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. Note however that the accelerated expansion of the Universe requires positive value of Λ and favors De Sitter Space dS4 instead of AdS4.
These observations give motivations for finding whether AdS4 or dS4 or both allow 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.
- The line element of M4 in spherical Minkowski coordinates (m,rM,θ,φ) reads as
ds2= dm2-drM2-rM2dΩ2 .
- 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) , rM= r ,
Θ = s(y) , Φ= ω× (t+f(y)) .
The non-trivial conditions on the components of the induced metric are given by
gtt= Λ2-x2sin2(Θ) = A(r) ,
gtr= 1/r0[Λ dh/dy -x2sin2(θ) df/dr]=0 ,
grr= 1/r02[(dh/dy)2 -1- x2sin2(θ)(df/dy)2- R2(dΘ/dy)2]= -1/A(r) ,
- For Θ(r) the equation for gtt 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 gtr one obtains
dh/dy = ( P/Λ)× df/dy ,
- The condition for grr 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)] .
(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. The only modification in the case of De Sitter space dS4 is the replacement of the function A= 1+y2 appearing in the metric of AdS4 with A=1-y2. Also now the imbedded portion of the metric is a spherical shell. This brings in mind TGD inspired model for the final state of the star which is also a spherical shell. p-Adic length scale hypothesis motivates the conjecture that stars indeed have onion-like layered structure consisting of shells, whose radii are consistent with p-adic length scale hypothesis. This brings in mind also Titius-Bode law.
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?".