## Deriving the equation for length contraction and e=mc^2

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angle.alpha
Posts: 21
Joined: Tue Mar 08, 2011 8:28 pm

### Deriving the equation for length contraction and e=mc^2

what am i doing wrong in deriving the equ of length contraction & why?
1st Method: Using Lorentz transformation #1, x' = (x-ut)/rt(1-(u^2)/(c^2))
I thought L_v = x'_2 - x'_1 simplifies to (x_2-x_1)/rt(1-(u^2)/(c^2)) = L_0/ rt(1-(u^2)/(v^2))
however for correct equ, L_0 = x'_2 - x'_1and L_v = x_2-x_1. why isnt the shifted FoR the frame of L', L_v?
In my 2nd method i just realised on typing out was sort of flawed, tried to fix itand made it worse...
here I measure length of a moving train AB by use of a laser gun.
inside the train L_0 = AB = ct_0
outside the train L_v = AB' (fired from A, in the time taken for light to travel, B has moved to B')
so L_v = c.t_0 - u.t_v
= L_0 - u.t_0/rt(1-(u^2)/(c^2))
= L_0.rt(1-(u^2/c^2)) - u.t_0

also, any thought experiments using which E=mc^2 can be derived? One method was found at http://www.davidbodanis.com/pages/promi ... hool2.html but it seems dodgy because it assumes photons have mass and mass m was the final m in E=mc^2.

angle.alpha
Posts: 21
Joined: Tue Mar 08, 2011 8:28 pm

### Re: Deriving the equation for length contraction and e=mc^2

in the 1st method, is it simply because of the notation? is x' is the position of the moving object from the moving frame and so that correlates to x_2- x_1 i.e. L_0?

joe
Posts: 745
Joined: Fri Aug 29, 2003 11:57 am
Location: Sydney

### Re: Deriving the equation for length contraction and e=mc^2

For relativistic length contraction, you can simply use time dilation and the speed of light: Define a light year in the normal way and, with invariant speed of light, these differ between observers. See chapter four of
http://www.phys.unsw.edu.au/einsteinlight

There's a section on the Lorentz transforms at
http://www.phys.unsw.edu.au/einsteinlight/jw/module4_Lorentz_transforms.htm
and it includes, I think, the ones that you need.

The relativistic version of the work-energy theorem (and hence that famous equation E = ...) takes a page or two. It's at
http://www.phys.unsw.edu.au/einsteinlight/jw/module5_dynamics.htm

Joe

angle.alpha
Posts: 21
Joined: Tue Mar 08, 2011 8:28 pm

### Re: Deriving the equation for length contraction and e=mc^2

could you please go into more detail regarding the light-year method of deriving length contraction? i don't seem to be able to bash the maths out, from my interpretation of your method. and apparently my train thought experiment apppears to be invalid and doesn't work; i need to use a mirror to reflect the light back in which case it is purely the horizontal arm of the MM exp and its all good.

joe
Posts: 745
Joined: Fri Aug 29, 2003 11:57 am
Location: Sydney

### Re: Deriving the equation for length contraction and e=mc^2

First, derive the time dilation, using a light clock, either parallel to or at right angles to the direction of motion.

So we get t' = gamma.t

Now let each observer define a unit of length. Let's say a light-nanosecond (which is conveniently about 30 cm). They can set their rulers to be one light-nanosecond long.

But each sees the other's clock to be running slow, so they see the other's rulers to be contracted, by factor gamma. See
http://www.phys.unsw.edu.au/einsteinlig ... htm#length

Try
http://www.phys.unsw.edu.au/einsteinlight
for both the explanation and the maths

Joe