Weighing vacuum energy

Start with two plates.

Consider parallel, perfectly conducting plates with area A and separation a, at zero temperature. Take their lateral dimensions to be much larger than the gap, so edge effects can be neglected. Relative to infinite separation, their electromagnetic interaction energy is negative: [1]

Here ℏ is the reduced Planck constant and c is the speed of light. The negative value is the energy difference relative to infinite separation; the plates and supports contribute additional energy.

Differentiating with respect to the gap gives the pressure. With positive force defined toward increasing separation, attraction has a negative sign:

The attraction acts between the plates. A balance supporting the whole assembly measures its gravitational load.

Surface force and gravitational load.

For a stationary apparatus in a weak, approximately uniform gravitational field, an energy change contributes to its weight. Define downward weight as positive and every difference as final minus initial. To leading order:

The total includes the plates, field, and supporting structure. Bimonte and colleagues calculate the Casimir contribution by referring all support forces to a common point in the gravitational field. They obtain gE_C/c². [2]

A more negative interaction energy contributes a smaller downward load. The weight change of an actual apparatus also depends on energy transferred to its supports, drive, or thermal reservoir while the cavity changes.

Area, gap, and signal size.

Double the plate area and the ideal energy doubles. Double the gap and its magnitude falls eightfold; the pressure falls sixteenfold. For N identical cavities whose interactions can be treated independently, the energies add to NE_C.

For an area of 1 cm² and a gap of 1 μm, equation (1) gives approximately −4.33 × 10⁻¹⁴ J. The pressure is about −1.30 × 10⁻³ Pa, corresponding to an attractive force of 1.30 × 10⁻⁷ N. At terrestrial gravity, the same interaction energy contributes only about −4.73 × 10⁻³⁰ N to the weight.

This example gives a substantial surface force alongside an extremely small gravitational contribution. Changing the gap from a₁ to a₂ changes the ideal Casimir energy in proportion to the difference of their inverse cubes. Equation (3) then relates that energy difference to its contribution to weight.

Changing the cavity’s response.

The 2014 Archimedes study proposes varying electromagnetic response instead of moving the plates. Its type-I example places a thin superconducting film between two normal-metal mirrors, with an insulating gap on either side. When the film changes state, the response of this double cavity changes with it. Its transition free-energy balance is: [4]

Here the difference is normal minus superconducting at a common temperature and zero applied field. G_tr denotes the transition Gibbs free energy, E_cond the film’s condensation-energy contribution, and ℱ_C the cavity’s Casimir free energy. The paper evaluates the latter using the layers’ frequency-dependent dielectric responses, with a finite-temperature extension. Its later high-temperature-superconductor estimates involve additional material assumptions. [4]

Lifshitz’s treatment connects fluctuation forces to material properties and temperature. Predicting the transition’s Casimir-energy change requires those material responses; the ideal-mirror formula in the interactive model supplies only a limiting case. [3]

Free energy and internal energy.

The Helmholtz free energy is F_H = U − TS, where U is internal energy, T temperature and S entropy. Rearranging this definition gives:

At equal endpoint temperatures, the final term becomes TΔS. If temperature changes, it is the difference between the final and initial values of TS. The weight calculation in equation (3) requires the total energy change, including this thermal contribution. Appendix B of the 2014 paper also treats the thermodynamic potential and field work for magnetic control. [4]

The beam-balance prototype.

In 2021, the Archimedes collaboration reported tests of a room-temperature beam balance at Virgo. Its torque sensitivity reached about 8 × 10⁻¹² N m/√Hz in the tens-of-millihertz region. These tests characterized the unloaded instrument: no samples were suspended at its ends, so the experiment had not yet measured a vacuum-related weight change. [5]

The quoted torque sensitivity is an amplitude spectral density. The lever arm converts it to force sensitivity, while the measurement bandwidth and noise spectrum determine the uncertainty after averaging.

The report identifies actuator and sensing noise as important limitations. Detecting the predicted weight modulation requires sufficient sensitivity at the modulation frequency, after accounting for the balance’s mechanical response. [5]