This question asks
Free electrons in a metal are attracted by gravity towards Earth. So why don't they lay down to the bottom of the conduit, like sediment at the bottom of a river?
The current answers there are about the overwhelming strength of the electric force compared to the gravitational force. That's fine for an intro-physics explanation, but there's a sneakier reason. A conducting metal is a lattice of positive ions and a gas of quasi-free electrons with thermal kinetic energies $kT \approx \rm25\,meV$. Those electrons have typical speeds
\begin{align} \frac{v^2}{c^2} &\approx \frac{2kT}{mc^2} \approx \frac{2\times25\rm\,meV}{500\rm\,keV} \\ \frac vc &\approx \frac 13\times10^{-3} \end{align}
This is somewhat faster than Earth's escape velocity $10^4\rm\,m/s$, so you would expect gravitationally-driven gradients in the density of the electron gas to have a scale height somewhat taller than Earth's atmosphere. A typical conductor fits within Earth's atmosphere, so you would expect uniform electron density even if the electrons didn't interact with each other.
My question: what happens if you make the electron gas cold enough that this scale height is reduced? I'm thinking along the lines of ultra-cold neutrons, which have energies below 100 nano-eV and a scale height of about two meters, so they can be "trapped" in a person-sized bucket with no lid.
In a sufficiently cold electrical conductor (or an electrical conductor with a sufficiently low electron temperature, which might not be quite the same thing), would the different gravitational interactions of the fixed-distance ion lattice and the quasi-free electron gas cause the positive and negative charge distributions to separate? Would the resulting gravito-electric field point up, or down? What is the magnitude of induced electric field to expect? Or would some hidden symmetry keep the positive and negative charge densities the same everywhere?
The experiment to test this would be to build an 800-meter-tall cryostat in one of the elevator shafts of the Burj Khalifa, and watch a potential difference of a few millivolts appear between the top and the bottom as the conductor in the cryostat gets cold — but probably disappear if the conductor goes superconducting and expels internal electric fields.
