the conservative absorbing core to full quantum field theory treatment with genuine quantum superposition of core states

Observational Tests

• Gravitational wave signatures: Deviations from classical predictions during core formation
• Black hole shadows: Subtle modifications in Event Horizon Telescope observations
• Late-time Hawking radiation: Information return after the Page time
• Primordial black hole constraints: Links between cosmology and quantum gravity

Summary

The temporal geometry framework provides a natural, elegant resolution to the black hole singularity problem. By recognizing that gravity is fundamentally about time's geometry and enforcing energy conservation at all scales, we find that singularities cannot form. Instead, finite absorbing cores emerge that avoid the loss of unitarity implied by classical singularities while maintaining all the successful predictions of classical black holes.

This isn't a mathematical curiosity. It's a concrete prediction about the nature of spacetime in extreme gravity. This predicts that black hole cores scale with mass in a universal way, tied to the same energy density that shaped the early universe. While observationally hidden today, this link to cosmology makes the proposal testable in principle.