Thermodynamics of Spacetime: The Einstein Equation of State
Ted Jacobson
Abstract and summary · read the original at the source
In one page
Ted Jacobson turns the usual argument upside down. Black holes were already known to obey laws that look exactly like thermodynamics — horizon area behaves like entropy, surface gravity like temperature — and that was derived from Einstein’s equation. Jacobson asks how classical general relativity could possibly have known, and answers by running the derivation backwards. Take two ingredients: entropy proportional to horizon area, and the plain thermodynamic relation that heat equals temperature times the change in entropy. Now demand that it hold not only for black holes but for the horizon of every accelerating observer, at every point in spacetime, with heat read as the energy crossing that horizon and temperature as the Unruh temperature an accelerated observer measures in the vacuum. Insisting on it everywhere forces spacetime to bend in exactly the way Einstein’s equation prescribes. So gravity is an equation of state, like the gas law — and quantising it may make as little sense as quantising the equation for sound in air.
Why it matters hereChapter 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.
What it claims
01The 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
Published and peer-reviewed02Heat, 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…’
Published and peer-reviewed03The 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…’
Published and peer-reviewed04The 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
Published and peer-reviewed05Because 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…’
What to watch06The 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
What to watch
Read it · abstract
Abstract
The Einstein equation is derived from the proportionality of entropy and horizon area together with the fundamental relation delta-Q equals T dS connecting heat, entropy, and temperature. The key idea is to demand that this relation hold for all the local Rindler causal horizons through each spacetime point, with delta-Q and T interpreted as the energy flux and Unruh temperature seen by an accelerated observer just inside the horizon. This requires that gravitational lensing by matter energy distorts the causal structure of spacetime in just such a way that the Einstein equation holds. Viewed in this way, the Einstein equation is an equation of state. This perspective suggests that it may be no more appropriate to canonically quantize the Einstein equation than it would be to quantize the wave equation for sound in air.
The way in
https://doi.org/10.1103/PhysRevLett.75.1260Published as Physical Review Letters 75, 1260 (1995) by Ted Jacobson of the University of Maryland, preprint UMDGR-95-114. The manuscript is free to read on arXiv as gr-qc/9504004, but that posting carries arXiv’s assumed licence for legacy submissions rather than a Creative Commons licence, and the journal version is under the APS default licence, so this page holds the summary, the claims and the author’s own abstract and sends the reader to the source.
How to cite it
Ted Jacobson (1995) Thermodynamics of Spacetime: The Einstein Equation of State. doi:10.1103/PhysRevLett.75.1260
Where it sits in the curriculum
Inertia and gravity from the vacuumWhat the vacuum isThe vacuum as a quantum fluidThe unified picture