Casimir, Electrostatic, and Material Backgrounds

At micrometre and submillimetre separations, the Newtonian force is usually not the largest interaction in the apparatus. A credible gravity search therefore predicts Casimir, electrostatic, patch, thermal, magnetic, and material forces from independent measurements, propagates their shared uncertainties, and tests whether a putative short-range signal survives changes that deliberately alter those backgrounds.

Required background. Weak-Field Gravity supplies the Newtonian signal model; Probability and Statistical Inference supplies covariance, nuisance-parameter, and model-checking methods. Helpful background. Classical Field Theory supplies electromagnetic boundary-value and Green-function reasoning.

Backgrounds have different physical inputs

For ideal, perfectly conducting parallel plates at zero temperature, the Casimir pressure is

$$P_{\rm C}(s)=-\frac{\pi^2\hbar c}{240s^4}.$$

The dimensions are force per area, and the negative sign denotes attraction. This formula is a benchmark, not a subtraction prescription. Real calculations require measured optical response, finite temperature, multilayer coatings, roughness, curvature, and edges. In the ideal sphere–plane proximity limit, $F_{\rm C}\simeq-\pi^3\hbar cR/(360s^3)$; convergence toward both ideal limits is a valuable code check.

A residual potential difference $\Delta V$ between parallel conductors gives

$$P_{\rm e}(s)=-\frac{\epsilon_0(\Delta V)^2}{2s^2}.$$

Spatially varying work functions produce patch fields even when the mean contact potential is nulled. Their force depends on the two-dimensional voltage power spectrum and can imitate a Casimir-like distance dependence for some patch scales. The analytic and numerical construction is developed in Speake and Trenkel 2003, pp. 160403-1–160403-4. Kelvin-probe maps, repeated after cleaning or coating changes, constrain that spectrum more directly than one fitted “patch amplitude.”

Other backgrounds have distinct controls: magnetic susceptibility responds to applied field; thermal radiation and expansion respond to temperature modulation; adsorbates and charging respond to preparation history; tilt and roughness alter the separation distribution; and density inhomogeneity changes both Newtonian and Yukawa templates. Distance-zero uncertainty is especially dangerous because differentiating a steep force gives $\delta F/F\simeq-n\,\delta s/s$ for $F\propto s^{-n}$.

A numerical scale comparison

At an illustrative gap $s=10\ \mu{\rm m}$, the ideal zero-temperature plate pressure is $P_{\rm C}\simeq-1.30\times10^{-7}$ Pa. A residual voltage of only $1$ mV gives $P_{\rm e}\simeq-4.43\times10^{-8}$ Pa. Two infinite gold slabs of density $1.93\times10^4\ {\rm kg\,m^{-3}}$ and thickness $1$ mm give Newtonian pressure $P_{\rm N}=-2\pi G(\rho t)^2\simeq-1.56\times10^{-7}$ Pa. All three toy pressures are comparable, although their physics and distance dependence differ. These are analytic scale estimates: real gold at room temperature requires the finite-temperature optical response, and finite plates require their actual geometry.

Voltage reversal by itself is not enough. Because the uniform electrostatic pressure is quadratic, $+\Delta V$ and $-\Delta V$ give the same force. A voltage sweep instead measures a parabola whose minimum estimates the contact potential; unresolved patches can leave a nonzero force at that minimum. This makes the control more informative than merely observing attraction.

A reproducible force-budget calculation

The analytic estimates above are a starting check, not a realistic material calculation. A numerical force budget uses measured topography and separation, layer thicknesses, complex dielectric response with an explicit low-frequency extrapolation, temperature sensors, Kelvin-probe maps, test-body geometry, and their joint calibration covariance. Record the software versions, material tables, surface maps, mesh, units, masks, and configuration so that the same input problem can be rerun.

For each separation:

1. integrate the Newtonian and unit-Yukawa kernels over the measured geometry;
2. evaluate the finite-temperature Lifshitz or scattering calculation over Matsubara frequency and transverse wave number;
3. solve the electrostatic boundary problem using the measured patch map and an independently parameterized unresolved spectrum;
4. repeat over topography, tilt, distance-zero, dielectric, temperature, and material-property samples drawn from their auxiliary likelihood; and
5. return force, gradient, covariance, and component-wise derivatives in a machine-readable table.

Use three mesh scales, for example $h$, $h/2$, and $h/4$, and three frequency tolerances. Require the signal-region force gradient to change by less than a declared fraction of the blinded short-range injection and show monotone or Richardson-consistent refinement. Cross-check the ideal parallel-plate and sphere–plane limits, then compare the production solver with a direct quadrature or boundary-element implementation. Nonconvergence, catastrophic cancellation between large components, an ill-conditioned covariance, or sensitivity to an unresolved patch spectrum is a diagnosed failure, not evidence for gravity.

Numerical convergence must be judged against the uncertainty required for the extra-force amplitude, not against the much larger total electromagnetic force. A solver that predicts a background to one percent may still be inadequate when the sought signal is a millionth of that background. No full material-response calculation or experimental reanalysis is supplied here; its reproduction requires the actual optical and surface inputs, along with the executable fit command.

Inference without subtraction bias

The evidence basis is current through 2026-08-19. A data result must name the primary force or torque data product, calibration version, surface and optical measurements, selection mask, retrieval date, and software commit. A useful forward model is

$$\mathbf d = \mathbf F_{\rm N} +\alpha\mathbf F_{\rm Y}(\lambda) +\mathbf F_{\rm C}(\boldsymbol\nu) +\mathbf F_{\rm patch}(\boldsymbol\nu) +\mathbf F_{\rm other}(\boldsymbol\nu) +\boldsymbol\epsilon.$$

The nuisance vector $\boldsymbol\nu$ is shared across components whenever the same distance, tilt, roughness, temperature, or material measurement enters them. The likelihood must therefore retain the full covariance. Subtracting best-fit backgrounds and treating the remainder as independent discards uncertainty and can sculpt a signal.

Selection also matters. State thresholds on sensor saturation, vibration, charging transients, contact events, temperature excursions, and unstable feedback, together with how much exposure each quality cut removes. Fit control-voltage and field-reversal runs jointly with science runs. Withhold some separations or modulation phases for posterior-predictive checks, and repeat the conclusion after broadening the patch and dielectric models.

The experimentally measured Casimir force and its use in new-force searches are discussed in Decca et al. 2005, Eqs. (1)–(5) and Figs. 1–3. The key lesson is not that a single background theory is exact; it is that every background is tied to an auxiliary observable and that the uncertainty envelope remains below a blinded signal across the tested range.

Exercises

At separation $s$, suppose the modeled force is $F(s)=As^{-4}+Bs^{-2}+\alpha Ye^{-s/\lambda}$. The separation zero has uncertainty $\delta s$. Explain why fitting $A$, $B$, and $\alpha$ after first subtracting fixed best estimates of the two backgrounds underestimates uncertainty.

Solution

The same displacement $\delta s$ changes all three components, with derivatives $-4As^{-5}$, $-2Bs^{-3}$, and $-\alpha Y e^{-s/\lambda}/\lambda$. Their errors are therefore correlated rather than independent. Fixing and subtracting $A$ and $B$ removes both their auxiliary uncertainty and their covariance with $\delta s$, forcing any remaining curvature toward $\alpha$. A joint likelihood instead fits the raw force with priors from calibration data and marginalizes $A$, $B$, and $\delta s$. Comparing the posterior with alternate patch and distance models then reveals whether the apparent Yukawa component is stable or merely background flexibility.

References

• Decca, R. S., et al., “Precision Tests of the Casimir Force and Constraints on Non-Newtonian Gravitation,” Physical Review Letters 94 (2005) 240401, DOI.
• Speake, C. C., and C. Trenkel, “Forces between Conducting Surfaces due to Spatial Variations of Surface Potential,” Physical Review Letters 90 (2003) 160403, DOI.

Further reading

• Decca, R. S., E. Fischbach, G. L. Klimchitskaya, D. E. Krause, D. López, and V. M. Mostepanenko, “Improved Tests of Extra-Dimensional Physics and Thermal Quantum Field Theory from New Casimir Force Measurements,” Physical Review D 68 (2003) 116003, DOI.