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We analyze the structure of scattering amplitudes of the Kaluza-Klein (KK)gravitons and of the KK gravitational Goldstone bosons in the compactified 5dGeneral Relativity (GR). Using a general $R_{\xi}$ gauge-fixing, we study thegeometric Higgs mechanism for the massive spin-2 KK gravitons. We newly proposeand prove a Gravitational Equivalence Theorem (GRET) to connect the scatteringamplitudes of longitudinal KK gravitons to that of the KK gravitationalGoldstone bosons, which formulates the geometric gravitational Higgs mechanismat the scattering $S$-matrix level. We demonstrate that the GRET provides ageneral energy-cancellation mechanism guaranteeing the $N$-point longitudinalKK graviton scattering amplitudes to have their leading energy dependencecancelled down by a large power factor of $E^{2N}$ ($N \geq 4$) up to any looporder. We propose an extended double-copy approach to construct the massive KKgraviton (Goldstone) amplitudes from the KK gauge boson (Goldstone) amplitudes.With these we establish a new correspondence between the two types of energycancellations in the four-point longitudinal KK amplitudes at tree level:$E^4\to E^0$ in the KK gauge theory and $E^{10} \to E^2$ in the KK GR theory.

Gravitational Equivalence Theorem and Double-Copy for Kaluza-Klein Graviton Scattering Amplitudes