relativistic mass controversy ... apparently physicists these days do not use the terms "rest mass" or "relatistic mass" ... even though that's what I learned and (I thought) was in vogue when I was doing physics in the 80's.
The formulas for conservation of energy and momentum in special relativity are not in dispute - just what you name the terms.
I was taught that it made sense to think of the rest mass of a particle as invariant, \( m_0 \) , and that the energy and momentum \( E = m c^2 \) and \(p = m v \) in relativity were \( E= \gamma m_0 c^2 \) and \( p = \gamma m_0 v \).
But apparently it's much more common these days to just treat these as the definitions of E and p, including \( gamma \), and just write mass "m" as our textbook does, meaning what I learned as "the rest mass" \( m_0 \).
See for example "two definitions of mass and why I use only one".
Our textbook also uses this language for mass and energy, which I found confusing at first but now see is consistent ... just make sure you know what you mean.
Using this terminology, "mass" means what I called "rest mass", and so
$$ E^2 - c^2 p^2 = [m c^2]^2 $$
where the m is the mass of the particle when it isn't moving. With this terminology, the mass of a particle never changes, photons do not have mass, and mass and energy are different properties, and \( E = m c^2 \) is only true for particles which not moving and therefore have no momentum. It's never true for photons, because photos have zero mass (with this definition of mass) but do have energy.
With this definition of mass, it is not mass which causes and is affected by gravity, it's energy. And it would be wrong to say that Einstein's big result was that mass is energy.
I see that wikipedia calls this the ladder paradox. Hmmph. Again not the way that I learned it ...
Other stuff you need to have on hand and/or be able to derive :
