# Quick relativity of simultaneity and length contraction thought experiment

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I was thinking about this as an example for another thread but thought it would only add confusion.

Say you have a train with proper length 100 m, traveling so fast that it contracts to 1 m according to the tracks' reference frame.

Suppose that in the train's frame, lightning strikes the front and back of the train simultaneously. How far apart are the lightning strikes on the track?

Can anyone answer this with just a few seconds thought? I can't, I have to figure out the details and calculations, but if I ask a different question...

Spoiler

Suppose the train goes through a tunnel and exactly fits inside it, in the train's frame. How long is the tunnel in the tracks' frame?

They're the same question, but only one seems intuitively obvious.

Spoiler

Is it a common to replace an abstract distance with a ruler, in thinking about things like this? Or is the first question intuitive with enough experience?

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Err, too boring? Or not intuitive enough? How about, "What proper length of tracks is contracted to 100 m in the train's frame, if the train with a proper length of 100 m is contracted to 1 m in the track's frame?"

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You said that the train's "rest length" is 100 m.   Then in the train's frame of reference the two lightening strikes are 100 m apart.  Of course, in the track's frame of reference, the two lightening strikes are 1 m apart.

Yes, that is the same question as the "tunnel" question.  In the train's frame of reference the tunnel is 100 m long.  In the track's frame of reference it is 1 m long.  (Who would build a tunnel one meter long?!)

I don't know what you mean by "abstract distance".  Any distance, pretty much by definition, can be measured, as well by a meter stick as any other way.

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3 hours ago, Country Boy said:

You said that the train's "rest length" is 100 m.   Then in the train's frame of reference the two lightening strikes are 100 m apart.  Of course, in the track's frame of reference, the two lightening strikes are 1 m apart.

[...] I don't know what you mean by "abstract distance".  Any distance, pretty much by definition, can be measured, as well by a meter stick as any other way.

Yes, they're 100 m apart in the train's frame. The strikes / tunnel can't be 1 m in the track frame unless that 1 m expands to 100 m in the train's frame, which it doesn't.

Yes, I used 'abstract' incorrectly. It's the added object (like a ruler, train platform, extra tunnel etc.), used to conceptualize the distance, that's abstract.

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10,000 meters in the track frame.  If the lightning strikes occur 100 m apart in the train frame, then scorch marks they leave on the tracks are 100 m apart as measured from the train frame.  Since the train measures the track as being length contracted by the same factor ( 1/100) as the track frame measures the train, then the proper distance between the scorch marks as measured from the track frame is 100m x 100 = 10,000 m.

Of course this means that the rear of the train ( being only 1 m long as measured from the tracks) reaches the point on the tracks where it and the tracks are hit by lightning while the front of the train is still 9,999 m short of where it will be struck by lightning, making the lightning strikes non-simultaneous in the track frame.

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1 hour ago, Janus said:

10,000 meters in the track frame.  If the lightning strikes occur 100 m apart in the train frame, then scorch marks they leave on the tracks are 100 m apart as measured from the train frame.  Since the train measures the track as being length contracted by the same factor ( 1/100) as the track frame measures the train, then the proper distance between the scorch marks as measured from the track frame is 100m x 100 = 10,000 m.

Of course this means that the rear of the train ( being only 1 m long as measured from the tracks) reaches the point on the tracks where it and the tracks are hit by lightning while the front of the train is still 9,999 m short of where it will be struck by lightning, making the lightning strikes non-simultaneous in the track frame.

I agree. Interesting that you reasoned it out right from the start, and transferred the events from the train to the tracks, at rest in their frame. Was that essential to the reasoning, or just because that's what the question asked? (If I could redo the question, I'd make them ships in space without the tracks, but would that make the question harder to reason?)

Reciprocality should become pretty intuitive given the first postulate, it seems like a major prerequisite to understanding relativistic effects.

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