A Unified Thermodynamic Identity

A Unified Thermodynamic Identity Bridging Energy Absorption, Radiation, and Gravitational Field Coupling

We propose the following covariant, universal identity that links a system’s absorbed energy, thermal radiation, and entropy-field interaction:

𝐸 (1 − α) = (π² kᴮ⁴)/(60 ℏ³ c²) · T⁴ − (c⁸)/(G² ℏ) · ∇Φ

Each term represents a physically grounded concept and maintains full dimensional consistency. This identity unifies Stefan–Boltzmann radiation, Tolman gravitational thermodynamics, Hawking radiation, and Landauer’s information entropy bound.

Term Definitions and Physical Interpretation
• 𝐸 (1 − α) — Absorbed Energy Flux:
𝐸 is incident energy per unit area per unit time (e.g., W·m⁻²); α is the reflectivity (albedo). Hence (1 − α) is absorptivity. For a perfect blackbody, α = 0 → 𝐸(1 − α) = 𝐸.
• (π² kᴮ⁴)/(60 ℏ³ c²) · T⁴ — Stefan–Boltzmann Radiative Term:
This is the thermal radiation emitted by a blackbody at temperature T. The prefactor is the Stefan–Boltzmann constant expressed in fundamental constants:
σ = (π² kᴮ⁴)/(60 ℏ³ c²)
• (c⁸)/(G² ℏ) · ∇Φ — Entropy–Field Coupling:
∇Φ is the gradient of the gravitational potential (or any acceleration field). The prefactor
γ = (c⁸)/(G² ℏ)
ensures dimensional consistency and scales with the Planck power per Planck area. This term encodes the impact of gravity (or acceleration) on thermal entropy flow.

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Physical Implications

This identity is consistent with:
• Stefan–Boltzmann Law:
When ∇Φ = 0 (no gravitational gradient), the identity reduces to:
𝐸 (1 − α) = (π² kᴮ⁴)/(60 ℏ³ c²) · T⁴
• Tolman–Ehrenfest Effect:
At thermal equilibrium in a gravitational field:
T(r)·√g₀₀(r) = const,
derived from the balance:
(π² kᴮ⁴)/(60 ℏ³ c²) · T⁴ = (c⁸)/(G² ℏ) · ∇Φ
• Hawking Radiation:
For a black hole with surface gravity gₕ:
T_H = (ℏ c³)/(8π G kᴮ M)
arises naturally from the balance between radiated energy and gravitational field term.
• Landauer’s Limit:
Information erasure cost (minimum energy dissipation):
E_min = kᴮ T ln 2
is consistent when the system dissipates energy thermally, with gravity modifying the effective dissipation threshold.
• Unruh Effect:
An observer undergoing proper acceleration a perceives a temperature:
T_U = (ℏ a)/(2π kᴮ c),
compatible with:
(π² kᴮ⁴)/(60 ℏ³ c²) · T⁴ = (c⁸)/(G² ℏ) · a

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Conclusion

We propose the identity:

𝐸 (1 − α) = (π² kᴮ⁴)/(60 ℏ³ c²) · T⁴ − (c⁸)/(G² ℏ) · ∇Φ

as a unifying thermodynamic law that governs systems from laboratory-scale heat transfer to black hole evaporation. It bridges general relativity, quantum field theory, thermodynamics, and information theory in a compact, invariant form.

This formulation provides a powerful tool for modeling entropy dynamics in curved spacetime, enabling new approaches to energy harvesting, gravitational propulsion, and entropy-based clocks.