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10 months ago in Astrophysics By Lily

Why does the schwarzschild metric give a weird distance for the radius?

In the Schwarzschild metric, the circumference around a black hole is 2πr, but the actual measured distance from the horizon to the center is more than r. How can a circle have a normal circumference but a stretched radius?

GeneralRelativity

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By Raghu Answered 5 months ago

You've put your finger on the essence of curvature. The key: coordinates are just labels, not tape measures. The circumference is defined using the angular part of the metric, which stays Euclidean. But the radial part is warped by gravity g?? > 1. So when you integrate the proper distance along the radius, you get a longer path than the coordinate difference suggests. The geometry isn't flat. That's not a paradox; it's the definition of curved space.

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