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We study the structure of scattering amplitudes of the Kaluza-Klein (KK) gravitons and of the gravitational KK Goldstone bosons in the compactified 5d general relativity (GR). We analyze the geometric ``Higgs'' mechanism for mass generation of KK gravitons under compactification with a general R_ gauge-fixing and prove that the massive KK graviton propagator is free from the van Dam-Veltman and Zakharov discontinuity. With these, we newly formulate a gravitational equivalence theorem (GRET) to connect the longitudinal KK graviton amplitudes to the corresponding KK Goldstone amplitudes, which is a manifestation of the geometric Higgs mechanism at S-matrix level. We directly compute the gravitational KK Goldstone amplitudes at tree level and show that they equal the corresponding longitudinal KK graviton amplitudes in the high-energy limit. We further use the double-copy method with color-kinematics duality to reconstruct the KK longitudinal graviton (Goldstone) amplitudes from the KK longitudinal gauge boson (Goldstone) amplitudes in the compactified 5d Yang-Mills (YM) gauge theory, under the high-energy expansion. From these, we reconstruct the GRET of the KK longitudinal graviton (Goldstone) amplitudes in the 5d GR theory from the KK longitudinal gauge boson (Goldstone) amplitudes in the 5d YM theory. Using either the GRET or the double-copy reconstruction, we provide a theoretical mechanism showing that the sum of all the energy power terms up to O (E^10) in the high-energy scattering amplitudes of four longitudinal KK gravitons must cancel down to O (E^2) as enforced by matching the energy dependence of the corresponding KK Goldstone amplitudes or by matching that of the double-copy amplitudes from the KK YM theory. With the double-copy approach, we establish a new correspondence between the two energy-cancellations in the four-particle longitudinal KK scattering amplitudes: E^4E^0 in the 5d KK YM theory and E^10E^2 in the 5d KK GR theory. We further analyze the structure of the residual term in the GRET and uncover a new energy-cancellation mechanism therein.
Hang et al. (Wed,) studied this question.