Is there even a relativistic mass or just relativistic momentum? How does one reason to prefer one over another? What is the problem with saying a hot gas will have more mass/inertia to it?
Answer
Relativistic mass is not a useful concept. It is the same thing as total energy, just in different units. Relativistic momentum on the other hand is useful, because there is no such duplicity. An object composed of fast moving constituents does have higher rest mass, because there is more energy in this object. But you don't need relativistic mass to calculate the rest mass of such object. Total energy is enough.
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