No-go theorem for gravivector and graviscalar on the brane

We prove the no-go theorem that the gravivector $(h_{5\mu })$ and graviscalar $(h_{55})$ cannot have any on-shell propagation on the Randall-Sundrum (RS) brane. For this purpose, we analyze all of their linearized equations with the de Donder gauge (5D transverse-tracefree gauge). But we do not introduce any matter source. We use the Z${}_2$-symmetry argument and their ($h_{5\mu },h_{55}$) compatibility conditions with the tensor $h_{\text{mn}}$-equation. It turns out that $h_{55}$ does not have any bulk (massive) and brane (massless) propagations. Although $h_{5\mu }$ has a sort of massive propagations, they do not belong to the physical solution. Hence we confirm that the Randall-Sundrum gauge suffices the on-shell brane physics.