Blog post: ajb: Energy is not frame independent

For a closed system we have energy conservation and this is deeply tied into the notion of the physics being unaltered under shifts in time. A salient point here is that we have fixed a frame in which to measure the energy. It is not true that the energy is the same measured in any inertial frame, what is true is that it is conserved.

 

 

 

 

A simple example

 

 

 

Let us consider a free particle of mass $latex m$ in one dimension. Let us pick some coordinates $latex (x,t)$. With respect to this frame we have that the kinetic energy

 

 

 

$latex E = frac{1}{2 m}p^{2}$,

 

where the momentum $latex p $ is given by $latex m v = m dot{x} $. The dot is the derivative with respect to time. (I am using the usuall abuses of notaion here, this should not confuse)

 

 

 

Now let us consider a Galilean transformation

 

 

 

$latex x' = x = ut$

$latex t' =t$.

 

Then we calculate how the momentum transforms

 

 

$latex p = m frac{d}{dt}left( x' - ut right) = m left( v' +u right)= p' + mu$.

 

Or $latex p' = p - mu$.

 

 

 

This is exactly what you would expect. So on to the energy...

 

 

 

$latex E = frac{1}{2m}p^{2} = frac{1}{2} left(p' + mu right)^{2}$

$latex = E' + frac{1}{2}mu^{2} + up' = E' - frac{1}{2}mu^{2}+ up$

 

so

 

 

 

$latex E' = E + frac{1}{2}mu^{2} -up$.

 

Clearly $latex E' neq E$

 

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Recap

 

 

Energy conservation should not be confused with energy being observer independent, it clearly is not.

 

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