Constraints on Nonmetricity from Bounds on Lorentz Violation
  Spacetime nonmetricity can be studied experimentally through its couplings to fermions and photons. We use recent high-precision searches for Lorentz violation to deduce first constraints involving the 40 independent nonmetricity components down to levels of order $10^{-43}$ GeV. 
Joshua Foster, Alan Kostelecky, Rui Xu
gr-qc hep-ph