Historical research guide · 0808 · 1995
Thermodynamics of Spacetime: The Einstein Equation of State
Ted Jacobson
In one minute
Ted Jacobson asks whether classical general relativity can be derived from the thermodynamic behavior of horizons rather than assumed. Take entropy proportional to horizon area and the relation delta-Q equals T dS. Demand that relation at every spacetime point for every local Rindler causal horizon, treating heat as energy crossing the horizon and T as the Unruh temperature of an accelerated observer. Jacobson shows the requirement fixes spacetime curvature in the Einstein-equation form. Newton’s constant is set by the entropy density per area, while the cosmological constant remains an integration constant. His equation-of-state language compares gravity to the gas law: a macroscopic relation that need not be the underlying quantum degrees of freedom. The derivation assumes local equilibrium, leaving non-equilibrium spacetime as the next question.
Connection
Chapter 3 argues that gravity is induced rather than fundamental, and this is the cleanest derivation of that position anyone has written: four pages, two thermodynamic inputs, Einstein’s equation out the other end, with Newton’s constant fixed by the horizon entropy density rather than put in by hand. It belongs to chapter 2 just as strongly, because the entropy and the temperature that drive the derivation are both properties of vacuum fluctuations — the field near the horizon is doing the work.
From the source
The Einstein equation can be derived from the proportionality of entropy and horizon area together with the fundamental relation connecting heat, entropy and temperature, demanded for all the local Rindler causal horizons through each spacetime point, with the heat read as the energy flux and the temperature as the Unruh temperature seen by an accelerated observer just inside the horizon.Abstract; derivation ending at Equation 6
Heat, in spacetime dynamics, is energy that flows across a causal horizon — it can be felt through the gravitational field it generates, but its particular form is unobservable from outside, so the system is separated from the outside world not by a diathermic wall but by a causality barrier.Paragraph beginning ‘In thermodynamics, heat is energy that flows…’
The entropy assigned to a horizon is entanglement entropy: the overwhelming majority of the hidden information sits in correlations between vacuum fluctuations just inside and just outside the horizon, and it is finite and proportional to horizon area only if there is a fundamental cutoff length, which consistency with thermodynamics requires to be of order the Planck length, ten to the minus thirty-three centimetres.Paragraphs beginning ‘That causal horizons should be associated with entropy…’ and ‘As we will see, consistency with thermodynamics…’
The constant of proportionality between entropy and horizon area determines Newton’s constant, so the strength of gravity is set by the horizon entropy density rather than being an independent input; the cosmological constant enters the derivation as an undetermined integration constant.Paragraph following Equation 6
Because the Einstein equation is born in the thermodynamic limit as a relation between thermodynamic variables, it may be no more appropriate to canonically quantise it than to quantise the wave equation for sound in air, even though the underlying degrees of freedom are quantum mechanical.Abstract; paragraph beginning ‘Given local equilibrium conditions…’
The derivation holds only under local equilibrium, and sufficiently high-frequency or large-amplitude disturbances of the gravitational field would fall outside the Einstein equation — not because the metric acquires an operator nature but because local equilibrium fails; Jacobson names an understanding of non-equilibrium spacetime as the goal of this line of inquiry.Final two paragraphs
The way in
https://doi.org/10.1103/PhysRevLett.75.1260How to cite it
Ted Jacobson (1995) Thermodynamics of Spacetime: The Einstein Equation of State doi:10.1103/PhysRevLett.75.1260