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



Moses Fayngold wrote:

1. If this is truely the case, and if we believe in E = mc2, then the photons have no energy of any kind either;

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.

2. In this case, if an electron and a positron annihilate producing, say, two photons having each neither mass nor energy of any kind, then where the mass and the energy of the electron-positron pair go to?

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.

3. Note that in this case (that is, assuming the invariant mass as the only legitimate entity) even this invariant mass would disappear entirely and with no traces.

It is, indeed, a good thing that isn't so!

--
John "Slo" Mallinckrodt

Professor of Physics, Cal Poly Pomona
<http://www.csupomona.edu/~ajm>

and

Lead Guitarist, Out-Laws of Physics
<http://www.csupomona.edu/~hsleff/OoPs.html>