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.
- The line element of M4 in spherical Minkowski coordinates (m,rM,θ,φ) reads as
ds2= dm2-drM2-rM2dΩ2 .
ds2=- R2(dΘ2+sin2(θ)dΦ2) .
ds2= A(r)dt2-(1/A(r))dr2-r2dΩ2 ,
A(r)= 1+y2 , y = r/r0 .
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) ,
x=Rω .
By some simple algebraic manipulations one can derive expressions for sin(Θ), df/dr and dh/dr.
- 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.
dh/dy = ( P/Λ)× df/dy ,
(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?”.
2012-12-10 22:42:08
Source: http://matpitka.blogspot.com/2012/12/is-there-connection-between-preferred.html
Source:
