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I recently read about Einstein's proposition that the spacetime (4 dimension) may be curved. In order to curve a plane (2D) we need a 3rd dimension in which to curve it. Would that mean that the spacetime is curved in a 5th dimension?

If yes, does it have a name? Does it represent something we can comprehend? Has it been observed in any other way?

Pier-Yves Lessard
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    There has been a lot of questions like this on this site; try searching. The short answer is that the beauty of general relativity is that there is no need to assume any extra dimensions - something deeply unintuitive that motivates a lot of the questions. – Anders Sandberg Aug 06 '21 at 13:26
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    Does this answer your question? Why does GTR not need a higher dimension to describe the bending of spacetime? – jacob1729 Aug 06 '21 at 13:29
  • No short answer is 'no' - you don't need to do that and if you wanted to there is no guarentee that a mere 5 dimensions would be enough. – jacob1729 Aug 06 '21 at 13:33
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    I agree with the others that the answer to your question is that we don't need an "embedding space" to talk about the geometry of curved spacetime. But another thought is that if you did want to embed our four dimensional spacetime in a higher-dimensional embedding space (for purely mathematical reasons), one extra dimension is not guaranteed to be enough. In general, you need $2d$ dimensions to embed a manifold of dimension $d$ according to the Whitney embedding theorem. For instance, you can't embed a Klein-Bottle in 3 dimensions. – Andrew Aug 06 '21 at 13:33
  • @jacob1729 Absolutely, thank you – Pier-Yves Lessard Aug 06 '21 at 13:33
  • Also see https://physics.stackexchange.com/q/90592/, https://physics.stackexchange.com/q/553938/, https://physics.stackexchange.com/q/26440/ – jng224 Aug 06 '21 at 13:38

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