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I was watching Hawking (2004 film) and in the movie, Hawking has a sudden insight that all the matter and energy observed in the known universe may, in fact, have sprung out from nothing (singularity) during the Big Bang event—an insight which later on goes to become the main subject of his thesis.

But this completely flies in the face of the laws of conservation of matter and energy. What gives?

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The link Qmechanic has suggested is a duplicate and does discuss the question you ask. However there is another point that is worth making here.

In general relativity we describe the universe as a manifold equipped with a metric, and the metric is the FLRW metric that desciribes expanding spacetime. However the FLRW metric does not include the point(s) at $t = 0$ i.e. the Big Bang itself. So we cannot say that:

everything all the matter and energy observed in the known universe may, in fact, have sprung out from nothing (singularity) during the Big Bang event

because general relativity tells us nothing about the Big Bang itself. As we wind back in time towards the Big Bang the matter/energy density gets higher and higher, but what happens at the moment of the Big Bang we can't say. So it isn't meaningful to say the current matter/energy content of the universe sprang out of nothing.

We hope that one day some theory of quantum gravity will explain exactly what happened at the Big Bang, but at the moment no such theory exists.

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