Mass That ClicksFinale · Technical Appendix ⑥ ／ The Einstein Equation from Finite Information (Jacobson 1995)

Your bet is realized in the literature ── but let's be honest, right down to "what is assumed"

The Einstein Equation
from Finite Information "The universe is finite information → gravity comes out of it" ── Ted Jacobson actually did this in 1995.
From area entropy + Unruh temperature + Clausius, the full Einstein equation comes out as an "equation of state."

Prerequisites: the finale (CKN/holography), Appendix ④ (induced gravity), Wall ③ Conclusion: derivable if you assume finite information ── but Wall ③ doesn't close

"You can derive the Einstein equation from a discrete/quantum substrate, right?" ── that intuition is the real thing, and it exists in the literature. Ted Jacobson (1995), starting from "horizon entropy ∝ area" = your finite information, derived the full Einstein equation as an "equation of state of spacetime." Not fake ── a known, beautiful result. Here we walk through it step by step ── and then draw an honest line around what is assumed, and what remains as Wall ③.

01The assumptions are just three (all from finite information = holography)

Starting point (imposed on every local Rindler horizon)

① Entropy ∝ area: \(\delta S=\eta\,\delta A\) (Bekenstein–Hawking = finite information)
② Unruh temperature: \(T=\dfrac{\hbar\kappa}{2\pi}\) (the horizon temperature seen by an observer with acceleration \(\kappa\))
③ Clausius relation: \(\delta Q=T\,\delta S\) (local thermodynamics)

02Derivation ── impose it on every local horizon, and Einstein comes out

Heat flux crossing the horizon (energy flux of matter). \(\chi^a\approx-\kappa\lambda k^a\), where \(k^a\) is the generator and \(\lambda\) the affine parameter
$$\delta Q=\int T_{ab}\chi^a d\Sigma^b\approx-\kappa\!\int\!\lambda\,T_{ab}k^ak^b\,d\lambda\,dA$$
Area change (generators focus via Raychaudhuri): \(d\theta/d\lambda\approx-R_{ab}k^ak^b\Rightarrow\theta\approx-\lambda R_{ab}k^ak^b\)
$$\delta A=\int\theta\,d\lambda\,dA=-\!\int\!\lambda\,R_{ab}k^ak^b\,d\lambda\,dA$$
Substitute into Clausius \(\delta Q=T\eta\,\delta A\). \(\kappa\) and the integral cancel, and demand it hold for every null \(k^a\)
$$T_{ab}k^ak^b=\frac{\hbar\eta}{2\pi}R_{ab}k^ak^b\quad(\forall\,k^a\ \text{null})$$
"Holds for all null" ⟹ a tensor equality (with an ambiguity \(\Phi\) from the \(g_{ab}k^ak^b=0\) piece)
$$T_{ab}=\frac{\hbar\eta}{2\pi}R_{ab}+\Phi\,g_{ab}$$
Local conservation \(\nabla^aT_{ab}=0\) + the Bianchi identity fix \(\Phi\): \(\Phi=-\frac{\hbar\eta}{2\pi}\frac{R}{2}+\Lambda\)

Putting in \(\eta=\dfrac{1}{4G\hbar}\) (= \(S=A/4G\hbar\)) gives \(\dfrac{2\pi}{\hbar\eta}=8\pi G\).

What came out ── the full Einstein equation
$$R_{ab}-\tfrac12 R\,g_{ab}+\Lambda\,g_{ab}=8\pi G\,T_{ab}$$

\(G\) comes naturally from the entropy–area coefficient (\(S=A/4G\)), and \(\Lambda\) as an integration constant.

Figure: a local Rindler horizon. When the matter energy flux \(\delta Q\) (amber) crosses the horizon, the generators (blue) focus and the area \(A\) (green) decreases. Increase the matter flux with the slider and the focusing gets stronger ── imposing Clausius \(\delta Q=T\delta S\) in all directions makes the curvature proportional to \(T_{ab}\) (= Einstein).
Move the matter flux and the horizon's focusing (= curvature) changes.
horizon generators (null) matter energy flux δQ area A (decreases with focusing)

03This is your bet itself

Input ① "horizon entropy ∝ area" = holography = finite information. So Jacobson's result is ── "the Einstein equation is the equation of state of the finite information living on the horizon." Your bet that "the universe is finite information → gravity comes out" is a genuine viewpoint that exists in the literature, as Jacobson (1995), Verlinde (2011), and the entanglement version (Van Raamsdonk, Faulkner et al.: the linearized Einstein equation from the first law of entanglement).

Analogy with the equation of state of a gas A gas's \(PV=nRT\) comes out of thermodynamics alone, without knowing the details of the atoms ── an "equation of state." Jacobson's claim is that the Einstein equation is the same ── it comes out of the horizon's thermodynamics (finite information) alone, without knowing the microscopic details of spacetime, as an "equation of state." Gravity may not be a fundamental force but a manifestation of the thermodynamics of information.
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04The honest line ── so did Wall ③ close? No

Here is the line of honesty. Look at what Jacobson assumed ── area entropy, Unruh temperature, Clausius. These are thermodynamic, holographic structures, and they were not derived from a quantum theory of a discrete substrate. So Wall ③ hasn't closed. The problem has just moved:

The problem moves

"Derive the Einstein equation" \(\longrightarrow\) "Why is the horizon entropy \(A/4G\) = which microscopic degrees of freedom are being counted?"

This derivation of the microscopic entropy = counting black-hole states is exactly the wall that remains (string theory achieves it for special BHs, loop quantum gravity has area quantization, causal sets partially ── none of them complete). The real job of a discrete substrate is to derive this area entropy ── and that is the unsolved frontier a chat can't crack.

The honest line

This derivation is the real thing (Jacobson 1995, a known result), not something I invented. And it shows that "if you assume finite information, Einstein comes out," not that "it comes out from a raw discrete substrate with zero assumptions." If anyone (AI included) hands you the latter as "finished" in a conversation, that is the fake to watch out for.

Also, this derivation relies on an "equilibrium" assumption, and there is discussion about higher-curvature = non-equilibrium corrections (Eling–Guedens–Jacobson), and about whether it's a derivation or a consistency argument. The entanglement version is the linearized Einstein equation; the fully nonlinear case depends on a holographic setup. The local speed of light is invariant; \(c\cdot t\) is a restatement of coordinates.

Questions to check
  1. In Jacobson's derivation, where does \(G\) come from?
    One answer
    From the entropy–area coefficient. Setting \(S=A/(4G\hbar)\) (\(\eta=1/4G\hbar\)) gives \(2\pi/\hbar\eta=8\pi G\), so the coefficient \(8\pi G\) on the right-hand side of the Einstein equation comes out. In other words, \(G\) appears as the reciprocal of "how many bits of information the horizon holds per unit area."
  2. Why doesn't this close Wall ③ (deriving GR from a discrete substrate)?
    One answer
    Because it assumes area entropy, Unruh temperature, and Clausius to derive Einstein, without deriving those thermodynamic, holographic structures from a quantum theory of a discrete substrate. The problem moves to "why does the horizon have entropy \(A/4G\) = which microscopic degrees of freedom are counted?" (state counting), and that remains unsolved.

Appendix ⑥ summaryEinstein comes out of finite information ── the literature's realization of your bet

Jacobson (1995), by imposing the horizon's area entropy (= finite information) + Unruh temperature + Clausius on every local Rindler horizon, derived the full Einstein equation \(R_{ab}-\frac12Rg_{ab}+\Lambda g_{ab}=8\pi G\,T_{ab}\) as an "equation of state." \(G\) comes from the area-entropy coefficient, and \(\Lambda\) as an integration constant. This is your bet of "finite information → gravity" itself ── a genuine viewpoint alongside Verlinde and the entanglement version.

But Wall ③ doesn't close ── it assumes area entropy, and its microscopic origin (state counting) remains. Derived (from finite information) / but not achieved from an assumption-free discrete substrate (Wall ③). Your intuition is correctly backed up, and you can point precisely at the one remaining piece ── a genuine "I tried it" you can do without producing a fake.

The most honest answer to "you can derive it, right?" I tried ── and it really can be derived. Grant finite information (the horizon's area entropy), and the Einstein equation comes out fully via Jacobson's procedure. Your bet existed in the literature, as a 1995 paper.
But that is not "solving a Millennium problem in a chat." Assuming finite information gives the form of GR, and the microscopic origin of that information (= counting horizon states = Wall ③) stays open. If I said "without even assuming area entropy, Einstein came out from a raw discrete substrate," that would be a fake ── so I won't say it.
Your intuition was winning. And you can see the remaining wall precisely. Holding a genuine derivation, with eyes that can tell a fake, and standing before the open door ── that is the most honest and proudest place this long journey arrives at.
This document is Appendix ⑥ of the "Mass That Clicks" series finale, a piece of reading for physics-loving high schoolers and undergraduates. Jacobson (1995) "Thermodynamics of Spacetime: The Einstein Equation of State" (that imposing entropy ∝ area, the Unruh temperature, and the Clausius relation \(\delta Q=T\delta S\) on local Rindler horizons derives the Einstein field equations via the Raychaudhuri equation and the Bianchi identity, with \(G\) appearing as the entropy–area coefficient and \(\Lambda\) as an integration constant) is an established result. Verlinde's entropic gravity (2011) and the derivation of the linearized Einstein equation from the first law of entanglement (Lashkari–Van Raamsdonk, Faulkner et al., Jacobson's 2016 entanglement equilibrium) are also real. These derive GR by assuming area-entropy / entanglement structure, and its microscopic origin (counting black-hole states) is unsolved (string theory's special supersymmetric BHs, loop quantum gravity's area quantization, etc. are partial). The equilibrium assumption, non-equilibrium corrections at higher curvature, and whether it is a derivation or a consistency argument are matters of discussion. This piece asserts no specific finished theory; it presents a known result with its assumptions and limits. \(c\cdot t=\text{const}\) is a restatement of coordinates and units; the local speed of light is invariant. ── Print / PDF: use your browser's "Print" and "Save as PDF" (in the print version the sliders and answers are frozen and hidden).

Print / save as PDF: Ctrl+P (⌘+P on Mac). On screen, change the matter flux with the slider and the horizon's focusing (= curvature = the right-hand side of Einstein) changes. "One answer" reveals the solution.