How to view time dilation?

Question

In a recent question that I asked, I received a response which presented time dilation as the result of a lack of simultaneity:

“It is better to think of time-dilation as being the result of a lack of simultaneity than vice versa. Imagine walking down a long corridor where every 10 metres there is a clock on the wall. If those clocks were set so that each was 1 second ahead of the previous one, then as you walked down you would see that the time on your watch was falling a second behind the clock every ten metres. You would not know if your watch was running slow or the clocks were out of synch- the two effects are inter-related” – Marco Ocram

This is a way of thinking about time dilation that I have never seen before. Previously, I have only thought of it in terms of the thought experiment of a light clock with a longer light path in the moving frame relative to the rest frame. Is this light clock perspective derived from the perspective of time dilation as the result of a lack of simultaneity? Are the two reconcilable ideas or completely different concepts? Thanks in advance.

Answer 1

The Lorentz transformation of a time coordinate is:

$$ t' = \gamma (t - \frac{vx}{c^2}) $$

so there are two effects. At $x=0$:

$$ t'=\gamma t$$

and the moving clock is ticking slower then the stationary one. (and vice versa). That's time dilation.

If you have a finite separation between the two inertial observers, then the relativity of simultaneity comes into play. Each clock sees the other ticking slower, but they have different definitions of the time at distant places: this is clock bias.

The best was to see this is how it resolves the twin paradox. If the space twin travels $L$ at $\beta$, the Earth twin sees it takes $t_0=L/\beta$, which the space twin only ticks of $t_0/\gamma$.

Meanwhile, the space twin see that only $t_0/\gamma^2$ has passed on Earth. (Hence the paradox): they're both younger then the other and see less time pass on each leg of the trip. In bound or outbound doesn't matter.

But the space twin has his imaginary lattice of synchronized clocks. When he reaches the turning point at $L$, his synchronized clocks back on Earth says it's $t_0/\gamma$ there (when Earth's clock reads $t_0/\gamma^2$....the twin sees Earth running slow).

Now he turns around, and that old lattice of clocks is junk. He needs a new lattice, and with that, his synchronized clock on Earth jumps into the future by $2t_0 - 2t_0/\gamma^2$, so that when he returns, Earth reads $2t_0$ and his space clock says $2t_0/\gamma$: he is younger.

Some people call that jump "gravitational time dilation" because if you make the turn around time finite with constant acceleration, it exactly matches the formula for gravitational time dilation. The problem with that is that the space twin can turn around again and make it go backwards, and time dilation never goes backwards.

Really, it's establishing a clock bias at a distant location, which depends on velocity. It can be positive or negative, and it is reversible.

The Lorentz transformation is a linear transformation, and we know a line has two things: a slope and an intercept. The slope is time dilation, and the intercept is clock bias.

Answer 2

Time dilation and the relativity of simultaneity are two facets of the same fact- you cannot have one without the other. For example, if your clock is running at nine-tenths of the rate of the clocks in another frame, then every nine seconds that passes on your clock you will pass a clock that has advanced ten seconds in the other frame. Suppose you synchronise your clock at 0 seconds with one of the clocks you pass- thereafter, every nine seconds you compare your clock with the time in the other frame: this is what you will see...

Your frame 0s, other frame 0s

Your frame 9s, other frame 10s

Your frame 18s, other frame 20s

Your frame 27s, other frame 30s

Your frame 36s, other frame 40s

and so on.

From your perspective, each of the clocks you pass, after another 9 seconds on your clock, will seem to be a second ahead of the last clock you passed- ie they will seem to you to be systematically out of synch. Conversely, from the perspective of the other frame, your clock seems to be losing a second in every ten, in other words to be time dilated.