Deriving Lorentz Transformation

[Arman777]

Homework Statement

How can we derive Lorentz Transformation using the length contraction and time dilation equations of relativity ?

Homework Equations

$γ = 1/ (\sqrt{1-u^2/c^2})$
$t = t_0γ$
$L = L_0/γ$

The Attempt at a Solution

[/B]
In position Lorentz Transformation calculations, simply I thought that the distance will get shorter since there's length contraction.
So in galilean transformation the position transformation is $x= x_0-ut$. So this "length" will get shorther by the amount of $γ$.
So we have $x = γ(x_0-ut)$

For the time part I am kind of stuck. I didnt understand where the $ux/c^2$ comes from.

 

[PeroK]

Homework Statement

How can we derive Lorentz Transformation using the length contraction and time dilation equations of relativity ?

Homework Equations

$γ = 1/ (\sqrt{1-u^2/c^2})$
$t = t_0γ$
$L = L_0/γ$

The Attempt at a Solution

[/B]
In position Lorentz Transformation calculations, simply I thought that the distance will get shorter since there's length contraction.
So in galilean transformation the position transformation is $x= x_0-ut$. So this "length" will get shorther by the amount of $γ$.
So we have $x = γ(x_0-ut)$

For the time part I am kind of stuck. I didnt understand where the $ux/c^2$ comes from.

That term is from the relativity of simultaneity. You need that as well to derive Lorentz.

 

[Arman777]

That term is from the relativity of simultaneity. You need that as well to derive Lorentz.

But how. How can I derive it ?

 

[PeroK]

But how. How can I derive it ?

You have to derive the "leading clocks lag" rule and using this along with time dilation and length contraction you can derive Lorentz.

 

[Arman777]

okay I manage to derive it thanks.