1. Smooth anomaly
No compact smoothable topological 4-manifold is known, which would allow only a single smooth structure. Even worse, the number of exotics is infinite in every known case! In the case of non-compact smoothable manifolds, which are physically of special interest, there is no obstruction against smoothness and they typically carry an uncountable family of exotic smooth structures.
One can argue that this is a catastrophe for classical general relativity since smoothness is an essential prerequisite for tensory analysis and partial differential equations. This also destroys hopes that the path integral formulation of quantum gravitation, involving path integral over all possible space-time geometries, could make sense. The term anomaly is certainly well-deserved.
Note however that for 3-geometries appearing as basic objects in Wheeler's superspace approach, the situation is different since for D≤3 there is only a single smooth structure. If one has holography, meaning that 3-geometry dictates 4-geometry, it might be possible to avoid the catastrophe.
The failure of the CCH is the basic message of Etesi's article. Any exotic R4 fails to be globally hyperbolic and Etesi shows that it is possible to construct exact vacuum solutions representing curved space-times which violate the CCH. In other words, GRT is plagued by causal anomalies.
Etesi constructs a vacuum solution of Einstein's equations with a vanishing cosmological constant which is non-flat and could be interpreted as a pure gravitational radiation. This also represents one particular aspect of the energy problem of GRT: solutions with gravitational radiation should not be vacua.
- Etesi takes any exotic R4, which has the topology of S3× R and has an exotic smooth structure, which is not a Cartesian product. Etesi maps maps R4 to CP2, which is obtained from C2 by gluing CP1 to it as a maximal ball B3r for which the radial Eguchi-Hanson coordinate approaches infinity: r → &infty;. The exotic smooth structure is induced by this map. The image of the exotic atlas defines atlas. The metric is that of CP2 but SU(3) does not act as smooth isometries anymore.
- After this Etesi performs Wick rotation to Minkowskian signature and obtains a vacuum solution of Einstein's equations for any exotic smooth structure of R4.
2. Holography of smoothness
In the TGD framework space-time is 4-surface rather than abstract 4-manifold. 4-D general coordinate invariance, assuming that 3-surfaces as generalization of point-like particles are the basic objects, suggests a fully deterministic holography. A small failure of determinism is however possible and expected, and means that space-time surfaces analogous to Bohr orbits become fundamental objects. Could one avoid the smooth anomaly in this framework?
The 8-D embedding space topology induces 4-D topology. My first naive intuition was that the 4-D smooth structure, which I believed to be somehow inducible from that of H=M4× CP2, cannot be exotic so that in TGD physics the exotics could not be realized. But can one really exclude the possibility that the induced smooth structure could be exotic as a 4-D smooth structure?
What does the induction of a differentiable structure really mean? Here my naive expectations turn out to be wrong.
- If a sub-manifold S⊂ H can be regarded as an embedding of smooth manifold N to S⊂ H, the embedding N→ S⊂ H induces a smooth structure in S (see this). The problem is that the smooth structure would not be induced from H but from N and for a given 4-D manifold embedded to H one could also have exotic smooth structures. This induction of smooth structure is of course physically adhoc.
It is not possible to induce the smooth structure from H to sub-manifold. The atlas defining the smooth structure in H cannot define the charts for a sub-manifold (surface). For standard R4 one has only one atlas.
- Could M8-H duality help and holography help? One has holography in M8 and this induces holography in H. The 3-surfaces X3 inducing the holography in M8 are parts of mass shells, which are hyperbolic spaces H3⊂ M4⊂ M8. 3-surfaces X3 could be even hyperbolic 3-manifolds as unit cells of tessellations of H3. These hyperbolic manifolds have unique smooth structures as manifolds with dimension D<4.
- One can ask whether the smooth structure at the boundary of a manifold could dictate that of the manifold uniquely. One could speak of holography for smoothness.
The implication would be that exotic smooth 4-manifold cannot have a boundary. Indeed, R4 does not have a boundary. Could this theorem generalize so that 3-surfaces as sub-manifolds of mass shells H3m determined by the polynomials defining the 4-surface in M8 take the role of the boundaries?
The regions of X4⊂ M8 connecting two sub-sequent mass shells would have a unique smooth structure induced by the hyperbolic manifolds H3 at the ends. These smooth structures are unique by D<4 and cannot be exotic. Smooth holography would determine the smooth structure from that for the boundary of 4-surface.
- However, the holography for smoothness is argued to fail (see this). Assume a 4-manifold W with 2
different smooth structures. Remove a ball B4 belonging to an open set U and construct a smooth structure at its boundary S3. Assume that this smooth structure can be continued to W. If the continuation is unique, the restrictions of the 2 smooth structures in the complement of B4 would be equivalent but it is argued that they are not.
The first layman objection is that the two smooth structures of W are equivalent in the complement W-B3 of an arbitrary small ball B3⊂ W but not in the entire W. This would be analogous to coordinate singularity. For instance, a single coordinate chart is enough for a sphere in the complement of an arbitrarily small disk. An exotic smooth structure would be like a local defect in condensed matter physics.
The second layman objection is that smooth structure, unlike topology, cannot be induced from W to W-B3 but only from W-B3 to W. If one a smooth structure at the boundary S3 is chosen, it determines the smooth structure in the interior as standard smooth structure.
- In fact, one could argue that the mere fact the 4-surfaces have boundaries as their ends at the light-like boundaries of CD, implies a unique smooth structure by holography. It is however possible that the mass-shells correspond to discontinuities of derivatives so that the smooth holography decomposes to a piece-wise holography. This would mean that M8-H duality is needed.
3. Can embedding space and related spaces have exotic smooth structure?
One can worry about the exotic smooth structures possibly associated with the M4, CP2, H=M4× CP2, causal diamond CD=cd× CP2, where cd is the intersection of the future and past directed light-cones of M4, and with M8. One can also worry about the twistor spaces CP3 resp. SU(3)/U(1)× U(1) associated with M4 resp. CP2.
The key assumption of TGD is that all these structures have maximal isometry groups so that they relate very closely to Lie groups, whose unique smooth structures are expected to determine their smooth structures.
- The first sigh of relief is that all Lie groups have the standard smooth structure. In particular, exotic R4 does not allow translations and Lorentz transformations as isometries. I dare to conclude that also the symmetric spaces like CP2 and hyperbolic spaces such as Hn= SO(1,n)/SO(n) are non-exotic since they provide a representation of a Lie group as isometries and the smoothness of the Lie group is inherited. This would mean that the charts for the coset space G/H would be obtained from the charts for G by an identification of the points of charts related by action of subgroup H.
Note that the mass shell H3, as any 3-surface, has a unique smooth structure by its dimension.
- Second sigh of relief is that twistor spaces CP3 and SU(3)/U(1)× U(1) have by their isometries and their coset space structure a standard smooth structure.
In accordance with the vision that the dynamics of fields is geometrized to that of surfaces, the space-time surface is replaced by the analog of twistor space represented by a 6-surface with a structure of S2 bundle with space-time surface X4 as a base-space in the 12-D product of twistor spaces of M4 and CP2 and by its dimension D=6 can have only the standard smooth structure unless it somehow decomposes to (S3× R)× R2. Holography of smoothness would prevent this since it has boundaries because X4 as base space has boundaries at the boundaries of CD.
- cd is an intersection of future and past directed light-cones of M4. Future/past directed light-cone could be seen as a subset of M4 and implies standard smooth structure is possible. Coordinate atlas of M4 is restricted to cd and one can use Minkowski coordinates also inside the cd. cd could be also seen as a pile of light-cone boundaries S2× R+ and by its dimension S2× R allows only one smooth structure.
- M8 is a subspace of complexified octonions and has the structure of 8-D translation group, which implies standard smooth structure.
See the article Intersection form for 4-manifolds, knots and 2-knots, smooth exotics, and TGD or the chapter Does M8 H duality reduce classical TGD to octonionic algebraic geometry?: Part II.
For a summary of earlier postings see Latest progress in TGD.
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