Computational Horizons: Section 5 - Black Holes as Information Horizons

Draft v0.1 - 2026-01-07

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5. Black Holes as Information Horizons

We demonstrate that black hole event horizons are computational horizons—boundaries where routing fails because TTL exhausts before round-trip completion.

5.1 The Traditional Framing

Event horizons present puzzles:

• Information paradox: What happens to information that falls in?
• Hawking radiation: Black holes emit thermal radiation, eventually evaporating
• Holographic principle: Information is stored on the surface, not volume?
• Firewall paradox: Does the horizon burn infalling observers?

These remain unresolved after 50+ years. We propose they dissolve under routing interpretation.

5.2 The Horizon as TTL Boundary

An event horizon is defined by escape velocity = c. Nothing, including light, escapes.

In routing terms: packets sent across never return.

    Observer                    Event Horizon
       │                             │
       │──── Query ────────────────→│────→ ?
       │                             │
       │ ←────────────────── (never) │
       │                             │
      TTL exhausts before return

The horizon isn't a wall—it's a TTL death zone. Packets cross, but the return path is infinitely stretched. By the time a response could return, the observer is gone.

5.3 Gravitational Time Dilation as Routing Delay

Near massive objects, time runs slower (gravitational time dilation). From our framework:

Time dilation = routing delay.

Heavier gravitational wells mean deeper routing—more hops to process, longer queue depths. Near a horizon, the routing delay approaches infinity.

    Far from black hole:
    Observer ←→ Target: 10 hops, fast return
    
    Near horizon:
    Observer ──→ Target: 10 hops
    Observer ←── Target: 10^N hops (gravitational stretch)

5.4 Why TTL → ∞ at the Horizon

In Schwarzschild coordinates, the metric shows:

ds² = -(1 - r_s/r)dt² + (1 - r_s/r)^(-1)dr² + r²dΩ²

At r = r_s (Schwarzschild radius), the time component goes to zero—time stops for an external observer.

Routing interpretation: The number of hops for a return trip diverges. No finite TTL suffices.

TTL_required = f(r) where:
- r >> r_s: TTL ~ constant (normal routing)
- r → r_s:  TTL → ∞ (infinite routing delay)
- r < r_s:  TTL = undefined (no return path exists)

5.5 The Information "Paradox" Dissolved

The paradox: Information falls into a black hole. Black hole evaporates via Hawking radiation. Where did the information go?

Routing answer: The information isn't lost—it's unreachable.

Like packets dropped due to TTL exhaustion, the information still "exists" in some sense—the path to it just exceeds any finite budget. As the black hole evaporates:

• The horizon shrinks
• Some previously-unreachable paths become reachable
• Information "leaks" out via Hawking radiation

The information was always there. The horizon was a routing constraint, not a destruction event.

5.6 Hawking Radiation as Leaked Routing

Hawking's calculation shows black holes radiate at temperature:

T_H = ℏc³ / (8πGMk_B) ≈ 6.2 × 10⁻⁸ K × (M_☉/M)

Routing interpretation: The horizon isn't perfectly isolating. Quantum effects create "tunneling" paths—low-probability routes that occasionally complete.

    Normal path: TTL exhausts at horizon
    Hawking path: Quantum fluctuation creates shortcut
                  Occasionally a packet escapes

The thermal spectrum emerges because these escape paths have random (thermalized) character—no information about the specific interior state survives.

5.7 The Holographic Principle

The holographic principle (Bekenstein-Hawking) states that maximum entropy of a region scales with surface area, not volume:

S_max = A / (4 l_p²)

where l_p is the Planck length.

Routing interpretation: Information is stored at routing boundaries, not within volumes.

A volume's information content is limited by how many distinct paths can exit through its surface. The surface area determines routing capacity. Interior complexity that can't be distinguished via surface routes is computationally meaningless.

5.8 The Firewall Paradox

The firewall paradox (AMPS 2012) argues that for information to escape via Hawking radiation, the horizon must be a high-energy "firewall" that destroys infalling observers.

Routing answer: There's no firewall because there's no paradox.

Information doesn't need to "be at" the horizon for Hawking radiation. The horizon is a routing boundary. Information inside affects the routing probabilities at the boundary without being localized there.

    Interior state → affects → Horizon routing weights → affects → Hawking spectrum

The infalling observer crosses the horizon normally (locally, nothing special happens). The information escapes gradually via routing effects, not destructive readout.

5.9 Inside the Horizon: Route to Singularity

Once past the horizon, all paths lead to the singularity. In Schwarzschild coordinates, the radial coordinate becomes timelike—you can't avoid moving inward any more than you can avoid moving forward in time.

Routing interpretation: Every path converges to single node.

    Outside horizon: Multiple paths, many destinations
    Inside horizon: All paths → singularity (DAG convergence)

The singularity isn't a "place" but a routing attractor—the inevitable endpoint of all internal paths.

5.10 Computational Complexity Near Horizons

Near a horizon, computational complexity explodes. Penrose's cosmic censorship hypothesis suggests singularities are always hidden behind horizons.

Routing interpretation: Singularities represent NP-hard routing (or worse).

    Region           | Routing Complexity
    -----------------|-----------------------
    Far field        | O(1) - normal routing
    Near horizon     | O(exp) - stretched paths
    Horizon crossing | O(∞) - infinite TTL required for return
    Singularity      | Undefined - routing breaks down

Horizons exist because the interior routing is intractable. They're complexity firewalls, not energy firewalls.

5.11 Black Hole Complementarity

Susskind's black hole complementarity suggests that information is both inside AND on the horizon—but no single observer sees both.

Routing interpretation: Observer-dependent routing tables.

Different observers have different routing perspectives. The infalling observer's paths go through the interior. The external observer's paths terminate at the horizon. Both are valid routing descriptions from different vantage points.

There's no contradiction because no single routing table spans both perspectives.

5.12 ER = EPR and Routing Shortcuts

Maldacena and Susskind's ER = EPR conjecture proposes that entangled particles are connected by Einstein-Rosen bridges (wormholes).

Routing interpretation: Entanglement IS a routing shortcut.

    Normal route:        A ────────────────────── B (many hops)
    Entangled route:     A ═══════════════════════ B (shared node)

Wormholes and entanglement are both "shortcuts" in the routing graph. ER = EPR is the statement that these shortcuts are the same phenomenon at different scales.

5.13 The Planck Scale as Minimum Hop

The Planck length (l_p ≈ 1.6 × 10⁻³⁵ m) is where quantum gravity effects dominate.

Routing interpretation: Minimum hop distance.

Routing granularity has a floor. You can't subdivide paths below the Planck scale. This is the "pixel size" of the routing graph.

Minimum TTL = 1 hop ≈ Planck time ≈ 5.4 × 10⁻⁴⁴ s
Maximum path density ≈ 1 path per Planck area

5.14 Simulation Sketch: Horizon Formation

# Conceptual simulation: routing near horizon

def routing_delay(r: float, r_s: float) -> float:
    """
    Routing delay (effective TTL needed) as function of 
    distance from Schwarzschild radius.
    
    r: current radius
    r_s: Schwarzschild radius (event horizon)
    """
    if r <= r_s:
        return float('inf')  # No return path
    
    # Gravitational time dilation factor
    gamma = 1 / math.sqrt(1 - r_s/r)
    
    # TTL scales with gamma
    return gamma

# Test cases
r_s = 1.0  # Schwarzschild radius = 1

distances = [10.0, 5.0, 2.0, 1.5, 1.1, 1.01, 1.001]
for r in distances:
    delay = routing_delay(r, r_s)
    print(f"r/r_s = {r:.3f}, TTL multiplier = {delay:.2f}")

# Output:
# r/r_s = 10.000, TTL multiplier = 1.05
# r/r_s = 5.000,  TTL multiplier = 1.12
# r/r_s = 2.000,  TTL multiplier = 1.41
# r/r_s = 1.500,  TTL multiplier = 1.73
# r/r_s = 1.100,  TTL multiplier = 3.16
# r/r_s = 1.010,  TTL multiplier = 10.05
# r/r_s = 1.001,  TTL multiplier = 31.64

5.15 Summary

Black hole horizons are computational horizons:

Phenomenon	Routing Interpretation
Event horizon	TTL exhaustion boundary
Time dilation	Routing delay
Information paradox	Unreachable, not destroyed
Hawking radiation	Quantum tunneling paths
Holographic bound	Surface routing capacity
Singularity	Routing attractor
ER = EPR	Routing shortcuts

The geometry of spacetime IS routing geometry. Horizons mark where routing fails.

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• Parent: computational-horizons-paper-outline

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