vol III: Development

chapter 4: Physics

page 5: Gravitation

Introduction

Here we take the view that gravitation is a quantum mechanical (ie computational) phenomenon which occurs prior to space-time and is fact responsible for the creation and annihilation of space-time. Einstein's general theory of relativity shows us in a kinematic way that the universe must either be expanding or contracting. Quantum mechanics adds the dynamics of the universal network to Einstein's kinematics. Misner, Thorne and Wheeler: Gravitation, pp 1203 sqq.

Aristotle and cosmology

Aristotle was one of the first thinkers to create a bridge between heaven and Earth, with his theory of potency and act. In the physical world, potency is matter, and act is form. Form informs matter to make it be what it is. The same bronze may be a sword or a ploughshare. Actus et potentia - Wikipedia, Hylomorphism - Wikipedia

Potency and act extended this idea of matter and form to the invisible world. Aristotle considered it axiomatic that no potency could actualize itself. From this he concluded that there must be a first unmoved mover which animates the world, since he understood motion to be the passage from potency to act. Aquinas used exactly the same argument in his first proof for the existence of God. Aquinas 13: Does God Exist?

Newton and gravitation

The next big link between the heavens and the earth came with Newton's theory of Universal gravitation and his application of this simple law to the movements of the planets and their satellites. Newton's law tells us that all mass points in the universe attract one another with a force which is equal to the product of the masses (M and m) divided by the square of the distance r between them:

F = GMm/r²

where G is the gravitational constant coupling force to mass and distance.

Newton based his derivation of this law on data collected by generations of astronomers and the laws of motion formulated by Galileo and Kepler. Astronomers long ago established a celestial reference frame in space and time for recording the positions and times of astronomical events. Celestial coordinate system - Wikipedia,

Newton assumed that space and time are independent of one another, so that the same time can prevail at all points in space. He also assumed that the gravitational interactions in the world are instantaneous: gravitational signals travel with infinite velocity.

Einstein and Special Relativity

Einstein developed two theories of relativity. The first, special relativity, was restricted to transformations between bodies in inertial motion. Inertial motion is the unaccelerated motion envisaged in Newton's first law: a body at rest remains at rest and a body in motion continues its motion in a straight line unless acted upon by a force. The second theory, general relativity, deals with all possible transformations between all states of motion, both inertial and accelerated. Inertial frame of reference - Wikipedia

One of the key questions in mathematical physics is the relationship between frames of reference and reality. The modern mathematical view is that motion is relative and it really makes no difference whether we imagine a fixed physical system in a moving reference frame, or vice versa. The essential feature of the motion is the transformation involved. In general a motion has a starting state, a transformation and a finishing state. We may think of the transformation as an algorithm or fixed point.

Einstein realized that the velocity of light is a local phenomenon described by Maxwell's equations, and must be the same for all freely falling observers. From this, with a bit of geometry, one can deduce the Lorentz transformation that carries us from one inertial observer to another. Since it takes signals time to travel a distance, Lorentz transformations affect both distances and times. If I watch you going past me at high velocity, I see your clocks are running slower than mine, your distances foreshortened relative to mine and your masses increased. You see the same changes looking back at me. Both of us see the same properties of standard things things like atoms and eggs in our own locality. Einstein's famous equation E = mc² is a further consequence of the Lorentz transformation. Maxwell's equations - Wikipedia, Lorentz transformation - Wikipedia, Time dilation - Wikipedia, Length contraction - Wikipedia, Mass in special relativity - Wikipedia

Einstein summed up special relativity in the following words:

Every general law of nature must be so constituted that it is transformed into a law of exactly the same form when, instead of the space-time variables x, y, z, t of the orginal co-ordinate system K, we introduce new space-time variables x', y', z', t' of a co-ordinate system K'. In this connection the relation between the ordinary and accented magnitudes is given by the Lorentz transformation. Or, in brief, general laws of nature are covariant with respect to Lorentz transformations.

This is a definite mathematical condition that the theory of relativity demands of a natural law, and in virtue of this, the theory becomes a valuable heuristic aid in the search for general laws of nature. . . . Einstein: Relativity, page 57

The notion of covariance arises from the belief that coordinate systems are created by physicists and engineers, and are not part of the system observed. Because this is so, we should expect the laws of nature to be indifferent to our choices of coordinate system. This is true in the classical physics of relativity. Quantum mechanics reveals that the reference system used to observe (the 'observable') plays a role in the result.

General relativity

Einstein realized that gravitation and accelerated motion are exceptions to special relativity:

In 1907 when I was working on a review of the consequences of special relativity . . . I realized that all the natural phenomena could be discussed in terms of special relativity except the law of gravitation. I felt a deep desire to understand the reason behind this . . . It was most unsatisfactory to me that although the relation between inertia and energy is so beautifully derived [in special relativity], there is no relation between inertial and weight. I suspected that this relationship was inexplicable by means of special relativity. Pais, page 179

The aim of general relativity is to find all the most general transformations which leave the real space-time distance between events unchanged. By observer we mean any particle capable of receiving and sending information. This led Einstein into complex mathematical issues that were resolved with the help of his friend Marcel Grossman. Grossman provided the crucial ingredient, Riemann's differential geometry, which enabled Einstein to formulate the general theory in mathematical language. Riemannian geometry - Wikipedia

Einstein said that the happiest thought in his life was the realization that a person in free fall would not feel his own weight. In other words it is possible to move inertially in a gravitational field. This path of free fall is called a geodesic. The gravitational field of the Earth, for instance, enables an object like a satellite to circle the Earth in free fall. Albert Einstein, Geodesics in general relativity - Wikipedia

The only way that such freely falling observers can detect the present of gravitation is by their movement relative to one another, a phenomenon called geodesic deviation. Because they are freely falling, they do not 'feel' gravitation. Freely falling observers thus became a reference system which Einstein used to explore the nature of gravitation.

A few years after he began to work on general relativity, Einstein realized that one cannot map space-time in terms of any independent frame of reference. Since we cannot look at the Universe from outside, we must look at it from inside. Also, we are confined to looking at things in our local environment. Our picture of the whole is built from local knowledge.

The solution to this problem was first seen by Gauss, who found a way to describe a surface not by referring it to some frame of reference but by considering the distances between points within the surface itself. This idea was taken up by Riemann who developed the differential geometry which became the mathematical foundation of general relativity. His desire to capture a universal space-time transformation introduced the notion that space-time is not a passive background for universal dynamics, but is part of the dynamics.

It is possible, using Riemann differential geometry, to implement the general principle of relativity: all bodies of reference K, K' etc are equivalent for the description of natural phenomena (formulation of the general laws of nature), whatever may be their state of motion. It required Einstein's insight to see how this could be done. His key was the principle of equivalence. Einstein: Relativity

The relationship between inertial mass and gravitational mass is mediated by the principle of equivalence tells us that no 'blind' observer can distinguish an acceleration from a gravitational field. A person in an enclosed container who feels an attraction to one side of the container cannot tell if this is because the container is being accelerated by an outside force or if it is being held stationary in a gravitational field.

Differential Geometry

Differential geometry describes a differentiable manifold. A differentiable manifold is a manifold in which it is possible to apply calculus. A manifold is a space that is locally flat or Euclidean. Calculus can be applied to spaces which are continuous. A continuous space can be mapped onto the real numbers. Differentiable manifold - Wikipedia, Calculus - Wikipedia

The general idea is that we can create a dynamic (curved) space by assembling infinitesimal portions of flat space in a way that permits differentiable transformations from one to another. We can define a path, called the geodesic, through this curved space, and the relationships between neighbouring geodesics reveals the curvature of space. The accelerated motion between neighbouring geodesics is interpreted as the force of gravity or acceleration, and controls the motions of moving bodies.

As in the Gaussian system, the shape of the space is determined by a metric which measures the distances between events in the space. Any gaussian coordinate system may be used to describe the a space as long as they all yield the same value for the metric.

The only mathematical constraint imposed on the structure of space by general relativity is that transformations be continuous and differentiable. The principal physical constraint is that the general theory predict approximately the same results as Newtonian theory in low energy regions of space-time like our Solar system.

Continuity and information

Continuous mathematics gives us the impression that it represents an infinite density of information. Cantor’s set theory has shown us how to treat continua as transfinite sets of discrete points. This theory is so convincing that we are inclined to treat such points as real symbols in a space of messages.

Quantum theory is worked out in multi-dimensional complex function spaces known as Hilbert spaces. Hilbert spaces are specified by their dimension, which ranges through the integers to the transfinite numbers. Although the coordinate bases of Hilbert spaces may be discrete and orthogonal, vectors in Hilbert space are described by continuous complex parameters. Quantum mechanical operations may be conceived as mathematically perfect parallel analogue computations. This has led the quantum computation community to believe that it may be possible to build quantum computers more powerful that classical turing machines. Quantum computing - Wikipedia

An alternative view of the information capacity of continua is provided by the mathematical theory of communication and algorithmic information theory.

For about 80 years much sweat, maybe some tears (and possibly a little blood) has been spent in the (so far) unsuccessful effort to quantize gravity. Here we interpret this situation as evidence that gravity may not be quantized, that is that it is something inherently continuous and therefore devoid of information.

This feeling arises from the notion that the universe may be modelled as a computer network. An important feature of stable networks is error free communication.

Shannon showed that we can approach error free transmission over a noisy channel by encoding our messages into packets so far apart in message space that the probability of confusion is minimal. We assume that the exchange of particles in the physical universe corresponds to the exchange of messages in the network model. Claude E Shannon

This correspondence enables us to interpret quantization as a consequence of the error free communication which (we assume) underlies the stability of universal structure. Conversely, we should not expect to find quantization where error is impossible, that is in a regime where every possible message is a valid message. Since gravitation couples universally to energy alone, and is blind to the particular nature of the particles or fields associated with energy, we can imagine that gravitation involves unconstrained and therefore intrinsically error free and non-quantized interaction.

Algorithmic information theory points in a similar direction. Working on the basis of Gödel's incompleteness result, Gregory Chaitin suggests that there can be no more information in a theory derived logically from a set of axioms than there is in the axioms themselves. An algorithm such as F = ma, therefore, even though the variables may be continuous quantities, contains no more information that that represented by the symbolic expression F = ma. Algorithmic information theory - Wikipedia, Chaitin: Information, Randomness & Incompleteness

From this point of view, Einstein’s principle of general covariance, that any gaussian coordinate system will do to describe a physical situation provided that any two coordinate representations same observable event can be joined by a continuous transformation also suggests that there is no information in a continuum. All gaussian representations express the same information in a different coordinate frame (or language), and the continuous transformation between them neither adds nor subtracts any information. General covariance - Wikipedia

This suggests that we can identify logical and geometric continuity. A logical continuum is equivalent to the deterministic argument (that is the computation) that binds the hypothesis of a theorem to its conclusion: if we accept Euclidian geometry, the interior angles of a triangle total 180 degrees. A logical continuum is a computable algorithm (turing machine) that, like a geometric continuum, transmits or transforms information without error from one representation to another. In this sense it is equivalent to a lossless codec. Codec - Wikipedia

Universality and the physical layer of the universal network

The general theory predicts that a geometrically consistent Universe must be either expanding or contracting. Current observations tell us that our Universe is expanding. Since the finite velocity of light enables us to look back in time, we can see the history of the Universe back to when it was a few hundred thousand years old and the photons we see as cosmic background radiation were emitted. Cosmic microwave background radiation - Wikipedia

As for as we know, gravitation is universal. Every form of energy in the Universe is believed to be coupled to gravitation so that every process can communicate with every other via the gravitational channel.

A corollary of the universality of gravitation is its antiquity. In the layered network model gravitation is the ultimate physical layer. We must assume that it is a property of the initial singularity since this is the origin of process of gravitational expansion and complexification that has led to the Universe we now observe. We assume here that energy, time, momentum and space are all early emergent properties of the Universe. This hypothesis couples gravitation to action, which we take to be the primordial property of the universe.

Our hypothesis is that gravitation existed before space and time emerged, and its sole reality is Einstein's field equation when the tensors are reduced to scalars. Einstein field equations - Wikipedia

Hawking and Ellis studied Einsteins theory and concluded that it predicted the existence of boundaries to space time which we call singularities. The formation of black holes is a time reversed microcosm of the big bang. We take the view that if there is a path from four dimensional space-time to a singularity there is formally a path back from a singuarity to four dimensional spacetime, as demonstrated by the expansion of the universe both in size and complexity. Hawking & Ellis: The Large Scale Structure of Space-Time

We are guessing here that gravitation is the root state of the Universe, the first fixed point to appear in the divine dynamics and, from a formal point of view, the root of all the states built from it. The gravitational state is one, and defines the whole structure of the Universe. From a network point of view, we may see gravitation as an unmodulated carrier, transmitting energy but no data.

Action

Gravitation sees only energy, that is the time rate of action. To apply the network model to physics we establish a correspondence between a quantum of action and the execution of Turing machine, ie a halting computation. Energy is then equivalent to the rate of completed computations, that is the frequency of network communication, and momentum to the spatial resolution of the message being transmitted. From a network point of view, this is equivalent to measuring traffic or bandwidth in symbols per second.

From physics to metaphysics

Aristotle's theory of potency and act served to build a bridge between the physics and metaphysics. In a more recent work, Bernard Lonergan developed this idea by interpreting potency and act as psychological terms. Potency is our desire to know; actuality is actual knowledge. The transformation between potency and act is the act of insight. Lonergan: Inaight

Insight defines as metaphysics as the integral heuristic structure of proportionate being. Here we wish to discard a presumption contained in this definition. Proportionate being is being which humans can understand. The definition implies a limit on human understanding, which Lonergan introduces to honour the Catholic dogma that God is an invisible mystery beyond human understanding. Here we consider restriction on human knowledge is unnecessary, given our assumption that the universe is divine and God is visible. Although we can understand our divine milieu, its complexity means that this task will never be complete.

A heuristic structure is in effect a map of the unknown used to guide the search for knowledge. The layering of the network model provides a natural heuristic structure, since we see that more complex processes are built from simpler ones. The simpler processes provide an heuristic guide to understanding the the more complex processes buit from them. From this point of view, metaphysics is the foundation of physics.

Gravitation and the Trinity

From a formal point of view there is nothing to distinguish the structureless initial singularity predicted by the general theory and the completely simple divinity proposed by medieval theologians. As mystics have long known, there is nothing to be said about this God. Nevertheless, we wish to argue that the enormously complex universe of our experience is this same God. The source of the complexity is the fixed points in the divine dynamics. These points are not outside the divinity, but part of it, fixed points in the dynamics.

Science begins when when the initial singularity splits into two, formally p and not-p. This step was taken long ago and is explicitly invoked in the opening words of John's Gospel: In the beginning was the Word. John's statement is a reflection of the ancient view that the intellectual word is a representation within us of what we know. This psychological idea, extended to the divinity, interprets God's understanding of itself as the Word of God, which is understood to be the second person of the Trinity. John: Gospel

This approach to understanding the Trinity was developed by Augustine and was further developed by Aquinas. More recently, Lonergan has developed the idea in books on the Word and the Trinity. Augustine: The Trinity, Aquinas 103402: Summa Theologiae I, 34, 2, Lonergan: Verbum, Lonergan: The Triune God

Self reference, gravitation and evolution

In 1912 Einstein realized that the gravitational field was non-linear because it acts upon itself. Gravitation sees only energy and gravity is a form of energy. Energy is the rate of action. Gravitation is action, and feeds back onto itself. Such self reference is the foundation for evolution and control.

From the network, point of view, gravitation is the protocol for sending messages. It does not look inside the messages, merely transmits them. From this point of view, it is isomorphic to a power transmission network which simply moves quanta of action with no reference to their meaning. It also applies to meaningful sets of action, like myself. I feel gravity in proportion to my energy and no other parameter.

The self reference of such a network lays the foundation for the evolution of messages with more complex structures and smaller constituencies. From our physical point of view, these messages correspond to the other three fundamental forces, the strong force, the weak force and electromagnetism. The rest of this chapter deals with the network origin of these fundamental universal symmetries.

We see the emergence of the complex world from the initial singularity as analogous to procession of the Word of God from the Father alluded to in John's Gospel. Gravitation is the first word of God, the first fixed point in the divine dynamics, upon which all else is built.

(revised 18 May 2016)

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Further reading

Books

Click on the "Amazon" link below each book entry to see details of a book (and possibly buy it!)

Augustine, Saint, and Edmond Hill (Introduction, translation and notes), and John E Rotelle (editor), The Trinity, New City Press 1991 Written 399 - 419: De Trinitate is a radical restatement, defence and development of the Christian doctrine of the Trinity. Augistine's book has served as a foundation for most subsequent work, particularly that of Thomas Aquinas.   Amazon   back

Chaitin, Gregory J, Information, Randomness & Incompleteness: Papers on Algorithmic Information Theory, World Scientific 1987 Jacket: 'Algorithmic information theory is a branch of computational complexity theory concerned with the size of computer programs rather than with their running time. ... The theory combines features of probability theory, information theory, statistical mechanics and thermodynamics, and recursive function or computability theory. ... [A] major application of algorithmic information theory has been the dramatic new light it throws on Goedel's famous incompleteness theorem and on the limitations of the axiomatic method. ...'  Amazon   back

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   Amazon   back

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.'  Amazon   back

Lonergan, Bernard J F, and Robert M. Doran, Frederick E. Crowe (eds), Verbum : Word and Idea in Aquinas (Collected Works of Bernard Lonergan volume 2) , University of Toronto Press 1997 Jacket: 'Verbum is a product of Lonergan's eleven years of study of the thought of Thomas Aquinas. The work is considered by many to be a breakthrough in the history of Lonergan's theology ... . Here he interprets aspects in the writing of Aquinas relevant to trinitarian theory and, as in most of Lonergan's work, one of the principal aims is to assist the reader in the search to understand the workings of the human mind.'  Amazon   back

Lonergan, Bernard J F, Insight : A Study of Human Understanding (Collected Works of Bernard Lonergan : Volume 3), University of Toronto Press 1992 '. . . Bernard Lonergan's masterwork. Its aim is nothing less than insight into insight itself, an understanding of understanding'  Amazon   back

Lonergan, Bernard J F, and Michael G Shields (translator), Robert M Doran & H Daniel Monsour (editors), The Triune God: Systematics, University of Toronto press 2007 Translated from De Deo Trino: Pars systematica (1964) by Michael G Shields. Amazon Product Description 'Buried for more than forty years in a Latin text written for seminarian students at the Gregorian University in Rome, Bernard Lonergan's 1964 masterpiece of systematic-theological writing, De Deo trino: Pars systematica, is only now being published in an edition that includes the original Latin along with an exact and literal translation. De Deo trino, or The Triune God, is the third great installment on one particular strand in trinitarian theology, namely, the tradition that appeals to a psychological analogy for understanding trinitarian processions and relations. The analogy dates back to St Augustine but was significantly developed by St Thomas Aquinas. Lonergan advances it to a new level of sophistication by rooting it in his own highly nuanced cognitional theory and in his early position on decision and love. Suggestions for a further development of the analogy appear in Lonergan's late work, but these cannot be understood and implemented without working through this volume. This is truly one of the great masterpieces in the history of systematic theology, perhaps even the greatest of all time.'  Amazon   back

Lonergan, Bernard J F, and Robert M Doran and H Daniel Monsour (eds), The Triune God: Doctrines (Volume 11 of Collected Works), University of Toronto Press 2009 Bernard Lonergan (1904-1984), a professor of theology, taught at Regis College, Harvard University, and Boston College. An established author known for his Insight and Method in Theology, Lonergan received numerous honorary doctorates, was a Companion of the Order of Canada in 1971 and was named as an original members of the International Theological Commission by Pope Paul VI.  Amazon   back

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. . . . '  Amazon   back

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.'  Amazon   back

Links

Actus et potentia - Wikipedia, Actus et potentia - Wikipedia, the free encyclopedia, 'The terms actus and potentia were used by the scholastics to translate Aristotle's use of the terms energeia or entelecheia, and dynamis. There is no single word in English that would be an exact rendering of either. Act, action, actuality, perfection, and determination express the various meanings of actus; potency, potentiality, power, and capacity, those of potentia.' back

Albert Einstein, The happiest thought of my life, '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) back

Algorithmic information theory - Wikipedia, Algorithmic information theory - Wikipedia, the free encyclopedia, 'Algorithmic information theory is a subfield of information theory and computer science that concerns itself with the relationship between computation and information. According to Gregory Chaitin, it is "the result of putting Shannon's information theory and Turing's computability theory into a cocktail shaker and shaking vigorously."' back

Aquinas 103402, Whether "Word" is the Son's proper name?, 'I answer that, "Word," said of God in its proper sense, is used personally, and is the proper name of the person of the Son. For it signifies an emanation of the intellect: and the person Who proceeds in God, by way of emanation of the intellect, is called the Son; and this procession is called generation, . . . Hence it follows that the Son alone is properly called Word in God.' back

Aquinas 13, Summa: I 2 3: Whether God exists?, I answer that the existence of God can be proved in five ways. The first and more manifest way is the argument from motion. . . . The second way is from the nature of the efficient cause. . . . The third way is taken from possibility and necessity . . . The fourth way is taken from the gradation to be found in things. . . . The fifth way is taken from the governance of the world. back

Calculus - Wikipedia, Calculus - Wikipedia, the free encyclopedia, 'Calculus (Latin, calculus, a small stone used for counting) is a discipline in mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern university education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem of calculus. Calculus is the study of change, in the same way that geometry is the study of shape and algebra is the study of equations.' back

Celestial coordinate system - Wikipedia, Celestial coordinate system - Wikipedia, the free encyclopedia, 'In astronomy, a celestial coordinate system is a system for specifying positions of celestial objects: satellites, planets, stars, galaxies, and so on. Coordinate systems can specify a position in 3-dimensional space, or merely the direction of the object on the celestial sphere, if its distance is not known or not important.' back

Claude E Shannon, A Mathematical Theory of Communication, 'The fundamental problem of communication is that of reproducing at one point either exactly or approximately a message selected at another point. Frequently the messages have meaning; that is they refer to or are correlated according to some system with certain physical or conceptual entities. These semantic aspects of communication are irrelevant to the engineering problem. The significant aspect is that the actual message is one selected from a set of possible messages.' back

Codec - Wikipedia, Codec - Wikipedia, the free encyclopedia, 'A codec is a device or computer program capable of encoding or decoding a digital data stream or signal. Codec is a portmanteau of coder-decoder or, less commonly, compressor-decompressor.' back

Cosmic microwave background radiation - Wikipedia, Cosmic microwave background radiation - Wikipedia, the free encyclopedia, 'The CMB is a snapshot of the oldest light in our Universe, imprinted on the sky when the Universe was just 380,000 years old. It shows tiny temperature fluctuations that correspond to regions of slightly different densities, representing the seeds of all future structure: the stars and galaxies of today.' back

Differentiable manifold - Wikipedia, Differentiable manifold - Wikipedia, the free encyclopedia, 'In mathematics, a differentiable manifold is a type of manifold that is locally similar enough to a linear space to allow one to do calculus. Any manifold can be described by a collection of charts, also known as an atlas. One may then apply ideas from calculus while working within the individual charts, since each chart lies within a linear space to which the usual rules of calculus apply. If the charts are suitably compatible (namely, the transition from one chart to another is differentiable), then computations done in one chart are valid in any other differentiable chart. back

Differential geometry - Wikipedia, Differential geometry - Wikipedia,the free encyclopedia, 'Differential geometry is a mathematical discipline that uses the techniques of differential calculus and integral calculus, as well as linear algebra and multilinear algebra, to study problems in geometry. The theory of plane and space curves and of surfaces in the three-dimensional Euclidean space formed the basis for development of differential geometry during the 18th century and the 19th century. Since the late 19th century, differential geometry has grown into a field concerned more generally with the geometric structures on differentiable manifolds.' 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

Frame of reference - Wikipedia, Frame of reference - Wikipedia, the free encyclopedia, 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 covariance - Wikipedia, General covariance - Wikipedia, the free encyclopedia, In theoretical physics, general covariance (also known as diffeomorphism covariance or general invariance) is the invariance of the form of physical laws under arbitrary differentiable coordinate transformations. The essential idea is that coordinates do not exist a priori in nature, but are only artifices used in describing nature, and hence should play no role in the formulation of fundamental physical laws.' 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

Geodesics in general relativity - Wikipedia, Geodesics in general relativity - Wikipedia, the free encyclopedia, 'In general relativity, a geodesic generalizes the notion of a "straight line" to curved spacetime. Importantly, the world line of a particle free from all external, non-gravitational force, is a particular type of geodesic. In other words, a freely moving or falling particle always moves along a geodesic. back

Hylomorphism - Wikipedia, Hylomorphism - Wikipedia, the free encyclopedia, 'Hylomorphism (Greek ὑλο- hylo-, "wood, matter" + -morphism < Greek μορφή, morphē, "form") is a philosophical theory developed by Aristotle, which analyzes substance into matter and form. Substances are conceived of as compounds of form and matter.' back

Inertial frame of reference - Wikipedia, Inertial frame of reference - Wikipedia, the free encyclopedia, 'In physics, an inertial frame of reference (also inertial reference frame or inertial frame or Galilean reference frame or inertial space) is a frame of reference that describes time and space homogeneously, isotropically, and in a time-independent manner' back

Johannes Kepler - Wikipedia, Johannes Kepler - Wikipedia, the free encyclopedia, 'Johannes Kepler (. . . December 27, 1571 – November 15, 1630) was a German mathematician, astronomer and astrologer. A key figure in the 17th century scientific revolution, he is best known for his eponymous laws of planetary motion, codified by later astronomers, based on his works Astronomia nova, Harmonices Mundi, and Epitome of Copernican Astronomy. These works also provided one of the foundations for Isaac Newton's theory of universal gravitation.' back

John, The Gospel according to John, 'In the beginning was the Word, and the Word was with God, and the Word was God. 2 He was with God in the beginning. 3 Through him all things were made; without him nothing was made that has been made.' back

Length contraction - Wikipedia, Length contraction - Wikipedia, the free encyclopedia, 'In physics, length contraction is the phenomenon of a decrease in length of an object as measured by an observer which is traveling at any non-zero velocity relative to the object. This contraction (more formally called Lorentz contraction or Lorentz–FitzGerald contraction after Hendrik Lorentz and George FitzGerald) is usually only noticeable at a substantial fraction of the speed of light.' back

Lorentz transformation - Wikipedia, Lorentz transformation - Wikipedia, the free encyclopedia, 'In physics, the Lorentz transformation or Lorentz-Fitzgerald transformation describes how, according to the theory of special relativity, two observers' varying measurements of space and time can be converted into each other's frames of reference. It is named after the Dutch physicist Hendrik Lorentz. It reflects the surprising fact that observers moving at different velocities may measure different distances, elapsed times, and even different orderings of events.' back

Mass in special relativity - Wikipedia, Mass in special relativity - Wikipedia, the free encyclopedia, 'Mass in special relativity incorporates the general understandings from the concept of mass–energy equivalence. Added to this concept is an additional complication resulting from the fact that mass is defined in two different ways in special relativity: one way defines mass ("rest mass" or "invariant mass") as an invariant quantity which is the same for all observers in all reference frames; in the other definition, the measure of mass ("relativistic mass") is dependent on the velocity of the observer.' back

Mass-energy equivalence - Wikipedia, Mass-energy equivalence - Wikipedia, the free encyclopedia, In physics, mass–energy equivalence is the concept that any mass has an associated energy and vice versa. In special relativity this relationship is expressed using the mass–energy equivalence formula E = mc2 where E is the energy of a physical system m is the mass of the system, and c = the speed of light in a vacuum . . . ' back

Maxwell's equations - Wikipedia, Maxwell's equations - Wikipedia, the free encyclopedia, 'In classical electromagnetism, Maxwell's equations are a set of four equations that describe the properties of the electric and magnetic fields and relate them to their sources, charge density and current density. Maxwell used the equations to show that light is an electromagnetic wave.' back

Quantum computing - Wikipedia, Quantum computing - Wikipedia, the free encyclopedia, 'Quantum computing studies theoretical computation systems (quantum computers) that make direct use of quantum-mechanical phenomena, such as superposition and entanglement, to perform operations on data.' 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 that varies smoothly from point to point. This gives, in particular, local notions of angle, length of curves, surface area, and volume. From those some other global quantities can be derived by integrating local contributions. . . . 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

Special relativity - Wikipedia, Special relativity - Wikipedia, the free encyclopedia, 'Special relativity . . . is the physical theory of measurement in an inertial frame of reference proposed in 1905 by Albert Einstein (after the considerable and independent contributions of Hendrik Lorentz, Henri Poincaré and others) in the paper "On the Electrodynamics of Moving Bodies". It generalizes Galileo's principle of relativity—that all uniform motion is relative, and that there is no absolute and well-defined state of rest (no privileged reference frames)—from mechanics to all the laws of physics, including both the laws of mechanics and of electrodynamics, whatever they may be. Special relativity incorporates the principle that the speed of light is the same for all inertial observers regardless of the state of motion of the source.' back

Speed of light - Wikipedia, Speed of light - Wikipedia, the free encyclopedia, 'The speed of light in vacuum, commonly denoted c, is a universal physical constant important in many areas of physics. Its value is 299,792,458 metres per second, a figure that is exact because the length of the metre is defined from this constant and the international standard for time.' back

Time dilation - Wikipedia, Time dilation - Wikipedia, the free encyclopedia, 'In the theory of relativity, time dilation is a difference of elapsed time between two events as measured by observers either moving relative to each other or differently situated from a gravitational mass or masses. An accurate clock at rest with respect to one observer may be measured to tick at a different rate when compared to a second observer's own equally accurate clocks. This effect arises neither from technical aspects of the clocks nor from the fact that signals need time to propagate, but from the nature of spacetime itself.' back

Tycho Brahe - Wikipedia, Tycho Brahe - Wikipedia, the free encyclopedia, 'Tycho Brahe (14 December 1546 – 24 October 1601), born Tyge Ottesen Brahe, was a Danish nobleman known for his accurate and comprehensive astronomical and planetary observations. Coming from Scania, then part of Denmark, now part of modern-day Sweden, Tycho was well known in his lifetime as an astronomer and alchemist.' back