Is there a connection between preferred extremals and AdS4/CFT correspondence?

The preferred extremals satisfy Einstein Maxwell equations with a cosmological constant and have negative curvature for negative value of Λ. 4-D space-times with hyperbolic metric provide canonical representation for a large class of four-manifolds and an interesting question is whether these spaces are obtained as preferred extremals and/or vacuum extremals.

4-D hyperbolic space with Minkowski signature is locally isometric with AdS₄. This suggests a connection with AdS₄/CFT correspondence of M-theory. The boundary of AdS would be now replaced with 3-D light-like orbit of partonic 2-surface at which the signature of the induced metric changes. The metric 2-dimensionality of the light-like surface makes possible generalization of 2-D conformal invariance with the light-like coordinate taking the role of complex coordinate at light-like boundary. AdS would presumably represent a special case of a more general family of space-time surfaces with constant Ricci scalar satisfying Einstein-Maxwell equations and generalizing the AdS₄/CFT correspondence.

For the ordinary AdS₅ correspondence empty M⁴ is identified as boundary. In the recent case the boundary of AdS₄ is replaced with a 3-D light-like orbit of partonic 2-surface at which the signature of the induced metric changes. String world sheets have boundaries along light-like 3-surfaces and space-like 3-surfaces at the light-like boundaries of CD. The metric 2-dimensionality of the light-like surface makes possible generalization of 2-D conformal invariance with the light-like coordinate taking the role of hyper- complex coordinate at light-like 3-surface. AdS₅× S⁵ of M-theory context is replaced by a 4-surface of constant Ricci scalar in 8-D imbedding space M⁴× CP₂ satisfying Einstein-Maxwell equations. A generalization of AdS₄/CFT correspondence would be in question.

These observations give motivations for finding whether AdS₄ allows an imbedding as vacuum extremal to M⁴× S²⊂ M⁴× CP₂, where S² is a homologically trivial geodesic sphere of CP₂. It is easy to guess the general form of the imbedding by writing the line elements of, M⁴, S², and AdS₄.

1. The line element of M⁴ in spherical Minkowski coordinates (m,r_M,θ,φ) reads as

ds²= dm²-dr_M²-r_M²dΩ² .

Also the line element of S² is familiar:

ds²=- R²(dΘ²+sin²(θ)dΦ²) .

By visiting in Wikipedia one learns that in spherical coordinate the line element of AdS₄ is given by

ds²= A(r)dt²-(1/A(r))dr²-r²dΩ² ,

A(r)= 1+y² , y = r/r₀ .

From these formulas it is easy to see that the ansatz is of the same general form as for the imbedding of Schwartschild-Nordstöm metric:

m= Λ t+ h(y) , r_M= r ,
Θ = s(y) , Φ= ω× (t+f(y)) .

The non-trivial conditions on the components of the induced metric are given by

g_tt= Λ²-x²sin²(Θ) = A(r) ,

g_tr= 1/r₀[Λ dh/dy -x²sin²(θ) df/dr]=0 ,

g_rr= 1/r₀²[(dh/dy)² -1- x²sin²(θ)(df/dy)²- R²(dΘ/dy)²]= -1/A(r) ,

x=Rω .

By some simple algebraic manipulations one can derive expressions for sin(Θ), df/dr and dh/dr.

1. For Θ(r) the equation for g_tt gives the expression

sin²(Θ)= P/x² ,

P= Λ² -A =Λ²-1-y² .

The condition 0≤ sin²(Θ)≤ 1 gives the conditions

(Λ²-x²-1)^1/2 ≤ y≤ (Λ²-1)^1/2 .

Clearly only a spherical shell is possible.

From the vanishing of g_tr one obtains

dh/dy = ( P/Λ)× df/dy ,

The condition for g_rr gives

(df/dy)² =[r₀²/AP]× [A^-1-R²(dΘ/dy)²] .

Clearly, the right-hand side is positive if P≥ 0 holds true and RdΘ/dy is small.
From this condition one can solved by expressing dΘ/dy using chain rule as

(dΘ/dy)²=x²y²/[P (P-x²)] .

One obtains

(df/dy)² = [Λ r₀²y²/AP]× [(1+y²)^-1 -x²(R/r₀)² [P(P-x²)]^-1)] .

The right hand side of this equation is non-negative for certain range of parameters and variable y.
Note that for r₀>> R the second term on the right hand side can be neglected. In this case it is easy to integrate f(y).

The conclusion is that AdS₄ allows a local imbedding as a vacuum extremal. Whether also an imbedding as a non-vacuum preferred extremal to homologically non-trivial geodesic sphere is possible, is an interesting question.

For details and background see the chapter The recent vision about preferred extremals and solutions of the modified Dirac equation of “Physics as Infinite-dimensional Geometry”, or the article with the title “Do geometric invariants of preferred extremals define topological invariants of space-time surface and code for quantum physics?”.

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