{"id":426,"date":"2011-02-23T07:39:29","date_gmt":"2011-02-23T12:39:29","guid":{"rendered":"http:\/\/www.journal-of-nuclear-physics.com\/?p=426"},"modified":"2011-02-28T05:03:46","modified_gmt":"2011-02-28T10:03:46","slug":"on-the-rise-of-gravitation-in-a-discrete-space-time","status":"publish","type":"post","link":"https:\/\/www.journal-of-nuclear-physics.com\/?p=426","title":{"rendered":"On the rise of Gravitation in a discrete space-time"},"content":{"rendered":"

by Chatzichri\u2019stos Aris<\/em><\/p>\n

Department of Electrical and Computer Engineering,
\nNational Technical University of Athens (NTUA), Athens, Greece.<\/em><\/p>\n

Direct Download<\/a><\/p>\n

<\/strong><\/p>\n

<\/strong><\/p>\n

Contents<\/strong><\/p>\n

Abstract
\nIntroduction
\n1. Basic Concepts and Principles
\n2. Conservation of Energy
\n3. Entropy and Space Expansion
\n4. Destination and Velocity
\n4.1 Movement
\n4.2 Destination
\n4.3 Velocity
\n<\/span><\/span><\/span><\/span>5. Newtonian Gravitation and Special Relativity
\n5.1 Gravitational Force
\n5.2 Special Relativity
\n6. General Relativity and Dark Matter
\n6.1 General Relativity
\n6.2 Dark Matter
\n7. Hubble\u2019s Law and Dark Energy
\n8. Inflationary Universe
\n9. Observable Predictions
\nConclusions
\nReferences<\/span><\/span><\/span><\/span><\/p>\n

Abstract<\/strong><\/p>\n

Deduction of a quantum model, with basic principles the Discrete nature of Space and the Second Law of Thermodynamics, which is compatible with Special Relativity in the absence of Gravity, while its linear edition is proven to be compatible with linear General Relativity (aka the first order terms of the two theories are basically identical). Finally, is being proven that the non-linear edition of this theory contains the phenomena of Dark Matter and Energy as its natural conclusions. \u00a0Also, some astronomical observations are proposed to test the theory.<\/p>\n

<\/p>\n

Introduction<\/strong><\/p>\n

According to General Relativity[1][2]<\/sup>, the mechanism of Gravitation is the curvature of space-time around massive bodies. This idea is exceptionally accurate at a macroscopic scale. Nevertheless, at a microscopic level it has been proven to be incompatible with the Standard Model of Quantum Mechanics. Therefore, it would be very useful for the pursuit of the unification of these two theories, if a model was to be found, that could be proven equivalent to the curvature of space-time, but without these characteristics that make Relativity incompatible with the theories of Quantum Mechanics.<\/p>\n

Moreover, a theory of Quantum Gravity should explain the phenomena of Dark Matter and Energy. According to the established theories, the expansion of the Universe was caused by the Big Bang and the total Gravitation of the Universe should slow it down. On the contrary, the astronomical observations of the last decade[3][4]<\/sup> prove that in fact this expansion accelerates. For the solution of this paradox, it has been assumed that the Universe is almost completely dominated by Dark Energy<\/em>[5]<\/sup>, a sort of exotic anti-gravitation of empty Space, which cancels the effects of normal Gravitation and thus accelerates the expansion of space-time. The nature of this Energy remains a mystery.<\/p>\n

In addition to the above stated problem, other astronomical observations imply that the total known mass of the galaxies is not enough for them to be stable and produce the extensive gravitational halos that are observed throughout the Universe. This unknown missing mass is named Dark Matter<\/em>[6]<\/sup>.<\/p>\n

The goal of this paper is to propose a theory that can explain the above phenomena. The point of view of this proposed model is close to \u201cLoop Quantum Gravity<\/strong>\u201d[7][8]<\/sup>, but with some crucial differences: On the one hand space is quantized<\/strong>, just like in LQG. But on the other hand there is no actual curvature<\/strong>. Space is discrete, but flat. All the phenomena that have to do with gravitation and the supposed curvature of space-time are being created by a different (but at the end equivalent) mechanism: The creation of new space around massive bodies<\/strong>.<\/p>\n

Let\u2019s enlighten this mechanism by using the following Gedank<\/em> experiment:<\/p>\n

Let two points A and B; between them there is a \u201cmoving walkway\u201d, like the ones that exist in every gym. When this walkway is \u201coff\u201d, it does not move and then an observer, starting from point A at some moment, calculates the amplitude of AB by moving with constant velocity from A to B and thus counting the time needed. Then obviously,<\/p>\n

|AB|=u\u00b7t<\/p>\n

Now, if the moving walkway runs with a constant velocity, smaller and opposite to the velocity u<\/strong> of the observer, then if he\/she moves from A to B with the same velocity as before, this observer will calculate a bigger value for the |AB|. Thus, one can argue that at the second occasion the line segment AB has been curved<\/strong>.<\/p>\n

One can say that when the moving walkway is on, then at the point B \u201cnew space\u201d is created, which moves afterwards in the opposite direction of the movement of the observer.<\/p>\n

Intuitively, one can argue that if massive bodies produce new space, which moves afterwards away from them, then perhaps this procedure can create a phenomenon that looks much alike a moving walkway and is equivalent to the curvature of space-time around masses<\/strong>.\u00a0\u00a0 <\/strong><\/p>\n

Thus, the basic compass\u2013question of this paper will be the following: Could the gravitational expansion of a given length (curvature) have a closer connection to the expansion of the Universe? <\/em><\/p>\n

Let us make the last thought more specific:<\/p>\n

Gravity is a Force, so it should be connected to a particle. If Gravity is produced by the expansion of space, then one concludes that space should be discrete<\/strong>. That will be the basic hypothesis, from which this theory of Quantum Gravity will be deduced. Empty space can be regarded as the particle with the smallest possible energy; therefore the energy of empty space is quantized too. We name this energy \u201cquantum of energy<\/strong>\u201d. In this model, quantum of energy can be consider as the sole \u201celementary particle\u201d in the sense that all other particles, including space, are systems of quanta of energy.<\/p>\n

When the Universe expands, new quanta of space should be created and thus, due to the Principle of Conservation of Energy, an equal amount of energy should be consumed somewhere. The source of this energy is obviously the massive bodies and the photons, i.e. the objects that contain energy. This means that new space is being created around great structures of mass and then is moving \u201coutwards\u201d. As a result, an observer stationed near a mass, notes the expansion of a given length, as light has to travel through this moving space, in addition to the classical constant space. The situation is much alike a three dimensional gym walkway, on which light , while moving with a constant velocity, needs more time in order to travel a given length, when a massive body is around.<\/p>\n

One might wonder how space can possibly be discrete without being observed so far. The answer is of course the order of magnitude. After all, mass is also discrete, but this fact is obvious only in scales of about 10-14<\/sup>m. The proposed quantum of space is expected to be much smaller. It is likely, for example, to be equal to Plank\u2019s length. As this is of the order of 10-35<\/sup>m, it can easily be understood why one cannot observe directly the discrete nature of Space.<\/p>\n

This paper studies gravitation, using the above considerations. It accomplishes the following:<\/p>\n

1)\u00a0\u00a0\u00a0\u00a0\u00a0 The construction of a \u201cgrid\u201d of quanta of space (similar to the \u201cspin network\u201d of Loop Quantum Gravity).<\/p>\n

2)\u00a0\u00a0\u00a0\u00a0\u00a0 The deduction of space expansion<\/strong> using only the Second Law of Thermodynamics in this space.<\/p>\n

3)\u00a0\u00a0\u00a0\u00a0\u00a0 The finding of Special Relativity<\/strong>, using only discrete mathematics and considerations similar to those of Zeno of Elea.<\/p>\n

4)\u00a0\u00a0\u00a0\u00a0\u00a0 The proof that the solution for the point-like source of gravity in this discrete space is completely equivalent to the Schwarzschild solution<\/strong>.<\/p>\n

5)\u00a0\u00a0\u00a0\u00a0\u00a0 The explanation of Dark Matter and Energy<\/strong>, without asking for other exotic structures. Not even a cosmological constant.<\/p>\n

6)\u00a0\u00a0\u00a0\u00a0\u00a0 The Inflationary era<\/strong>: Why it started and why it ended.<\/p>\n

7)\u00a0\u00a0\u00a0\u00a0\u00a0 The proposal of some possible experiments and observations for the test of the theory.<\/p>\n

1. <\/strong>Basic Concepts and Principles<\/strong><\/p>\n

If one accepts the basic principle of the Discrete nature of Space, as it follows from the introduction, then one should first of all solve the problem of the topology of such a Space. As space is made of particles, this problem is identical to the basic problem of crystallography<\/strong>, with the obvious replace of atoms with quanta of space. Thus, we will search for a similar solution:<\/p>\n

[1.1] First of all, let an infinite mathematical grid<\/strong>. Like all grids, its edges (or nodes) are countably infinite and therefore they can be defined by a vector with integer coordinates. As this grid is mathematical, it does not exist physically, it is considered merely as a base for the formalism and the better understanding of the theory. Thus, its existence will not be inserted axiomatically. However, the number of dimensions of this grid is important to the Universe itself; therefore one must accept the following postulate:<\/p>\n

1st<\/sup> Axiom:<\/strong> The grid is 3-dimentional<\/em>.<\/p>\n

(Dimensional Principle)<\/p>\n

[1.2] That grid should satisfy some obvious crystallographic prerequisites:<\/p>\n