Re: [Phys-l] Relativity Question

[John Denker <jsd@av8n.com>]

On 05/07/2007 10:38 PM, Hugh Haskell wrote:

> Since the electrical forces that are released when one allows the 
> spring to expand are all internal, perhaps this is an example of 
> something that can be thought of as a complete system in itself and 
> thus has the energy stored within it, so we can think of the mass of 
> the spring as increasing when it is compressed and decreasing when it 
> expands.

Quite so.  I can't imagine any doubt or controversy about this.

It's required by conservation of energy.

Conservation of energy is most precisely expressed in terms of
continuity of world lines.  Using the Stokes theorem, you can
switch back and forth between thinking in terms of divergence
at a point, or flow across the boundary of a region.
  http://www.av8n.com/physics/conservative-flow.htm

To the extent that the spring has a well-defined boundary, or
can be surrounded by a well-defined boundary of your choosing,
you can apply conservation of energy to that region.

That tells you the energy.

In all generality, the mass is related to the [energy,momentum]
4-vector by
           m^2 c^4 = E^2 - ps^2 c^2           [1]
or
           m^2 = E^2 - ps^2
which can be simplified to E = m c^2 if you choose a frame
comoving with the center of mass of the spring.

Yes, equation [1] applies just fine to complex systems.   How
could it not?  Whatever X is, if X has an [energy,momentum]
4-vector it is possible (and often useful) to apply equation [1].

Energy is energy.  If you change the energy of a spring (or
anything else) it must affect the mass in accordance with
equation [1].