The AdS-Schwarzschild black hole solution

Question

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???

Answer 1

I know this question is 6 years old, but for those wandering here, the limit $z_h\to\infty$ is not the limit of a black hole of infinite size. Remember that in these coordinates, $z\to0$ corresponds to the boundary of (asymptotic) AdS. The black hole singularity sits at $z\to\infty$.

Answer 2

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