Mass in special relativity? 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? 
 A: 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.
A: No, $E=mc^2$ covers only the mass energy of the object. Requirement of gravitational potential or thermal energy correctly require additional terms to the energy equation.
It's worth noting they are usually neglected because they are so small compared with the mass energy. Consider a rock of mass 1kg a few metres above the surface of the Earth:


*

*Mass energy = $mc^2$ $\approx 10^{17}$J

*GPE $\approx$ $mgh$ $\approx$ $50$J

*Thermal energy = $mTq$ $\approx$ $300$kJ .

A: No, this equation only contains the mass and not the other forms of energy. In order for them to be added into the the equation and into the total energy (E), they have to be added separately but since they are negligible as compared to the mass energy, they are usually ignored. This means that their addition will not make much impact so generally they are not there in the formula. 
A: There's no doubt that the energy of an object increases when the object is heated. Therefore there can't be any doubt that the mass of an object increases when the object is heated.
There's no doubt that the energy of an object increases when the object is heated by first lifting it up and then dumping it down on the hard ground. Therefore there can't be any doubt that the mass of an object increases when the object is heated by first lifting it up and then dumping it down.
Notice the pattern there: Undisputed increase of energy -> undisputed increase of mass.
When exactly does the energy of an object increase when the object is heated by first lifting it up and then dumping it down? One possible answer is: When the moving ground hits the still standing object. So therefore we can say that lifting an object does not change its energy or mass.
