  Superconductivity, the amazing phenomenon of lossless transmission of
electric current through metallic wires, requires cooling of the wire to low
temperatures. Metallic hydrogen is considered as the most likely candidate for
superconductivity at very high temperatures, possibly even room temperature.
However, as a result of various approximations used, conflicting theoretical
predictions exist for the range of temperatures where superconductivity is
expected to occur. Here we avoid those approximations and confirm that metallic
hydrogen is indeed a superconductor, but this is limited to temperatures far
below previous estimates. We exploit the "jellium" model proposed in 1966 by De
Gennes, where superconductivity is caused by the combination of Coulomb
repulsion between the electrons and Coulomb attraction between the protons and
the electrons. We find that the superconducting order develops over an energy
range far exceeding the characteristic phonon energy, and that the phase of the
order parameter flips 180 degrees at the characteristic phonon energy above and
below the Fermi energy.
