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I've read some high school "derivations" of ##E=m\cdot c^2## that all considered single photons with momentum ##p=E/c## that are absorbed or emitted from some massive object, changing its mass. So they actually only showed the incremental

$$\Delta E=\Delta m\cdot c^2 .$$

Most of those books simply generalized this to ##E=m\cdot c^2##, only one author said that this formula is obtained by integration and dropping constants. It's unclear to me why the integration constants simply can be set to zero. Maybe they don't matter in SR, but wouldn't they change things in GR? Or could this integration constant be absorbed in Einstein's cosmological constant?

Also, one of those books used ##\Delta E=\Delta m\cdot c^2## to compute that the mass of a spring with spring constant ##k## increases by

$$\Delta m=\frac{\Delta E_S}{c^2}=\frac{\frac{1}{2}k\cdot s^2}{c^2}$$

when compressed by a distance ##s##. How is such a generalization possible? The derivations critically relied on ##p=E/c## for photons which are not present in this example.