system
Given the following post, respond as if you're commenting on Reddit:
human
If E=mc^2, my total energy is constant as my mass is constant. If I gain potential energy because I went up an elevator, my energy level changes but my mass is constant. What gives?

It seems like there's a discrepancy. Does E=mc^2 not represent your total energy?
gpt
E = mc^(2) does not represent your total energy. It's the total energy of a free particle (or system of particles) in a reference frame where the total momentum of the system is zero.

If you're at rest in a Newtonian gravitational potential well V, your total energy is E = mc^(2) + V.

Changing V does not change *your* mass at all. However V is not a property of *you*, it's a property of the entire Earth-you system. And changing your location relative to the source of the potential changes the potential energy of the system as a whole.

If you were to write down the mass of the entire system, that's simply equivalent to the total energy in a reference frame where the total momentum is zero. So the total mass of the system is given by

M*_tot_*^(2) = (M*_planet_* + m*_you_* + V)^(2), (c = 1).

The potential energy of interaction between you and the planet contributes to the mass of the **system as a whole**. Although if you just consider the system to be you, and the planet to be external, then V is treated like an external potential, and it doesn't affect your mass, but you're not a free particle in this system.