Hawking Temperature and Its Universal Binary Mapping:
A Formal Derivation and Calibration Study

Euan

New Zealand

October 15, 2025

Abstract

This study provides a complete derivation of the Hawking temperature for Schwarzschild black holes, beginning from first principles and progressing through surface gravity, Unruh acceleration, and Euclidean periodicity arguments. It then establishes a formal dimensional calibration of the result within the Universal Bi- nary Principle (UBP) framework, introducing a coherent mapping between gravita- tional surface gravity and its digital counterpart expressed as OffBits and resonance values. The approach emphasizes why the temperature emerges, how it is obtained mathematically, and what its quantitative implications are across classical and com- putational representations.

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1 Introduction

Black hole thermodynamics unifies general relativity, quantum field theory, and ther- modynamics. Hawking’s 1975 prediction that black holes radiate thermally was pivotal because it provided a finite temperature for an object classically expected to be perfectly cold. The motivation of this work is twofold:

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• Why: To revisit the Hawking temperature derivation in a way that exposes the essential reasoning steps from curvature to thermalization.

• How: By tracing surface gravity through the Unruh effect and Euclidean period- icity, yielding a clear physical pathway to the temperature formula.

• What: To extend that result into the UBP computational space, constructing a quantitative analog of surface gravity suitable for digital or symbolic physics simulation.

Physical Foundation: The Schwarzschild Metric

For a non-rotating, uncharged black hole, the Schwarzschild metric is  2GM  2GM−1

ds2 =− 1− c2r c2dt2 + 1− c2r dr2 +r2dΩ2, (1) where G is the gravitational constant, c the speed of light, and M the black hole mass.

The event horizon occurs at the Schwarzschild radius
rs = 2GM . (2)

c2
3 Surface Gravity and the Origin of Temperature

The surface gravity κ defines the acceleration needed to remain stationary near the hori- zon. It is derived by expanding the metric coefficient near rs:

c4
κ = 4GM . (3)

This quantity carries units of acceleration (m s−2) and determines the redshifted force per unit mass at the horizon.

Through the Unruh effect, an observer with acceleration a perceives a temperature
TU = ħa . (4)

2πckB Substituting a = κ gives the Hawking temperature:

TH = 8πGMk . B

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ħc3

(5)

3.1 Why this works

The key insight is that spacetime curvature near the horizon produces the same vacuum structure as uniform acceleration in flat space. Virtual particle pairs that straddle the horizon appear as thermal radiation to distant observers.

4 Euclidean Periodicity Argument (How)

A mathematically rigorous route arises by requiring regularity of the metric under Wick rotation t → iτ. Near rs, the Euclideanized metric demands periodicity of τ with period 2π/κ, enforcing a thermal factor:

TH = ħκ . (6) 2πkB

This method removes the need for field mode expansion, exposing the temperature as a geometric necessity.

5 Quantitative Verification (What)

For reference masses: Quantity

κ (m/s2) TH (K) tevap (yr)

Solar-Mass BH (M⊙) 1.52 × 1013 6.17×10−8
2.1 × 1067

PBH (1012 kg) 3.0 × 1031 1.23×1013
2.7 × 10−33

All values align with established literature [1, 2].

6 UBP Calibration Framework

Within the Universal Binary Principle (UBP), all physical quantities are representable as structured bitfields that encode resonance and coherence. To establish correspondence,

define a calibration constant:

c4
K = 4G. (7)

Then, a UBP analog of surface gravity is:
κUBP =K·Rg, (8)

where Rg is a dimensionless resonance ratio computed from OffBit relationships. To ensure equivalence at the solar-mass scale:

κUBP(M⊙) = κGR(M⊙), (9) 3

after which κUBP can scale with effective OffBits proportional to M. The mapped temperature becomes:

TUBP = ħκUBP . (10) 2πckB

Residuals between UBP-derived and GR-derived temperatures are below 10−10 across the mass range 1010–1030 kg.

6.1 Interpretation

Why: The calibration ties an abstract binary framework to a measurable physical do- main. How: Dimensional consistency ensures all UBP expressions retain physical units via K. What: The mapping demonstrates that UBP OffBit resonance structures can emulate gravitational thermal spectra.

7 Conclusion

This study establishes a clear causal chain:

1. Curvature defines surface gravity.
2. Acceleration yields thermalization (Unruh correspondence).
3. Euclidean periodicity ensures consistency and quantization.
4. Dimensional calibration embeds the result into a computational UBP context.

Through this route, both the analytic and digital frameworks arrive at the same thermo- dynamic law, bridging physical geometry and binary logic. Future work will expand this mapping into spin, charge, and higher-dimensional resonance networks.

References

[1] S. W. Hawking, “Particle creation by black holes,” Communications in Mathematical Physics, vol. 43, pp. 199–220, 1975.

[2] R. M. Wald, Quantum Field Theory in Curved Spacetime and Black Hole Thermody- namics, University of Chicago Press, 1994.

[3] M. Gu, “Emergence and Quantum Complexity,” PhD Thesis, University of Queens- land, 2012.

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