- In QM frequencies scale linearly with energy, but in GR they scale exponentially. These can't both be true.
The exact formula for gravitational time dilation is [tex]T_d = exp(m\phi / mc^2)[/tex]. (We usually cancel out the ms, but I want to show this as a ratio of energies.) Since [tex]m\phi[/tex] is the potential energy, this says that the rates or frequencies of things in GR depend exponentially on their energy. On the other hand, [tex]E = h\nu[/tex] implies that the frequencies of things in QM are linearly proportional to their energy. It's quite obvious that these things cannot both be universally true. So, which one is wrong, and when?