The interference paradox

Now from the point of view of the observer on ground what will happen? If the signals don’t arrive simultaneously the device must not explode.

Constructive interference occurs at a given event when the phase, $\phi$, of the two waves differs by an even multiple of $\pi$ at that event. Destructive interference occurs when they differ by an odd multiple of $\pi$. So, the question is whether or not the phase is a relativistic invariant. If it is an invariant then all frames will agree whether or not device is triggered because all frames will agree if there is constructive interference.

Now, we introduce the four-position $R^{\mu}=(ct,x,y,z)$ and the four-wavevector $K^{\mu}=(\omega/c,k_x,k_y,k_z)$ (see Four-vector in Wikipedia for details). These are both four-vectors so their product is an invariant. Expanding their product in an inertial coordinate system we get $g_{\mu\nu}R^{\mu}K^{\nu}=\omega t-k_x x-k_y y-k_z z = \phi$. We immediately recognize this quantity as the phase of a wave.

Therefore, because the product of two four-vectors is manifestly invariant and because the phase can be written as the product of two four-vectors then we conclude that the phase is an invariant. This in turn implies that any device whose function is based on the phase, such as the detector in this scenario, will be predicted to behave identically in all frames.