What’s gone wrong with my derivation of the Lorentz Factor?

[ryan2]

Homework Statement

Trying to derive the Lorentz factor

Relevant Equations

Lorentz Factor

https://ibb.co/Wy9Pq8j

I’ve gotten something that looks almost correct, but the expression in the root is 1 + v^2/c^2 instead of minus. I understand intuitively why my answer is wrong, but don’t know where mathematically. Could someone please help me see where I went wrong? Thank you!

 

[ryan2]

If any clarifications are needed, I’m more than happy to make them! I understand my work isn’t the cleanest looking.

 

[BvU]

If any clarifications are needed

You might reveal which variable is in which frame of reference ...

 

[ryan2]

You might reveal which variable is in which frame of reference ...

View attachment 349017

The left hand side pertains to the stationary one, and the right to the moving one. The variable x represents the Lorentz factor(it should have been on the right). D1 is common between the two, and I simplified it to c*t.

 

[Sagittarius A-Star]

I simplified it to c*t.

What is t ?

 

[ryan2]

What is t ?

The time it takes for light to make it from the bottom to the top of the vertical distance, from a stationary frame. Is it incorrect to assume it would remain the same for the moving frame?

 

[Steve4Physics]

The time it takes for light to make it from the bottom to the top of the vertical distance, from a stationary frame. Is it incorrect to assume it would remain the same for the moving frame?

It is incorrect.

In your first (left) diagram, light travels a distance D at speed v [edit - that should of course say 'at speed c'] So you can find how long it takes.

In your second (right) diagram, light travels a distance bigger than D - but at the same speed (since the speed of light is constant). So the time taken is longer than for the first diagram.

Read about 'time dilation'.

 

[Sagittarius A-Star]

Is it incorrect to assume it would remain the same for the moving frame?

Yes. The light moves with $c$ along the diagonal path. In a light-clock the light-pulse moves forth and back between the mirrors within one period of the clock, so I call 1/2 of this period of the moving clock $t_m$. Analog for the stationary clock: $t_s$.

$c^2{t_s}^2 + v^2{t_m}^2 = c^2 {t_m}^2$ (theorem of Pythagoras)

$\gamma = t_m / t_s$