The AdS-Schwarzschild black hole solution My question is from an AdS/CFT review:  http://arxiv.org/abs/1112.5403 
The AdS_5 metric in the article is written
$$ ds^2=\frac{l^2}{z^2}(-dt^2+dz^2+dx^2), $$
where I'm denoting collectively three dimensions by $dx^2$.
Then the article says Schwarz black hole solution with a horizon radius $z_h$ in this spacetime is 
$$ ds^2=\frac{l^2}{z^2}\Big[-\big(1-\frac{z^4}{z_h^4}\big)dt^2+\frac{1}{1-\frac{z^4}{z_h^4}}dz^2+dx^2\Big]. $$
I'm observing some weird things here: 
when the black hole radius $z_h$ approaches infinity, the Schwarz metric becomes AdS,
so an infinitely large AdS-Schwarz black hole is AdS???
 A: It is known that a Schwarz blackhole in asymptotic AdS space is an Einstein vacuum solution,which at spatial infinity tends to AdS space time.Also to a Kerr metric as a generalisation of Schwarz metric.
.Further generalization can conform to electrovacuum Einstein solution represented by Einstein.Maxwell.dirac[emd] equation,as a starting point for quantum state of a vector potential in a compactified dimension,as a generalised dimensional reduction of field theories,in a geometric way.This conforms to a gauge field,affording a gravity interpretation in a gauge theory duality perspective,as a maximally extended Schwarz blackhole in asymptotic AdS space time,generalised by the Einstein.Maxwell Dirac metric.
This can be extended to explain particles like electrons and to QGP confinement /deconfinement models[witten].....thus pointing to a heuristics for a unified metric,combined metric tensor,gauge field in an extended Hilbert space and in an AdS/CFT correspondence context.
The generalized eigen function theory base of extended Hilbert space as a pair of super structure and a dense subspace,has an inclusion map as a homeomorphism for binary operations for computations that are consistent in large structure and substructure.This can study spectral theory combining eigenvector bound functions and continuum ones.The dense subspace is a topological vector space for a vector potential,analogous to a quantized energy field in compactified dimension,conforming to a QED vacuum,in a simplectic form,flowing from the compact space,as a generalised dimensional reduction of field theories,corresponding to an AdS_5 metric,but as a gauge U(1)field,having no requirement to conform to CY manifold,though analogous to string like theories..
The model explains quantization of energy field in compact space,allowing momenta of a discrete nature,which has a gravity interpretation in dual space time as an AdS space time,with a thermal state corresponding to maximally extended Schwarz blackhole in asymptotic AdS space time...Thermal state corresponding t blackhole on conformal boundary,in a cft/ads duality.
The invariance of change in scale (gauge)allows introduction  of a complex quantity to transform scale change as phase change.Radiation likened to blackhole[Hawking/Page].,with phase transitions at critical thermal state threshold.
Prof.Suresh Kumar.S,formerly Chief Scientist CSIR
