# How different observers measure time?

Suppose I have a coordinate system, call it $$S$$, in which an observer $$O$$ is not moving, and $$O'$$ is moving with constant velocity and another coordinate system $$S'$$ where $$O'$$ is not moving and $$O$$ is moving in a constant velocity, they both have a stop watch and measure the time of the movement of $$O'$$, what time does each measure?

Since in $$S$$, $$O$$ is not moving then he measures the time of $$O$$ by using the Minkowsky metric, when I change to $$S'$$ then $$O'$$ is not moving and he measures time by using a different metric, that is the metric I get after I change coordinates using the Lorrentz transformation.

My question is this idea correct, have I misunderstood something? If yes can you explain what is my mistake?

• I'm not sure if I understand what you mean by "the observer measures time using a metric". Why wouldn't you simply use a clock? Jan 22 at 13:41
• A Lorentz transformation, by definition, preserves the metric. Jan 22 at 15:58
• By saying "measure time using a metric", I mean he actually uses the metric to compute the equations of motion. Jan 22 at 17:48
• @WillO the distances are preserved, but the metric looks different in a different coordinate system, although lengths remain unchanged. I am not talking about proper time. Jan 22 at 17:54

Should be correct, except in special relativity people use the term "frame" rather than "metric", which is used more often in general relativity. You can say $$u$$ boosted from frame $$S$$ to frame $$S'$$ with a Lorentz boost.