The Main Solutions of Einstein Field Equations
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MISTAKES:
There are some imprecisions in this video, so we want to clarify them here. Make sure to check the linked PDF, where all explanations are more precise, and where we can correct typos and update the material on our website whenever necessary.
7:10 — A true (physical) singularity occurs only when r = 0. What happens at r = r_S is not a physical singularity.
7:43 — The infinity symbol in the radial component does not imply a true physical singularity at r = r_S. This is a coordinate singularity, meaning it appears due to our choice of coordinates. The spacetime curvature remains finite here, and the metric can be extended smoothly through r_S using better coordinates.
8:45 — We say that this metric (written in this form) “turns out to be a consistent physical model,” which is not exactly true. The infinity symbol in the radial component still has no physical meaning. What we should have said is: if a body with the mass of the Earth occupies a volume with radius less than r_S ≈ 8.87mm, then we can apply the Schwarzschild exterior metric outside the object. However, even then, better coordinates would be needed to ensure the metric components have clear physical interpretation at the horizon.
9:03 — This is indeed a formula for producing a black hole: compressing a mass below the volume corresponding to a sphere with its Schwarzschild radius. However, to be precise, this doesn’t automatically make the object a black hole, since the radius is not zero. But at that point, gravity becomes strong enough that the object would collapse into itself, ultimately forming a singularity.
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