- The first one corresponds to a stationary metric having interpretation in terms of interior of an object with constant mass density. The line element reads
ds2 =Adt2-Bdr2-r2dΩ2 ,
A =1-(r/l)2 , B= 1/A .
l has natural interpretation as outer boundary of the object in question. It will be found that TGD suggests 2-fold covering of this metric.
- Second coordinatization has interpretation as simplest possible inflationary cosmology having flat 3-space:
ds2= dt12-e2t1/l dr2-r12dω2 .
- The two coordinatizations are related to each other by the formulas deducible from the general transformation property of metric tensor:
t =t1+ log[1+(r1/l)2e2t1/l]/2 ,
r =et1/lr1 .
In TGD framework also the imbedding of space-time as surfaces matters besides the metric which is purely internal property. The most general ansatz for the imbedding of De Sitter metric into M4× CP2 is as a vacuum extremal for for Kähler action with the understanding that small deformation carries energy momentum tensor equal to Einstein tensor so that Einstein's equations would old true in statistical sense.
- The general ansatz for the stationary form of the metric is of same general form as that for Schwartchild metric. One can restrict the consideration to a homologically trivial geodesic sphere S2 of CP2 with vanishing induced Kähler form and standard spherical metric. This means that CP2 is effectively replaced with S2. This imbedding is a special one but gives a good idea about what is involved.
Denoting by (m0,rM,θ,φ) the coordinates of M4 and by (Θ, Φ) the coordinates of S2, a rather general ansatz for the imbedding is
m0= t+ h(r) , rM=r ,
Rω × sin(Θ (r))= +/- r/l , Φ= ω t+ k(r) .
- The functions h(r), k(r), and Θ (r) can be solved from the condition that the induced metric is the stationary metric. For Schwartschild metric h(r) and k(r) are non-vanishing so that the imbedding cannot be said to be stationary at the level of imbedding space since t=constant surfaces correspond to m0 h(rM)=constant surfaces.
De Sitter metric is however very special. In this case one can assume h(r)=k(r)=0 for Rω=1. The imbedding reduces simply to an essentially unique imbedding
sin(Θ(r))=+/- r/l= rM/l , Φ= t/R= m0/R .
This imbedding is certainly very natural and would describe stationary non-expanding cosmology with constant mass density. Not that the imbedding is defined only for rM<l. Unless one allows 3-space to have boundary, which for non-vacuum extremals does not seem plausible option, one must assume double covering
sin(Θ(r))= sin(π-Θ(r))= +/- rM/l
Stationarity implies that there is no Big Bang.
- The transition to the inflationary picture looks in TGD framework very much like a trick in which one replaces radial Minkowski coordinate with r1 =exp(-t1/l) rM and in these new coordinates obtains Big Bang and exponential expansion as what looks like a coordinate effect at the level of imbedding space. Also the transition to radiation dominated cosmology for which the hyperbolic character of M4+ metric ds2=da2 a2(dr2/(1+r2) +r2dΩ2) is essential, is difficult to understand in this framework. The transition should correspond to a transition from a stationary cosmology at the level of imbedding space level to genuinely expanding cosmology.
The cautious conclusion is that sub-manifold cosmology neither excludes nor favors inflationary cosmology and that critical cosmology is more natural in TGD framework. In TGD Universe De Sitter metric looks like an ideal model for the interior of a stationary star characterized by its radius just like blackhole is characterized by its radius. It seems that TGD survives the new findings at qualitative and even partially quantitative level.