It is a old and forgotten question, please see York paper attached. Wheeler: "What is fixed on the boundary in the action principles of general relativity?”

Spacial and general relativity are compatible with any boundary condition minimizing a relativistic action at the boundary. But new phenomena should arise when we add boundary conditions to space-time. My result, rigorously proven in peer-reviewed papers, is that it is possible to quantize relativistic particles by using such relativistic compatible boundary conditions. Indeed boundary conditions imply quantization if we think for instance to a particle in a box.

Article Boundary terms in the action principles of general relativity

General Relativity tells us only that the universe can be represented as a 4 dimensional manifold, and it seems accurate for large area. Bondaries depend on the system considered. An observable system needs observers, and the system must be defined as a part of the universe. The most sensible way to represent this part is by a fiber bundle, based on R (for the time) with typical fiber a 3 dimensional riemannian manifold (the space). Then the system is defined by the area spanned by the initial 3 dimensional manifold when it moves with the time of the observers. And the initial conditions are defined on the initial 3 dimensional manifold. We have a manifold with boundary, but the boundary depends on the system. It is not too difficult that for usual models the conditions for well behaved solutions are met.

Cosmologists consider the whole universe. And this is a totally different story, not least because it is difficult to conceive of an observer. However general topological considerations lead generaly to extend the previous model (a warped universe) but the initial manifold is actually a singularity (the big bang).

I may be naive but it's not clear to me what the difficult is. For the Schartzschild solution the boundary condition is that the spacetime be asymptotically flat. The cosmological solutions need no boundary conditions because the region considered (the whole of spacetime) is not bounded. The situation for GR is no different in principle from situations encountered in Newtonian dynamics. The vibrational modes of a membrane in the form of a flat disc present a problem needing boundary conditions to take account of the stiffness of the edge of the disc. Compare this to vibrations of a spherical surface. No boundary therefore no boundary conditions.

Dear jean claude Dutailly and Eric Lord,

I understand perfectly what you mean, the fiber bundle of infinite manifold is the description generally given in literature (e.g. Quantum Gravity). Even leaving aside for the moment quantum mechanics, this is however not very satisfying. In relativity space-time is a dynamical object. Its dynamics are determined by the Einstein equation. But where and what are the boundary conditions of this differential equation? Or using Wheeler words (e.g. see York paper given above), "What is fixed on the boundary in the action principles of general relativity?” Indeed, if we try to associate an Action to the Einstein equation we find that there is an ambiguity in the boundary terms. The action of general relativity is defined modulo boundary terms (Hilbert action, Einstein action, etc...) so we don't know which boundary conditions are associated to relativistic space-time. In ordinary problems one typically assumes that fields are zero at the space-time infinity and space-time is flat at infinity. But does space-time exist before the big bang? Is the universe infinite? The big bang and other singularities may be regarded as boundary conditions for space-time. Consider for instance models such as cyclic universe (R. Penrose) and eternal recurrence: cosmological space-time has periodic boundary conditions. Also it is not very satisfying saying that the system determines the boundary conditions of space-time: if you put a particle in a space-time box doesn't mean that you instantaneously cut the whole space-time of the universe to a compact space-time with the dimension of your box....

The fact that the general relativity action can be defined modulo boundary conditions means that one can play with boundary conditions of space-time without compromising the differential structure of space-time, i.e. the GR architecture. On the other hand, these boundary conditions should give rise to new phenomena with respect to the ordinary (non-quantum) description of infinite space-time. As originally proposed by Einstein himself boundary conditions ("boundary values" as he said) of space-time can be used to try for a simple ("but not simpler") unification of relativity and quantum mechanics (e.g. see A. Paris "The science and life of A. Einstein")...

Thus the problem of the boundary conditions of space-time is absolutely non-trivial.

1. Contrary to what is commonly said, the structure of 4 dimensional manifold of the universe does not depend on the dynamic of energy-matter. A manifold has no curvature per se. The gravitational field is usually represented by a connection on a fiber bundle, which itself induces a metric for the frames of observers.

2. To stay on scientific ground we have to define what is observed : if there is no observation there is no science. Any observable system is embedded in a finite area of the universe, and everything in this area (matter and field) is the system. The area has an initial boundary (defined by the observer) and so intitial conditions. As initial conditions there are given, and not subject to specific conditions. This seems a bit paradoxal, as we may assume that the intial data are themselves subject to some conditions, but this is the way one conceives science. The implementation of the principle of least action, which is by itself defined in very general term, leads to some integral on the whole of the area, so the initial values are accounted for, but if the integral is with an absolutely continuous measure their weight is null.

3. Cosmology is very different, and I believe that, in order to claim any scientific status, it should establish a strong epistemological basis prior to assert anything. For me claims such as what happens very close to the beginning of the universe is pure metaphysics (if not b...).

The key point is as usual to define clearly what is the subject of your study, how you assume that it can be observed, and what could be the relations between your model and your measures.

Dear Jean,

please don't tell me what science is. I know it vary well and I assure you that I am fighting to defend the scientific method against who likes conjectures and to increase the confusion or myths. Science means to produce objective undeniable evidence s(through experiments or mathematic demonstrations) of results. At the same time scientists has the duty to check results of other scientists and to accept or rule out them if they are correct or wrong. Contrarily to what you say, science is not conceived as "paradoxal", and if there is something paradoxical is because there is something that must be better understood.

If you want a subject and an observation you can consider, for instance, a particle and the observation of its energy and momentum. Even though you meant probably something completely different, I must agree with what you said: if you prepare a particle with a given energy E in a laboratory you set the boundary conditions of the space-time for that particle, in the sense that you fix the periodicity of space-time for that particle as L = h / p and T = h / E, i.e its wavelength and frequency according to de Broglie Einstein relations. This implicitly contains the Heisenberg uncertainty relation.

Dolce ~

I'm still struggling to understand your viewpoint. Thank you for the reference to Abraham Pais's book. It is fascinating. I've been looking at the relevant chapters on Einstein's views about reconciling QM with his world-view (which, in a nutshell, was" nature should be describable by differential equations"). But I cannot find anything there about "boundary conditions". What are you referring to? Anyway, I see from a preliminary glance at "cyclic spacetime dimensions" that you are attempting to realize Einstein's intuition that QM should come out of essentially "classical" physics. But that is a matter of "periodicity" conditions rather than "boundary" conditions (related but not the same, which led me to misunderstand your original question). You mentioned the Hilbert variational principle and quoted Wheeler. Quite frankly, I don't know what Wheeler is talking about; my understanding of the variational principle applied to Lagrangian field theories is that the field equations correspond to a minimization of the action integral; in other words, the action in any region should be invariant under arbitrary (continuous, differentiable) infinitesimal changes in the field - in particular, invariant under changes chosen to vanish on the boundary of any arbitrary region. Thus "boundaries" are not relevant in this context. Boundary conditions, or periodicity conditions, or some other imposed conditions, come into the picture when the equations are applied to solve a particular problem. They are a characteristic of the particular system under investigation, not of the equations that determine it's behaviour. As I understand it, you are aiming to get away from that convention in order to explain quantum theory by giving a more fundamental role to periodicity in classical field theories such a GR. Am I correct?

Sorry Dolce,

I see that you use freely the Plank equation, periodicty of something called, I assume, a photon, conservation of energy and momentum. Myself I stay at something more basic, I do not worship all these laws, I have never been much of a true believer. I try only to understand what is space, time and the geometry of the universe, without going to the Pious Scriptures and comments from Brothers Feynman or Dirac.It is complicated enough.

Dear Eric Lord (and jean claude Dutailly),

thank you for your comments, and for considering my paper. I will try to answer you in detail, so I divide the answer in three posts.

Reply concerning Einstein's proposal of unification of quantum and relativistic mechanics described in Pais's book. In the chapter that you mention (I translate from my Italian edition) this point is discussed in 26b and 26c, but I remember that similar statements are also reported elsewhere in the book (even more explicitly). By the way, according to Pais, "Einstein believed that a unified description would be possible by starting from a classical [relativistic] field theory and *conditions* imposed on the theory itself". Then Pais refers to Einstein's paper (title in German " [Classical-relativistic] Field theory offers the possibility to solve the quantum problems") in which "Einsten describes the successes of relativity in electrodynamics by using [classical-relativistic] differential equations and initial conditions [i.e. boundary conditions] for the space-[time]". These kind of boundary conditions are those meant by Jean Claude Dutailly in his post above. Then, from the book, "according to Einstein, the discrete orbits of the Bohr atom show that such *initial conditions* cannot be chosen arbitrarily". Then he addresses this non-arbitrary (i.e. intrinsic in the quantum system) initial conditions as "overdetermination" condition of the relativistic differential equation of the classical fields. Now, if you think about Bohr atom you see that the atomic orbits are determined by a conditions of periodicity, i.e. of closed orbits in space-time according to the Bohr-Sommerfeld quantization condition. In other words the atomic orbitals can be seen as the deformation, due to the Coulomb potential, of a particle in a box. As in the particle in a box, the atomic orbital can be seen as the vibrational harmonics associated to boundary conditions (similarly to a vibrating string). Thus, Einstein's central concept of "superdetermined" conditions, is a kind of boundary conditions for the field differential equations, intrinsic in the quantum system.

Remarkably, the wave particle duality "superdetermines" the boundary conditions of a quantum system. In fact, every elementary quantum system, e.g. an elementary particles, is a periodic phenomena in which space and time variables enter as angular variable of de Broglie - Planck lengths. According to my results, the superdetermination of intrinsic periodicity imposed to elementary particles is mathematically equivalent to ordinary quantum mechanics in both the canonical and Feynamn formulations. in other words my results proves that Einstein was absolutely right in his thought about quantum mechanics. Is it possible to unify quantum and relativistic mechanics by superimposed (intrinsic) boundary conditions to relativistic differential equations.

Dear Eric Lord

Concerning the second part of your comment, obviously Periodic Boundary Conditions (i.e. periodicity) are Boundary Conditions. In that paper I use Periodic Boundary Conditions because I mainly investigate bosons. As I explain in my paper "Compact time and determinism" one can assume more general boundary conditions such as combination of Dirichlet and Neumann boundary conditions (from which Periodic boundary conditions can be inferred) and these describes different kind of particles. For instance Fermions are describe by anti-Periodic Boundary Conditions in order to get spin-statistic and spin 1/2 (i.e. from spin up, after a period you have spin down).

Space and time separately have a well defined physical meaning characterized by two well defined units of measurement. The space-time is a metaphysical quantity that doesn't have a defined standard of measurement. Consequently it is very hard (or impossible) to define the boundary of space-time as it is impossible to define the boundary of any metaphysical quantity. When then boundary conditions of the space-time have been fixed for a particle, the following conditions of periodicity have been considered: L=h/p for space and T=h/E for time. These conditions are valid just for space and time separately and not for the metaphysical space-time.

The considered uncertainty doesn't derive from the Heisenberg Uncertainty Principle but it is already present in the only definition of equivalent wavelength for a particle derived from the de Broglie relation. In fact p=mv and consequently the equivalent wavelength of a particle depends on the speed of particle and it just generates indetermination. I would advise to make use of the Compton wavelength L=h/mc that is independent of the speed.

The inconsistency of the Heisenberg Uncertainty Principle is based on the inadequate use of another mathematical model for photons and consequently for massive particles.

Dear Eric Lord

I reply to the last part of your comment: the variational principle. What I am going to say is also described in the first section of my paper "Compact time and determinism..." (see also references therein) and it is well-known in the high energy physics since it is at the base of the description of compact extra-dimensional theory and string theory. It is a pity that these aspects of the variational principle are not generally known.

Consider an action defined in a finite or infinite volume, for instance the action of a free bosonic field. To calculate the variation you must integrate by parts the terms containing the derivative of the variation of the field. You obtain the integral of the Euler-Lagrange equation multiplied by the arbitrary variation of the field, plus a boundary terms containing the derivative of the field times the variation of the field evaluated at the boundary of the volume. The variational principle says that the classical solution is that minimizing the action in every point of the volume. The minimization the action inside the volume implies the Euler-Lagrange equations, i.e. the equations of motion of the system. The boundary terms of the action variations are in general non vanishing. To minimize the action in every point you must therefore assume some boundary conditions. Typically, for an infinite volume, this boundary conditions are given by the represented by the assumption that the field at the spatial infinity must be zero, so the boundary terms are zero and the variation is zero everywhere. For a finite volume, to fulfill the variational principle, i.e. to have zero variation of the boundary terms, you can assume several boundary conditions. Since the boundary terms are the derivative of the field times the variation of the field you can assume that: the field has fixed or zero value on the boundary (namely Dirichlet boundary conditions), the derivative of the field is zero (namely Neunman boundary conditions), or you can assume periodicity or anti-periodicity so that the boundary terms at the ends of the integration region cancel each other out, etc....

In other words, if you define an action, the variational principle, besides the equations of motion, also imposes you the boundary conditions that are compatible with your action. Similarly, If you try to write the action associated to the Einstein equations of general relativity, you see that the variation principle also imposed some boundary conditions for the space-time metric. These boundary conditions are not trivial and depends on possible boundary terms of the actions....

Article Compact Time and Determinism for Bosons

Dear Jean Claude Dutailly,

like you, I am also trying to understand the nature of space-time, but this inevitably implies to understand quantum mechanics and vice versa. The results that I have obtained are extremely simple, after all it is a "Columbus egg". If you like Feynman point of view, please read his book "QED, the strange theory of light and matter". The idea presented in that book is at the base of my description of elementary space-time cycles. You will see that Feynman describes elementary particles as clocks following different paths. The higher probability to find a particle in a given point is where these clocks are in phase.

regards,

Donatello

Dear Daniele Sasso,

I do not understand what you mean when you say that space and time have a well defined physical meaning but space-time has a metaphysical meaning. Minkowski, at the beginning of the past century announces (I do not remember the exact words) "from today space and time are not separated entities but they are interconnected". This new entity was named space-time. The meaning of this claim is clear if you think to the Lorentz transformations from which, something that is purely temporal in a reference frame appear to be both temporal and spatial: es c t = \gamma c t' - \gamma \beta x'.

Indeed in my papers I use extensively the Compton periidicity which is independent on the reference frames and is fixed by the mass T_C = h / M c. From this Compton periodicity you can get, through Lorentz trasformation, the resulting de Broglie Planck periodicities in a generic reference frame c T_C = c \gamma T - \gamma \beta L, where T = h / E and L = h / p, being E and p the energy and momentum of the particle in that reference frame.

Dear Docle

Can I understand from your phrase"The idea presented in that book is at the base of my description of elementary space-time cycles. You will see that Feynman describes elementary particles as clocks following different paths. The higher probability to find a particle in a given point is where these clocks are in phase. " That your theory implied that the space time you dealing with is indeed a dynamic space time.

Dynamic space time means the clocks and sticks are not static as it was supposed in SR but are moving with the particle movements?

Dear Dolce,

Thanks you for your answer. Myself I have been perplexed by QM. This is a very muddled topic. So I tried to understand at least one part, the one which is related to the celebrated axioms (Hilbert space, observables, eigen values and so on). I discovered that they have nothing to do with physics : they are pure mathematical results, true whatever the scale, and even valid also in Economics (see my paper on Systems and Hilbert space). These results are pure mathematics and theorems. But QM is more than that and encompasses the "duality matter / /field". I did some research on the subkect, and I have a good understanding of why material bodies can be represented by spinor fields (and an explanation for spinors), but the converse (from field to bosons) is less obvious.

However I do not see why space-time would be a metaphysical concept. This is a theory of the geometry of universe, and it has solid ground. At least as solid as QM, the concept of particles, energy,... We need a theory about the geometry of the universe because we need to tell how to compute the data when one goes from one observer to another observer (at another location, in space or time). Without the possibility to check data from different observers there is no science, because the laws would depend of the observer. And as any measure comes from measures of lengths and times, the theory must account for both. Relativity provides such a theory, substituted to the galilean geometry. It can be checked and so it is a scientific theory. Perhaps it will be replaced by something else, but so far this the best thatwe have.

About the principle of least action, there is much to tell ! The Euler Lagrange equations are valid only under some conditions, notably that the integral is defined by an obsolutely continuous measure on the whole of the area. And then the values on the boundary do not matter. They are used in the differential equations which result from the stationary conditions. The fact that some quantties in the action are derivatives is not an issue, as any lagrangian can be defined on jet prolongation of a fiber bundle.

Dear Sadeem Fadhil,

happy to hear about you again. Feynman in his book described the path integral in terms of clock phases. The periods of the clock are given by the energy according to Planck relation T = h / E. He provided nice pictures about that, see for instance:

https://nige.files.wordpress.com/2011/01/feynman-qed-1985.gif

or the nice animation from wikipedia

https://en.wikipedia.org/wiki/File:Feynmans_QED_probability_amplitudes.gif

The model described in my papers is based on the assumption of intrinsic periodicity. It enforces the wave-particle duality. Since every particle is a periodic phenomenon, the Planck time periodicity implies that every free particle (persistent periodicity) is a reference clock. This means that from the point of view of an elementary particle of Planck time periodicity T=h / E, the path ending at time t and the paths ending at time t +/- n T are equivalent. See my papers:

http://arxiv.org/pdf/0903.3680v4 fig.4,5,6

http://arxiv.org/pdf/1111.3319.pdf

Thus the boundary conditions of space-time in this description are determined by the wave-particle duality. That is it is possible to have a full consistent description of physics by assuming that the space-time variables are cyclic, with de Broglie- Planck periods, and the manifestation of these periodic boundary conditions is quantum mechanics with all its phenomenology. In my paper I have unequivocally proven that all the foundational aspects of quantum mechanics (Hilbert space, Shcroedinger equation, Hamiltonian operator, commutation relations, Born rule) can be derived from intrinsic periodicity. The classical evolution of such periodic phenomena are exactly described by the Feynman Path Integral, and this is clear if you read the mentioned Feynman Book "QED, the strange theory of light and matter". The amazing thing here is that this idea seems to be systematically ignored by the community, despite the undeniable proves of its feasibility.

In particular, at rest a particles has an intrinsic periodicity equal to the Compton time T_c = h / M c, whereas in a generic frame the time periodiity is T = h / E as said in the post above about the Lorentz transformation of the Compton periodicity. From this also follows that the periodicity of the clocks varies with interactions, i.e. variations of energy, so they are dynamical in both special relativity and general relativity meanings; that is they are described by geometrodynamics of space-time boundaries (notice the analogy with the holographic principle).

Dear Docle

Thank you for your answer. I have another questions which are related to the first. the first question Is the periodicity in particles' time represents an indication of periodicity in space time? if yes

Does your theory describe it?

jean claude Dutailly,

Indeed the outcomes of my work is that physics, including quantum mechanics, has a simple geometric description. QM mechanics can be derived by space-time geometrodinamics as soon as you assume boundary of space-time. These boundary condition are dictated by the de Broglie- Planck relation. In this way undulatory mechanics can be directly encoded into the space-time geometry. This yields an exact derivation of QM mechanics from relativistic dynamics. The interpretational problem of QM is that we have not understood its underlying physical meaning, and this mystery is hidden behind the "de broglie periodic phenomenon", i.e. behind the wave-particle duality. The principle of last actions implies assumption of boundary conditions, which is typically assumed to be stationary boundary conditions. But it is known that this boundary conditions are not fully physically constant: even preparing a particle with a stationary value at the beginning of the path, who says to the particle that it must have a stationary value just at the end on the path? Try to find on the literature about this problem of stationary boundary conditions. What I have found is that QM (and gauge interactions) can be derived from the classical principle of last actions, applied to a relativistic theory, as soon as you assume the right boundary conditions, that is to say the boundary conditions of space-time implicit in undulatory mechanics. Please try to read my papers from rigorous demonstrations.

Dolce,

The fundamental axioms of QM (Hilbert space,...) can be proven from some very general hypotheses, by ure mathematics. There is absolutely no physics in them. As for the duality wave / particle one must start to define what is a particle and what is a wave or field. You start from the de Broglie/Planck relation, but one does not know to which objects the energy or the wave lengths are related. Historically it has been written for particle and plan waves.

Non constant boundary conditions ? Non Constant with respect to what ? In relativity we have an area of the 4 dimensional space time, and whatever it contains, which is constant by definition. The trajectory of a particle is a map and it is taken care of in the lagrangian. The issue of "preparing" a particle in a given state is QM verbiage. When particles collide in an accelerator the physicists have no such qualms. They know they trajectories and even their speed.

Dear Dutailly,

Sorry. It was a typo. I meant consistent, in the sense described in the post.

The Hilbert space has absolutely a physical meaning. It was invented to describe vibrating string (sound waves), i.e. object vibrating in compact dimensions. The harmonics of a vibrating string form a complete set, so they can be used to define a base of an Hilbert space with related inner product. the analogous of the Hamilton operator is the frequency operator which returns the frequency if applied to a give harmonic. In this analogy the time evolution of a vibrating string (wave equation) written in the Hilbert space corresponds to the Schrodinger eq. As you seee, by assuming waves in compact dimensions you get analogies with quantum systems. The assumption of compact space-time dimensions with BCs yields an exact equivalence to ordinary QM.

Dear Dolce,

We do not need analogies. Take a material body, its trajectory can be represented in some fixed frame by a map giving a vector X(t) . This map is square integrable, so it belongs to a well known Hilbert space. There is not need for the magic of QM. And of course when you want to observe this map X, you have to estimate a map x from a finite number of data, and there is an operator which goes from X to x which is self adjoint, the value which is measured is an eigen vector, and the probability to get one value is linked to the square of X. There is absolutely no mystery, no exotic assumption about space, time, matter or anything, only common sense and a bit of mathematics.

Dear Sadeem,

thank you for your question. It goes to the core of the problem that I want to rise. The answer is yes: the space-time periodicity of a particle, prescribed by undulatory mechanics, is the manifestation of periodicities of the relativistic space-time dimensions. If in a given point of space-time you have energy E and momentum p, then in that point you have a periodicity of space-time T = h / E and L = h / p. In my model this is described by associating to every space-time point local periodic boundary conditions with periodicities T and L. These periodicities transforms according to the relativistic laws (Lorentz transformations). Indeed relativity describes how the periodicity and lengths of reference clocks and distances vary with change of reference frame and interactions. Relativistic causality is given by the fact that the energy, and thus the periodicity of space-time, from different points varies according to the retarded potential.

To have a fully consistent description however we must consider the role of intermediator of interactions, that is the responsible of the transmission of the energy and thus of the retarded variations of periodicities. Indeed the photon can have infinite space-time periodicities, in fact a photon has zero mass and therefore it has infinite Compton periodicity (contrarily to matter particles which have finite Compton periodicity). These infinite space-time periodicities of light can be thought of as defining the reference, emphatically non compact, space-time of the ordinary description of relativity. In other words this defines the relational structure of relativistic space-time cycles.

It is a matter of fact that particles are periodic phenomena whose space-time periodicity are related to their kinematical state. This is the wave particle duality in which the periodicity in time and space of a particle is related to its energy and momentum T = h / E and L = h / p. In the ordinary picture we assume infinite space-time and that we put these coordinates in phasor, e.g. the wave (quantum number n = 1) associated to a particle is sin(t / T - x / L) - or more generally exp(-i(t / T - x / L)). In my model these periodicities are directly encoded in the space-time geometrodynamics. The harmonics of these vibrations of space-time associated with relativistic, dynamical, periodicities (similarly to vibrating string of particles in a box) reproduces exactly, i.e. are mathematically equivalent, to ordinary quantum mechanics. This is proven, above any doubt, in my peer-reviewed papers (but it is systematically ignored from those people who base their academic careers on increasing confusion and myths in physics).

Dear Docle

Thank you for your answer

I agree with you about the periodicity of time and position. But this involves the applicability of boundary conditions by one of the following two choices:

1. The boundary conditions of space time are already fixed at the points of space,

or 2. the space time it self is dynamic and repeat itself periodically; in other words the space points are moving in a periodic manner.

which choice you prefer?

Thanks

Dear Sadeem,

there are aspects of both options that applies to my model.

1) space-time boundary conditions at a given point are fixed by the energy and momentum (energy momentum tensor) if that point according to T = h / E and L = h / p.

2) The energy and momentum (energy momentum tensor) propagates according to the relativistic law. This implies that the space-time periodicities dynamically varies according to the relativistic causality and general relativity (variations of space-time metric). If you assume that periodicity is equal every where (energy and momentum constant in every space-time point) and you imagine to vary the energy-momentum in a given point, that the variation propagates so that the points in causal connection with this point vary space-time periodicity with the variation of energy momentum. Distant points, e.g. outside the light cone, are not affected by this variation so in these points the periodicity reman the same.

In this way you see that causality, which at a first fight may seem to be not satisfied by assuming space-time periodicity, is perfectly satisfied in this description. This is very related to the notion of time, as described in my papers. In particular, the assumption of intrinsic periodicity implies that every free particle (constant energy) is a reference clock. As in a calendar (think to defined time in terms of the phases of periodic phenomena such as the motion of the sun, moon and planets) the flow of time is described by the relational combination and retarded modulations of all these clocks (now is 19:27 of 23 March, that is, approximatively the instant of time corresponding to 2014.5 cycles around the sun from a reference year 0, nearly 6 moon cycles from the beginng of the year and 5/6 of earth rotation, ...)

That's good stuff. In my opinion I think the second answer is closer to reality and causality.

Thanks for your answer.

The idea of space-time arise in the human mind by way of delusion. This is identical to mistaking a rope for a snake. The human mind superimposes the idea of snake on a rope and behaves exactly as if it has seen a snake. We can present this argument in a little more scientific language. When a particle wave is presented to a physicist, instead of seeing the oscillating energy, what he does is, superimposes the idea of wavelength and period on this wave and sees the space-time. All the geometrical theories in physics are founded upon such delusion. So the boundary of space-time is determined by the extent of the oscillating energy you are looking at.

Please read the paper, may be helpful for you.  A new approach to the correction of

Galilean transformation.