### Singularatities of space-time geometry in TGD approach

Clifford Johnson wrote to Asymptotia a posting with title Beyond Einstein: Fixing Singularities in Spacetime. The discussion was quite reasonable with the point being that stringy mathematics has provided nice insights to the understanding of geometric singularities. The reason is basically that stringy physics carries more information about the background geometry than the physics of pointlike particles for the obvious reason that string is 1-dimensional object. Complexification of the moduli space for the solutions of Einstein's equations containing singular geometries as isolated points is the key idea inspired by string model. In this approach the solution space of field equations of general relativity is extended to that for a more general theory and if one believes on this kind of theory one can be happy.

Lubos Motl echoed Clifford Johnson with his Singularities and stringy geometry . Lubos bravely extrapolates from the mathematics of string theory to a much wider human context by adding to the discussion a lot of "crackpots", the usual not so friendly comments about those among us who still believe on climate change, and the not so cautious conclusion which deserves empty lines around it:

**One thing is obvious. Whoever claims that he is a theoretical physicist and he or she can answer these fundamental questions without learning string theory is simply a crackpot.**

Obviously the posting of Lubos represents the determined and impatient voice of the Empire. People usually find it better to agree with Empire and I do not know whether anyone is interested on listening to the much less determined and almost timid voices of those outside the hegemony. In any case, I collect all my courage and try to articulate my own views in the following.

** 1. TGD as a generalization of string model**

In general relativity blackhole and primordial cosmology represent the basic singularities. Inside the blackhole the matter is concentrated at single point. For radiation dominated cosmology mass density which becomes infinite at the moment of Big Bang. Probably most theoreticians agree that these singularities demonstrate that something goes wrong with General Relativity at very short distances and when the density of gravitational mass becomes very high.

In TGD basic objects are not strings but 3-D lightlike surfaces of certain non-dynamical 8-D space-time whereas spontaneous compactification leads to 10-D or 11-D (or 12-D or....) dynamical space-times identified as a higher-D counterpart of 4-D space-time in the original superstring approach. In M-theory space-time could also correspond to a brane of particular kind.

Since the super-conformal symmetries of string models are extended dramatically in TGD framework, TGD like theory could have been the next unavoidable step after it had become clear that string models do not work without the introduction of the ad hoc notion of spontaneous compactification, which was unfortunately not yet quite enough and led later to the introduction of branes. It is now clear that even this was not yet quite enough and we can only guess what next.

As a matter fact, although it took quite a long time to discover the formulation of TGD utilizing light-like 3-surfaces, I dare say that the situation had become ripe for the replacement of strings with 3-D objects within few years after the revolution but due to certain "general reasons" this step was not taken and within few years string theory community became the Empire ("general reasons" has a particular flavor in the history of finnish politics, and applies also to - and not only finnish - science politics).

TGD generalizes string models. Therefore it is not surprising that string like objects belong to the basic stuff of TGD Universe. There are also other kinds of objects, so called CP_{2} type vacuum extremals serving as correlates for elementary particles. Also "massless extremals" or "topological light rays" for which the analogy would be a laser beam with strong non-linear interactions implying that signals are channelled and move in single direction only and there is not dispersion.

String like objects are 3-dimensional surfaces like the strings of the real world, being very thin in the dimensions transversal to the string. Whereas strings of string theory bear no direct connection to the world that we observe, string like objects of TGD have a clear physical interpretation with direct connections to the anomalies of astrophysics, cosmology, etc..., even of biology. One highly non-trivial application is nuclear string model.

The most dramatic prediction of basic TGD is topological field quantization. A given physical system, say my computer or me, is a 3-D surface with an outer boundary: my personal universe literally ends up at my skin. Also fields decompose into topological field quanta so that the notion of field body emerges: magnetic flux tubes, also string like objects, are key building blocks of the field body. These notions make no sense in General Relativity where space-times are abstract 4-manifolds rather than 4-surfaces.

** 2. Blackholes and big bang in TGD framework**

In TGD framework the solution of the singularity problem of General Relativity relies on the fact that I just mentioned: space-time is a 4-D surface, not an abstract 4-manifold.

- There is a large number of imbeddings of Schwartshild metric as a vacuum extremal but everyone of them fails at some radius above or below Schwartschild radius. Thus blackhole as a pointlike singularity is not possible in TGD framework even as a limiting case. The notion of many-sheeted space-time allows to guess what happens. Below the critical radius space-time surface must be something different and there are many possibilities. TGD based model of hadrons and nuclear string model lead to a model of blackhole like structures.
- In the model of hadron new degrees of freedom assignable to super-canonical symmetries - super-conformal symmetries which are not present in string models - are responsible for about 75 per cent of nucleon mass with quarks and gluons contributing the rest. These degrees of freedom do not couple to electro-weak fields or ordinary gluons. They however have color interactions besides gravitational interactions. Recently direct evidence for this kind of exotic particles from cosmic ray spectrum emerged. Thus even hadronic strings would have a blackhole like aspect.
- Blackhole like object is identified as a highly entangled string obtained by fusing hadronic strings to for a single very long string. In the ideal case that electro-weakly interacting particles are not present, only these degrees of freedom would contribute to the mass of the blackhole and blackhole would have no electro-weak interactions. As a matter fact, Schwharthchild metric is always accompanied by a long range electro-weak gauge field which can be arbitrarily weak but never completely vanishing so that the ideal blackhole is not possible. The string tension of string like objects (Kähler magnetic energy density per unit length) can vary so that a rich spectrum of black hole like objects looking like the black holes of general relativity outside Schwartschild radius is predicted.

- In TGD framework the first candidate for the early cosmology is the Robertson-Walker metric representing quantum critical universe with critical mass density. The critical cosmology is highly unique from imbeddability so that the details of dynamics do not affect the geometry. In inflationary scenarios one must introduce a lot of ad hoc stuff in an attempt to reproduce observations. The flatness of 3-space is a manifestation of quantum criticality (there are no dimensional parameters in critical dynamics so that curvature scalar must vanish). That macroscopic quantum coherence is present in astrophysical scales means that the physics of this model differs profoundly from that of inflationary models.
Critical cosmology fails to be imbeddable as a whole and a transition to -say- radiation dominated cosmology must take place. The mass per co-moving volume approaches to zero at the initial moment so that Big Bang actually becomes a "silent whisper amplified to a relatively big bang". In this sense one gets rid of the singularity. Of course, one can well ask whether space-time sheets make sense at this limit and my proposal is that one can only speak about soup of TGD counterparts of cosmic strings at this limit. This soup of string like objects would represent the ultimate blackhole like object in the sense that the string like objects would have maximum possible string tension and being analogs of cosmic strings of GUTs. At this limit string model would be an excellent approximation.

**3. Can one allow metrically singular manifolds?**

One can ask what one really means with "singular". The notion of smooth manifold codes partially for our prejudice about what it is to be non-singular. Smooth manifold is obtained by gluing together pieces of Euclidian space together smoothly so that there are no corners or other singularities.

One can also consider a more stringent notion of singularity by introducing the notion of metric. It is this kind of notion of singularity which is appropriate in general relativity. For nonsingular metric the inverse of the metric tensor exists and the metric dimensionality of the space-time is same as its topological dimension. Light-like surfaces represent a basic example of singularity in the metric sense. In TGD basic dynamical objects are light-like surfaces and thus singular in the conventional sense of the world. One ends up with this picture in following manner.

- In general relativity the metric of space-time is assumed to have a global Minkowskian signature although Euclidian space-times have been used in the path integral approach as a mathematical trick in a attempt to make some sense of quantization of General Relativity. In TGD framework the realization of Poincare symmetries as those of 8-D imbedding space allows space-time surfaces to have both Euclidian (-1,-1,-1,-1) and Minkowskian (1,-1,-1,-1) signature.
- Since elementary particles correspond to CP
_{2}type extremals, their gluing to Minkowskian background means that the resulting space-time surface possesses a light-like 3-surface, "wormhole throat", for which the metric has signature (0,-1-1-1). These light-like surfaces are singular as Riemann manifolds. The most important implication of their identification as fundamental dynamical objects (general coordinate invariance in 4-D sense allows this) requires the formulation of quantum TGD as almost-topological Quantum Field Theory based on the analog of Chern-Simons action and its fermionic variant fixed by super-conformal symmetry (its form is not quite what one might guess). The very notion of light-likeness however brings the notion of metric into the theory so that one does not lose the notion of mass and obtains a genuine physical theory.

**4. Can one allow manifolds which are not smooth?**

A good example about a non-smooth manifold is an ordinary book. The pages of the book are smooth 2-D manifolds but the back of the book at which the pages meet represents a singularity. Obviously these structures are completely well-defined geometrically and it is not clear whether these kind of manifolds are "singular" in the sense that they would be "nonphysical". A book like structure indeed emerges in TGD framework as a non-dynamical structure via the generalization of the notion of imbedding space. Also TGD counterparts of Feynman diagrams involve branching, the basic singularity of book topology.

- The realization of the idea about a hierarchy of Planck constant motivated by several intriguing observations, in particular the finding that planetary orbits satisfy in reasonable approximation Bohr rules with gigantic value of Planck constant leads to the necessity of generalizing the notion of imbedding space. Without going to details one can say that generalized imbedding space is obtained by gluing infinite number of almost copies of the original imbedding space together somewhat like pages of book. Planck constant has different values at different pages of this Big Book. Matter can reside at any page of this book and matter at given page of book sees the matter at other pages of book as dark matter.
- TGD leads to a generalization of the notion of Feynman diagram. Generalized Feynman diagrams are obtained by replacing the lines of Feynman diagram with light-like 3-surfaces glued together at their 2-D ends so that the resulting 3-manifolds are singular in the same sense as Feynman diagrams are singular as 1-manifolds. Notethat the 2-D vertices are smooth as 2-manifolds. Stringy diagrams are smooth as 2-manifolds but vertices are not singular as 1-D manifolds. This represents a fundamental difference between TGD and string models. Also string diagrams are possible but the physical interpretation for stringy loops is completely different.

**5. Can one allow even local topology to vary?**

The notion of singularity discussed above relies on the assumption that the local manifold topology is induced from the topology of real numbers. p-Adic topology and smoothness represent a totally different kind of view about what it is to be non-singular.

- Second generalization of the notion of imbedding space relates to the need to fuse real physics and various p-adic physics (one for each prime p and infinite number of them if algebraic extensions are allowed) relying on the notion of p-adic imbedding space containing p-adic space-time sheets as basic objects. The only approach which seems to work is based on the generalization of the number concept: various p-adic number fields and reals are glued together along common rationals (and common algebraics too) playing the role of the back of book. This generalization of number concept implies also a generalization the notion of manifold and of surfaces and one can speak about space-time sheets which are real and p-adic and which meet along common rational points.
- p-Adic physics can be assumed to be smooth but smoothness understood is in the sense of p-adic topology. From real point of view p-adic smoothness implies what I call p-adic fractality: long range correlations and local chaos. From the basic properties of p-adic norm it follows that p-adic space-time sheets have literally infinite size in the real sense. The interpretation would be that our "cognitive bodies" (kind of thought bubbles;-)) have literally infinite size and duration. Hence quantum classical correspondence raises the question whether we are eternal also as conscious entities (also with respect to subjective time).

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