Time & Gravity · T.08 · Where a causal boundary turns a pure vacuum into a thermal bath · Step 9 of 13 · 14 min read
A Horizon Has a Temperature
On T.07 a choice of clock split a stationary quantum state into an observer and a system. When acceleration or gravity makes that split geometric, a causal horizon hides half the vacuum modes from any clock that stays outside. Tracing out the hidden partner turns the pure vacuum into a Planck thermal state at the Unruh or Hawking temperature, and integrating the first law of black-hole mechanics fixes the horizon entropy at one quarter of its area in Planck units.
Human LineagePioneers of Step T.08 · Penzias & Wilson (1965) · Jacob Bekenstein (1973) · Stephen Hawking (1975) · Joseph Bisognano & Eyvind Wichmann (1975) · William Unruh (1976) · John Mather & COBE FIRAS (1996) · Jeff Steinhauer (2016)Honor the humans ↓
Fixed boundaryNo theory of everything here: six evidence labels, one open question per page.
Nothing on this page is a theory of everything, and this site proposes none. Every claim carries one of six labels, each tied to a ledger row: MEASURED, DERIVED, BOUND, PROPOSED, OPEN, or SINGLE RESULT. Preprints never rise above PROPOSED. Each page ends with one open question that the next page takes up.
Plate T.08-V · Left: the Rindler causal horizon X = ±cT and the accelerated worldline hyperbola, with one mode partner outside and one hidden behind the horizon. Right: the reduced state's Fock occupation bars matching the Planck thermal curve. Drawn by timegravity/make_plates.py from tools/timegravity_kernel.py.About this illustrationThis plate is a deterministic drawing, not an AI-generated image and not a photograph. The curve is the kernel's own trajectory; the shading exaggerates a clock-rate difference of about one part in 10^16 per metre so that it can be seen at all.
Instrument · Tracing out the hidden wedge turns a pure vacuum into a Planck bath
Left: the Rindler spacetime wedge (cT against X), with the causal horizon X = ±cT, the accelerated clock's hyperbola X² − c²T² = (c²/a)², and the paired mode partners straddling the horizon. Right: the occupation probabilities pₙ of the reduced state ρ_R = Tr_L(|Ψ⟩⟨Ψ|) on a 16-level Fock space against the Planck curve (1 − e^{−ξ})e^{−nξ}, alongside the Hawking temperature T_H and Bekenstein–Hawking entropy S_BH / k_B = A / (4 ℓ_P²) for the selected black-hole mass. The sliders set the Boltzmann exponent ξ = βħω and the black-hole mass M / M_☉. The knockout severs entanglement across the horizon (setting the squeezing to zero, |0_L⟩ ⊗ |0_R⟩): the thermal tail vanishes, S(ρ_R) drops to zero, and the verifier rejects the run. Keyboard: Tab and arrows. With reduced motion on, nothing animates.
Predict before you revealBefore you check the readout: how does the Hawking temperature of a one-solar-mass black hole compare with the 2.725 K cosmic microwave background?
Horizon entropy S_BH / k_B = A / (4 ℓ_P²)1.049e+77
Entangled across the horizon: tracing out the hidden left partner L turns the pure vacuum into a Planck thermal bath on the right wedge R.
I · Two measured anchors: redshifted clocks and a Planck sky
Clocks at different heights or accelerations tick at different rates, with fractional shift Δf/f = g Δz / c²: the 22.5-metre tower measurement of 1960 and the 10,000-kilometre suborbital hydrogen-maser flight of 1980 confirmed Einstein's gravitational redshift to 7×10⁻⁵.And a thermal radiation bath follows Planck's Bose–Einstein occupation law n̄(ω) = 1/(e^{ħω/k_B T} − 1): discovered as an isotropic microwave background in 1965 and measured by COBE FIRAS in 1994 to match a 2.726 K Planck blackbody within 0.03 percent of peak brightness. What follows on this page is DERIVED from these two pieces of physics—proper-time redshift and quantum field theory—pushed to a causal horizon.
In Plain Words · Extra-Easy Explanation
Imagine two dancers holding hands across a curtain. Together they are completely still and orderly: if you see both, there is zero randomness in their pose. Now draw the curtain so you are forbidden, by the speed of light itself, from ever seeing the dancer on the left. Watching only the dancer on the right, every shared motion looks like random thermal jiggling.
A horizon is that curtain. A quantum vacuum is not empty; its field modes are entangled across every boundary in space. When a rocket accelerates forever, or when a black hole forms, a horizon hides the inside partner of every pair. To any clock outside, the half it can still see is a warm bath with a precise Planck temperature set by the steepness of the horizon.
Everyday Analogy & Controls Primer
Everyday analogy: a stereo recording where the left channel cancels the hiss of the right channel when heard together. Unplug the left speaker and you hear pure thermal static from the right speaker alone, even though the master tape holds a clean, noiseless signal.
How the controls work in plain words: the first slider changes ξ = ħω / (k_B T), the ratio of mode energy to horizon temperature; smaller ξ means a hotter horizon and a longer thermal tail across the Fock bars. The second slider sets a black hole's mass in solar masses and updates its Hawking temperature and area entropy. The knockout unplugs the cross-horizon entanglement: both halves sit in their separate ground states, the thermal bars collapse into n = 0, and the entropy drops to zero.
Plate T.08-D · Visual Intuition Bridge: tracing over the hidden Rindler wedge turns a pure two-mode state into an exact Gibbs thermal state (left and center), and integrating the first law recovers the quarter-area entropy (right).
Archival Tribute · The Hands Behind the Instruments
The Humans Behind Step T.08 · Causal Horizons, Rindler Wedges, and the Cosmic Microwave Sky
Linking the steepening clock slope at a horizon to an exact thermal bath brought together quantum-field theorists studying accelerated observers and radio astronomers measuring the 2.725 K cosmic microwave background that bathes every astrophysical black hole.
1973–1976 · Princeton, Cambridge, Berkeley & Vancouver
Jacob D. Bekenstein · Stephen W. Hawking · William G. Unruh · Joseph Bisognano & Eyvind Wichmann
Princeton University, University of Cambridge, UC Berkeley & University of British Columbia
Jacob Bekenstein proposed in 1973 that a black hole carries entropy proportional to its horizon area; in 1975 Stephen Hawking fixed the proportionality at 1/4 by showing that a horizon radiates thermally at T_H = ħκ/(2πck_B). Within months Joseph Bisognano and Eyvind Wichmann proved the modular boost theorem on a Rindler wedge, and in 1976 William Unruh showed that an accelerated detector in flat vacuum sees T_U = ħa/(2πck_B).
1965 & 1996 · Holmdel, New Jersey & NASA Goddard
Arno A. Penzias & Robert W. Wilson · John C. Mather & the COBE FIRAS Team
Bell Labs Crawford Hill Horn Antenna & Cosmic Background Explorer
In 1965 Arno Penzias and Robert Wilson detected the isotropic microwave background at 4,080 MHz with the Bell Labs horn reflector at Crawford Hill, and in 1996 John Mather and the COBE FIRAS team measured its Planck spectrum at 2.728 ± 0.004 K—nearly forty-four million times warmer than the 61.7 nK Hawking temperature of a solar-mass black hole.
2016 · Haifa, Israel
Jeff Steinhauer
Technion – Israel Institute of Technology · Acoustic BEC Horizon
Because astrophysical Hawking radiation is buried beneath the 2.725 K sky, Jeff Steinhauer swept a laser step potential through a rubidium-87 Bose–Einstein condensate at the Technion in 2016 to create a supersonic acoustic horizon, observing density correlations between partner phonons across the sonic boundary—an analogue kinematics test without gravitational dynamics.
II · From Rindler wedges to Hawking's quarter
T_U = ħa / (2π k_B c), T_H = ħc³ / (8π G M k_B), S_BH = k_B c³ A / (4 G ħ) = k_B A / (4 ℓ_P²)
A clock with uniform proper acceleration a follows the hyperbola X² − c²T² = (c²/a)², bounded by a causal horizon at X = ±cT where its proper-time rate dτ = (aξ/c²) dt vanishes; Bisognano and Wichmann proved in 1975 that the Minkowski vacuum restricted to the right Rindler wedge is a thermal state with respect to Lorentz boosts, and Unruh showed in 1976 that an accelerated detector responds as if immersed in a Planck bath at T_U = ħa / (2π k_B c). At 1g the Unruh temperature is 4.0×10⁻²⁰ K; reaching 1 K requires 2.5×10²⁰ m/s². On a 16-level Fock space per wedge at ξ = 2.5, the verifier embeds the partial trace ρ_R = Tr_L(|Ψ⟩⟨Ψ|) into a dense symmetric matrix, diagonalises it with the track's pure-Python Jacobi solver, and matches the Planck weights (1 − e^{−ξ})e^{−nξ} within 1.7×10⁻¹⁵.
Near a Schwarzschild horizon of radius r_s = 2GM/c² and area A = 4πr_s², the redshifted surface gravity κ = c⁴/(4GM) plays the role of proper acceleration: Hawking showed in 1975 that quantum fields on that geometry radiate thermally at T_H = ħκ / (2π k_B c) = ħc³ / (8π G M k_B), which combined with the 1973 Bardeen–Carter–Hawking first law d(Mc²) = (κc² / 8πG) dA fixes Bekenstein's horizon entropy coefficient at S_BH = k_B A / (4 ℓ_P²). Integrating d(S/k_B) = c² dM / (k_B T_H(M)) by Gauss–Legendre quadrature from 0 to 1 M_☉ reproduces A / (4 ℓ_P²) = 1.049×10⁷⁷ to 2.5×10⁻¹⁶, with T_H = 61.7 nK.
In 2019 Muñoz de Nova, Golubkov, Kolobov and Steinhauer reported a thermal spectrum of spontaneous phonon pairs emitted across a sonic horizon in a rubidium Bose–Einstein condensate, at a temperature matching the acoustic surface gravity. This is a fluid-mechanics analogue, not a gravitational horizon: it tests Bogoliubov mode mixing across a sonic flow boundary, not Einstein's field equations or Bekenstein–Hawking area entropy, and stays SINGLE RESULT.
§ II.1 · The derivation the verifier runs, in four stages
A clock at constant proper acceleration a follows X(τ) = (c²/a) cosh(aτ/c), cT(τ) = (c²/a) sinh(aτ/c). Across 81 rapidity steps the kernel checks X² − c²T² = (c²/a)² and four-velocity normalisation to < 1e-15.
Stage 2 · Two-mode squeezed state & partial trace
|Ψ(ξ)⟩ = Z_d(ξ)^{−1/2} Σ_{n=0}^{15} e^{−nξ/2} |n_L⟩ ⊗ |n_R⟩ on 16×16 = 256 Fock states. Tracing out L gives ρ_R, which is rotated by an orthogonal DCT-II matrix into a dense 16×16 symmetric matrix and diagonalised with cyclic Jacobi.
Stage 3 · Planck check & first-law quadrature
At ξ = 2.5 (where truncation leakage e^{−16ξ} = 4.25e-18), Jacobi eigenvalues match (1 − e^{−ξ})e^{−nξ} within 1.7e-15; across ξ ∈ [0.5, 4.0] they match the exact truncated Gibbs weights within 4.1e-15. Five-point Gauss–Legendre quadrature of ∫ c² dM / T_H(M) matches k_B A / (4 ℓ_P²) within 2.5e-16.
Stage 4 · Planted defect knockout
Setting cross-horizon squeezing to zero replaces |Ψ(ξ)⟩ with the product state |0_L⟩ ⊗ |0_R⟩: S(ρ_R) collapses to 0 and the Planck spectrum check fails with deviation 0.082, rejecting the run.
III · What this page does not say
Neither Hawking radiation from a gravitational black hole nor the thermal Unruh bath on a macroscopic accelerated thermometer has been directly measured: at one solar mass T_H ≈ 61.7 nK sits 4.4×10⁷ times below the 2.725 K cosmic microwave background, and what microstates of spacetime geometry are counted by A / (4 ℓ_P²) remains an open question. While the radiation itself is far too cold to see in the sky, the classical horizon geometry and area law behind T_H and S_BH can be tested directly when two black holes collide and ring down—the subject of the next page.
Grounding ledger rows cited on this page
Every row below resolves in the site's offline Crossref cache. The lint derives what each set of rows can carry; a person decided the label on each claim above. Rows pass a review date and the build fails until someone rereads the source.
Scite column: [observed: Scite ML citation classification, accessed 2026-11-08, 2026-11-09, 2026-11-10] via Scite's public tallies endpoint. The counts are a classifier's output over indexed citation statements, an evidence-synthesis habit borrowed from medicine; they never replace reading the source, and a row with more contrasting than supporting statements is flagged for a human re-read in the label checklist.
AI contribution to the cited results: none. No AI system contributed to any measurement or derivation cited on this page. The page text, instrument and verifier were drafted with AI assistance and checked by the site maintainer against the primary sources listed below.
Rebuild this instrument locally and run the verifier as the acceptance gate: python skills/timegravity/t08/make_t08.py then python skills/timegravity/t08/verify_t08_horizon_thermodynamics.py (and --planted-defect, which must fail). Then python tools/check_timegravity_labels.py.
Show the SKILL.md source
---
name: timegravity-t08-horizon-thermodynamics
description: Rebuild and verify the Time & Gravity T.08 toy - a two-mode Rindler thermofield double on 16 Fock levels whose partial trace over the left wedge yields a Planck thermal density matrix on the right wedge (verified via a pure-Python Jacobi eigensolver), alongside Rindler proper-time redshift and Gauss-Legendre integration of dS = c^2 dM / T_H(M) = k_B c^3 A / (4 G hbar) - with a planted defect (r = 0 product state with zero cross-horizon entanglement).
license: MIT
compatibility: Claude Code, GitHub Copilot, Cursor, Google Antigravity, OpenAI Codex, Ollama (Qwen 3.8 / Gemma 4)
---
# A Horizon Has a Temperature (T.08 · Time & Gravity)
> **Fixed Boundary (`EN`):** Nothing on this page is a theory of everything, and this site proposes none. Every claim carries one of six labels, each tied to a ledger row: MEASURED, DERIVED, BOUND, PROPOSED, OPEN, or SINGLE RESULT. Preprints never rise above PROPOSED. Each page ends with one open question that the next page takes up.
>
> **Frontera Fija (`ES-MX`):** Nada en esta página es una teoría del todo, y este sitio no propone ninguna. Cada afirmación lleva una de seis etiquetas, cada una ligada a una fila del libro de registro: MEDIDO, DERIVADO, COTA, PROPUESTO, ABIERTO o RESULTADO ÚNICO. Las prepublicaciones nunca suben de PROPUESTO. Cada página termina con una pregunta abierta que la siguiente página retoma.
## 1. Page, canonical URL and governing equations
- **Track page**: `T.08` (Time & Gravity, reading order 9 of 11, dated `2026-11-10`)
- **Canonical URL**: https://informatico.ai/timegravity/t08/ (Spanish: https://informatico.ai/es-mx/timegravity/t08/)
- **Rindler wedge & redshift**: `X² − c²T² = ξ² = (c²/a)²`, `dτ = (aξ/c²) dt`.
- **Unruh & Hawking temperatures**: `T_U = ħa / (2π k_B c)`, `κ = c⁴ / (4GM)`, `T_H = ħκ / (2π k_B c) = ħc³ / (8π G M k_B)`.
- **Thermofield double & Bekenstein-Hawking entropy**: `|0_M⟩ = Z_d^{-1/2} Σ_{n=0}^{d-1} e^{-nξ/2} |n⟩_L ⊗ |n⟩_R`, `ρ_R = Tr_L |0_M⟩⟨0_M|`, `S_BH = k_B c³ A / (4Għ)`.
- **Central numbers**: at `ξ = βħω = 2.5` (`d = 16` Fock levels), Jacobi eigenvalues of `ρ_R` match the infinite-dimensional Planck weights `(1 − e^{-ξ})e^{-nξ}` within `1.67e-15` (`< 1e-12`) and match the exact truncated Gibbs distribution across `ξ ∈ [0.5, 4.0]` within `4.11e-15`; Gauss–Legendre integration of `∫ c² dM / T_H(M)` matches `k_B A / (4 l_P²)` within `2.45e-16`; for `1 M_☉`, `T_H = 6.170 × 10^{-8} K` and `S_BH / k_B = 1.049 × 10^{77}`.
- **Generator**: `python skills/timegravity/t08/make_t08.py`
- **Verifier**: `python skills/timegravity/t08/verify_t08_horizon_thermodynamics.py`
- **Knockout check**: `python skills/timegravity/t08/verify_t08_horizon_thermodynamics.py --planted-defect` (must exit non-zero: `r = 0` product vacuum `|0⟩_L ⊗ |0⟩_R`, `S(ρ_R) = 0`)
- **Label lint**: `python tools/check_timegravity_labels.py` (and `--planted-defect`, `--mutants`)
## 2. Ledger rows this page may cite (timegravity/ledger.json)
| Row | Label | Source | DOI |
| --- | --- | --- | --- |
| `T01-PR1960` | MEASURED | Pound & Rebka 1960, gravitational redshift | https://doi.org/10.1103/PhysRevLett.4.337 |
| `T01-GPA1980` | MEASURED | Vessot et al. 1980, Gravity Probe A | https://doi.org/10.1103/PhysRevLett.45.2081 |
| `T08-PENZIAS1965` | MEASURED | Penzias & Wilson 1965, cosmic microwave background | https://doi.org/10.1086/148307 |
| `T08-MATHER1994` | MEASURED | Mather et al. 1994, COBE FIRAS Planck blackbody spectrum | https://doi.org/10.1086/173574 |
| `T08-BCH1973` | DERIVED | Bardeen, Carter & Hawking 1973, four laws of black hole mechanics | https://doi.org/10.1007/BF01645742 |
| `T10-BEKENSTEIN1973` | DERIVED | Bekenstein 1973, black-hole entropy proportional to horizon area | https://doi.org/10.1103/PhysRevD.7.2333 |
| `T10-HAWKING1975` | DERIVED | Hawking 1975, particle creation by black holes (`T_H`, `A/4`) | https://doi.org/10.1007/BF02345020 |
| `T08-BW1975` | DERIVED | Bisognano & Wichmann 1975, modular boost generator and KMS state | https://doi.org/10.1063/1.522605 |
| `T08-UNRUH1976` | DERIVED | Unruh 1976, accelerated detector thermal response (`T_U`) | https://doi.org/10.1103/PhysRevD.14.870 |
| `T08-MUNOZ2019` | SINGLE RESULT | Muñoz de Nova et al. 2019, acoustic BEC sonic horizon (analogue only) | https://doi.org/10.1038/s41586-019-1241-0 |
## 3. Tool-neutral agent packet and security boundary (`security-runtime`)
1. **Rebuild locally**: `python skills/timegravity/t08/make_t08.py`; standard library only; writes only under `src/research/timegravity/t08/files/`.
2. **Acceptance gate**: `python skills/timegravity/t08/verify_t08_horizon_thermodynamics.py` must print `"status": "PASS"` with `partial_trace_recovers_planck: PASS`; `--planted-defect` must exit non-zero. Then `python tools/check_timegravity_labels.py`.
3. **Security boundary (`security-runtime`)**: no shell outside the repository, no home-directory access, no outbound network for this task.
## 4. Deterministic verification and falsification rules
1. At `ξ = 2.5` (`d = 16` Fock levels), the Jacobi eigenvalues of `ρ_R = Tr_L |0_M⟩⟨0_M|` match `(1 − e^{-ξ})e^{-nξ}` within `1e-12`, and match the exact truncated Gibbs distribution across `ξ ∈ [0.5, 4.0]` within `1e-12`.
2. Every Rindler worldline point satisfies `X² − c²T² = (c²/a)²` and `dτ/dt = aξ/c²` within `1e-12`.
3. 16-point Gauss–Legendre integration of `∫ (c² / T_H(M')) dM'` matches `k_B c³ A / (4Għ)` and `dM c² = (κ c² / 8πG) dA` within `1e-12`.
4. Planted defect: setting the squeezing parameter `r = 0` yields the product state `|0⟩_L ⊗ |0⟩_R` with `S(ρ_R) = 0`; rule 1 fails and `--planted-defect` exits non-zero.
5. `t08_reference.json` matches `manifest.json` and regenerates byte for byte.
## 5. What the instrument shows (reader side)
- Left panel: Rindler spacetime diagram (`cT` vs `X`) showing the null horizons `X = ±cT`, accelerated hyperbolas, and behind-the-horizon mode partner `L`. Right panel: Fock-level occupation bar chart of `ρ_R` against the Planck thermal curve, with readouts for `T_U`, `T_H(M)`, and `S_BH / k_B`.
- Knockout: sever cross-horizon entanglement (`r = 0`, product vacuum `|0⟩_L ⊗ |0⟩_R`); the thermal tail collapses to `n = 0` and `S(ρ_R)` drops to `0`.
- Predict before you reveal: whether a solar-mass black hole is hotter or colder than the `2.725 K` cosmic microwave background.
## 6. Exact T0–T3 truth boundary
### In the repository today (T0/T1)
- Kernel `tools/timegravity_kernel_horizon.py`, generator, verifier, ledger rows above, and the ES module `src/research/modules/timegravity-slope.mjs`.
### Not on this page (T2/T3)
- Direct detection of astrophysical Hawking radiation (`61.7 nK` for `1 M_☉` is eight orders of magnitude colder than the `2.725 K` CMB) or the full Unruh effect for a macroscopic thermometer is OPEN.
- Acoustic BEC horizons (`T08-MUNOZ2019`) test linear dispersion on a fluid background, not Einstein's field equations or Bekenstein–Hawking entropy.
Show the verifier verify_t08_horizon_thermodynamics.py
#!/usr/bin/env python3
"""Standalone stdlib verifier and planted-defect knockout for Time & Gravity · T.08 (A Horizon Has a Temperature).
Canonical URL: https://informatico.ai/timegravity/t08/
Central number: tracing the behind-the-horizon Rindler partner L out of the pure Minkowski two-mode squeezed state
produces a reduced density matrix rho_R on the right wedge whose Jacobi eigenvalues match the Planck thermal Gibbs
weights (1 - exp(-xi)) exp(-n xi) within 1e-12 at xi = 2.5 (d = 16 Fock levels) and match the exact truncated Gibbs
distribution across xi in [0.5, 4.0] to machine precision; integrating dS = c^2 dM / T_H(M) recovers A / (4 l_P^2)
within 1e-12. Planted defect: a product state |0>_L x |0>_R with zero cross-horizon entanglement (r = 0) gives pure
zero entropy and fails the thermal spectrum check; `--planted-defect` exits non-zero with FAIL on stderr.
Boundary: Nothing on this page is a theory of everything, and this site proposes none.
"""
from __future__ import annotations
import hashlib
import json
import sys
from pathlib import Path
ROOT = Path(__file__).resolve().parents[3]
sys.path.insert(0, str(ROOT / "tools"))
from timegravity_kernel_horizon import HORIZON_TOL, check_horizon # noqa: E402
FILES_DIR = ROOT / "src" / "research" / "timegravity" / "t08" / "files"
def verify_reference_files() -> dict:
manifest = json.loads((FILES_DIR / "manifest.json").read_text(encoding="utf-8"))
out = {}
for entry in manifest["files"]:
raw = (FILES_DIR / entry["filename"]).read_bytes()
assert len(raw) == entry["bytes"], f"{entry['filename']}: byte count mismatch"
digest = hashlib.sha256(raw).hexdigest()
assert digest == entry["sha256"], f"{entry['filename']}: sha256 mismatch"
out[entry["filename"]] = digest[:12]
ref = json.loads((FILES_DIR / "t08_reference.json").read_text(encoding="utf-8"))
assert ref["max_dev_from_planck"] < HORIZON_TOL, "reference Rindler trace misses Planck weights"
assert ref["hawking_bekenstein"]["quadrature_vs_area_quarter_rel_err"] < HORIZON_TOL, "reference first-law integral misses A/(4 l_P^2)"
return out
def main(argv: list[str] | None = None) -> int:
args = argv if argv is not None else sys.argv[1:]
planted = "--planted-defect" in args
try:
res = check_horizon(planted_defect=planted)
assert res["planck_ok"], (
f"reduced state rho_R misses Planck thermal weights by {res['max_dev_from_planck']:.3e} (> {HORIZON_TOL:g}); "
"without cross-horizon entanglement the wedge is not thermal"
)
assert res["worldline_ok"], f"Rindler redshift / hyperbola invariant error {res['max_hyperbola_invariant_err']:.3e}"
assert res["first_law_ok"], f"Hawking-Bekenstein first-law quadrature error {res['quadrature_vs_area_quarter_rel_err']:.3e}"
assert res["defect_rejected"], "planted zero-entanglement product state was accepted as thermal"
files = verify_reference_files()
except AssertionError as exc:
print(json.dumps({"status": "FAIL", "folio_id": "T.08", "error": str(exc)}), file=sys.stderr)
return 1
print(json.dumps({
"status": "PASS",
"folio_id": "T.08",
"canonical_url": "https://informatico.ai/timegravity/t08/",
"central_number": {
"max_dev_from_planck": res["max_dev_from_planck"],
"entropy_dev_from_planck": res["entropy_dev_from_planck"],
"quadrature_vs_area_quarter_rel_err": res["quadrature_vs_area_quarter_rel_err"],
"solar_hawking_temp_k": res["solar_hawking_temp_k"],
"solar_s_bh_over_kb": res["solar_s_bh_over_kb"],
"tolerance": HORIZON_TOL,
},
"geometry_checks": {
"partial_trace_recovers_planck": "PASS",
"truncated_gibbs_sweep": "PASS",
"rindler_redshift_hyperbola": "PASS",
"first_law_area_quarter_quadrature": "PASS",
"planted_defect_knockout": "REJECTED_AS_EXPECTED",
"planted_entropy_nats": res["planted_entropy_nats"],
},
"reference_files_sha256_12": files,
}, indent=2))
return 0
if __name__ == "__main__":
raise SystemExit(main())
Vessot, R. F. C., et al. (1980). Test of relativistic gravitation with a space-borne hydrogen maser. Physical Review Letters, 45(26), 2081–2084. DOI DOI 10.1103/PhysRevLett.45.2081[Scite ↗]
Penzias, A. A., & Wilson, R. W. (1965). A measurement of excess antenna temperature at 4080 Mc/s. The Astrophysical Journal, 142, 419–421. DOI DOI 10.1086/148307[Scite ↗]
Mather, J. C., et al. (1994). Measurement of the cosmic microwave background spectrum by the COBE FIRAS instrument. The Astrophysical Journal, 420, 439–444. DOI DOI 10.1086/173574[Scite ↗]
Bardeen, J. M., Carter, B., & Hawking, S. W. (1973). The four laws of black hole mechanics. Communications in Mathematical Physics, 31(2), 161–170. DOI DOI 10.1007/BF01645742[Scite ↗]
Hawking, S. W. (1975). Particle creation by black holes. Communications in Mathematical Physics, 43(3), 199–220. DOI DOI 10.1007/BF02345020[Scite ↗]
Bisognano, J. J., & Wichmann, E. H. (1975). On the duality condition for a Hermitian scalar field. Journal of Mathematical Physics, 16(4), 985–1007. DOI DOI 10.1063/1.522605[Scite ↗]
Muñoz de Nova, J. R., Golubkov, K., Kolobov, V. I., & Steinhauer, J. (2019). Observation of thermal Hawking radiation and its temperature in an analogue black hole. Nature, 569, 688–691. DOI DOI 10.1038/s41586-019-1241-0[Scite ↗]
This page ends with one open question
Does a real horizon have the hairless geometry that this thermodynamic law assumes?
