Understanding the Lorentz Transformation Equation for Time

[cr41g]

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.

 

[Ibix]

vt? Are you sure?

 

[cr41g]

Sorry vt'

Do you know the second part or even where I should start?

 

[Ibix]

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?

 

[cr41g]

X' I think. Sorry if I sound a bit stupid. I'm in my first year and this is only my second question.

 

[phyzguy]

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.

 

[cr41g]

Ok I've done what you have said and got-

t'=(x+γ(γx-γvt))/γv

Now I am competent lost

 

[cr41g]

Completely*

 

[phyzguy]

This is correct, you just need to simplify it. What is γ equal to?

 

[Ibix]

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.

 

[cr41g]

Can anyone actually show me this step as I am completely thrown. I just can't make sense of it.

 

[Ibix]

You have a sign wrong in your expression for t', I just noticed. That might be your problem.

 

[cr41g]

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?

 

[Ibix]

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 $\beta=v/c$ to save ink.

 

[cr41g]

Ok. Well just got into bed so I will give it a go before I go to university tomrrow. Thanks for your help guys.

 

[Chestermiller]

When you studied algebra, did they teach you how to solve 2 linear algebraic equations in two unknowns?

 

[cr41g]

Yeah as far as I'm aware no hate when algebra is explained in words. But yeah I think I did.

 

[Chestermiller]

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

 

[cr41g]

Thank you