I wrote one day ago about synchronization of clocks and found that the clocks distributed at the hyperboloids of light-cone assignable to CD can in principle be synchronized in Lorentz invariant manner (see this). But what about actual Lorentz invariant synchronization of the clocks? Could TGD say something non-trivial about this problem? I received an interesting link relating to this (see this). The proposed theory deals with fundamental uncertainty of clock time due to quantum-gravitational effects. There are of course several uncertainties involved since quantum theory of gravity does not exist (officially) yet!
The difficult problem is the identification of Minkowski coordinates unless one regards GRT as QFT in Minkowski space. In astrophysical scales QFT picture one must consider solutions of Einstein’s equations representing astrophysical objects. For the basic solutions of Einstein’s equations the identification of Minkowski coordinates is obvious but in general case such as many-particle system this is not anymore so. This is a serious obstacle in the interpretation of the classical limit of GRT and its application to planetary systems.
What about the situation in TGD? Particle physicist inside me trusts symmetry based thinking and has been somewhat reluctant to fill space-time with clocks but I am ready to start the job if necessarily! Since I am lazy I of course hope that Nature might have done this already and the following argument suggests that this might be the case!
The imbeddability of space-time as to CD× CP2 ⊂ H= M4× CP2 allows the proper time coordinate a2 =t2-r2 near either CD bouneary as a universal time coordinate, “cosmic time”. At a= constant hyperboloids Lorentz invariant synchronisation is possible. The coordinate a is kind of absolute time near a given boundary of CD representing the perceptive field of a particular conscious observer and serves as a common time for all space-time surfaces in the superposition. Newton would not have been so wrong after all.
Also adelic vision involving number theoretic arguments selects a as a unique time coordinate. In p-adic sectors of adele number theoretic universality (NTU) forces discretization since the coordinates of hyperboloid consist of hyperbolic angle and ordinary angles. p-Adicallhy one cannot realize either angles nor their hyperbolic counterparts. This demands discretization in terms of roots of unity (phases) and roots of e (exponents of hyperbolic angles) inducing finite-D extension of p-adic number fields in accordance with finiteness of cognition. a as Lorentz
invariant would be genuine p-adic coordinate which can in principle be continuous in p-adic sense. Measurement resolution however discretizes also a.
This discretization leads to tesselations of a=constant hyperboloid having interpretation in terms of cognitive representation in the intersection of real and various p-adic variants of space-time surface with points having coordinates in the extension of rationals involved. There are two choices for a. The correct choice
corresponds to the passive boundary of CD unaffected in state function reductions.
Suppose that tesselation defines a clock network. What synchronization could mean? Certainly strong correlations between the nodes of the network Could the correlation be due to maximal quantum entanglement (maximal at least in p-adic sense) so that the network of clocks would behave like a single quantum clock? Bose-Einstein condensate of clocks as one might say? Could quantum entanglement in astrophysical scales predicted by TGD via hgr= heff=n× h hypothesis help to establish synchronized clock networks even in astrophysical scales? Could Nature guarantee Lorentz invariant synchronization automatically?
What would be needed would be not only 3-D lattice but also oscillatory behaviour in time. This is more or less time crystal (see this and this)! Time crystal like states have been observed but they require feed of energy in contrast to what Wilzek proposed. In TGD Universe this would be due to the need to generate large heff/h=n phases since the energy of states with n increases with n. In biological systems this requires metabolic energy feed. Can one imageine even cosmic 4-D lattice for which there would be the analog of metabolic energy feed?
I have already a model for tensor networks and also here a appears naturally (see this). Tensor networks would correspond at imbedding space level to tesselations of hyperboloid t2-r2=a2 analogous to 3-D lattices but with recession velocity taking the role of quantized position for the point of lattice. They would induce tesselations of space-time surface: space-time surface would go through the points of the tesselation (having also CP2 counterpart). The number of these tesselations is huge. Clocks would be at the nodes of these lattice like structures. Maximal entanglement would be key feature of this network. This would make the clocks at the nodes one big cosmic clock.
If astrophysical objects serving as clocks tend to be at the nodes of tesselation, quantization of cosmic redshifts is predicted! What is fascinating is that there is evidence for this: for TGD based model for this see (see this and this)! Maybe dark matter fraction of Universe might have taken care of the Lorentz invariant synchronization so that we need not worry about that!
For a summary of earlier postings see Latest progress in TGD.
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