Energy-Time uncertainty: observer's clock or observed's?

[Polama]

How do time-energy uncertainty and special relativity's rules about observers seeing time different for each other interact?

Heisenberg's uncertainty principal applies to time and energy: a system existing for a short duration of time has more uncertainty in its energy than one that exists for longer.

When I measure a system moving at a greatly different speed than myself, relativity kicks in, right? So I observe the system as evolving substantially slower than it perceives itself.

So which delta-t do we use? Am I able to observe the energy of a fast moving system more accurately then I could if I was travelling the same speed as it? Is it only the system's clock that matters? Or is energy-uncertainty relative? But what would that mean? If I perform a measurement of energy, and a scientist moving at high speeds makes a measurement, what would we see when we compared notes?

[Bill_K]

Let's say there's an atom on board with a spectral line of energy E. You on Earth will see this line Doppler shifted to a lower energy, E/γ. Likewise the width of the line dE will be Doppler shifted to dE/γ. So yes, you will see a smaller spread in energy. But of course this doesn't buy you anything!