Re: [Phys-l] E=mc^2 because E=mc^2?

[John Denker <jsd@av8n.com>]

On 04/25/2007 12:33 PM, Fayngold, Moses wrote:

>   In this respect I agree with Sciama, not with John. 

I don't know what Sciama said about this, but my usage agrees
with Misner/Thorne/Wheeler, with Weinberger, ..... and with
Einstein.

  Reference:   Gary Oas, ``On the abuse and use of relativistic mass''
    http://arxiv.org/PS_cache/physics/pdf/0504/0504110.pdf


> The photons 
> are massless - in the sense that they have zero rest mass; 
> however, they have a non-zero relativistic mass m.

In a supposedly-introductory article, if they are going to
use "mass" in a way contrary to decades-old convention, it
might have been nice to explain that a little bit.

>   Moreover, there are situations when a single photon in free space
> has A NON-ZERO REST MASS! Such a situation can be created in a quantum
> optics experiement when a single photon is passed through a 
> beam-splitter and no attempt is made to locate path chosen by the 
> passed photon; in this case the meaningful characteristic of the photon
>  is its average momentum, which is, in this state, NOT equal to E/c. 

1) I didn't say it was equal to E/c, and
2) This is not even remotely relevant to the article in question ...
 ... so why bring it up?

======================

In any case,
  A) There are lots easier ways of getting to E=mc^2, and
  B) That's not where you want to get to, anyway.

The recommended modern (i.e. post-1908) approach is to write
  m^2 c^4 = E^2 - ps^2 c^2                  [1]
or simply
  m^2 = E^2 - ps^2

where m is "the" mass (which is an invariant Lorentz scalar)
and where ps is the 3-momentum i.e. the spatial part of
the 4-momentum.

We see that -m^2 is just p·p i.e. the dot product of the
4-momentum with itself.  The 4-momentum is also known as
the [energy,momentum] 4-vector.

Equation [1] has the advantage that it includes E=mc^2 as a
special case, and makes it clear that mc^2 is the /rest energy/
(not the total energy).

Another advantage is that it builds upon and reinforces what
the students know about vectors.

Next year is the 100th anniversary of spacetime.  All the
students have heard of spacetime;  it's well established as
part of pop culture.  Isn't it about time we taught them
what spacetime really is, and how to use it?