Why "There is No Such Thing As Relativistic Mass"

In Newtonian mechanics, the relation between mass, velocity and momentum
is p = mv.  In Special Relativity, it is p = mv/sqrt(1 - v^2/c^2).  
It was once thought that the way to present this to people learning
relativity is to define a "relativistic mass", m(rel) = m/sqrt(1-v^2/c^2),
and thereby keep the form of the equation  p = mv, by having it become
p = m(rel) v.

By itself, this step is pretty harmless.  The difficulty is that from this
step, many people conclude that to go from a non-relativistic equation to
a relativistic one, all you have to do is plug in relativistic mass every
time mass appears and you're done.  This happens not to be the case.

Consider kinetic energy.  In Newtonian mechanics, KE = 1/2 m v^2.  In
SR is it 1/2 m(rel) v ^ 2?  No.  It's mc^2 /(1/sqrt(1-v^2/c^2) - 1).  What
about F=ma?  Can we get a relativistic dynamics by plugging in relativistic
mass here?  Again, no.  In fact, the amount of acceleration you get for
a given force depends on the *direction* of the force.  Accelerating
opposite the direction of motion is easier than transverse to the direction
of motion, and that's easier than accelerating along the direction of
motion.

So you do NOT (in general) get relativistic equations by plugging in
relativistic mass for mass in Newtonian equations.  That is, there is nothing
physical about grouping "m" and 1/sqrt(1-v^2/c^2) together into a single
quantity.  In particular, you can't plug relativistic mass into 
F = ma and F = -G M m/r^2 and get the right answers out in all cases.


A Few comments and objections I have heard:

1. But isn't the relativistic mass formalism easier to understand?

I don't think so.  It's easier to *misunderstand*, and to believe
things that aren't so to be true.  But I don't think learning the wrong
thing faster is really understanding.

2. Where can I go to read more about this?

_Spacetime Physics_, by Ed Taylor and John Archibald Wheeler.