Solving Relativity Problem: Prove c^2τ1τ2 is ∆s^2

[SHawking01]

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!

 

[Goddar]

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?

 

[SHawking01]

Sorry, I am not the smartest. We have to use the Lorentz transformation to find that, right?

 

[phinds]

Sorry, I am not the smartest.

Well then, you are not living up to your log-on name now are you? :D

 

[Goddar]

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?

 

[SHawking01]

Is it (τ1+τ2)/2?

 

[Goddar]

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?

 

[SHawking01]

r=c(τ2-τ1)/2

 

[Goddar]

Yep. Now the first answer was your Δτ, the second your Δr. So what's the invariant interval?

 

[SHawking01]

∆s^2=c∆t^2-|∆r|^2 (this is the invariant interval right?)

 

[Goddar]

Yes. Don't forget to square c as well...

 

[SHawking01]

Okay, well I think I have it from here. Thank you!