The Wikipedia page on anti-de Sitter space defines such a space as the hypersurface

$$- X_{1}^{2}-X_{2}^{2} + \sum_{i=3}^{n+1}X_{i}^{2} = \frac{(n-1)(n-2)}{2\Lambda}$$

in an $n+1$-dimensional flat space with metric signature $(-,-,+, \dots, +)$.

How can you show, directly from the constraint, that such a spacetime satisfies Einstein's equations with a negative value of $\Lambda$?

  • $\begingroup$ The wikipedia article you have linked works out the curvature tensor for such a hypersurface and it is manifest that it satisfies the vacuum Einstein equations with a negative cosmological constant. $\endgroup$ – Dvij Mankad Jun 27 '17 at 13:28

The hypersurface may be defined, as the OP stated, through the algebraic constraint,

$$-X^2_1 - X^2_2 + \sum_{i=3}^{n+1} X^2_i = \frac{(n-1)(n-2)}{2\Lambda}$$

involving $n+1$ coordinates $\{X\}_{1}^{n+1}$ in $n+1$-dimensional flat space with signature $(n-1,2).$ In order to show the hypersurface satisfies the Einstien equations with a negative cosmological constant, we need to solve the constraints for $X$ in terms of $n$ coordinates.

Denoting the $n$ coordinates on the submanifold $\sigma$, the induced metric on the submanifold can be expressed as,

$$\gamma_{ab} = \eta_{\mu\nu}\frac{\partial X^\mu}{\partial \sigma^a}\frac{\partial X^\nu}{\partial \sigma^b}$$

which is related by a projection operator to the first fundamental form, $h_{\mu\nu}$. From this metric, one can now directly compute the curvature tensors and arrive at the Einstein equations.


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