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One of the Lorentz equations is $$t ′ = \gamma \left( t − \frac{v x}{c^2} \right) $$

This shows that clocks that are synchronized in the inertial reference frame will be offset in the observer reference frame. Clocks displaced positively in the direction of travel will appear to be running behind and clocks in the other direction will appear to be running ahead.

Other relativistic effects such as time dilation or length contraction have a name that neatly describes them. I'm aware of a number of terms that dance around the concept but don't reference the specific phenomenon of (for lack of a better term) time shift:

  • Time Dilation: I believe this refers only to the slowing of time in the inertial frame and not a shift of observed time in the direction of travel.
  • Relativity of Simultaneity: This is perhaps the closest term to what I want, but it refers specifically to simultaneous events rather than observed time.
  • Local Time: Refers specifically to a time value as viewed from the inertial frame, and not to the greater phenomenon.
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SR (special relativity) states the coordinates transformation laws between inertial reference frames. In each reference frame clocks are synchronized.

Time shift is not a terminology which is used.

Time dilation occurs between the proper time of a moving reference frame and the coordinate time of the observer. The observer time runs faster than the moving frame proper time independently whether the moving frame is approaching or is going far from the observer. Of course, the relation is symmetrical as measured by the moving frame vs. the observer, according to the first principle of SR that states that every inertial frame is equivalent.

Relativity of simultaneity means that two events which are simultaneous in a reference frame, are not in another frame.

In SR each reference frame has its proper time, as measured by a clock comoving with the frame. I think that is the local time.

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