"Can negative velocity states in non-inertial reference frames yield observational equivalence to superluminal motion through relativistic Doppler shift?"

I'm investigating whether the mathematical framework of special relativity permits a loophole through negative velocity states. Specifically:

Consider an observer O and object A in relative motion. While |v| < c must hold universally, can we construct a reference frame transformation where v < -c becomes mathematically valid? If we analyze the relativistic Doppler shift formula:

f' = f √[(1 - β)/(1 + β)] where β = v/c

For negative velocities approaching -c, the observed frequency shift could theoretically mimic superluminal recession. This wouldn't violate causality if the "faster-than-light" motion exists only as an observational artifact in specific reference frames.

Key questions: 1. Does the Lorentz transformation group mathematically permit such negative velocity states? 2. Could this reconcile with the cosmic recession velocities exceeding c in comoving coordinates? 3. What experimental signatures would distinguish this from actual FTL motion?

I'm particularly interested in whether this approach offers new insights into the interpretation of relativistic constraints. Has anyone explored similar frameworks or can point to theoretical obstacles I may be overlooking?

#SpecialRelativity #LorentzTransformation #DopplerEffect #TheoreticalPhysics #ReferenceFrames