Re: [Phys-l] Relativity Question

[John Denker <jsd@av8n.com>]

On 05/06/2007 03:28 PM, David Abineri wrote:
> I am looking for a high school level explanation for the following question.
>
> Does a compressed spring have more mass than a relaxed spring (at its 
> natural length) by virtue of it's having more energy while compressed?

Yes.

> Or, perhaps more generally, how does potential energy figure in to a 
> relativistic point of view.

PE fits in as part of the mass muuuch better than KE does.

> I know this is a little vague but perhaps someone who understands my 
> question might attempt a response appropriate to a high school level class.

I don't see anything vague about it.

In the spacetime view of things, where length means invariant length,
time means invariant time, and mass means invariant mass, the following
holds as an immediate consequence of the geometry of 4-vectors:

The dot product of the [energy,momentum] 4-vector with itself is:

      m^2 c^4 = E^2 - ps^2 c^2
or
      m^2     = E^2 - ps^2          in units where c=1

where ps is the 3-momentum, i.e. the spatial piece of the 4-momentum.

This makes it clear that mc^2 is the /rest energy/ in a particular frame,
i.e. the energy when the 3-momentum is zero in that frame.  It is clearly
not the total energy.

Potential energy is necessarily included in the rest energy, and therefore
in the mass.

This is routinely and directly observable in the _mass deficit_ associated
with the binding energy of nuclides and suchlike.