Time dilation is one of the fundamental consequences of the Lorentz transformations in Special Relativity. It describes how the passage of time depends on the relative motion of observers.
In particular, a clock moving at a constant velocity relative to an observer will appear to tick more slowly than a clock at rest with respect to that observer.
Let us imagine a train car moving at a constant velocity v along the x-axis, as observed from a stationary observer in what we call the "lab frame".
Inside the train car (the "moving frame"), there is a light clock: a laser that emits a light pulse vertically towards a detector on the ceiling.
Let:
In the rest frame of the clock (), the light travels vertically up.
Thus, the time measured in the moving frame (the proper time), denoted , is:
In the lab frame, the train is moving horizontally at speed .
From the perspective of an observer in , the light pulse follows a diagonal path — because while the light moves vertically, the clock is moving horizontally.
The light path thus forms the hypotenuse of a right triangle.
For the one-way trip (emitter detector), the horizontal distance covered is:
The vertical distance remains .
Thus, the total path length for the one-way trip is:
Since light always travels at speed , the time for the one-way trip is:
Now solve for .
Recall that in the moving frame (the proper frame of the clock), the time for the full cycle is:
Comparing with our result for :
We define the Lorentz factor as:
Thus:
Conclusion: Moving clocks appear to run more slowly by a factor .
This is the phenomenon of time dilation:
Suppose a jet in an AGN moves at relative to the observer.
Compute :
Thus, time in the moving frame is dilated by a factor of — an observer will see processes in the jet taking 7 times longer than in the jet’s own frame.