CALCIFER: tracing the inputs to a horizon-temperature calculation

Bee Rosa Davis

Research note · Version 0.3 · Updated 8 October 2026

CALCIFER organizes horizon calculations through a “curvature cost” built from metric components. The interesting question is which parts of the temperature formula follow from geometry and which physical identifications enter along the way.

TL;DR

The question: Can one metric-built quantity organize the calculation of horizon temperatures across black holes, cosmology, accelerated observers and laboratory analogs—and which physical inputs does it still require?

What the work does: A derivation manuscript covering five spacetimes and three analog media, a numerical test suite checking formula routes, and CALCIFER, a Rust-with-Python tool that locates analog horizons and computes temperature.

What is still open: A dependency table separating geometric steps from supplied inputs (ℏ, entropy normalization, first law), and a case where the procedure predicts something standard calculations do not.

THE RESEARCH IN CONTEXT

In 1974 Hawking showed that black holes should glow faintly, at a temperature set by their surface gravity—roughly, how strongly the horizon pulls. Related results assign temperatures to cosmological horizons and to observers who accelerate uniformly. The temperature can be obtained several ways: from quantum fields near the horizon, or from requiring a smooth geometry when time is made imaginary. Because astrophysical Hawking radiation is far too faint to observe, laboratories build analog horizons in flowing condensates, water channels and optical fibres, where waves cannot travel upstream, and try to measure the corresponding temperature.

The manuscript applies one extraction procedure—take the field’s divergence structure at the horizon, obtain surface gravity, then temperature—to Schwarzschild, Kerr, Reissner–Nordström, de Sitter and Rindler spacetimes. It extends the field to analog media, where an effective version diverges where flow speed equals wave speed. CALCIFER implements this for experiments: it accepts density and velocity, channel depth or refractive-index profiles, locates analog horizons by root-finding, and predicts a temperature. The repository holds a Rust core with Python bindings; the manuscript reports 210 tests across eight files, and an earlier README describes a 29-test suite.

The tests largely check internal consistency. Computed temperatures match textbook formulas, Kerr and charged solutions reduce to Schwarzschild at zero spin or charge, and two routes to the Schwarzschild temperature agree from one to a million solar masses. One test confirms that only a one-quarter entropy coefficient reproduces Hawking’s temperature—the coefficient is selected by that requirement. The manuscript’s ‘What has not been shown’ section states that the work does not derive Planck’s constant, and that it applies only to static or stationary horizons.

AVAILABLE RECORD

Local derivation manuscript and consistency tests; different source versions make different normalization claims.

TECHNICAL FIGURE

source-informed technical atlas

Source: thermal-geometry v2. Euclidean regularity and the Schwarzschild expression are mathematical constructions; the temperature identification retains ℏ explicitly.

Metric inputs and the temperature formula

Manuscript equation table

Spacetime / chart | Lapse α | Radial spatial coefficient | K = √γ / α | Surface gravity κ

Schwarzschild exterior | √(1 − rs/r) | γrr = (1 − rs/r)⁻¹ | (1 − rs/r)⁻¹ | c⁴ / (4GM)

De Sitter static patch | √(1 − r²/RH²) | γrr = (1 − r²/RH²)⁻¹ | (1 − r²/RH²)⁻¹ | c² / RH

Rindler wedge, α > 0 | 1 + gz/c² | γzz = 1 | (1 + gz/c²)⁻¹ | g

All three temperature expressions use T = ℏκ / (2πckB). The Rindler row follows the manuscript’s explicit flat-spatial metric, not its overbroad earlier statement that f = h for every nonrotating case. K and c/K depend on coordinates; locally measured light speed remains c. This table organizes assumptions, not a quantum-free derivation.

thermal_geometry_paper_v2.tex — Unification table and explicit Rindler metric

A coordinate-dependent field and a physical temperature

The reviewed manuscript defines K = √γrr / α using the spatial metric coefficient and ADM lapse. It studies expressions for this field, surface gravity, and temperature in Schwarzschild, Kerr, Reissner–Nordström, de Sitter, and Rindler settings. K depends on the chosen foliation; the temperature claim requires an appropriately specified surface gravity and normalization.

The proposed chain links the regularity period of a Euclidean horizon to T = ℏκ/(2πckB). The geometric period and the thermodynamic interpretation of that period are distinct steps. Planck’s constant still enters the temperature relation.

Where the quarter factor enters

An earlier README attributes the entropy coefficient 1/4 to four causal domains. The later website account instead says it is fixed by consistency with the first law, rather than independently derived. This change matters: choosing the entropy coefficient to reproduce a known temperature is a consistency calculation, not an independent prediction of that coefficient.

The local tests compare formula routes across masses and spacetime examples. Agreement is useful for catching algebra or implementation mistakes. If the routes share the same thermodynamic input, their agreement does not establish that the input has been derived twice independently.

THE NEXT RESEARCH STEP

The representative paper needs a dependency diagram for each route: metric and normalization, regularity condition, quantum scale, entropy assumption, and first-law identification. That would expose precisely what the framework adds beyond a re-expression of the familiar result.

TEACHING CONSTRUCTION (not project measurements)

Write the derivation as a directed graph from premises to intermediate expressions to the target relation. Track the definitions of surface gravity, Euclidean periodicity, entropy normalization, and thermodynamic inputs separately. Preserve the unit convention throughout.

premises → periodicity / normalization → target relation

Premise: A supplied assumption or previously established result.

Consistency check: Agreement after the stated inputs are supplied.

Independent prediction: A consequence not already fixed by the premises being tested.

If a calculation assumes y=2x and independently assumes x=3, obtaining y=6 checks substitution. It does not derive the factor 2. This elementary dependency example is the logic behind asking where a temperature coefficient or entropy factor entered.

SOURCE RECORD

local-project/calcifer/README.md — local source reviewed; not republished

local-project/calcifer/thermal_geometry_paper_v2.tex — local source reviewed; not republished

Source versions: https://www.davisgeometric.com/research/account-source-records.json

No new experiments are reported in this research note.

Illustrated study: https://www.davisgeometric.com/research/figures/draft-horizon-thermodynamics.html