Vacuum quantum fluctuations in curved space and the theory of gravitation
Andrei D. Sakharov
Summary and citation · read the original at the source
In one page
Andrei Sakharov’s note runs to three pages and four formulas, and it made one of the most consequential suggestions in twentieth-century physics: that gravity is not a fundamental force at all. It emerges from quantum field theory, he argued, in roughly the way hydrodynamics or the elasticity of a solid emerges from molecular physics — gravity as an elasticity of the spacetime medium. The mechanism is simple to state. Take a curved spacetime, assume nothing whatever about how its geometry moves, quantise the matter fields living on it, and work out the leading quantum correction. What comes back already contains a cosmological-constant term, an Einstein-Hilbert term proportional to the curvature, and curvature-squared terms. The equations of gravity fall out of the vacuum’s response to curvature, uninvited. Sakharov’s own reading was to set every tree-level constant to zero, let that leading correction dominate, and put a cut-off at the Planck energy — which yields Newton’s constant as a formula in the cut-off and the particle spectrum.
Why it matters hereThis is the paper chapter 3 leans on when it says gravity is induced rather than fundamental, and it is the reason chapter 5 can treat the vacuum as a medium with material properties: if Sakharov is right, the stiffness of spacetime is a property of the field filling it, and therefore a quantity in principle open to engineering.
What it claims
01Gravity is not fundamental in the sense particle physics uses the word. General relativity emerges from quantum field theory in roughly the same sense that hydrodynamics or continuum elasticity theory emerges from molecular physics — Sakharov’s own picture is of gravity as an elasticity of the spacetime medium.Sakharov 1967, the whole three-page note; restated in Visser, arXiv gr-qc/0204062, Section 1
What to watch02Take a Lorentzian manifold, make no assumption about the dynamics of its geometry, quantise everything except gravity itself, and the leading quantum correction to the action automatically contains three terms: a constant, a term proportional to the curvature scalar — the Einstein-Hilbert action — and curvature-squared terms.Sakharov’s central observation; Visser, Eqs. 1 and 2
Settled physics03Sakharov’s interpretation is a specific set of assumptions: set all tree-level constants to zero, assume the one-loop physics dominates, take the dimensionless numbers to be of order one, and impose an explicit cut-off at the Planck scale. Newton’s constant is then induced, with its reciprocal proportional to the square of the cut-off multiplied by a supertrace over the particle spectrum.Sakharov 1967; Visser, Section 4, ‘one-loop dominance’, Eq. 29
What to watch04The ambition of the programme is to relate the observed value of Newton’s constant, and of the cosmological constant, to the spectrum of particle masses — which would make the strength of gravity a consequence of what matter exists rather than an independent input.Sakharov 1967; Barceló, Liberati and Visser, ‘Analogue Gravity’, Section 7.10
What to watch05The same mechanism induces a cosmological term scaling as the fourth power of the cut-off, so without some counterbalancing contribution induced gravity predicts an enormous cosmological constant. That mismatch is the open question the note leaves on the table, and the condensed-matter analogues suggest where to look: in a liquid-like system at equilibrium the internal pressure is zero, which forces the corresponding term to be small automatically.Barceló, Liberati and Visser, ‘Analogue Gravity’, Section 7.9, ‘The cosmological constant problem’
What to watch
The way in
https://link.springer.com/article/10.1023/A:1001947813563The 1967 Doklady note, its 1968 English translation and the 2000 General Relativity and Gravitation reprint are all held by their publishers, so this page carries a summary. The original bibliographic record is free at mathnet.ru (dan33444), and Matt Visser’s open-access restatement of the argument, arXiv gr-qc/0204062, is the clearest way in.
How to cite it
Andrei D. Sakharov (1967) Vacuum quantum fluctuations in curved space and the theory of gravitation. doi:10.1023/A:1001947813563
Where it sits in the curriculum
Inertia and gravity from the vacuumWhat the vacuum isThe vacuum as a quantum fluid