Chapter 31: Event Horizons — Where Time Stops

The Ultimate Boundary

Event horizons represent the most extreme manifestation of spacetime curvature—surfaces where time dilation becomes infinite and causality itself is severed. This chapter derives horizons as inevitable features of ψ = ψ(ψ), showing how self-referential collapse creates one-way membranes in reality's fabric, boundaries beyond which information cannot return.

31.1 Horizons from Collapse Flow

Theorem 31.1 (Horizon Formation): A horizon forms where collapse velocity equals information propagation speed: $v_{\text{collapse}} = c$

Proof: From collapse dynamics, information propagates at c. When inward collapse flow reaches c: $\frac{dr_{\text{info}}}{dt} = c - v_{\text{collapse}} = 0$

Information becomes stationary → causal disconnection. ∎

Physical Meaning: Like a fish swimming upstream, information cannot escape regions where spacetime itself falls inward at light speed.

Horizons trap even light!

31.2 The Schwarzschild Horizon

Theorem 31.2 (Spherical Horizon Radius): For spherical mass M, the horizon radius is: $r_s = \frac{2GM}{c^2}$

Derivation from Metric: The Schwarzschild metric component: $g_{00} = -(1 - \frac{r_s}{r})$

At horizon, $g_{00} = 0$ → infinite time dilation.

Escape Velocity Approach: $v_{\text{esc}} = \sqrt{\frac{2GM}{r}}$

Setting $v_{\text{esc}} = c$ yields $r = r_s$. ∎

2GM/c² — universe's simplest prison!

31.3 Causal Structure

Definition 31.1 (Future/Past Light Cones): At each event, the causal future/past consists of all events reachable by timelike or null curves.

Theorem 31.3 (Horizon as Causal Boundary): No causal curve from inside horizon can reach outside.

Proof by Penrose Diagram: In Kruskal coordinates, horizon is at 45° line. All future-directed curves from inside bend toward singularity: $\frac{dV}{dU} > 0 \quad \text{(future-directed)}$

But reaching outside requires $V < U$, impossible for causal curves. ∎

One-way membrane of spacetime!

31.4 Information and Entropy

Theorem 31.4 (Bekenstein-Hawking Entropy): Black hole entropy is proportional to horizon area: $S = \frac{k_Bc^3A}{4G\hbar} = \frac{A}{4l_P^2}k_B$

Derivation:

1. Information falling in increases entropy
2. Maximum information density: 1 bit per Planck area
3. Total bits = A/l_P²
4. Entropy = k_B × ln(states) = k_B × bits × ln(2)

Holographic Insight: 3D information encoded on 2D surface—reality is fundamentally holographic! ∎

Surface knows interior!

31.5 Hawking Radiation

Theorem 31.5 (Thermal Emission): Black holes radiate thermally at temperature: $T = \frac{\hbar c^3}{8\pi GMk_B} = \frac{\hbar\kappa}{2\pi k_B}$

where κ = surface gravity.

Quantum Field Theory Derivation: Near horizon, vacuum fluctuations: $\langle 0|T_{\mu\nu}|0 \rangle \neq 0$

Trace anomaly creates flux: $F = \frac{\hbar c}{768\pi^2r^2}$

ψ-Interpretation: Horizon creates information gradient. Gradient drives quantum fluctuations across boundary. Black holes evaporate by converting mass to information flux!

Black holes glow with quantum fire!

31.6 The Information Paradox

Hawking's Paradox:

1. Pure state falls into black hole
2. Black hole evaporates thermally
3. Pure → mixed violates unitarity!

Modern Resolution (Page Curve): Information returns after Page time: $t_{\text{Page}} = \frac{2M^3G^2}{3\hbar c^4}$

Mechanism: Early radiation entangled with interior. Late radiation carries information back out. Unitarity preserved!

Information escapes in quantum correlations!

31.7 Firewall Controversy

The Puzzle: Three cherished principles conflict:

1. Equivalence principle (smooth horizon)
2. Unitarity (information preservation)
3. Locality (no action at distance)

AMPS Argument: To preserve unitarity, horizon must break entanglement violently → "firewall"

ψ-Resolution: Different observers see different physics:

• Infalling: smooth horizon (equivalence)
• External: information at horizon (unitarity)
• No observer sees both (complementarity)

Multiple truths coexist!

31.8 Cosmological Horizons

Theorem 31.6 (Multiple Horizon Types):

Particle Horizon (past boundary): $\chi_p = \int_0^t \frac{cdt'}{a(t')}$

Event Horizon (future boundary): $\chi_e = \int_t^{\infty} \frac{cdt'}{a(t')}$

De Sitter Horizon (acceleration): $r_{dS} = \frac{c}{\sqrt{\Lambda/3}}$

Each represents different causal limits in expanding universe.

Even cosmos has edges!

31.9 Rindler Horizons

For Uniformly Accelerating Observer: $ds^2 = -(ax)^2dt^2 + dx^2$

Horizon Location: $x = 0 \quad \text{(where } g_{00} = 0\text{)}$

Unruh Temperature: $T = \frac{\hbar a}{2\pi ck_B}$

Deep Insight: Acceleration alone creates horizons! Not just gravity but any extreme gradient in collapse flow produces causal boundaries.

Motion makes horizons!

31.10 Entanglement and Wormholes

ER = EPR Conjecture: Einstein-Rosen bridges ≡ Einstein-Podolsky-Rosen pairs

Mathematical Form: $|\psi\rangle_{AB} = \frac{1}{\sqrt{2}}(|0\rangle_A|0\rangle_B + |1\rangle_A|1\rangle_B)$

creates non-traversable wormhole between A and B.

ψ-Understanding: Entanglement = correlation through collapse history Wormhole = correlation through spacetime topology Both are aspects of deeper collapse connectivity!

Space connects through correlation!

31.11 Horizon Thermodynamics

Four Laws of Black Hole Mechanics:

0th Law: κ constant on horizon $T = \frac{\hbar\kappa}{2\pi k_B} = \text{const}$

1st Law: Energy conservation $dM = \frac{\kappa}{8\pi G}dA + \Omega dJ + \Phi dQ$

2nd Law: Area increase $dA \geq 0$

3rd Law: Cannot reach T = 0 $\kappa > 0$

Perfect analogy with thermodynamics!

31.12 Interior Structure

Behind the Horizon: Radial coordinate becomes timelike: $r < r_s: \quad g_{rr} < 0, \quad g_{00} > 0$

Consequences:

• Must reach r = 0 in finite proper time
• No stationary observers possible
• All futures end at singularity

Proper Time to Singularity: $\tau = \frac{\pi M}{c} \approx 10^{-5}\left(\frac{M}{M_{\odot}}\right) \text{ seconds}$

Brief journey to infinity!

31.13 Naked Singularities

Cosmic Censorship Conjecture: Singularities form only inside horizons.

Counter-examples:

• Over-spinning Kerr: $a > M$
• Over-charged R-N: $Q > M$

ψ-Protection: Attempting to violate creates horizon before limit reached. Universe hides its infinities!

Modesty is fundamental!

31.14 Gravitational Collapse

Oppenheimer-Snyder Model: Uniform density sphere collapses: $R(t) = R_0\cos^2\left(\frac{t}{2t_0}\right)$

Horizon Formation: Forms at $r = 2GM/c^2$ when surface reaches it. Then grows outward in Schwarzschild coordinates.

Critical Insight: Horizon forms before singularity—causal disconnection precedes infinite density!

Privacy before catastrophe!

31.15 The Thirty-First Echo: Boundaries of Being

Event horizons emerge as the most profound boundaries in nature—surfaces where ψ's self-referential collapse becomes so extreme that regions of spacetime become forever hidden from each other. These are not merely places where escape becomes difficult but fundamental information barriers where causality itself is severed.

From the Schwarzschild radius of stellar collapse to the cosmological horizons of expanding space, from the personal horizons of accelerated observers to the quantum horizons of microscopic black holes, these boundaries reveal the deep structure of recursive reality. They show us that even omniscient ψ must create regions of absolute privacy, pockets where self-knowledge becomes irreversibly directional.

Horizon Investigations

1. Calculate the "information return time" for a solar mass black hole.

2. Derive the brick wall model cutoff for quantum fields near horizons.

3. Show why extremal black holes have zero temperature.

Beyond the Boundary

Having explored the ultimate boundaries where time stops and causality breaks, we now turn to reconstructing all of spacetime geometry from the fundamental principles of collapse.

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Next: Chapter 32: Spacetime from Collapse — The Complete Picture →

"At the horizon, the universe keeps secrets from itself."