I thought about gravitational field. In Newtonian mechanics, gravitational energy between two matter is $U=-G\frac{M_1 M_2}{R^2}$ when mass of each matter is M1 and M2, having a distance R. With this equation, I was able to obtain the energy density of the gravitational field. But since Newtonian mechanics are good approximation of relativity, it should be used only in non-relativistic situation. Anyway, I obtained the formula that if the strength of gravitational field(which also means gravitational acceleration) is g, the energy density of field was $u_g=-\frac{1}{8\pi G}g^2$.
Now here is my question.
- Does gravitational field really has negative energy density? Or is my miscalculation due to an error between Newtonian mechanics and relativity?
- If so, can a system have a negative net energy?
edit) In the link from comment, there was solution using Schwarzschild raidus. Then what is the net energy difference between black hole with mass $M$ and system with same mass $M$, but nearly no potential energy due to large distane between particles. (Lets assume that black hole is Schwarzschild Black Hole for simple calculation :)
