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Within the gauge symmetry framework, the [Formula: see text] symmetry of electrodynamics is violated in the presence of gravity with spacetime translational gauge symmetry in inertial frames. For a light ray, an eikonal equation with effective metric tensors is derived in the geometric-optics limit. Under these conditions, the angle of the deflection of light by the sun is calculated to be [Formula: see text] in inertial frames without requiring a gauge condition such as [Formula: see text]. In contrast, if the theory is [Formula: see text] gauge invariant, one can impose the gauge condition [Formula: see text] in the derivation of the eikonal equation. In this case, one obtains a slightly different effective metric tensor and a different angle of deflection [Formula: see text]. However, because the precision of experiments in the last century using optical frequencies has been no better than (10–20)% due to large systematic errors, one cannot unambiguously rule out the result [Formula: see text]. We hope that the precision of these data can be improved in order to test Yang–Mills gravity.

In 1955, Lee and Yang discussed a new massless gauge field based on the established conservation of baryon number. They predicted the existence of a repulsive force between baryonic matter, just as the conservation of electron–lepton number was later shown to imply the existence of a repulsive force between electrons. Although Eötvös experiments showed the force to be undetectably small at that time, such a force may be related to the dark-energy-induced acceleration of the expansion of the universe. If the gauge invariant Lagrangian involves a spacetime derivative of the field strength, the resultant potential has properties similar to that of the "dark energy" implied by the cosmological constant in the Einstein's equation.

In the calculations of the static gauge fields in (14) and (22)–(24), the normalization constant of the color singlet charmonium state ( 1/ √ 3 ) δij |cicj〉 should have been taken into account in the delta function sources related to three color quarks. As a result, the factor g s involved in the results (17)–(21) and (25)–(27) should be replaced by g s/3. Moreover, the potential energies VXst, VY st and VZst associated with color carrying confions, X , Y and Z, should also be included in the total potential energy because these contributions are consistent with color singlet state of the charmonium. Thus, the strong confining potential energy Vst of the charmonium should be