Historical research guide · 1066 · 2021
Modeling inertia through the interaction with quantum fluctuations
Jaume Giné · Giuseppe Gaetano Luciano
In one minute
Jaume Giné and Giuseppe Gaetano Luciano take up the question Newton left open: where does inertia actually come from? Their model starts from two assumptions — that any body can be pictured as a collection of resonant parts each about a Planck length across, and that inertia is what happens when those parts interact with the quantum fluctuations boiling around them. They then reach the same answer twice, by two different routes. The first applies Heisenberg’s uncertainty principle directly: a fluctuation carrying more energy can travel less far, so only the nearby ones reach the body, and adding up the energy they deliver to each Planck-sized part reproduces the body’s mass exactly. The second revisits Mike McCulloch’s quantised inertia: accelerate, and a Rindler horizon appears behind you, shading the fluctuations arriving from that side, so the unshaded pressure from ahead pushes back in proportion to the acceleration. Both routes give the same inertial mass, agreeing up to a numerical factor of about 1.2 which they fix by requiring the two to match.
Connection
Chapter 3 is built on the proposition that inertia is the vacuum’s reaction rather than a property matter simply has, and this paper is one of the cleanest recent statements of it: two independent derivations, one from the uncertainty principle and one from a horizon-shaded Casimir imbalance, landing on the same mass. It also gives chapter 2 something concrete — an explicit reason why the vacuum energy density has to be counted in Planck-sized cells rather than in a continuum.
From the source
The model rests on two stated assumptions, and the authors are explicit that they are assumptions. First, any body — elementary or composite — can be conceived as a collection of resonant parts of Planck size, generalising the corpuscular picture of black holes to ordinary matter. Second, inertia is the result of those components interacting with vacuum fluctuations, which at the Planck level become strong enough to produce macroscopic effects in the same way the Casimir force does. Equation 4 in the source counts the parts: the number of resonant parts is the body’s Schwarzschild radius divided by the Planck length, which equals twice the mass divided by the Planck mass.Introduction, and Inertia from Heisenberg Uncertainty Principle, equations 4 and 5
The first derivation gets inertia straight out of the uncertainty principle. A vacuum fluctuation of energy delta-E can propagate only a distance of about the reduced Planck constant times the speed of light divided by twice that energy, so the more energetic the fluctuation the shorter its reach and only fluctuations born nearby arrive at the body. Differentiating that energy with respect to distance gives the force one fluctuation exerts on one resonant part, and integrating it from the Planck length out to an effective cut-off radius gives the energy transferred to each part as the reduced Planck constant times the speed of light divided by twice the Planck length. Multiplying that by the number of resonant parts and dividing by the speed of light squared returns exactly the mass of the body.Inertia from Heisenberg Uncertainty Principle, equations 1 to 3 and equation 7
The second derivation is a horizon effect. A body accelerating to the right has a Rindler horizon at a distance of the speed of light squared divided by the acceleration on its left, because information from beyond it can never catch up; fluctuations arriving from that side are therefore damped while those from the right are not. Working out the difference in radiation pressure over all angles gives a net force, equation 17 in the source, equal to pi squared minus four, over ninety-six, times the radiation energy density, times the peak Unruh wavelength, times the intercepted area, times the acceleration, divided by the speed of light squared — always opposing the acceleration. The mass of each resonant part comes out as about six tenths of the Planck mass divided by a factor k.Inertia from Rindler-scale Casimir effect, equations 12 to 21
The two routes agree, and that agreement is the paper’s main result. Requiring the Rindler-horizon mass to match the uncertainty-principle mass fixes the free factor at k of about 1.2, which in turn says the wavelength of the fluctuation spectrum doing most of the work is about 1.2 times the peak Unruh wavelength — consistent, the authors note, with their own argument that it should be the longer of the two. They add that the model is conceptually similar to the Higgs mechanism, in that mass stops being a primary quality of matter and becomes the outcome of an interaction with a field.Inertia from Rindler-scale Casimir effect, equation 21, and Conclusions and outlook
Two corrections to the earlier quantised-inertia literature are recorded here. The angular integration in McCulloch’s and in Giné and McCulloch’s papers is taken over the azimuthal angle across a range that yields an incorrect numerical factor, which this paper repairs. More substantially, the authors argue that the volume in the energy-density calculation is not the volume of the particle, as assumed in that earlier work, but the volume of the space with respect to which the density is measured — and that this volume cannot be taken arbitrarily small, because the energy comes from quantum fluctuations and those cannot occur at every point of a continuum. A minimum length, and therefore a Planck-sized minimum volume, is required; Causal Set theory supplies one without breaking Lorentz invariance.Inertia from Rindler-scale Casimir effect, footnote to equation 17 and the discussion following equation 19
What the authors leave on the bench. They have treated only linear acceleration; circular motion is expected to be much harder, if only because the existence of a rotational analogue of the Rindler horizon is itself unsettled. They ask how the model meshes with the equivalence principle. And they point to the Generalized Uncertainty Principle predicted by several quantum-gravity models: a GUP-modified expression for inertia could in principle be turned around to bound the GUP parameter using current limits on equivalence-principle violation. They also flag the several recent results finding non-thermal behaviour in the radiation an accelerated observer sees, and ask whether that deviation from thermality changes the picture.Conclusions and outlook
The way in
https://doi.org/10.1016/j.rinp.2021.104543How to cite it
Jaume Giné, Giuseppe Gaetano Luciano (2021) Modeling inertia through the interaction with quantum fluctuations doi:10.1016/j.rinp.2021.104543