Q:

Would relativistic LIGO hear different chirp?

Imagine a binary blackholes merger about 100 Mpc away and LIGO's relativistic twin is orbiting Earth at close to speed of light, I wonder would both LIGO and its twin show different results? For this event assume both instruments alignment are the same when gravitational wave hits. 

A:

This question was answered here for the case where LIGO was boosted in the same direction the GW was travelling.  In this case, the chirp's strain amplitude remains the same but the chirp is stretched in time (redshifted) if LIGO is moving away from the source or compressed (blueshifted) if LIGO is moving toward the source.
If LIGO is boosted transverse to the z travel direction of the GW wave, you can modify the elements in the boost matrices of the previous answer to see what happens.  A boost in the x or y directions does not commute with strains in the xy plane, and results in a space-time rotation.  In the frame of the boosted LIGO, the boosted LIGO will no longer just experience the xy strain of its mirrors, but will also experience an x,t or y,t rotation which is new to me, and I don't know how to interpret right now.

