jpo said:
Scenario A:------------
1A) A box with mass M contains mass m, their weight is g(m+M)
2A) the mass m is (somehow) converted to energy E=mc^{2}
3A) at this moment, the box still has weight g(m+M)
Let me try to specify these scenarios more carefully so that the underlying issues in analyzing them will be clearer. I think what is really meant here is:
1A) A box with invariant mass M contains some substance with invariant mass m, which is at rest with respect to the box. (Note that we specify "invariant mass" to make it clear what we are talking about.) When sitting at rest on the surface of the Earth, where the local "acceleration due to gravity" is g, the box with its contents registers a weight g(m + M) on a scale. (However, note carefully that this "weight" is due to the box being accelerated, i.e., not in free fall. If we dropped the box from a height in vacuum, along with the scale, the scale would read zero.)
2A) The substance inside the box is converted to energy, say in the form of radiation. The net motion of the radiation is still zero with respect to the box, so it still has invariant mass m considered as a single system (even though the individual particles of radiation, the photons, each have zero invariant mass--the fact that they are bouncing around inside the box instead of escaping to travel freely is what gives the total system a nonzero invariant mass).
3A) As long as the radiation remains confined inside the box, and the box remains sitting at rest on the surface of the Earth, then yes, the reading on the scale will be unchanged.
jpo said:
Scenario B: ----------
1B) A box with mass M has weight gM
2B) external force imparts kinetic energy E to the box, such, that E=mc^{2}
3B) at this moment, the box has weight g(m+M)
This one is more complicated, but *not*, as you will see, because of any effect of kinetic energy:
1B) Again, we have a box with invariant mass M sitting at rest in a gravitational field g, so it registers a weight gM on a scale.
2B) We apply a force to the box that imparts a kinetic energy E to it (where we write m = E/c^2 to give a "mass equivalent" to this energy), as measured in its original rest frame (i.e., as measured in a frame at rest with respect to the surface of the Earth). You didn't specify in what direction the box moves, and different cases will give different answers for the weight:
2Bi) The box moves horizontally. In this case, the reading on the scale is unaffected by the box's motion, since the motion is perpendicular to the direction of gravity. 3Bi) So the scale still reads gM, even though the box's total energy, as measured by an observer at rest with respect to the Earth, is Mc^2 + E--meaning its "mass equivalent" is M + m.
2Bii) The box is lifted upward by an elevator at a constant speed, relative to the Earth. (We'll assume the upward distance is small enough that the acceleration due to gravity, g, is unchanged.) In this case, the reading on the scale *is* affected by the box's motion, but not in the way you appear to think. 3Bii) When the box first *accelerates* to its constant upward speed, the reading on the scale *increases*. Assuming the upward acceleration is a, the scale reads (g + a)M during the period of acceleration. Once the box reaches its constant upward speed, however, the reading on the scale goes back to gM. (And if we bring the box to a stop again, with acceleration - a, the scale will read (g - a)M during the period of deceleration.)
Note, by the way, that in the second case, the total energy of the box is *more* than just M + m, because it is being lifted to a higher gravitational potential energy. Its total energy *increases* during the lifting, even if it moves upward at a constant speed.
But of course the key point is that in *neither* case above does the kinetic energy of the box affect what the scale reads. Now let me describe a scenario that I think captures what you were really trying to ask about whether "kinetic energy has weight":
1C) A box has a bunch of stuff at rest inside it; the total invariant mass of box plus stuff is M. The box is sitting at rest on the surface of the Earth, in a gravitational field g. The box with all the stuff inside registers weight gM on a scale.
2C) The stuff inside the box is somehow put in motion (perhaps it is dust that is induced to fly around inside the box; perhaps it is small marbles that are jiggled around). The total kinetic energy of the stuff, as measured in the rest frame of the box, is m. The box as a whole remains at rest on the surface of the Earth, with the (now moving) stuff contained inside. The scale now reads g(M + m).
