Is the mass of a object at rest defined by $$E=mc^2$$ where $m$ is the rest mass. I.e. does the rest mass include every thing from thermal to gravitational potential energy and every other possible energy that it could have at rest. And thus if we write the following: $$total\ energy=mc^2+potential\ energy+thermal\ energy $$ are we double counting the potential energy and the thermal energy?
Answer
The mass term includes all internal "energies". Heating up a body increases the internal kinetic energy. Binding energies also contribute to the mass (when nuclear fission occur, energy is freed and the products of the reaction are lighter than the original element), and this include any bond due to the fundamental forces of nature (which include, e.g., gravitational interaction, but only for parts within the body). Any extra energy coming from the interaction of the body as a whole with an external field doesn't contribute to the rest mass, which is a relativistic invariant.
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