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General relativity with fermions has two independent symmetries: general coordinate invariance and local Lorentz invariance. General coordinate invariance is implemented by the Levi-Civita connection and by Cartan's tetrads both of which have as their action the Einstein-Hilbert action. It is suggested here that local Lorentz invariance is implemented not by a combination of the Levi-Civita connection and Cartan's tetrads known as the spin connection, but by independent Lorentz bosons ${{L}^{ab}}_{i}$ that gauge the Lorentz group, that couple to fermions like Yang-Mills fields, and that have their own Yang-Mills-like action. Because the Lorentz bosons couple to fermion number and not to mass, they generate a static potential that violates the weak equivalence principle. If a Higgs mechanism makes them massive, then the static potential also violates the inverse-square law. Experiments put upper bounds on the strength of such a potential for masses ${m}_{L}l20\text{ }\text{ }\mathrm{eV}$. These upper limits imply that Lorentz bosons, if they exist, are nearly stable and contribute to dark matter.

Is the local Lorentz invariance of general relativity implemented by gauge bosons that have their own Yang-Mills-like action?