Gravitational Potential Energy & Mass Change: Andrew's Question

[andrew s 1905]

TL;DR Summary

Does the work done in separating two masses increase their mass in accordance with e=mc^2

If I start with two, otherwise isolated, masses M and m initially together and do work to separate them then the work done, I assume, goes into the gravitational binding energy between them. Will the system of mass M and m have increased in mass due to this in accordance with e=mc^2?

I believe yes as it is broadly analogous to chemical or nuclear binding.

If I am wrong please provide some pointers in the right direction.

Thanks Andrew
PS this is not a homework question I am 70 and not formally studying it's for personal interest.

 

[Ibix]

I think for this question to make sense you need to be far enough away from the masses for them to be treated as a single point mass at all times. Otherwise you can't really characterise the gravitational field in terms of "the mass", since it depends on the mass distribution even in Newtonian gravity.

But then you run into the problem of where the energy came from to move the masses apart. Was it already there, e.g. in the form of rocket fuel? Or was it supplied from outside, e.g. by shining a laser on to a solar sail on one of the masses? In the first case I would not expect the overall mass to change because no energy was added to the system, but in the second I would expect it to change because energy was supplied.

 

[andrew s 1905]

Thanks, it was intended to be very simplified and I did intend the energy to be supplied from outside the system. Regards Andrew

 

[Mister T]

Will the system of mass M and m have increased in mass due to this in accordance with e=mc^2?

Assuming we are working with point masses, the mass of the system is greater than $M+m$ and it increased by an amount $W/c^2$, where $W$ is the work you did separating them.

 

[jbriggs444]

Assuming we are working with point masses, the mass of the system is greater than $M+m$ and it increased by an amount $W/c^2$, where $W$ is the work you did separating them.

If we are treating gravitational potential energy has having a mass equivalent then...

The mass of a gravitationally bound system will be less than the sum $M+m$. But after being separated by the applied work, the energy deficit ("binding energy") is reduced by an increment of $W/c^2$.