Simultaneity is relative.

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Simultaneity is relative.

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1richardbsmith
Dic 25, 2011, 3:54pm

It is always tough for me to grasp the idea of the relativity of simultaneity. Any help or helps will be appreciated.

2daschaich
Dic 28, 2011, 12:10am

This is a great topic, since relativity of simultaneity underlies lots of apparent paradoxes (and associated confusion) in special relativity. A couple of weeks ago I was speaking with colleagues about how they teach special relativity, since one of them will be doing so in the next semester. One claimed that whenever his students devise some complicated apparent-paradox, his immediate response is: "I don't yet know exactly how this is resolved, but I'm confident it will involve relativity of simultaneity."

I'm afraid I can't think of a book where this issue is explained well. Instead, let me relate one of my favorite examples I still remember from my school days: imagine a meter-long snake moving past you at a relativistic speed. Suppose you don't like snakes and happen to have a cleaver in each hand, which you slam down simultaneously (from your point of view), 0.8 meter apart, right as the snake passes. The snake lives: from your perspective, it was length-contracted to be less than 0.8 meter long, so one of your cleavers hit (and, we imagine, bounced right back up again) just in front of it, the other just behind it.

I believe this was actually a homework exercise, and we were first asked to calculate how fast the snake had to be moving to be Lorentz-contracted to this extent. We were then presented with an apparent paradox: from the snake's point of view, it was still a full meter long, while the distance between your cleavers was contracted to be less than 0.8 meter -- so how did the snake get through? The resolution is that from the snake's perspective, the cleaver in front of it came down just before the snake reached it, while the cleaver behind it came down some time later, just after the snake passed: simultaneity is relative, so the two cleaver blows that had appeared simultaneous from your perspective were not simultaneous in the snake's reference frame.

We often visualize this sort of thing with space-time diagrams like Figure 1 in this document (which explains how to resolve the famous "twin paradox"). Space-time diagrams are discussed briefly in that document, and any good "modern physics" textbook should also present them in more detail. I'm afraid I can't recall whether the textbook I used in school, Modern Physics by Kenneth S. Krane, is a good enough text to do so. (My colleague who will soon be teaching special relativity is considering writing his own textbook for the course, since he doesn't think any of those on the market are very good.)

3richardbsmith
Dic 28, 2011, 7:02am

dashaich,
I am glad you are still willing to comment on my posts.

Simultaneity is relative then because of lenght contraction and time dilution? It seems to me that the paradox of simultaneity was established before the solution of special relativity was provided. Most examples that I find try to explain it before length and time contraction is explained, as if it follows from the invariability of c.

Each explanation seems to be an optical illusion of perspective. At least your example suggests that there is a real result from relativity of simultaneity - both perspectives leave the snakeophobe with a live snake.

I may come back with some specific questions about specific examples. I may even try your homework assignment.

Thanks,
Richard

4daschaich
Dic 28, 2011, 12:41pm

I wouldn't say that the relativity of simultaneity is a result of length contraction and time dilation, but rather that all three are related phenomena that do follow from the invariance of the speed of light c. (One could argue that relativity of simultaneity is the most difficult of these to grok, and also the most important.)

The key is that all inertial reference frames are equally valid. Observers in all inertial frames must agree about all outcomes (for example, that the snake lives in the story above). The explanations of observers in different frames may differ (length-contraction vs. non-simultaneity), but each explanation can be translated to the others with Lorentz transformations (which may be visualized through space-time diagrams). It is a matter of perspective, but not an illusion or a paradox, just something unfamiliar from everyday experience.