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volume II: Synopsis

section III: Modern Physics

page 24: Meaning

The Universe, as we see it, has grown enormously over the last century. Modern cosmology reveals a Universe which is very large and very old, and appears to have evolved from a structureless point, the initial singularity). Gravitational singularity - Wikipedia

This scenario is modelled using the general theory of relativity published by Einstein in 1916. The special theory of relativity tells us how things look in a Universe where everything is moving at a uniform velocity. There is (by definition) no force (and no acceleration) between inertial frames. General relativity - Wikipedia

Einstein wished to extend this theory to include force and acceleration. The seed of the general theory was for Einstein 'the happiest thought of my life': 'if a person falls freely he will not feel his own weight. I was startled. This simple thought made a deep impression on me. It impelled me toward a theory of gravitation'. In 1907, Einstein realized that all natural phenomena could be discussed in terms of special relativity except gravitation. Wikiquote - Einstein's happiest thought

Putting this idea into mathematical form was not easy, and required a change from the Cartesian coordinates used by Newton to Gaussian coordinates. Newton imagined an absolute space and time coordinate system which enabled him to give a name ('coordinate') to every point in the solar system. Einstein, following Gauss, imagined that the Universe is in effect its own coordinate system, so that we work with 'intrinsic' coordinates. This opened the way to treating the geometry of the world as a dynamic rather than a static entity. Gaussian curvature - Wikipedia The mathematical theory needed to develop this view came from the geometry of dynamic spaces invented by Riemann. Riemannian geometry - Wikipedia

The study of the Universe requires the identification of fixed points or invariances that stay the same as the system around them changes. As I walk, lots of muscles move, but the lengths of my leg bones remains constant. Gaussian coordinates enable us to define the structure of a space by measuring the distances between observable events in the space.

Although Gaussian coordinates are intrinsic, they still involve the duality of knower and known. We use coordinates as a known system in order to study an unknown one. The relationship between them we call meaning or mapping. The coordinate system in general relativity Einstein calls the 'reference mollusc' by which I understand him to mean the soft and flexible parts of a mollusc rather than the hard and rigid parts. Einstein

After a lot of work, Einstein found the gravitational field equation, which in its simplest form reads G = T. G is the Einstein tensor which encodes the metric structure of space. T is the 'stress energy' which encodes the distribution of mass-energy in space. Space-time intervals in the Universe depend on the energy density in the neighbourhood. Einstein field equations - Wikipedia, Pais

Although Einstein's equations are rather rich and complex, they result from a very simple mathematical constraint: the transformation between any two systems of Gaussian coordinates should be continuous and differentiable and leave the the interval between any two points unchanged. This idea is called 'general covariance'. General covariance is a statement about meaning or mapping. Broadly, to truly represent the world as languages (coordinate systems) change, the words used to express the representation (another coordinate system) must also change in order to respect the invariance of a reality independent of coordinates. Metric tensor - Wikipedia

Mathematically, a function is said to be continuous if its input and output are coupled so that small changes in input yield small changes in output, input and output tending to zero together. The differential equations of the general theory of relativity represent fixed points in the continuous dynamics of the Universe. Relativity is a classical theory and has proven very hard to reconcile with quantum mechanics. Continuous function - Wikipedia

The general theory is the theory of dynamic space, and it tells us that the Universe does not have a fixed size: it is either expanding or contracting. Observation shows that it is expanding, and careful mathematical analysis of the theory suggests that the Universe started from a structureless point. But we have a problem here. Gravitation is always attractive so how does it make the Universe expand? Hawking & Ellis

On this site we are trying to model the Universe as a computer network, so we ask ourselves how do we interpret the classical continuity that lies at the heart of general relativity in terms of communication and computation? Continuity means symmetry, the equivalent of nothing happening, no change. This is consistent with the observation that gravitation sees everything simply as a quantity of energy.

Energy is conserved,which in quantum mechanical terms means that no matter when we look at the Universe, the overall rate of activity, measured in quanta of action per unit of time, is the same.

Let us guess that quantum mechanics does not see the difference between potential and kinetic energy. Gravitation, which emerges from quantum mechanics with the origin of space, does. We can interpret Einstein's equation to mean that the potential energy of a spatial structure is equal to the kinetic energy that exists in that space.

Now we assume that the Universe has evolved by natural selection since the very beginning. Natural selection picks out those systems which are optimized for survival. Let us further guess that the efficacy of the Lagrangian method in physics is a consequence of this optimization. Natural selection - Wikipedia, Lagrangian - Wikipedia

Hamilton's principle tells us that Nature tends to favour situations where the action (which can be measured in units of Planck's contant) is stationary. The action is the time integral of the Lagrangian. Let us assume here that the action of the Universe is stationary because the Lagrangian of the Universe is zero, implying that its kinetic and potential energy are (integrated over time) equal. This is the simplest interpretation of Einstein's field equation. Action (physics) - Wikipedia, Hamilton's principle - Wikipedia

Gravitation introduces interaction between inertial frames, thereby differentiating the frames and creating space. The invariant that defines the structure of quantum mechanics is the quantum of action. The differentiation of the Universe into space and time introduces the measures of rate of action per unit time (frequency) and action per unit space (momentum). In special relativity space and time and momentum and energy transform in the same way: they are mathematically indistinguishable. This situation describes life in an inertial frame.

We can imagine the initial singularity as inertial by definition, since there is nothing else to perturb it. Now let it communicate with itself, thus copying itself. For the copy to exist, it must be differentiated from the orignal and let this differentiation be by putting the copy in a different place, which in the inertial world means moving at a different velocity. The change of velocity implies force and changes in the kinetic and potential energy of the child frames. The symmetry of the initial inertial frame is broken to make two interacting frames. We imagine this process to continue indefinitely subject only to conservation of action, energy and momentum to give us the spacetime manifold we now enjoy.

We can express this 'classical' description in network terms. Networks grow by copying themselves. The copies then communicate, the network equivalent of a force. As the network continues to proliferate, the rate of communication between the different sources may vary, giving rise to different fores between them.

Symmetry breaks to create differentiation, communication and meaning. The classical theological version of this model is the model of the Trinity developed by Thomas Aquinas. Aquinas 160

To support this picture we have the second law of thermodynamics: overall, entropy never decreases (and mostly increases). What is entropy? A count of states, that is a count of differentiation. As the Universe differentiates, generating more fixed points , its entropy increases. Second law of thermodynamics - Wikipedia

We can communicate with one another because we share a common code or language. When I say 'I feel sad' you know what I mean because you have felt sad too. And so with all the communications that we have with one another. Meaning is made possible by shared experience. It may be one of the complex feelings that surround our reproductive activity, the simple statement 'this is a brick' or the most general protocol: I follow the rules of quantum mechanics.

[revised 24 May 2013]

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Further reading

Books

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Einstein, Albert, and Robert W Lawson (translator) Roger Penrose (Introduction), Robert Geroch (Commentary), David C Cassidy (Historical Essay) , Relativity: The Special and General Theory, Pi Press 2005 Preface: 'The present book is intended, as far as possible, to give an exact insight into the theory of relativity to those readers who, from a general scientific and philosophical point of view, are interested in the theory, but who are not conversant with the mathematical apparatus of theoretical physics. ... The author has spared himself no pains in his endeavour to present the main ideas in the simplest and most intelligible form, and on the whole, in the sequence and connection in which they actually originated.' page 3  
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Hawking, Steven W, and G F R Ellis, The Large Scale Structure of Space-Time , Cambridge UP 1975 Preface: Einstein's General Theory of Relativity ... leads to two remarkable predictions about the universe: first that the final fate of massive stars is to collapse behind an event horizon to form a 'black hole' which will contain a singularity; and secondly that there is a singularity in our past which constitutes, in some sense, a beginning to our universe. Our discussion is principally aimed at developing these two results.' 
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Misner, Charles W, and Kip S Thorne, John Archibald Wheeler, Gravitation, Freeman 1973 Jacket: 'Einstein's description of gravitation as curvature of spacetime led directly to that greatest of all predictions of his theory, that the universe itself is dynamic. Physics still has far to go to come to terms with this amazing fact and what it means for man and his relation to the universe. John Archibald Wheeler. . . . this is a book on Einstein's theory of gravity. . . . ' 
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Pais, Abraham, 'Subtle is the Lord...': The Science and Life of Albert Einstein, Oxford UP 1982 Jacket: In this ... major work Abraham Pais, himself an eminent physicist who worked alongside Einstein in the post-war years, traces the development of Einstein's entire ouvre. ... Running through the book is a completely non-scientific biography ... including many letters which appear in English for the first time, as well as other information not published before.' 
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Smolin, Lee, The Life of the Cosmos, Oxford University Pres 1997 Jacket: 'Smolin posits that a process of self-organisation like that of biological evolution shapes the universe, as it develops and eventually reproduces through black holes, each of which may result in a big bang and a new universe. Natural selection may guide the appearance of the laws of physics, favouring those universes which best reproduce. . . . Smolin is one of the leading cosmologists at work today, and he writes with an expertise and a force of argument that will command attention throughout the world of physics.' 
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Links
Action (physics) - Wikipedia Action (physics) - Wikipedia, the free encyclopedia 'In physics, action is an attribute of the dynamics of a physical system. It is a mathematical functional which takes the trajectory, also called path or history, of the system as its argument and has a real number as its result. Action has the dimension of energy × time, and its unit is joule-seconds in the International System of Units (SI). Generally, the action takes different values for different paths. Classical mechanics postulates that the path actually followed by a physical system is that for which the action is minimized, or, more strictly, is stationary. The classical equations of motion of a system can be derived from this principle of least action. The stationary action formulation of classical mechanics extends to quantum mechanics in the Feynman path integral formulation, where a physical system follows simultaneously all possible paths with amplitudes determined by the action.' back
Aquinas 160 Summa: I 27 1 Is there procession in God? 'Our Lord says, "From God I proceeded" (Jn. 8:42).' back
Continuous function - Wikipedia Continuous function - Wikipedia, the free encyclopedia 'IIn mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be a "discontinuous function". A continuous function with a continuous inverse function is called "bicontinuous".' back
Einstein field equations - Wikipedia Einstein field equations - Wikipedia, the free encyclopedia 'The Einstein field equations (EFE) or Einstein's equations are a set of ten equations in Albert Einstein's general theory of relativity which describe the fundamental interaction of gravitation as a result of spacetime being curved by matter and energy. First published by Einstein in 1915[ as a tensor equation, the EFE equate spacetime curvature (expressed by the Einstein tensor) with the energy and momentum within that spacetime (expressed by the stress-energy tensor).' back
Gaussian curvature - Wikipedia Gaussian curvature - Wikipedia, the free encyclopedia 'In differential geometry, the Gaussian curvature or Gauss curvature of a point on a surface is the product of the principal curvatures, κ1 and κ2, of the given point. It is an intrinsic measure of curvature, i.e., its value depends only on how distances are measured on the surface, not on the way it is isometrically embedded in space. This result is the content of Gauss's Theorema egregium.' back
General relativity - Wikipedia General relativity - Wikipedia, the free encyclopedia 'General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916.[1] It is the current description of gravitation in modern physics. General relativity generalises special relativity and Newton's law of universal gravitation, providing a unified description of gravity as a geometric property of space and time, or spacetime. In particular, the curvature of spacetime is directly related to the four-momentum (mass-energy and linear momentum) of whatever matter and radiation are present. The relation is specified by the Einstein field equations, a system of partial differential equations.' back
Gravitational singularity - Wikipedia Gravitational singularity - Wikipedia, the free encyclopedia 'A gravitational singularity (sometimes spacetime singularity) is, approximately, a place where quantities which are used to measure the gravitational field become infinite. Such quantities include the curvature of spacetime or the density of matter. More accurately, a spacetime with a singularity contains geodesics which cannot be completed in a smooth manner. The limit of such a geodesic is the singularity.' back
Hamilton's principle - Wikipedia Hamilton's principle - Wikipedia, the free encyclopedia 'IIn physics, Hamilton's principle is William Rowan Hamilton's formulation of the principle of stationary action (see that article for historical formulations). It states that the dynamics of a physical system is determined by a variational problem for a functional based on a single function, the Lagrangian, which contains all physical information concerning the system and the forces acting on it. The variational problem is equivalent to and allows for the derivation of the differential equations of motion of the physical system. Although formulated originally for classical mechanics, Hamilton's principle also applies to classical fields such as the electromagnetic and gravitational fields, and has even been extended to quantum mechanics, quantum field theory and criticality theories.' back
Lagrangian - Wikipedia Lagrangian - Wikipedia, the free encyclopedia 'The Lagrangian, L, of a dynamical system is a function that summarizes the dynamics of the system. It is named after Joseph Louis Lagrange. The concept of a Lagrangian was originally introduced in a reformulation of classical mechanics by Irish mathematician William Rowan Hamilton known as Lagrangian mechanics. In classical mechanics, the Lagrangian is defined as the kinetic energy, T, of the system minus its potential energy, V. In symbols, L = T - V. ' back
Metric tensor - Wikipedia Metric tensor - Wikipedia, the free encyclopedia 'In the mathematical field of differential geometry, a metric tensor is a type of function defined on a manifold (such as a surface in space) which takes as input a pair of tangent vectors v and w and produces a real number (scalar) g(v,w) in a way that generalizes many of the familiar properties of the dot product of vectors in Euclidean space. In the same way as a dot product, metric tensors are used to define the length of, and angle between, tangent vectors. . . . back
Natural selection - Wikipedia Natural selection - Wikipedia, the free encyclopedia 'Natural selection is the process by which favorable traits that are heritable become more common in successive generations of a population of reproducing organisms, and unfavorable traits that are heritable become less common. Natural selection acts on the phenotype, or the observable characteristics of an organism, such that individuals with favorable phenotypes are more likely to survive and reproduce than those with less favorable phenotypes. If these phenotypes have a genetic basis, then the genotype associated with the favorable phenotype will increase in frequency in the next generation. Over time, this process can result in adaptations that specialize organisms for particular ecological niches and may eventually result in the emergence of new species.' back
Riemannian geometry - Wikipedia Riemannian geometry - Wikipedia, the free encyclopedia 'Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a Riemannian metric, i.e. with an inner product on the tangent space at each point which varies smoothly from point to point. . . . Development of Riemannian geometry resulted in synthesis of diverse results concerning the geometry of surfaces and the behavior of geodesics on them, with techniques that can be applied to the study of differentiable manifolds of higher dimensions. It enabled Einstein's general relativity theory, made profound impact on group theory and representation theory, as well as analysis, and spurred the development of algebraic and differential topology.' back
Second law of thermodynamics - Wikipedia Second law of thermodynamics - Wikipedia - The free encyclopedia 'The second law of thermodynamics is an expression of the universal law of increasing entropy, stating that the entropy of an isolated system which is not in equilibrium will tend to increase over time, approaching a maximum value at equilibrium.' back
Wikiquote - Einstein's happiest thought Albert Einstein 'I was sitting in a chair in the patent office at Bern when all of sudden a thought occurred to me: If a person falls freely he will not feel his own weight. I was startled. This simple thought made a deep impression on me. It impelled me toward a theory of gravitation. Einstein in his Kyoto address (14 December 1922), talking about the events of "probably the 2nd or 3rd weeks" of October 1907, quoted in Why Did Einstein Put So Much Emphasis on the Equivalence Principle? by Dr. Robert J. Heaston in Equivalence Principle – April 2008 (15th NPA Conference) who cites A. Einstein. “How I Constructed the Theory of Relativity,” Translated by Masahiro Morikawa from the text recorded in Japanese by Jun Ishiwara, Association of Asia Pacific Physical Societies (AAPPS) Bulletin, Vol. 15, No. 2, pp. 17-19 (April 2005).' back

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