User:Josephnew

The True Concept of Mass

The concept of mass in modern physics is mostly important for fundamental understanding of relativity. The common mistake of dependence of mass to velocity is common pedagogical virus specially in undergrad text books. Both energy and momentum are velocity dependent but not mass. Mass is a Lorentz invariant and does not depend on reference frame. The terms "rest mass" and "relativistic mass" are redundant and misleading. There is only one mass in physics, m, which does not depend on the reference frame. By rejecting the relativistic mass there is no need to call the other mass the "rest mass" and to mark it with index 0. Einstein in a letter to Lincoln Barnett, 19 June 1948, said "It is not good to introduce the concept of the mass as $$M = m/\sqrt{1 - v^2/c^2}$$ of a moving body for which no clear definition can be given. It is better to introduce no other mass concept than the 'rest mass' m". We know that the "Length" of momentum four vector is invariant in Lorentz transformation. By using c = 1 units:


 * $$ k^2 (invariant)\,\!$$ = $$ (E,\vec{p})^2 = E^2\,\!$$ - $$ \vec{p}^2 =  \gamma^2m^2\,\!$$ -  $$ \gamma^2m^2\beta^2\,\!$$ = $$ \gamma^2m^2\,\!$$ ( 1 - $$ \beta^2\,\!$$) = $$ m^2\,\!$$
 * $$ k^2 = m^2 \,$$ [ $$mass^2 \,$$ is invariant ]

Mass is invariant quantity that doesn’t depend on the choice of inertial reference frame! Writing mass in the form:
 * $$ m =  {{m_0 }\over \sqrt{1-\beta^2}},\!$$

is completely wrong. As we said before mass is Lorentz invariant and so does not change in different reference frames. Also Einstein's famous equation had been written in several different forms in different books:
 * $$E_0= mc^2 \,$$,
 * $$E_0= m_0c^2 \,$$,
 * $$E= m_0c^2 \,$$,
 * $$E= mc^2 \,$$,

But the only correct form of it as he mentioned later is:
 * $$E_0 = m c^2 \,$$

In conclusion we can say, both energy and momentum depend on velocity, but mass does NOT.
 * $$ E = \gamma m c^2 \,$$
 * $$ P = \gamma m v \,$$
 * $$ m = m(v) \,$$ is a "  Pedagogical Virus  "

See: e.g. L.B. Okun, 2006, “The Concept of Mass in the Einstein Year” http://arxiv.org/abs/hep-ph/0602037