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Relativity Problem

  • Thread starter SHawking01
  • Start date
  • #1
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Homework Statement



An inertial observer O bounces a radar signal off an arbitrary event P. If the signal is emitted and received by O at times τ1 and τ2 respectively, as indicated by O’s clock, prove that the squared interval ∆s 2 between O’s origin event (i.e., its spatial origin at time τ = 0) and P is c 2 τ1τ2.

Homework Equations


∆s^2=ct^2-|r|^2

The Attempt at a Solution


u'=γ(u1-βu4)?? I am not even sure if I need to do a Lorentz transformation. Please give me some direction!
 

Answers and Replies

  • #2
205
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Hi. I assume your solution is supposed to read: c2τ1τ2...
So:
What is the distance between O and P?
At what time on the clock does the signal reach P?
 
  • #3
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Sorry, I am not the smartest. We have to use the Lorentz transformation to find that, right?
 
  • #4
phinds
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Sorry, I am not the smartest.
Well then, you are not living up to your log-on name now are you? :D
 
  • #5
205
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No: you're looking at everything from O's perspective, and O is not moving so try to see this without thinking about relativity.
- You start the clock at 0;
- At τ1 you send a radar signal (what does that tell you about the speed of your signal?);
- At τ2 the signal is back to you after having bounced from P.
So now: at what time did the signal get to P? How far did it travel from you?
 
  • #6
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Is it (τ1+τ2)/2?
 
  • #7
205
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If that's the answer to the first question: yes. Do you see why? (The trip to P takes (τ2–τ1)/2 but the clock started at 0 so you have to add τ1.)
Now what's the speed of your signal? How far is P, then?
 
  • #8
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r=c(τ2-τ1)/2
 
  • #9
205
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Yep. Now the first answer was your Δτ, the second your Δr. So what's the invariant interval?
 
  • #10
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∆s^2=c∆t^2-|∆r|^2 (this is the invariant interval right?)
 
  • #11
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Yes. Don't forget to square c as well...
 
  • #12
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Okay, well I think I have it from here. Thank you!
 

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