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Assume a space contains one black hole and two electrons $e_1$ and $e_2$ only. The distance between the black hole and electron $e_1$ is $10^6$ km, and the distance between the black hole and $e_2$ is $10^{15}$ km (about 100 light years). Relative velocities of two electrons to the black hole are zero. There is no other interaction between the black hole and two electrons, except gravitational force.

My question is : Is $e_2$'s mass is greater than $e_1$'s mass because of gravitational potential energy difference?

Because I am an engineer, not a physicist, my question may have some defects. What I want to know is : if the potential energy is indeed in the object, the mass of the object can vary because of the potential energy difference?

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Potential energy certainly contributes to mass. For example the mass of the hydrogen atom contains proton and electron masses as well as their kinetic and potential energy. However an external potential energy does not contribute to the mass. Mass is simply the energy of a system at rest. The interaction with another object is not included. The interaction does contribute to the mass of the combined object of black hole plus electron.

By the way, why bring up black holes at the risk of starting a discussion on general gravitation? The electron could also be in an electric field, which is more close to your expertise.

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  • $\begingroup$ I heard that potential energy is real and exist in an object. So I think that potential energy can be measured by mass or something else. Am I rightly understanding? (Why black hole? No particular reason. I just want an object can generate very large potential energy) $\endgroup$ – pdh0710 May 26 '18 at 10:57
  • $\begingroup$ My example of the hydrogen illustrates how the mutual interaction between parts of an object contributes potential energy to the mass. Also the kinetic energy of the parts contributes to it. $\endgroup$ – my2cts May 26 '18 at 10:59
  • $\begingroup$ What I understand through your hydrogen atom example is : proton has potential energy generated by electron, and the electron has potential energy generated by the proton. These potential energies increase the hydrogen atom's mass, but not increase mass of the proton and the electron ....... Am I rightly understanding? $\endgroup$ – pdh0710 May 26 '18 at 11:31
  • $\begingroup$ That is correct. Note that you should count the mutual potential energy only once. $\endgroup$ – my2cts May 26 '18 at 11:56
  • $\begingroup$ Thank you. But still, something is unclear to me. Then, the mass of our solar system is greater than the sum of mass of sun and planets? (assume that our solar system consists of sun and planets only) And gravity of our solar system is greater than the sum of gravity of sun and planets? $\endgroup$ – pdh0710 May 26 '18 at 12:49

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