If the mass of a neutron star in its collapse becomes a singularity, then the resting energy of this gravitational field must be $E = mc^2$ ($m$ = star mass).
Is this possibility wrong?
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The rest energy of a star which collapsed to form a black hole is not represented by the gravitational field before and after the collapse. The gravitational field has no energy density and hence is not a source of gravity. It is caused by the mass in the singularity of the black hole.
Let's consider the idealized case, where there's no radiation in any form. Furthermore, we usually even assume that this neutron star (and later black hole) is the only thing in the universe. Basically, this just means that we ignore the influence of any distant things so that we can get mathematically exact answers.
So you start off with a nice, quiet neutron star, and assume that we can ignore any radiation it's giving off. Just to make things extra simple (though this isn't strictly necessary), let's assume that it's not even spinning. If you look at the gravitational field outside of this star, Birkhoff's theorem tells us that it should look exactly like the Schwarzschild metric. And we define the mass of the neutron star to be the $M$ found in that metric.
Now there's a second way of defining the mass in this simple universe, called the Bondi mass. It turns out that, numerically, this mass is just the same as the $M$ we defined above. But the nice thing about Bondi mass is that there's a conservation law that says that the initial Bondi mass, $M$, is equal to the final Bondi mass plus whatever mass-energy that's radiated.
So if we assume that this neutron star collapses to a black hole without ejecting any matter, or radiating any type of mass-energy, then this conservation law tells us that the final Bondi mass is the same as the initial Bondi mass, $M$. And if the final object is a black hole, this means that the mass that goes into its Schwarzschild metric.
Now, as usual, the mass I'm talking about here is just the rest mass. And through Einstein's mass-energy equivalence, we have $E = M\, c^2$ — both before and after the collapse. Of course, all of this really relies on the assumption that absolutely nothing is radiated, which isn't realistic. Especially with rotation, there's likely to be some mass ejection, and even more likely to be some electromagnetic radiation, which would both reduce the final mass-energy of the black hole.
But one important point is that the total mass does not depend on what form that mass takes. Whether in the puffy matter hypergiant star, super-dense neutron star matter, a black hole singularity, or a truly enormous collection of squeaky clown noses, mass is mass all the same.