>[!warning] >This content has not been peer reviewed. # Einstein's equations from thermodynamics **Jacobson (1995):** If one assumes that (1) each local Rindler horizon has an entropy proportional to its area, and (2) the Clausius relation $\delta Q = T \, dS$ holds for matter flux across the horizon, then **Einstein’s field equations** follow as an equation of state (Jacobson 1995). So gravity is not only analogous to thermodynamics; it can be **derived** from thermodynamic principles applied to horizons. This is established work in semiclassical gravity. --- ## What it is - **Setup:** A local acceleration horizon (Rindler) has temperature $T = \hbar a/(2\pi c k_B)$ (Unruh 1976) and entropy $S = A/(4 l_P^2)$ (Bekenstein 1973; Hawking 1975). - **Result:** $\delta Q = T \, dS$ plus the assumption that $S \propto A$ yields $G_{\mu\nu} + \Lambda g_{\mu\nu} = (8\pi G/c^4) T_{\mu\nu}$ (Jacobson 1995). So **Einstein’s equations = thermodynamics of horizons**. - **Implication:** Gravity can be seen as an **emergent** phenomenon — the macroscopic law that holds when the underlying “substrate” (vacuum, quantum fields) satisfies thermodynamic relations at horizons. That supports the idea of a **relational substrate** whose statistics (entropy, temperature) give rise to geometric law. --- ## How RRT/RST uses it - **RST as substrate:** **[[Relational Substrate Theory (RST)]]** posits a **[[Format]]** (A1) — a finite, noisy, relational medium. If that format has horizon-like structure (boundaries, causal patches) and obeys an **[[Energy Floor]]** (Landauer, dissipation), then the **macroscopic** dynamics can be thermodynamic in the Jacobson sense. So RST does not contradict Jacobson; it **generalises** the “underlying system” from “quantum fields + horizon” to “relational substrate + format.” The resulting gravity sector (e.g. MOND-like in the weak field) is then the equation of state of that substrate. - **Hard knowledge:** Jacobson is the **established** result that “gravity = thermodynamics of horizons.” RST uses it to argue that any theory whose substrate has entropy and temperature at effective horizons will have a geometric limit; the **form** of that limit (Einstein, MOND, or interpolating) depends on the substrate’s scaling (e.g. $n$, $a_0$). --- ## Links | Concept | Note | |:---|:---| | Full RST framework | **[[Relational Substrate Theory (RST)]]** | | Cost per bit, dissipation | **[[Energy Floor]]** | | Substrate, capacity | **[[Format]]** | --- ## References - Unruh, W. G. (1976). *Notes on black-hole evaporation.* Phys. Rev. D **14**, 870–892. [DOI](https://doi.org/10.1103/PhysRevD.14.870) (Unruh temperature $T = \hbar a/(2\pi c k_B)$.) - Bekenstein, J. D. (1973). *Black holes and entropy.* Phys. Rev. D **7**, 2333–2346. [DOI](https://doi.org/10.1103/PhysRevD.7.2333) - Hawking, S. W. (1975). *Particle creation by black holes.* Commun. Math. Phys. **43**, 199–220. [DOI](https://doi.org/10.1007/BF02345020) - Jacobson, T. (1995). *Thermodynamics of spacetime: The Einstein equation of state.* Phys. Rev. Lett. **75**, 1260–1263. [DOI](https://doi.org/10.1103/PhysRevLett.75.1260) [arXiv:gr-qc/9504004](https://arxiv.org/abs/gr-qc/9504004)