| Trying to Make sense of: “with an AdS3 radius which is of order of the AdS5” Maldacena Feel free to correct me, and also if you think there’s something terribly wrong with the formula, feel free to share along. Now, AdS3 is a three-dimensional space with negative curvature, which is used in string theory as a model for the three-dimensional world. The AdS5 radius is the distance from the center of AdS3 to its boundary. Maldacena is saying that the AdS3 radius is of the same order as the AdS5 radius, meaning that they are both large compared to the Planck length. Is this supported by empirical evidence? No, this is not supported by empirical evidence. This is based on the fact that the AdS3 radius is much larger than the Planck length, while the AdS5 radius is only slightly larger than the Planck length. This means that the two spaces are of the same order of magnitude, and so the AdS3 radius is of the same order as the AdS5 radius. The Gibbons-Hawkings metric is a metric on the space of all Riemannian manifolds. It is defined where S is the action of the manifold and R is the Ricci scalar curvature. If we consider S to be an action under the Gibbons-Hawkings metric, where R is the Ricci scalar curvature of the manifold, In that case, S would be the action of the AdS3 manifold, which is where R is normed curvature of the AdS3 manifold. This is the Ricci scalar curvature: R = \frac{1}{2} \mathrm{tr}\left( R_{\mu \mu} \right) = \frac{1}{2} \mathrm{tr}\left( \frac{\partial{{\partial x^{\mu} \omega_{ {\mu} - \frac{\partial{\partial {x^{\mu} \omega_{\mu} + \omega_{\mu} \omega_{ u} - \omega_{ u} \omega_{\mu} \right) where R_{\mu u} is the Ricci tensor and \omega_{\mu} is the connection 1-form. This is the Ricci scalar curvature: R = \frac{1}{2} \mathrm{tr}\left( R_{\mu u} \right) = \frac{1}{2} \mathrm{tr}\left( \frac{\partial}{\partial x^{\mu}} \omega_{ u} - \frac{\partial}{\partial x^ u} \omega_{\mu} + \omega_{\mu} \omega_{ u} - \omega_{ u} \omega_{\mu} \right) There is not enough empirical evidence to determine whether or not the AdS3 radius and AdS5 radius are of the same order of magnitude. However, based on the results of the calculation, it seems that the AdS3 radius is of the same order of magnitude as the AdS5 radius. |