Regarding the Casimir effect, is gravitational attraction too weak in order to explain the attraction between the plates?
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asked Dec 17, 2018 at 10:59
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$\begingroup$ This question is developped notably here : quora.com/Why-cant-the-Casimir-Effect-be-explained-by-gravity $\endgroup$– J.ACommented Dec 17, 2018 at 13:40
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In terms of order of magnitude, as noted in the reference in the comment (or also https://physics.info/gravitation-extended/practice.shtml), the gravity field is proportional to $\mathbf{g}\sim \rho l G$, where $l$ is the thickness of the plates, $\rho$ is the density of the plates, and $G$ the gravity constant, so the force per unit area $A$ is $F_G/A \sim G \rho^2 l^2 $.
Regarding Casimir force, the derivation (https://en.wikipedia.org/wiki/Casimir_effect) shows that $F_C/A \sim 10^{-2} h c r^{-4} $, with $h$ the Planck constant and $c$ the speed of light and $r$ the distance between the two plates.
If we compare them : $\frac{F_C}{F_G}=\frac{10^{-2} h c r^{-4}}{ G \rho^2 l^2}$. You can see when gravity becomes insignificant with respect to Casimir effect.
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$\begingroup$ Still, the zero point energy of the hypothetical graviton should be $\frac{h\omega} {2}$like that of a photon. $\endgroup$– my2ctsCommented Dec 17, 2018 at 22:43