Relativistic Mass and Volume Change in a Rapidly Expanding Sphere

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Frogeyedpeas
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suppose I have a 10 kg sphere with homogenous distribution of mass (aka density is same everywhere) that is 10 m^3. Now suppose I rapidly increased its radius at a speed of 1/2c m^3 (and because its homogenous the volume increased correspondingly). Now my question is whether that would cause the "relativistic mass" of the box to increase. Clearly some particles are moving at high velocity while one in the middle is stationary. Would it be possible to treat this entire system as opposed to each individual particle of mass?
 
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The sphere as a whole system is not moving and would not have any relativistic mass. However, let's do away with the term "relativistic mass", as it is confusing and doesn't need to be used any more. Instead we can use the term "momentum", as Einsteins equation e=mc^2 is actually an incomplete portion of the larger equation which uses momentum as well.

In a real world scenario this expansion would require a source of energy, and assuming this source of energy already existed inside the sphere then the total energy/mass of the system would not increase nor would it's momentum.
 
See, if one just considers the outer layer of the sphere; then, yes it is gaining momentum (ie becoming heavier) due to the lorentz factor, but at the same time as the volume expands this mass gets distributed around so the sphere is definitely becoming "heavier". Relativity says it has to. I was wondering if this could be measured by some lorentz-volume factor that measures the increase or decrease in the quantity (m/(1-v^2/c^2)^(1/2). But instead of using v one uses d(volume)/dt
 
Frogeyedpeas said:
so the sphere is definitely becoming "heavier". Relativity says it has to.
This is not correct if, as Drakkith mentioned, the source of energy was inside the sphere.

If the energy comes from outside the sphere then the mass of the sphere increases by the additional energy divided by c^2. If the energy comes from inside then the mass is unchanged.