I have a simple but technical problem:
How to calculate the extrinsic curvature of boundary of AdS2?
The boundary of AdS2 metric
ds2=dt2+dz2z2
is given by (t(u),z(u)).
The induced metric on the boundary is
ds2bdy=gαβdxαdxβ=gαβ∂xα∂ya∂xβ∂ybdyadyb==gαβeαaeβbdyadyb=habdyadyb
For
Ads2 case,
ds2bdy=huududu where
huu=z′2+t′2z2
My calculation is the following:
1) compute normal vecotr (nt,nz)
From the orthogonal relation eαanα=o and unit norm condition gαβnαnβ=1, we have nt=zz′√t′2+z′2,nz=−zt′√t′2+z′2
2) compute the extrinsic curvature K=∇αnα
K=∇αnα=1√g[∂t(√gnt)+∂z(√gnz)]=1√g[1t′∂u(√gnt)+1z′∂u(√gnz)]
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.