It also applies to things like gravitational potential energy, or velocity. Climb a flight of stairs, and you gain energy. Gaining potential energy really means that you've gained mass, and so you're heavier at the top of the stairs than at the bottom. The amount that you are heavier depends on the amount of energy you gained, converted into mass. Similarly, as you gain velocity, you gain kinetic energy, which also makes you heavier. If you wave your hand in front of your face, your hand's motion causes it to gain mass. Velocity also dilates time, so time passes ever so slightly slower than for the rest of your body.
This typically only matters in a relativistic context, and doesn't impact day to day life. The difference in passage of time can be measured by an atomic clock if you put it on a rocket into space, but is otherwise insignificant. It's unlikely any scale could measure your weight-gain from climbing stairs, since the amount of energy you gain is insignificant when converted into mass. You can compute the mass gain by solving for "m" in the formula E=m*c^2, so m=E/(c^2).
The speed of light c is a really big number, and so to compute the mass you gain, you're dividing the energy by c^2, which is a much bigger number. Thus a gain to kinetic or potential energy does not noticeably affect your mass in day to day situations. Conversely, if you can convert any meaningful part of your mass into energy, then it's an absolutely tremendous amount of energy: atomic weapons.
> Gaining potential energy really means that you've gained mass
I've never heard this before. Can you link to some further explanation? Intuitively, if anything, you'd lose mass, because you're in a place now where space is less curved than where you were before.
The system composed of you + planet gains mass. You can not measure a different rest mass for yourself at either situation, so you'll probably attribute the extra mass to the planet (and the planet to you).
And yes, there's also some change due to changes in gravity. I'd expect that to be much smaller.
Mass is constant in this equation. It is absolutely incorrect to talk about a "rest" mass and a "relativistic" mass, only energy and momentum are relativistic, while mass is always a constant. Algebraically it may make a tiny bit of sense, but there is no physical meaning behind it.
Basically, there are two schools of thought. An outdated one, which merges γ into m, and the current one (e.g., the Landau lineage) which leaves m alone. You won't find any "relativistic mass" in any decent source published since the famous theoretical minimum ( https://en.wikipedia.org/wiki/Course_of_Theoretical_Physics ).
There is a very good reason for this approach. In the simplest, most classic way of deriving the special relativity from the first principles, the 4-velocity is introduced before the 4-momentum. I.e., γ and m are coming from different places and are not connected in any ways whatsoever.
"In general, relativistic and rest masses are equal only in systems which have no net momentum and the system center of mass is at rest". This is the case of a system being heated. I'm still led to believe that the mass is still increased by the increased motion of the particles in that system. https://en.wikipedia.org/wiki/Mass_in_special_relativity
As I said, algebraically both approaches are equivalent, so there is not much harm in using this notion. But the "relativistic mass" does not have physical meaning and does not make any sense because of the way relativity theory is defined. And I would not count wikipedia as an authoritative source, anyway. 2nd volume of the Course is a bit more legit.
