Distances in SR

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This is a worked out problem in the book 'Spacetime Physics'. The answer obtained by working it out using the alternate method is different. Will someone explain which is correct ?
Thank you in advance.

Problem : A command centre gets message that a meteor has just passed one of its outposts located 100 light seconds away, the meteor is travelling at 1/4 light speed and headed towards the command centre. How long does the command centre have for evasive action (radio messages travel at light speed)
Ans (in text book) : The message takes 100 light seconds to reach the command centre. As the metero is travelling at 1/4 light speed, it will take a total of 400 light seconds. Hence time available for evasive action = 300 light seconds.

Alternate calculation : Since message takes 100 light seconds to reach earth, distance from outpost to centre = 3e8 / 100 = 3e10 m
For the meteor this distance appears as 3e10*sqrt(1-0.25^2) = 2.9e10 m due to length contraction.
hence time for meteor to reach = 2.9e10 / 0.25*3e8 = 387 light secs
Therefore time for evasion = 287 light secs
 

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Dale
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Why do you care what the distance is in the meteor frame? Nobody on the meteor is trying to get ready.
 
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Janus
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This is a worked out problem in the book 'Spacetime Physics'. The answer obtained by working it out using the alternate method is different. Will someone explain which is correct ?
Thank you in advance.

Problem : A command centre gets message that a meteor has just passed one of its outposts located 100 light seconds away, the meteor is travelling at 1/4 light speed and headed towards the command centre. How long does the command centre have for evasive action (radio messages travel at light speed)
Ans (in text book) : The message takes 100 light seconds to reach the command centre. As the metero is travelling at 1/4 light speed, it will take a total of 400 light seconds. Hence time available for evasive action = 300 light seconds.

Alternate calculation : Since message takes 100 light seconds to reach earth, distance from outpost to centre = 3e8 / 100 = 3e10 m
For the meteor this distance appears as 3e10*sqrt(1-0.25^2) = 2.9e10 m due to length contraction.
hence time for meteor to reach = 2.9e10 / 0.25*3e8 = 387 light secs
Therefore time for evasion = 287 light secs
With the alternate calculation (from the meteor's point of view) there are a few more things to factor in. Since we are considering how long the Command center has by its clock, we have to take time dilation, the relativity of simultaneity and the time it takes for the message to travel the distance between outpost and command center according to the meteor.

First you need to consider the Relativity of simultaneity. If we assume that clocks at the command center and outpost both read 0 at the moment the meteor passes the outpost, in the rest frame of the command center, then the outpost clock reads 0 at that moment according to the meteor, but the command center clock already reads 25 sec past zero according to the meteor.

Now we look at the time it would take for the message to reach the command center according to the meteor. it will be ~ 96.8 light sec. The signal is traveling at c relative to the meteor, and the command post is closing in on the meteor at 1/4c, so the time it would take for the command center and message to meet by the meteor's clock would be 96.8 ls/(c+0.25c) ~77.5 sec.

The command center clock will be time dilated and will accumulate 77.5*0.986 = 75 sec.
This, plus the 25 sec it read at the moment the message was sent equals 100 sec. Thus the command clock reads 100 sec upon getting the message.

The meteor and command center are ~96.8 light sec apart, and will meet in 96.8 ls/0.25c = 387.2 sec by the meteor clock, during which time, the command center clock advances 375 sec due to time dilation. This, plus the 25 seconds it started at, gives 400 sec for the command center clock reading when the meteor reaches it.
Thus according to the meteor, the command center has 400-100 = 300 sec by the command center clock to avoid the meteor. The same answer as you get when working from the command center frame.
 

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