64 Chapter 2: Motivation yields Compare this to the metric of a p + 2 dimensional hyperboloid embedded in a flat space with SO(2,p+1) isometries If we reparameterize the hypersurface with coordinates y α = ( t, r, ⃗x ), r > 0 such that then the induced metric is obtained from the pullback [319] ds 2 = r 2 R $_{2}$( − dt 2 + dx 2 1 + dx 2 2 + dx 2 $_{3}$) + R 2 r 2 dr 2 + R $^{2}$d Ω 2 $_{5}$. (2.50) X 0 2 + X p +2 2 − p +1 ∑ i =1 X i 2 = R 2 (2.51) ds 2 = − dX 0 2 − dX p +2 2 + p +1 ∑ i =1 dX i $^{2}$. (2.52) X 0 = 1 2 r ( R 2 + r 2 R $_{2}$( R 2 + ⃗x 2 − t $^{2}$)$^{)}$ , (2.53) X i = rx i R , ( i = 1 . . . p ) (2.54) X p +1 = 1 2 r ( R 2 − r 2 R $_{2}$( R 2 − ⃗x 2 + t $^{2}$)$^{)}$ , (2.55) X p +2 = rt R , (2.56) ( y $^{∗}$g )$_{αβ}$ = dx µ dy α dx ν dy $_{β}$g$_{µν}$ (2.57) = r 2 R $_{2}$( − dt 2 + p +1 ∑ i =1 dx $^{i}$) + R 2 r 2 dr $^{2}$. (2.58)