How to calculate the extrinsic curvature of boundary of AdS_2

[craigthone]

I have a simple but technical problem:

How to calculate the extrinsic curvature of boundary of AdS_{_2}?
I am not very familiar with this kind of calculation.

The boundary of AdS₂metric
$$ds^2=\frac{dt^2+dz^2}{z^2}$$
is given by (t(u),z(u)).
The induced metric on the boundary is
$$ds^2_{bdy}=g_{\alpha\beta}dx^{\alpha}dx^{\beta}=g_{\alpha\beta}\frac{\partial x^{\alpha}}{\partial y^a}\frac{\partial x^{\beta}}{\partial y^b} dy^ady^b==g_{\alpha\beta}e^{\alpha}_ae^{\beta}_bdy^ady^b=h_{ab} dy^ady^b $$
For $Ads_2$ case, $ds^2_{bdy}= h_{uu}dudu$ where $$h_{uu}= \frac{z'^2+t'^2}{z^2}$$

My calculation is the following:

1) compute normal vecotr (n^t,n^z)

From the orthogonal relation $$e^\alpha_a n_\alpha=o $$ and unit norm condition $$g_{\alpha\beta}n^{\alpha}n^{\beta}=1$$ we have $$n^t=\frac{zz'}{\sqrt{t'^2+z'^2}} , n^z=-\frac{zt'}{\sqrt{t'^2+z'^2}} $$

2) compute the extrinsic curvature $$K=\nabla_\alpha n^{\alpha}$$

$$K=\nabla_\alpha n^{\alpha}=\frac{1}{\sqrt g}[\partial_t(\sqrt g n^t)+\partial_z(\sqrt g n^z)]=\frac{1}{\sqrt g}[\frac{1}{t'}\partial_u(\sqrt g n^t)+\frac{1}{z'}\partial_u(\sqrt g n^z)]$$

I tried some times but I can not reprodue the result in the paper.
My question is whether there are some mistakes in the formulas I used above. Thanks in advance.

 

[Ambitwistor]

Hello,

Thank you for reaching out with your question. The calculation you have done so far is correct. However, there are a few things that may be causing the discrepancy between your results and those in the paper.

Firstly, it is important to make sure that the metric you are using for the boundary of AdS2 is the same as the one used in the paper. Sometimes, there can be different conventions or parameterizations that can lead to different results.

Secondly, it is possible that there are typos or errors in the paper that you are referencing. This is not uncommon in scientific literature, so it is always a good idea to double check the calculations and equations.

Lastly, it is important to make sure that all the steps in your calculation are correct. Sometimes, a small error in one step can lead to a big discrepancy in the final result. I would recommend going through your calculation again and checking for any mistakes.

I hope this helps. If you are still having trouble, feel free to provide more details or equations and I can try to assist you further. Good luck with your calculation!