Historical research guide · 0058 · 1967
Vacuum quantum fluctuations in curved space and the theory of gravitation
Andrei D. Sakharov
In one minute
Andrei Sakharov’s three-page 1967 note proposes that gravity is not fundamental but induced by quantum fields, much as elasticity emerges from molecular physics. Start with curved spacetime, do not specify its dynamics, quantise the matter fields on it, and calculate the leading quantum correction. Sakharov finds a cosmological-constant term, an Einstein-Hilbert curvature term and curvature-squared terms in the effective action. On his interpretation, the vacuum’s response to curvature supplies the gravitational equations. He then explores a particular assumption set: zero tree-level constants, dominant one-loop physics and a Planck-scale cut-off. Under those assumptions Newton’s constant is tied to the cut-off and particle spectrum. The paper leaves the cosmological-term problem exposed: the same cut-off produces a vastly large contribution unless another effect offsets it.
Connection
This 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.
From the source
Gravity 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
Take 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
Sakharov’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
The 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
The 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’
The way in
https://link.springer.com/article/10.1023/A:1001947813563How to cite it
Andrei D. Sakharov (1967) Vacuum quantum fluctuations in curved space and the theory of gravitation doi:10.1023/A:1001947813563