# How to derive the AdS spacetime metric

So I have been working on AdS/CFT for a while now and realized that I have never actually seen the derivation for the metric. In every literature, introductory or advanced, they just give you the AdS metric.

The AdS metric is a solution to Einstein's field equations. But my question is, where can I see this explicitly? Maybe someone can point me in a direction where someone actually derives the AdS metric from scratch? I was not able to find it in any book and the old original papers are usually not very easy to read.

• What are you looking for exactly? A derivation of the AdS metric from Einstein's equation? Oct 28, 2021 at 21:49

The defining characteristic of AdS is being $$\color{red}{\text{maximally symmetric}}$$ with $$\color{blue}{\text{negative curvature}}$$. It therefore has $$\color{red}{\frac12n(n+1)}$$ linearly independent Killing vector fields in $$n$$-dimensional spacetime. You van verify$$\underbrace{\color{red}{X_a\partial_b-X_b\partial_a}}_{\frac12n(n-1)},\,\underbrace{\color{red}{X_a\partial_U+U\partial_a}}_n$$are such KVFs for $$\color{blue}{ds^2=\sum_{a=1}^nX_a^2-dU^2}$$ on $$\color{blue}{\sum_{a=1}^nX_a^2-U^2=-1}$$. You can get the full details here.