Re: [Phys-l] Relativisitic mass vs Invariant mass

["Fayngold, Moses" <fayngold@ADM.NJIT.EDU>]

 John Mallinckrodt wrote:

> "The fact that that object appears to another observer to 
asymptotically approach the speed of light is the "observer's"
> problem"

 According to Relativity, all observers enjoy equal rights and
may deserve equal attention. If we engage into dismissing something
as an observer's problem, we should not do Physics.
  Later, answering my posting, John wrote:  


> "I, for one, don't "believe in" applying E = mc^2 (which I will write 
as E = m_rel c^2) although I know what it is intended to mean and I 
EVEN know to use it if push comes to shove!  The relativistic mass, 
m_rel, is just gamma times the invariant mass, m, where gamma = 
1/(1-(v/c)^2)^(1/2) so that

   E = gamma m c^2

Photons are a real challenge for this equation because they have zero 
invariant mass and infinite gamma.

Better, then, to use the fact that the (invariant) mass of a system 
is the magnitude of its energy-momentum four vector, (E/c^2, 
p_3vector/c).  That is, m = [ (E/c^2)^2 - (p/c)^2 ]^(1/2) in units 
where c = 1.  A more useful form is what I will call the "master 
equation"

   E^2 = (pc)^2 + (mc^2)^2.

Now one sees at a glance that the energy of a massless system is just 
its momentum times the speed of light.
...
 I will assume for simplicity that the electron and positron are 
essentially at rest when they annihilate.  Conservation of momentum 
tells us that the two photons have equal and opposite momenta. 
Conservation of energy then tells us that they each have an energy 
E_gamma = m_electron c^2.  Since neither has mass, the master 
equation tells us that

    p_gamma = E_gamma/c = m_electron c

Considered together, however, the two photon *system* has no 
momentum.  The master equation tells us that its total energy is 
equal to its invariant mass times the square of the speed of light. 
That is,

   2 E_gamma = m_sys c^2

or

   m_sys = 2 m_electron

as expected!  NB: A multi-photon *system* can and usually does have 
> invariant mass"
                               ****************

  This was precisely my point. The system of the two photons moving 
in the opposite directions has a nonzero invariant mass. John 
determines this mass referring to pre-existing electron-positron
system. This is perfectly OK, but what if I do not know about 
prehistory? I may only detect the produced photons themselves, and 
it is reasonably then to express the rest mass of the two-photon 
system in terms of the constituting photons only, without any 
reference to prehistory. In this case John's description only
illustrates limitations of the concept of invariant mass alone.  
The only way to remove these limitations is to invoke the concept
of relativistic mass. The system of photons has the nonzero invariant
mass only because each individual photon has a nonzero relativistic 
mass. 
  As I said before, it is OK to say that the invariant mass of the 
system of non-interacting particles is the sum

    M_ = m1_ gamma(v1) + m2_ gamma(v2) + ....             (1)

But writing this as the sum of relativistic masses 

               M_ = m1 + m2 + ...,                           (2)
  
where mi = mi_ gamma(vi), i = 1, 2, 3, ..., - looks definitely
more elegant. This is especially true in a simple case of say, two
photons in the system of center of mass. As John has admitted, 
using (1) in this case is already getting a little awkward, since 
gammas are explicit infinities, and m_s are explicit zeros: 

                 M_ = 0 inf + 0 inf                        (3) 

Please, do not tell me now of the preexisting massive particles as
the means for determining M_! As an observer detecting photons, 
I have a right (and the means!) to determine M_ using the 
constituents of current system in their own right. You may formally
try to do this, too, still without relativistic masses, but invoking
a limiting procedure using first the Eq. (1) and then letting mi_'s 
go to zero, and vi-s in their respective gammas go to c in a certain 
pre-coordinated way, and this may be OK mathematically but not
physically, because we (thank God!) do not have a continuous variety
of paticles with invariant masses all the way down to zero. You can 
make another try by just writing

                         M_ = E_/c2                          (4)

where E_ is the total energy in the system in its center of mass. 
But, again, in real situations you may have no other way to find
the total energy as to measure and then sum the individual energies.
Then you have to write
                         E_ = E1 + E2                         (5)

And finally you define the invariant mass in terms of Ei-s by 
combining (4) and (5):
                         M_ = (E1 + E1)/c2                    (6)

Here the situation is literally begging you to complete this 
by writing obvious 
                        M_ =  E1/c2 + E2/c2 ,                  (7)

but you stop short of doing this, or else you would come to my
Eq. (2) and concept of relativistic mass.
  I think anyone who really appreciates elegancy, would admit that 
it is far more elegant to introduce the concept of relativistic
mass and use the Eq. (2), rather than go through indignities of 
Eq. (1) and especially (3). 
  

Moses Fayngold,
NJIT