I saw a solved example in a book (Concepts of Physics by H.C. Verma, volume 2), where there is a body near surface of the earth, the problem is to calculate the increase in mass of the body when it is lifted 1 meter on the surface of earth. The book assumes that the increase in potential energy goes to the mass of the body, and applies Einstein's formula. I am attaching a link to the screenshot of the book. Check question number 7. (Somehow the uploader on this site says file type not supported)

However, the potential energy can also be thought of as integral over all space of the function $\frac { -g_1 \cdot g_2}{4\pi G}$, where $g_1$ and $g_2$ are the gravitational field of the earth and the body, respectively, and the integral can be shown to be equal to the interaction in potential energy (there is one similar problem with electrostatic potential energy in Introduction to Electrodynamics by Griffiths).

Why do we assume that the increased energy is stored as increase in mass of the body, instead of being stored in the field?

Also, why do we assume that only the mass of the body increases, not that of earth?

  • 3
    $\begingroup$ Would be good to see some context from the book. It could just be sloppy language. Einstein's equation tells us that if you increase the internal energy of an object, you will also increase its mass - hence the $E=mc^2$ equation. In this situation, the increase in mass should correspond to an increase in the total mass of the Earth-body system treated as a single bound object. $\endgroup$ – enumaris May 2 '18 at 18:49
  • $\begingroup$ @enumaris I added a link to the question $\endgroup$ – Archisman Panigrahi May 2 '18 at 19:24
  • $\begingroup$ The text seems intentionally ambiguous to me. It says "the increase in mass", not "the increase in mass of the body" like your question seems to state. I see nothing in the text's question or answer that is incompatible with assigning the increased mass to the entire system. $\endgroup$ – BowlOfRed May 2 '18 at 21:25

Mass corresponds to its internal energy as @enumaris has pointed out in his comment. The internal energy of a mass depends on the kinetic and potential energy of its constituents. This means that the mass as a whole doesn't change if its potential energy changes.

As an aside, mass is invariant and thus the same in all frames of reference.

  • $\begingroup$ What about the energy stored in the field? It also increases $\endgroup$ – Archisman Panigrahi May 3 '18 at 3:04
  • $\begingroup$ If you mean the gravitational energy of a mass m due to an attractive mass M at distance r that is $U=-GMm/r$ which equals the work done to bring m to this point. Otherwise see energy of gravitational field $\endgroup$ – timm May 3 '18 at 7:42
  • $\begingroup$ EDIT it should read: gravitational potential energy. $\endgroup$ – timm May 3 '18 at 14:25
  • $\begingroup$ Yes, this potential energy is stored in the field. The integral of $\frac { -g_1 \cdot g_2}{4\pi G}$can be shown to be equal to total interaction potential energy. So why are we assuming that it is stored as mass, not in the field? This was my question. Your answer is not very clear to me $\endgroup$ – Archisman Panigrahi May 4 '18 at 6:33
  • $\begingroup$ "mass is invariant" in newtonian physics and galilean transformations, it should be "rest mass is invariant" $\endgroup$ – anna v May 4 '18 at 7:45

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