Black Holes as Radii of Stopped Time: A Geometric Resolution to Information and Gravity Paradoxes

J. Rogers, SE Ohio, 21 July 2025, 0203

---

Abstract

We propose that a black hole is fundamentally a radius of stopped time—a boundary where the flow of time terminates absolutely. This perspective eliminates the need for an "interior" spacetime, reinterpreting gravity as the propagation of this time-stop discontinuity into the surrounding universe. By treating the event horizon as a time-frozen information membrane, we resolve the black hole information paradox, eliminate firewall and singularity pathologies, and provide a geometric origin for mass and entropy without invoking quantum gravity. The model is consistent with general relativity, holography, and observational data from black hole mergers and shadows.

---

1. Introduction

The conventional view of black holes—as regions of spacetime with an "interior" containing a singularity—introduces paradoxes (e.g., information loss, firewalls) and relies on untested quantum gravity conjectures. Here, we argue that:

1. A black hole has no interior. The event horizon is a time-stop surface, where $g_{tt}=0$ permanently.

2. Gravity is not a "force" from the hole but the universe’s response to this time-stop discontinuity.

3. Hawking radiation is a boundary effect—it does not originate from "inside" the hole.

This framework simplifies black hole thermodynamics, preserves unitarity, and aligns with holographic principles.

---

2. The Time-Stop Horizon

2.1. Geometric Definition

In Schwarzschild coordinates, the metric at $r=2GM\mathrm{/}{c}^2$ has:

$${\mathrm{<span\; style="background-color:\; white;">g</span>}}_{\mathrm{<span\; style="background-color:\; white;">t</span>}\mathrm{<span\; style="background-color:\; white;">t</span>}}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">1</span>}<span\; style="background-color:\; white;">-</span>\frac{\mathrm{<span\; style="background-color:\; white;">2</span>}\mathrm{<span\; style="background-color:\; white;">G</span>}\mathrm{<span\; style="background-color:\; white;">M</span>}}{\mathrm{<span\; style="background-color:\; white;">r</span>}{\mathrm{<span\; style="background-color:\; white;">c</span>}}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}}<span\; style="background-color:\; white;">\to </span>\mathrm{<span\; style="background-color:\; white;">0</span>}<span\; style="background-color:\; white;">,</span>$$

signifying time freezing. Unlike a coordinate artifact, this is a physical boundary condition:

• All infalling worldlines terminate here for external observers.

• No signals can propagate outward from $r\le 2GM$.

2.2. No Interior Spacetime

The "interior" ($r<2GM$) is a mathematical fiction. The true physical object is the horizon as a time-frozen 2D surface:

• Mass $M$ is encoded holographically in the horizon area $A=16\pi G^2{M}^2\mathrm{/}{c}^4$.

• Singularity is replaced by the time-stop condition—no infinite curvature is needed.

---

3. Gravity as Propagated Time-Stop

3.1. The Time Gradient = Curvature

The external gravitational field is the gradient of stopped time:

$$\mathrm{<span\; style="background-color:\; white;">\nabla </span>}{\mathrm{<span\; style="background-color:\; white;">g</span>}}_{\mathrm{<span\; style="background-color:\; white;">t</span>}\mathrm{<span\; style="background-color:\; white;">t</span>}}<span\; style="background-color:\; white;">\sim </span>\frac{\mathrm{<span\; style="background-color:\; white;">G</span>}\mathrm{<span\; style="background-color:\; white;">M</span>}}{{\mathrm{<span\; style="background-color:\; white;">r</span>}}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}{\mathrm{<span\; style="background-color:\; white;">c</span>}}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}}<span\; style="background-color:\; white;">,</span>$$

which matches Newtonian gravity. This gradient:

• Curves geodesics (free-fall paths).

• Replaces "gravitons" with pure geometry.

3.2. Apparent Mass Without Matter

The black hole’s gravitational influence is entirely determined by the horizon’s time-stop radius:

$$\mathrm{<span\; style="background-color:\; white;">M</span>}<span\; style="background-color:\; white;">=</span>\frac{{\mathrm{<span\; style="background-color:\; white;">c</span>}}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}}{\mathrm{<span\; style="background-color:\; white;">2</span>}\mathrm{<span\; style="background-color:\; white;">G</span>}}{\mathrm{<span\; style="background-color:\; white;">r</span>}}_{\mathrm{<span\; style="background-color:\; white;">h</span>}}<span\; style="background-color:\; white;">,</span>$$

where $r_h=2GM\mathrm{/}{c}^2$. No "mass inside" is required—only the boundary condition matters.

---

4. Resolving Paradoxes

4.1. Information Paradox Solved

• No information enters the hole: All data is projected onto the horizon (Bekenstein-Hawking entropy $S=k_{B}A\mathrm{/}4{\mathrm{\ell }}_P^2$).

• Hawking radiation is emitted from the horizon’s quantum vacuum, not the interior. Unitarity is preserved.

4.2. No Firewall, No Singularity

• Firewall paradox vanishes: With no interior, there’s no contradiction at $r=2GM$.

• Singularity is replaced: The time-stop horizon is the fundamental object—no $r=0$ pathology exists.

4.3. Entropy Without Quantum Gravity

The horizon’s entropy arises from classical information capacity:

$$\mathrm{<span\; style="background-color:\; white;">S</span>}<span\; style="background-color:\; white;">\sim </span>\text{<span style="background-color: white;">Area</span>}\mathrm{<span\; style="background-color:\; white;">/</span>}{\mathrm{<span\; style="background-color:\; white;">\ell </span>}}_{\mathrm{<span\; style="background-color:\; white;">P</span>}}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}<span\; style="background-color:\; white;">,</span>$$

without invoking quantum mechanics.

---

5. Observational Consistency

5.1. Black Hole Shadows

The EHT image of M87* confirms the horizon size $r_h=2GM\mathrm{/}{c}^2$, matching the time-stop radius.

5.2. Gravitational Waves

LIGO detects quasinormal modes—vibrations of the horizon’s time-frozen surface—not "internal" dynamics.

5.3. Hawking Radiation

Thermal spectrum arises from horizon boundary effects, not interior thermodynamics.

---

6. Implications

1. Quantum gravity is unnecessary for black hole physics—GR + holography suffice.

2. Gravity is emergent from time-stop geometry, not particle exchange.

3. Cosmology: The Big Bang singularity may similarly resolve as a past time-stop.

---

7. Conclusion

A black hole is not an object with an interior but a radius where time stops, projecting its gravitational influence outward via spacetime curvature. This model:

• Resolves information/firewall paradoxes.

• Eliminates singularities.

• Unifies gravity, thermodynamics, and holography.

Future work: Extend to Kerr (rotating) black holes, where frame-dragging modifies the time-stop surface.

---

References

1. Bekenstein (1973) – Black hole entropy.

2. Hawking (1975) – Horizon particle creation.

3. Jacobson (1995) – Thermodynamics of spacetime.

4. EHT Collaboration (2019) – M87* shadow.