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  1. While writing the metric for AdS Space, why are we starting with a five dimensional Flat space and embedding a hyperboloid in it?

  2. Does it have to do with the fact that the cosmological constant being negative?

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  • $\begingroup$ FWIW, a positive cosmological constant leads to a similar construction for de Sitter space. $\endgroup$ – Qmechanic Nov 15 '17 at 6:23
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Embedding the AdS 4D spacetime as a hypersurface in a 5D manifold is just a convenient way of studying it. There is no physical meaning to the 5D manifold - it is just a mathematical device.

Any 4D spacetime can be embedded in a manifold of higher dimensionality, though it turns out to be surprisingly complicated to specify what dimensionality the manifold has to be. See this question in the Math Overflow for more on this.

So, no, the embedding is not specifically related to a negative cosmological constant. De Sitter space can be embedded in a 5D manifold in the same way, and indeed as Robert Greene's work tells us any spacetime can be embedded, though in general the dimensionality required will be higher than five.

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  • $\begingroup$ Any 4D vacuum solution of GR can be embedded in a flat 5D canonical space with a cosmological constant. There is a number of publications on this topic. This paper specifically mentions the implications for the cosmological constant in AdS: arxiv.org/ftp/arxiv/papers/1011/1011.0214.pdf - And this one specifies the effective cosmological constant in general: arxiv.org/ftp/arxiv/papers/0705/0705.0067.pdf $\endgroup$ – safesphere Nov 15 '17 at 7:40
  • $\begingroup$ Then, while driving the metric where do we use the fact that the cosmological constant is negative? $\endgroup$ – Khushal Nov 15 '17 at 8:22
  • $\begingroup$ @Khushal: I'm not sure what you're asking. Assuming you mean deriving rather than driving, the AdS metric is the vacuum solution to the Einstein equation with a negative value of $\Lambda$. $\endgroup$ – John Rennie Nov 15 '17 at 8:38

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