For a fleeting moment I thought that for the twistor space of Minkowski space the 2-D fiber could be hyperbolic sphere H2 (t2-x2-y2 =-RH2) rather than sphere S2 as it is for CP2 with Euclidian signature of metric. I however soon realized that the infinite area of H2 implies that 6-D Kähler action is infinite and that there are many other difficulties.
The correct manner to define Minkowskian variant of twistor space is by starting from the generalization of complex and Kähler structures for M4= M2+ E2 of local tangent space to longitudinal (defined by light-like vector) and to transversal directions (polarizations orthogonal to the light-like vector. The decomposition can depend on point but the distributions of two planes must integrated to 2-D surfaces. In E2 one has complex structure and in M2 its hyper-complex variant. In M2 has decomposition of replacing complex numbers by hyper-complex numbers so that complex coordinate x+iy is replaced with w=t+ie, i2=-1 and e2=-1.
It took time to realize I have actually carried out this generalization years ago with quite different motivations and called the resulting structure Hamilton-Jacobi structure! The twistor fiber is defined by projections of 4-D antisymmetric tensors (in particular induced Kähler form) to the orthogonal complement of unique time direction determed by the sum of light-like vector and its dual in M2. This part of tensor could be called magnetic. Th magnetic part of the tensor defines a direction and one has natural metric making the space of directions sphere S2 with metric having signature (-1,-1). This requires that twistor space has metric signature (-1,-1,1,-1,-1,-1) (I also considered seriously the signature (1,1,1,-1,-1,-1) so that there are three time-like coordinates) .
The radii of the spheres associated with M4 and CP2 define two fundamental scales and the scaling of 6-D Käler action brings in third fundamental length scale. On possibility is that the radii of the two spheres are actually identical and essentially equal to CP2 radius. Second option is that the radius of S2(M4) equals to Planck length, which would be therefore a fundamental length scale.
The radius RD of the 2-D fiber of twistor space assignable to space-time surfaces is dynamical. In Euclidian space-time regions the fiber is sphere: a good guess is that its order of magnitude is determined by the winding numbers of the maps from S2(X4)→ S2(M4) and S2(X4) → S2(CP2). The winding numbers (1,0) and (0,1) represent the simplest options. The question is whether one could say something non-trivial about cosmic evolution of RD as function of cosmic time. This seems to be the case.
Before continuing it is good to recall how the cosmological constant emerges from TGD framework. The key point is that the 6-D Kähler action contains two terms.
- The first term is essentially the ordinary Kähler action multiplied by the area of S2(X4), which is compensated by the length scale, which can be taken to be the area 4π R2(M4) of S2(M4). This makes sense for winding numbers (w1,w2)=(n,0) meaning that S2(CP2) is effectively absent but S2(M4) is present.
- Second term is the analog of Kähler action assignable assignable to the projection of S2(M4) Kähler form. The corresponding Kähler coupling strength αK (M4) is huge - so huge that one has
αK (M4)4π R2(M4)== L2 ,
where 1/L2 is of the order of cosmological constant and thus of the order of the size of the recent Universe. αK(M4) is also analogous to critical temperature and the earlier hypothesis that the values of L correspond to p-adic length scales implies that the values of come as αK(M4) ∝ p≈ 2k, p prime, k prime.
- The Kähler form assignable to M4 is not assumed to contribute to the action since it does not contribute to spinor connection of M4. One can of course ask whether it could be present. For canonically imbedded M4 self-duality implies that this contribution vanishes and for vacuum extremals of ordinary Kähler action this contribution is small.Breaking of Lorentz invariance is however a possible problem. If αK(M4) is given by above expression, then this contribution is extremely small.
Given the parameter L2 as it is defined above, one can deduce an expression for cosmological constant Λ and show that it is positive. One can actually get estimate for the evolution of RD as function of cosmic time if one accepts Friedman cosmology as an approximation of TGD cosmology.
- Assume critical mass density so that one has
ρcr= 3H2/8π G .
- Assume that the contribution of cosmological constant term to the mass mass density dominates. This gives ρ≈ ρvac=Λ/8π G. From ρcr=ρvac one obtains
Λ= 3H2 .
- From Friedman equations one has H2= ((da/dt)/a)2, where a corresponds to light-cone proper time and t to cosmic time defined as proper time along geodesic lines of space-time surface approximated as Friedmanncosmology. One has
in Robertson-Walker cosmology with ds2= gaada2-a2 dσ32.
- Combining this equations with the TGD based equation
- Assume that quantum criticality applies so that L has spectrum given by p-adic length scale hypothesis so that one discrete p-adic length scale evolution for the values of L. There are two options to consider depending on whether p-adic length scales are assigned with light-cone proper time a or with cosmic time t
T= a (Option I) , T=t (Option II).
Both options give the same general formula for the p-adic evolution of L(k) but with different interpretation of T(k).
L(k)/Lnow= T(k)/Tnow , T(k)= L(k) = 2(k-151)/2× L(151) , L(151)≈ 10 nm .
Here T(k) is assumed to correspond to primary p-adic length scale. An alternative - less plausible - option is that T(k) corresponds to secondary p-adic length scale L2(k)=2k/2L(k) so that T(k) would correspond to the size scale of causal diamond. In any case one has L ∝ L(k). One has a discretized version of smooth evolution
L(a) = Lnow × (T/Tnow) .
- Feeding into the formula following from two expressions for Λ one obtains an expression for RD(a)
RD/lP= (8/3)1/2π× (a/L(a)× gaa1/2
This equation tells that RD is indeed dynamical, and becomes very small at very early times since gaa becomes very small. As a matter of fact, in very early cosmic string dominated cosmology gaa would be extremely small constant (see this). In late cosmology gaa→ 1 holds true and one obtains at this limit
RD(now)= (8/3)1/2π× (anow/Lnow) × lP ≈ 4.4 ×(anow/Lnow) × lP .
- For T= t option RD/lP remains constant during both matter dominated cosmology, radiation dominated cosmology, and string dominated cosmology since one has a∝ tn with n= 1/2 during radiation dominated era, n= 2/3 during matter dominated era, and n=1 during string dominated era (see this). This gives
RD/lP=(8/3)1/2π× at (gaa1/2(t(end)/L(end)) = (8/3)1/2π×(1/n)(t(end)/L(end)) .
Here "end"> refers the end of the string or radiation dominated period or to the recent time in the case of matter dominated era. The value of n would have evolved as RD/lP∝ (1/n (tend/Lend), n∈ [1,3/2,2}. During radiation dominated cosmology RD ∝ a1/2 holds true. The value of RD would be very nearly equal to R(M4) and R(M4) would be of the same order of magnitude as Planck length. In matter dominated cosmology would would have RD ≈ 2.2 (t(now)/L(now)) × lP .
- For RD(now)=lP one would have
Lnow/anow =(8/3)1/2π≈ 4.4 .
In matter dominated cosmology gaa=1 gives tnow=(2/3)× anow so that predictions differ only by this factor for options I and II. The winding number for the map S2(X4)→ S2(CP2) must clearly vanish since otherwise the radius would be of order R.
- For RD(now)= R one would obtain
anow/Lnow =(8/3)1/2π× (R/lP)≈ 2.1× 104 .
One has Lnow=106 ly: this is roughly the average distance scale between galaxies. The size of Milky Way is in the range 1-1.8 × 105 ly and of an order of magnitude smaller.
- An interesting possibility is that RD(a) evolves from RD ∼ R(M4) ∼ lP to RD ∼ R. This could happen if the winding number pair (w1,w2)=(1,0) transforms to (w1,w2)=(0,1) during transition to from radiation (string) dominance to matter (radiation) dominance. RD/lP radiation dominated cosmology would be related by a factor
to that in matter dominated cosmology. Similar factor would relate the values of RD/lP in string dominated and radiation dominated cosmologies. The condition RD(rad)/RD(mat) =lP/R expressing the transformation of winding numbers would give
L(now)/L(rad,end) =(4/3) (lP/R) (t(now)/t(rad,end)) .
One has t(now)/t(rad,end)≈ .5× 106 and lP/R =2.5× 10-4 giving L(now)/L(rad,end)≈ 125, which happens to be near fine structure constant.
- For the twistorial lifts of space-time surfaces for which cosmological constant has a reasonable value , the winding numbers are equal to (w1,w2)=(n,0) so that RD=n1/2 R(S2(M4)) holds true in good approximation. This conforms with the observed constancy of RD during various cosmological eras, and would suggest that the ratio t(end)/L(end) characterizing these periods is same for all periods. This determines the evolution for the values of αK(M4).
For a summary of earlier postings see Links to the latest progress in TGD.