I Reconciling two observers in two frames

[Freixas]

It looks like @SlowThinker has already given a solution that satisfies you, but I'm going to go ahead and work it the way I would work this type of problem. Actually I'll work it two ways.

Thanks, PeterDonis! I appreciate all your time and effort.I apologize for any frustration you may have experienced.

I'll give a couple of examples of things you might find clear and that confused me: In your reply for example, you consider A2 and A3 as "events" even though, if people associate a location with an event, it is the location at which the event occurred. In physics perhaps you define it differently. Event A3 seems completely irrelevant anyway and I'm still scratching my head as to why it's there. Why do you give Bob's values as A1, A2 and A2 instead of B1, B2 and B3 (or: "why is there an 'A' prefix at all?"). This particular reply was detailed enough that I could figure out what you meant, but sometimes people provide answers with embedded assumptions and not enough detail to decode.

Among responses, "Go study the relativity of simultaneity" is a correct, but unproductive answer. "Use the Lorentz transform" misses the entire point since I was trying to cross-check the transform. Using it (properly) gives the correct answer, but without any insight. "You forgot to account for the fact that the entry and exit clocks don't appear synchronized to Bob" provides the insight and completely answers my question. It doesn't even require that many words. Credit goes to PeroK--we should put a gold star next to his reply. SlowThinker's calculations were icing on the cake since I was interested in cross-checking numbers and that was what he did.

I'm sure I phrased my question poorly. Putting up a diagram at the start would have been better. That PeroK was able to decipher my question and address the key point succinctly is amazing.

 

[Freixas]

I don't seem to be able to edit my response with the diagram, so here's the corrected diagram.

 

[PeroK]

In other words, if Alice has a row of synchronised clocks At rest relative to her at regular intervals; and Bob is moving relative to Alice in the direction of the row of clocks; then, in Bob's reference frame the clocks are closer together, but still regularly spaced, out of sync (with leading clocks lagging) and running slow (all at the same rate).

The Lorentz Transformation effectively encapsulates that difference between the two reference frames.

 

[Dale]

In your reply for example, you consider A2 and A3 as "events" even though, if people associate a location with an event, it is the location at which the event occurred. In physics perhaps you define it differently.

In physics an event is a location and a time.

 

[robphy]

To summarize @PeterDonis 's careful analysis of your original problem,
( https://www.physicsforums.com/threa...vers-in-two-frames.952195/page-2#post-6032350 )
here's a spacetime diagram on rotated graph paper
(for the case of $\beta=3/5$, which leads to easier arithmetic... then for the case of $\beta=1/2$).

By the way, $\gamma(1+\beta)=\sqrt{\frac{1+\beta}{1-\beta}}=k$ (the Doppler Factor).
For $\beta=3/5$, we have $k=2$.
For $\beta=1/2$, we have $k=\sqrt{3}\approx 1.73205$.

In my rotated graph paper diagrams,
the "light-clock diamonds" (the "ticks") have the same area
and the "aspect-ratio"="width"/"height"=$k^2$ (which is easier to see for $\beta=3/5$).

 

[SlowThinker]

"Use the Lorentz transform" misses the entire point since I was trying to cross-check the transform. Using it (properly) gives the correct answer, but without any insight.

You can work with various light clock scenarios to get qualitative insights (which clock is ahead or behind, what's shorter and what's longer, and even derive the Lorentz transformation itself), but since you came asking for actual numbers, there's really no way around LT.
It's not my first choice, I prefer to guess what the result is going to be, but LT is a reliable cross-check, and most importantly a quantitative tool.

I recommend that you spend some time looking at what are events A1 and A2, what are their x and t coordinates in Bob's frame and in Alice's frame, and what they mean.

 

[PeterDonis]

In your reply for example, you consider A2 and A3 as "events" even though, if people associate a location with an event, it is the location at which the event occurred.

"Event" in relativity means a point in spacetime--in a particular frame it is a point in space (a "location") at an instant of time. Note that all of my events have both $x$ and $t$ coordinates; that's how coordinates work in relativity, spacetime is 4-dimensional. (Two dimensions are left out here because they don't matter for this particular problem, so we just have one dimension of space and one of time.)

Event A3 seems completely irrelevant anyway and I'm still scratching my head as to why it's there.

You can leave it out, it's not really necessary. I just included it to illustrate how elapsed time on the clock that is actually co-located with Alice works.

Why do you give Bob's values as A1, A2 and A2 instead of B1, B2 and B3

Because the "A" events are the key ones to look at, since you said you wanted to figure out how the events on Alice's track look in Bob's frame. The "B" events were events associated with Bob's track, which you said you weren't really interested in. If you want to look at the "B" events, a similar analysis can be done for those.

"Go study the relativity of simultaneity" is a correct, but unproductive answer.

"You forgot to account for the fact that the entry and exit clocks don't appear synchronized to Bob" provides the insight and completely answers my question.

And, as I noted, the second statement ("clocks not synchronized") is saying the same thing as the first ("relativity of simultaneity"), just in different words. They're not two different concepts; they're just two different ways of stating the same concept. That's why I said that, if "clocks not synchronized" works better for you as a way to describe that concept, you should substitute that wherever you see "relativity of simultaneity". That is important because "relativity of simultaneity" is a much more common way of referring to this concept, so you need to be aware of what that phrase refers to.