Gravity violating the conservation of energy in a closed system?

[logos]

I have devised a simple thought experiment which leads me to an absurd conclusion and I feel I’m missing something obvious but I can't see where I’m wrong and I hope you could help point out my error.

I start with an empty space initially containing two masses that are at rest relative to each other, the total energy of this system would be the rest mass (E=mc²) right?

But as time starts gravity will start accelerating these masses towards each other giving me the energy of the rest mass + a non 0 kinetic energy. This would in my mind imply either a decrease in the mass of the objects or an increase of the total energy of the closed system, neither of which seem logical?

Now I realize I’m probably missing something obvious here but I can't for my life figure out what…

 

[Pengwuino]

When you put the first mass in the system, all is fine and dandy. However, by placing the second mass in the system, you obviously impart a potential energy to each mass. This potential energy is what accounts for the missing energy in your thought experiment.

 

[cesiumfrog]

What is the rest mass of a gravitationally bound system?

Presumably it is less than that of its components, by the amount of the escape energy. Might not be trivial to prove the difference in the system's inertia.

 

[logos]

When you put the first mass in the system, all is fine and dandy. However, by placing the second mass in the system, you obviously impart a potential energy to each mass. This potential energy is what accounts for the missing energy in your thought experiment.

Ah, of course. As I thought it’s obvious now that somebody pointed it out. I blame not noticing it myself on having substituted sleep with caffeine for to long now during exam season…

Anyway thanks so much for the help.

 

[Riemannliness]

Be careful with those caffeine substitutions - they work for math but not so much for thought experiments in physics.

To add to the above:
Law of conservation of energy (simplified)
For this system -> E_total = E_K + E_G
Where E_K is the kinetic energy of the system and E_G is the gravitational potential energy.

E_total is constant in a closed system like the one you described, so you can see that as your masses are drawn together by the force of gravity, E_G decreases as E_K increases. :P