Re: Mass: Corrections,etc.

[Hugh Logan <hloganPHY@CFL.RR.COM>]

Hugh Logan wrote:

> . Neither can I find
> "relativistic mass" in Einstein's
> popular book, _Relativity, The Special and General Theory_, Crown
> Publishers. His "m" is rest mass. He discusses how the rest mass of a
> body is changed by an amount E_0/c^2 when the body gains or loses energy
> E_0 (not the rest energy in his notation.) Einstein gives the total
> energy mc^2/sqrt(1-v^2/c^2), expanding it  to compare it with the
> classical kinetic energy, only later interpreting the rest energy, mc^2.
> He often writes only the right side of the equation, so there is no
> notation for rest mass or total mass in this book

The last sentence was intended to be:

> He often writes only the right side of the equation, so there is no
> notation for rest energy or total energy in this book.

As mentioned, there is no mass other than rest mass m in Einstein's book.

I also suggested that the fact that the relativistic kinetic energy
reduces to the classical limit, (1/2)*m*v^2, made it reasonable that the
m in the energy component of the energy-momentum 4-vector is inertial
mass, as that is the role of m in the classical expression for kinetic
energy.
The same could have been said for the three momentum components when
v<<c. As John D. pointed out, the norm of the energy-momentum 4-vector,
(E, p_x, p_y, p_z),  is the mass m -- the rest mass. In the 4-vector
formulation,  m does not depend on the velocity or the speed of a
particle or object. Although relativity tells us that m is essentially
the internal energy of the particle, it is also the measure of
mechanical inertia of the particle that would be used in classical
mechanics.

One might ask if the now largely abandonned formulation of relativity
which uses "relativistic mass,"
m=gamma*m_0 where m_0 is rest mass, doesn't provide a more intuitive
representation of inertia thought of as a resistance to a change in
motion. It would seem that a particle  moving at a speed of
0.99*c would be more difficult to accelerate than one at or nearly at
rest corresponding to its greater (relativistic) mass.

I suspect that one reason the 4-vector theory is favored is that it fits
in better with general relativity, where local regions of spacetime can
be thought of as approximately flat  much as localized regions of the
earth's surface are considered to be flat.  The 4-vector theory is
neater, and brings out the idea of invariance more clearly. I suspect
the formulation with relativistic mass could be used equally well in
practical work such as the design of particle accelerators. I think one
can extract the same useful formulas such as E^2=E_0^2+p^2c^2 from
either, and it hardly matters, as far as practical results are
concerned, whether one writes E=mc^2 (with m=gamma*m_0) or
E=gamma*m*c^2  (with m= rest mass).

Hugh Logan