What happens to the gravitational potential energy of the objects at the singularity of a black hole? As they are 0 distance from the center of the gravitational body, do they have no potential energy? Does this mean that the singularity is at the lowest energy state possible?
Sorry if this sounds stupid. I only know the concepts of advanced physics, and the only proper physics I know is classical.
 A: In general relativity, the gravitational potential energy of an object is dependent on the gravitational time dilation it is experiencing:
$$U_G=mc^2\left(t_d-1\right)=mc^2\left(\sqrt{1-\frac{2GM}{rc^2}}-1\right)$$
$$Classical\ U_G=\lim_{r\to \infty}{\left(mc^2\left(\sqrt{1-\frac{2GM}{rc^2}}-1\right)\right)}=-\frac{GMm}{r}$$
Inside of a black hole, time dilation probably becomes imaginary. Assuming the theory that all the matter inside a black hole is concentrated at its center is true, we can keep using our equation below to find out what the gravitational time dilation should be at the center of a black hole.
$$t_d=\sqrt{1-\frac{2GM}{0*c^2}}=\sqrt{1-\frac{2\infty GM}{c^2}}\approx\sqrt{-\frac{2\infty GM}{c^2}}=\frac{i}{c}\sqrt{2\infty GM}$$
Which is essentially an imaginary infinity large time dilation. Plugging that into our equation we get:
$$U_G=mc^2\left(\frac{i}{c}\sqrt{2\infty GM}-1\right)\approx\ mci\sqrt{2\infty GM}$$
Which is also an infinitely large gravitational potential energy.
So the gravitational potential at the singularity is an infinitely large imaginary number, not 0. Hope this helped.
