Nonmetricity theories and aspects of gauge symmetry

Note on the absence of the second clock effect in Weyl gauge theories of gravity

“…This is, of course, at variance with what other authors have typically assumed, but is nevertheless a strong conclusion of this work. Moreover, since we have shown that this leads to the absence of a second clock effect, the objections presented in [33] to our previous claims regarding the SCE are certainly not valid for the case we have discussed above.…”

“…It has recently been claimed, however, that our previous analysis contains a flaw and that the SCE does indeed occur in WGTs [33]. This difference of opinion results partly from a lack of clarity in our original presentation, but more significantly from the analysis in [33] not properly taking into account the requirement in the second part of our argument for the introduction of the scalar compensator field. As we will now show, when one carefully considers this aspect of our full argument, one finds that the SCE does not, in fact, occur, thereby confirming our original findings.…”

“…which is equivalent to equation (180) in [33], although the latter is derived at somewhat greater length. As noted in [33], equation (3) implies that the inner product of two non-orthogonal parallel-transported vectors is path independent only if…”

“…Thus, the norm of the parallel-transported tangent vector u µ = dx µ /dλ itself is unchanged, and so the original basis for suggesting the existence of a SCE is removed, as it is this norm that was taken to represent the clock rate. Since we were concerned in [19] exclusively with (coordinate) vectors of weight w = −1, however, we did not clarify -as correctly pointed out in [33] -that (2) holds only if the scalar g µν v µ z µ has weight w = 0 (recalling that w(g µν ) = 2), although the first equality in (2) implicitly contains this assumption. More generally, if the weights of the vectors v µ and z µ are w v and w z , respectively, and denoting v • z ≡ g µν v µ (λ)z ν (λ) and u • B = u µ (λ)B µ for brevity, one immediately has…”

“…It is then further claimed in [33], however, that this implies that the SCE does occur, since one may take v µ and z µ to coincide with the 4-momentum p µ of a particle, which has weight w(p µ ) = −2, so that the mass (squared) m 2 = g µν p µ p ν of an atom, and hence the frequencies of its spectral lines, will depend on its past history. We now show that this further claim is unjustified on account of the requirement in the second part of our original argument presented in [19] that one must introduce a scalar compensator field; this results in the 4-momentum of a particle not being parallel transported along its worldline, even if it is in free-fall, so that (3) does not apply.…”