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Special Relativity question

  • Thread starter cr41g
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  • #1
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Question
Suppose an inertial frame of reference S’ moves at a constant velocity v in the
positive x-direction with respect to a second inertial frame S. The Lorentz
transformation from S to S’ for the x coordinate of displacement is given by:
x′ =γ (x − vt)
Write down a corresponding expression for the inverse transformation, i.e. from
S’ to S, giving x in terms of x’ and t’.
Use these two expressions to derive the Lorentz transformation equation for time:

t'=γ(t-vx/c^2)


I think I have the first part, I answered x =γ (x' + vt). But the second part I have no idea I have been looking online and even watching lectures on youtube.
Thanks in advance.
 

Answers and Replies

  • #2
Ibix
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vt? Are you sure?
 
  • #3
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Sorry vt'

Do you know the second part or even where I should start?
 
  • #4
Ibix
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You've got x' in terms of x and t, and x in terms of x' and t'. Which variable don't you want to appear in the final expression?
 
  • #5
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X' I think. Sorry if I sound a bit stupid. I'm in my first year and this is only my second question.
 
  • #6
phyzguy
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Take your answer for the first part, substitute in for x' in terms of x and t, then solve for t' in terms of x and t.
 
  • #7
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Ok I've done what you have said and got-

t'=(x+γ(γx-γvt))/γv

Now I am competent lost
 
  • #8
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Completely*
 
  • #9
phyzguy
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This is correct, you just need to simplify it. What is γ equal to?
 
  • #10
Ibix
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Expand the Lorentz gammas and collect terms. Courage! This one always looks a mess to me until it all clicks into place at the end.
 
  • #11
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Can anyone actually show me this step as I am completely thrown. I just can't make sense of it.
 
  • #12
Ibix
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You have a sign wrong in your expression for t', I just noticed. That might be your problem.
 
  • #13
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Yeah I found that when I started from scratch. But I'm still at a loss. It just looks a mess. Do you expand all the gammas?
 
  • #14
Ibix
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Collect your x and t terms. The t term should fall out straight away. That leaves the x term. I'd suggest that if in doubt, expand, is a good rule of thumb here. You might want to use [itex]\beta=v/c[/itex] to save ink.
 
  • #15
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Ok. Well just got into bed so I will give it a go before I go to university tomrrow. Thanks for your help guys.
 
  • #16
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When you studied algebra, did they teach you how to solve 2 linear algebraic equations in two unknowns?
 
  • #17
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Yeah as far as I'm aware no hate when algebra is explained in words. But yeah I think I did.
 
  • #18
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Yeah as far as I'm aware no hate when algebra is explained in words. But yeah I think I did.
If that is the case, that's all you have to do in this problem. Your two unknowns are x and t. Solve for 'em.

Chet
 
  • #19
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Thank you
 

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