Chapter 4: A new beginning
4.1 From Judaism to Christianity
Yahweh the God of the Hebrews speaks but never appears in person. At Mount Sinai Yahweh arranged with Moses to introduce themself to their people:
Moses led the people out of the camp to meet God, and they stationed themselves at the foot of the mountain. Now Mount Sinai was completely enveloped in smoke, because the LORD had come down upon it in fire. The smoke rose from it as though from a kiln, and the whole mountain trembled violently. . . .. Then God spoke all these words: I am the LORD your God, who brought you out of the land of Egypt, out of the house of slavery. You shall not have other gods beside me. . . .. For I, the LORD, your God, am a jealous God. (Exodus (19:17–20:5)
Christianity is a child of Judaism. This time Yahweh appeared in Judea in person as Jesus of Nazareth and brought them a new commandment. First he recited the worlds of Yahweh recorded in Exodus and Deuteronomy:
Hear, O Israel! The LORD is our God, the LORD alone! Therefore, you shall love the LORD, your God, with your whole heart, and with your whole being, and with your whole strength (Dt 6:4–5).
To this Jesus added:
This is the first and great commandment. “And the second is like unto it, Thou shalt love thy neighbour as thyself”. On these two commandments hang all the law and the prophets. (Matthew 22:37–40).
He was politically active and incurred the wrath of the scribes and chief priests by criticizing the vast and confusing structure of law they had built around these simple commands. So they conspired with the occupying Romans to kill him.
In time the writers of the New Testament developed the idea that the murder of Jesus was human a sacrifice made to the Father to expiate the sins committed by the first humans in the Garden of Eden.
A complex theology of revelation, baptism, grace, redemption, the Church, an apocalyptic reconstruction of the damaged world and the resurrection of all the dead has been developed to explain the fact that we sinners could not save ourselves but needed the grace of God, distributed to baptized members of the Church, to get to heaven in an afterlife. All of this bureaucracy is managed by a carefully selected and trained hierarchy of clergy answering to a supreme ruler, the Pope, who claims to be God's representative on Earth, receiving infallible guidance from heaven. A dream of power which has made the the Catholic Church very wealthy. Pope John Paul II (1992): The Catechism of the Catholic Church
4.2: From Christianity to reality
Neither Thomas Aquinas nor Bernard Lonergan proved that God is other than the world. It is a matter of faith. Aristotle, on the other hand, was on the right track when he argued for the existence of a dynamic unmoved mover to move Plato’s forms. Like mathematics and other formal structures, these ideas have no life of their own. They are dead kinematic puppets and need something alive to move them.Aristotle described his idea of God in the Metaphysics:
Such, then, is the first principle upon which depend the sensible universe and the world of nature. And its life is like the best which we temporarily enjoy. It must be in that state always (which for us is impossible), since its actuality is also pleasure. . . . . If, then, the happiness which God always enjoys is as great as that which we enjoy sometimes, it is marvellous; and if it is greater, this is still more marvellous. Nevertheless it is so. Moreover, life belongs to God. For the actuality of thought is life, and God is that actuality; and the essential actuality of God is life most good and eternal. We hold, then, that God is a living being, eternal, most good; and therefore life and a continuous eternal existence belong to God; for that is what God is. Aristotle: Metaphysics book XII: The life of God: (1072b14 sqq)
Millennia later Einstein’s general theory of relativity led us back to this ancient conclusion. Einstein’s first major work was the special theory of relativity which shows that space and time are an inseparable unity. Shortly after it was published, Hermann Minkowski stated Einstein’s discovery in very plain language:
Gentlemen! The views of space and time which I want to present to you arose from the domain of experimental physics, and therein lies their strength. Their tendency is radical. From now onwards space by itself and time by itself will recede completely to become mere shadows and only a type of union of the two will still stand independently on its own. Hermann Minkowski (1908): Hermann Minkowski, Wikiquote
The space of special relativity is now called Minkowski space. It is the structural framework from which general relativity is built. General relativity, in its turn, led us to understand the existence of an initial singularity which is identical to the Christian God developed by Aquinas. This serves as my new beginning. The foundation for a new theology that implies that the Universe is itself divine. Hawking & Ellis (1975): The Large Scale Structure of Space-Time
In a nutshell, our feeling of the presence of gravity is the feeling of the presence of God, and the world that we derive from it is identical to the physical mind of God. In effect, the world, seen from the point of view of Western theological history, is the third incarnation of this God: Old Testament, New Testament, modern science. In a divine world, modern science is the foundation of true theology.
4.3: The general theory of relativity
Einstein was led to special relativity by a subtle point in the classical theory of electromagnetism. A moving magnetic field causes a current in a wire. A moving current in a wire causes a magnetic field; but the traditional theory treated these two cases differently. Einstein saw that all that really matters is the relative motion between the electric current (a flow of electrons) and the magnetic field. Albert Einstein (1905): On the Electrodynamics of Moving Bodies
He solved this problem not with a modification of electrodynamics, but with an even deeper change, to the structure of spacetime. The key idea, implicit in Maxwell’s equations for the propagation of light, is that every observer in uniform motion sees the same speed of light. Lorentz transformation - Wikipedia
This is true of uniform motion, that is of motion that obeys Newton’s first law: bodies at rest remain at rest, and bodies in motion continue to move in a straight line at constant speed unless they are acted upon by a force. The next question for Einstein was: what about bodies that are accelerating, that is, in Newton’s terms, being acted upon by a force? This was not an easy problem, and it took Einstein about ten years to solve it. Much of the trouble, he said, was getting the ideas of classical dynamics out of his mind.
Two ideas helped Einstein along. The first is that a person in free fall does not feel their own weight. Astronauts orbiting the Earth are, like the Moon, in free fall, weightless. Nevertheless they are following a curved path. Newton explained the orbit of the Moon as the balance of two forces. One is the gravitational attraction between Earth and Moon. The other is the centrifugal force arising from the Moon’s curved path. In free fall, this is wrong. A falling person is accelerating, but they feel no force. Their space is curved.
The second idea is that there is no perceptible difference between gravitation and acceleration. If you are sitting in a closed box and feel yourself being pushed to one side you cannot tell whether this is gravity because the box is sitting on the Earth, or a force resulting from someone pushing on your box to make it accelerate. The force you feel pushing you against the seat as your vehicle accelerates is indistinguishable from gravity.
How did Einstein put these two ideas together to arrive at the general theory of relativity? The first step was to get rid of the frames of reference, or more precisely, to assume that any frame of reference would do. The idea is that nature goes its own way regardless of how we look at it. In physics this idea is called general covariance. If you change your point of view, you change what you see. This is a difference for you, but not for reality, so you must take the frame of reference into account when you are describing what you see.
In special relativity, the Lorentz transformation does not change reality, it simply accounts for velocity differences between for a different point of view. What Einstein needed was a transformation which nullified not just the effects of inertial motion, but of any motion.
The answer, you might guess, is mathematical. It used to be called absolute differential calculus, we now call is tensor calculus, and the first glimpse of it was caught by Carl Friedrich Gauss (1777–1855). Gauss found a theorema egregium (remarkable theorem) to measure the curvature of a surface using measurements taken in the surface rather than from a frame of reference surrounding the surface. Theorema Egregium - Wikpedia
Newton was looking at the Universe from God’s eye, outside the Universe. Einstein was seeking to describe the Universe from within. Gauss’s idea was extended from two dimensional surfaces to spaces of any dimension by Bernhard Riemann (1826–1866) using a mathematical structure called a differential manifold. Each point in the curved manifold of general relativity has the metric of flat Minkowski space. These Minkowski spaces are tangents to the curved space, coupled together like chain mail by differentiation. Freely moving particles, like the Moon in orbit around the Earth, follow a path called a geodesic which is a curved line of flat Minkowski spaces. In general relativity the orbit of the Moon is a geodesic. The Moon is moving inertially on a straight path in each infinitesimal Minkowski spaces but the overall space is curved by the presence of the massive Earth. Misner, Thorne and Wheeler have described the mathematics of relativity in exhaustive detail in a 1300 page tome where you can find all the details. Differentiable manifold - Wikipedia, Misner, Thorne & Wheeler (1973): Gravitation
4.4: The initial singularity
comprehensive theory of everything. It gives us the big picture. It appears to be capable of describing the whole Universe. Hawking and Ellis showed that the mathematics of the general theory predict the existence of singularities or points outside spacetime, the edges to the Universe. They reach this conclusion by showing that in certain conditions geodesics in space-time can come to an end, meaning that they have come to the end of spacetime. There is no further to go. Nevertheless, we may imagine that Hilbert space and quantum mechanics may exist outside or prior to spacetime.Hawking & Ellis (1975): The Large Scale Structure of Space-Time
One class of these singularities is called black holes. These result when the local density of matter becomes so great that the gravitational potential destroys the local material structure and also, perhaps, the structure of spacetime itself. Black holes are surrounded by an event horizon which acts as a boundary between accessible and inaccessible spacetime. If you cross the boundary you cannot escape from the singularity. You are inside a Penrose closed trapped surface. Trapped surface - Wikipedia
Black holes do not emit light so they cannot be seen, but because they are massive, they attract matter near them, including photons. The Event Horizon Telescope Collaboration have recently made images of the light orbiting the black hole estimated to have the mass of billions Suns at the centre of the galaxy M87 50 million light years away. These pictures were made with radio telescopes around the Earth synchronized so that they acted as one telescope with a dish as big as the planet. Event Horizon Telescope Collaboration (2019): First M87 Event Horizon Telescope Results. IV: Imaging the Central Supermassive Black Hole
The other boundary is believed to lie at the beginning of the Universe and is called the initial singularity. Hawking and Ellis propose that the big bang, which is thought to have begun the Universe, may be imagined as a time reversed black hole. This does not make a lot of sense. The decay of black holes due to Hawking radiation is very slow, nothing like an explosion. Hawking radiation - Wikipedia
Another apparent difficulty with the classical big bang theory is that it seems to imply that all the energy of the Universe is contained in a structureless initial singularity. This is difficult to understand, since energy and momentum exist in spacetime which is a product rather than the source of the big bang. I will outline an alternative approach to the emergence of the Universe within the initial singularity as we go along.
4.5: Naked gravitation
Here we imagine that the initial singularity described by Hawking and Ellis is similar to gravitation but lacking the Minkowski space which is the foundation of general relativity. Minkowski space and the detailed structure of the Universe created the quantum processes will begin to emerge in a few chapters’ time.Let us call this singularity naked gravitation and assume that it is identical to the traditional Christian God, a necessary, eternal, living, absolutely simple and omnipotent being, lacking omniscience. Omniscience will be encoded in the complex Universe that quantum mechanics creates within this singularity. Consequently we can call the Universe the mind of God or the organ of God’s mind, their brain.
Although the singularity is introduced here as a new beginning, what we might call the idea of a singular creator existed in other cultures at about the same as the Hebrew book of Genesis and the other books of the Hebrew Torah were being written. In the Greek world Thales of Miletus and Parmenides were writing about the same period. The richest and most detailed description of the ancient idea of God comes from Aristotle in the the twelfth book of the Metaphysics. There he writes about an immaterial subsistent mind enjoying its reflection upon itself (see above). Torah - Wikipedia, Thales - Wikipedia, Parmenides - Wikipedia
This is my new beginning replacing the traditional God of Genesis, and in the following pages we will develop an hypothesis to explain how it created the present Universe within itself.
Here my idea of a new beginning is theological, a new interpretation based on scientific hypothesis rather than purely imaginative fiction about the nature and creation of ourselves and our world. Since nothing comes from nothing, the source of the world must be eternal, that is outside time. The resulting theological and scientific stories have a lot in common.
1. The initial singularity in the Catholic story and in this story are almost the same: omnipotent, eternal and structureless. Structurelessness means that the Thomistic singularity could not have been omniscient from a modern point of view.
2. The only constraint on the random actions of the singularity is the requirement of consistency: we have the randomness and selection necessary for evolution and we can assume (since they exist in the present) that this system gave rise to stable structures.
3. This structure is validated right back to the beginning because when we collide massive units of energy together we get the same spectrum of particles every time, but with varying probabilities. Some, like the Higgs boson, have required a large number of trials to manifest themselves. Higgs boson - Wikipedia
4. I wish to introduce the Platonic psychological approach which gives us kinematic forms which are puppets and cannot move themselves as Aristotle pointed out. These forms transformed into real dynamic particles by energy drawn from gravitation. We work by analogy to swith Aristotle's hylomorphism which uses matter (scilicet energy) to make the forms real and alive (self moving).
5. Spacetime is a consequence of the quantum mechanical bifurcation of particles into bosons and fermions by something like a superselection rule.
6. We will use the Dirac equation to describe the way in which bosons and fermions create Minkowski space. Dirac noted that the 4D Schrödinger equation does not obey special relativity so he took the square root of the momentum operator and linearized the result using the gamma matrices and arrived at 4 linear 2D Schrödinger equations which can be added together (because of their linearity [and dynamism]) to give the metric of Minkowski space, aka quaternions and Clifford Algebra. Attiyah: The Dirac equation in geometry in Goddard. Dirac equation - Wikipedia, Gamma matrices - Wikipedia, Minkowski space - Wikipedia, Quaternion - Wikipedia, Clifford algebra - Wikipedia, Peter Goddard (1998); Paul Dirac, The Man and His Work
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Notes and references
Further reading
Books
Goddard (1998), Peter , and Stephen Hawking, Abraham Pais, Maurice Jacob, David Olive, and Michael Atiyah, Paul Dirac, The Man and His Work, Cambridge University Press 1998 Jacket: Paul Adrien Maurice Dirac was one of the founders of quantum theory and the aithor of many of its most important subsequent developments. He is numbered alongside Newton, Maxwell, Einstein and Rutherford as one of the greatest physicists of all time.
This volume contains four lectures celebrating Dirac's life and work and the text of an address given by Stephen Hawking, which were given on 13 November 1995 on the occasion of the dedication of a plaque to him in Westminster Abbey. In the first lecture, Abraham Pais describes from personal knowledge Dirac's character and his approach to his work. In the second lecture, Maurice Jacob explains not only how and why Dirac was led to introduce the concept of antimatter, but also its central role in modern particle physics and cosmology. In the third lecture, David Olive gives an account of Dirac's work on magnetic monopoles and shows how it has had a profound influence in the development of fundamental physics down to the present day. In the fourth lecture, Sir Michael Atiyah explains the widespread significance of the Dirac equation in mathematics, its roots in algebra and its implications for geometry and topology.'
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Hawking (1975), 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 (1973), 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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Links
Albert Einstein (1905), On the Electrodynamics of Moving Bodies, An english translation of the paper that founded Special relativity. 'Examples of this sort, [in the contemporary application of Maxwell's electrodynamics to moving bodies] together with the unsuccessful attempts to discover any motion of the earth relatively to the ``light medium,'' suggest that the phenomena of electrodynamics as well as of mechanics possess no properties corresponding to the idea of absolute rest. They suggest rather that, as has already been shown to the first order of small quantities, the same laws of electrodynamics and optics will be valid for all frames of reference for which the equations of mechanics hold good.' back |
Aristotle: Metaphysics book XII, The life of God: 1072b14 sqq, 'Such, then, is the first principle upon which depend the sensible universe and the world of nature. And its life is like the best which we temporarily enjoy. It must be in that state always (which for us is impossible), since its actuality is also pleasure. . . . .If, then, the happiness which God always enjoys is as great as that which we enjoy sometimes, it is marvellous; and if it is greater, this is still more marvellous. Nevertheless it is so. Moreover, life belongs to God. For the actuality of thought is life, and God is that actuality; and the essential actuality of God is life most good and eternal. We hold, then, that God is a living being, eternal, most good; and therefore life and a continuous eternal existence belong to God; for that is what God is.' back |
Book of Genesis - Wikipedia, Book of Genesis - Wikipedia, the free encyclpedia, ' Genesis is part of the Torah or Pentateuch, the first five books of the Bible. Tradition credits Moses as the Torah's author. It was probably composed around the 5th century BC, although some scholars believe that primeval history (chapters 1–11), may have been composed and added as late as the 3rd century BC. Based on scientific interpretation of archaeological, genetic, and linguistic evidence, most mainstream Bible scholars consider Genesis to be primarily mythological rather than historical.' back |
Clifford algebra - Wikipedia, Clifford algebra - Wikipedia, the free encyclopedia, 'In mathematics, Clifford algebras are a type of associative algebra. As K-algebras, they generalize the real numbers, complex numbers, quaternions and several other hypercomplex number systems. The theory of Clifford algebras is intimately connected with the theory of quadratic forms and orthogonal transformations. Clifford algebras have important applications in a variety of fields including geometry, theoretical physics and digital image processing. They are named after the English geometer William Kingdon Clifford. back |
Differentiable manifold - Wikipedia, Differentiable manifold - Wikipedia, the free encyclopedia, ' In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One may then apply ideas from calculus while working within the individual charts, since each chart lies within a vector 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 |
Dirac equation - Wikipedia, Dirac equation - Wikipedia, the free encyclopedia, 'In particle physics, the Dirac equation is a relativistic wave equation derived by British physicist Paul Dirac in 1928. In its free form, or including electromagnetic interactions, it describes all spin-1⁄2 massive particles such as electrons and quarks, for which parity is a symmetry, and is consistent with both the principles of quantum mechanics and the theory of special relativity, and was the first theory to account fully for special relativity in the context of quantum mechanics. It accounted for the fine details of the hydrogen spectrum in a completely rigorous way.' back |
Event Horizon Telescope Collaboration (2019), First M87 Event Horizon Telescope Results. IV: Imaging the Central Supermassive Black Hole, ' We present the first Event Horizon Telescope (EHT) images of M87, using observations from April 2017 at 1.3 mm wavelength. These images show a prominent ring with a diameter of ∼40 μas, consistent with the size and shape of the lensed photon orbit encircling the “shadow” of a supermassive black hole. The ring is persistent across four observing nights and shows enhanced brightness in the south. To assess the reliability of these results, we implemented a two-stage imaging procedure. In the first stage, four teams, each blind to the others’ work, produced images of M87 using both an established method (CLEAN) and a newer technique (regularized maximum likelihood). This stage allowed us to avoid shared human bias and to assess common features among independent reconstructions. In the second stage, we reconstructed synthetic data from a large survey of imaging parameters and then compared the results with the corresponding ground truth images. This stage allowed us to select parameters objectively to use when reconstructing images of M87. Across all tests in both stages, the ring diameter and asymmetry remained stable, insensitive to the choice of imaging technique. We describe the EHT imaging procedures, the primary image features in M87, and the dependence of these features on imaging assumptions.' back |
Gamma matrices - Wikipedia, Gamma matrices - Wikipedia, the free encyclopedia, 'In mathematical physics, the gamma matrices, {γ0, γ1, γ2,γ3}, also known as the Dirac matrices, are a set of conventional matrices with specific anticommutation relations that ensure they generate a matrix representation of the Clifford algebra Cl1,3(R). When interpreted as the matrices of the action of a set of orthogonal basis vectors for contravariant vectors in Minkowski space, the column vectors on which the matrices act become a space of spinors, on which the Clifford algebra of space time acts. This in turn makes it possible to represent infinitesimal spatial rotations and Lorentz boosts. Spinors facilitate space-time computations in general, and in particular are fundamental to the Dirac equation for relativistic spin-1/2 particles.' back |
Hawking radiation - Wikipedia, Black hole - Wikipedia, the free encyclopedia, ' Hawking radiation is dependent on the Unruh effect and the equivalence principle applied to black-hole horizons. Close to the event horizon of a black hole, a local observer must accelerate to keep from falling in. An accelerating observer sees a thermal bath of particles that pop out of the local acceleration horizon, turn around, and free-fall back in. The condition of local thermal equilibrium implies that the consistent extension of this local thermal bath has a finite temperature at infinity, which implies that some of these particles emitted by the horizon are not reabsorbed and become outgoing Hawking radiation.' back |
Hermann Minkowski (1908), Hermann Minkowski, Wikiquote, ' The views of space and time which I wish to lay before you have sprung from the soil of experimental physics, and therein lies their strength. They are radical. Henceforth, space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality.' Address to the 80th Assembly of German Natural Scientists and Physicians, (Sep 21, 1908) back |
Higgs boson - Wikipedia, Higgs boson - Wikipedia, the free encyclopedia, ' The Higgs boson is an elementary particle in the Standard Model of particle physics, produced by the quantum excitation of the Higgs field, one of the fields in particle physics theory. It is named after physicist Peter Higgs, who in 1964, along with five other scientists, proposed the Higgs mechanism to explain why particles have mass. This mechanism implies the existence of the Higgs boson. The boson's existence was confirmed in 2012 by the ATLAS and CMS collaborations based on collisions in the LHC at CERN. On December 10, 2013, two of the physicists, Peter Higgs and François Englert, were awarded the Nobel Prize in Physics for their theoretical predictions. Although Higgs's name has come to be associated with this theory (the Higgs mechanism), several researchers between about 1960 and 1972 independently developed different parts of it. ' 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 |
Minkowski space - Wikipedia, Minkowski space - Wikipedia, the free encyclopedia, ' By 1908 Minkowski realized that the special theory of relativity, introduced by his former student Albert Einstein in 1905 and based on the previous work of Lorentz and Poincaré, could best be understood in a four-dimensional space, since known as the "Minkowski spacetime", in which time and space are not separated entities but intermingled in a four-dimensional space–time, and in which the Lorentz geometry of special relativity can be effectively represented using the invariant interval x2 + y2 + z2 − c2 t2.' back |
Parmenides - Wikipedia, Parmenides - Wikipedia, the free encyclopedia, 'Parmenides of Elea (early 5th century BC) was an ancient Greek philosopher born in Elea, a Greek city on the southern coast of Italy. He was the founder of the Eleatic school of philosophy, his only known work is a poem which has survived only in fragmentary form. In it, Parmenides describes two views of reality. In the Way of Truth, he explained how reality is one; change is impossible; and existence is timeless, uniform, and unchanging. In the Way of Opinion, he explained the world of appearances, which is false and deceitful. These thoughts strongly influenced Plato, and through him, the whole of western philosophy.' back |
Pope John Paul II (1992), The Catechism of the Catholic Church, The text of the Apostolic Constitution Fidei Depositum Prologue: '... 11 This catechism aims at presenting an organic synthesis of the essential and fundamental contents of Catholic doctrine, as regards both faith and morals, in the light of the Second Vatican Council and the whole of the Church's Tradition. Its principal sources are the SacredScriptures, the Fathers of the Church, the liturgy, and the Church's Magisterium. It is intended to serve "as a point of reference for the catechisms or compendia that are composed in the various countries. ...' back |
Quaternion - Wikipedia, Quaternion - Wikipedia, the free encyclopedia, ' In mathematics, the quaternions are a number system that extends the complex numbers. They were first described by Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. A feature of quaternions is that multiplication of two quaternions is noncommutative. Hamilton defined a quaternion as the quotient of two directed lines in a three-dimensional space or equivalently as the quotient of two vectors.' back |
Speciation - Wikipedia, Speciation - Wikipedia, the free encyclopedia, ' Speciation is the evolutionary process by which populations evolve to become distinct species. The biologist Orator F. Cook coined the term in 1906 for cladogenesis, the splitting of lineages, as opposed to anagenesis, phyletic evolution within lineages. Charles Darwin was the first to describe the role of natural selection in speciation in his 1859 book On the Origin of Species. He also identified sexual selection as a likely mechanism, but found it problematic. There are four geographic modes of speciation in nature, based on the extent to which speciating populations are isolated from one another: allopatric, peripatric, parapatric, and sympatric. Whether genetic drift is a minor or major contributor to speciation is the subject of much ongoing discussion. back |
Thales - Wikipedia, Thales - Wikipedia, the free encyclopedia, 'Thales of Miletus (Greek: Θαλῆς (ὁ Μιλήσιος), Thalēs; c. 624 – c. 546 BC) was a pre-Socratic Greek philosopher from Miletus in Asia Minor, and one of the Seven Sages of Greece. Many, most notably Aristotle, regard him as the first philosopher in the Greek tradition. Aristotle reported Thales' hypothesis about the nature of matter – that the originating principle of nature was a single material substance: water. |
Theorema Egregium - Wikpedia, Theorema Egregium - Wikipedia, the free encyclopedia, 'Gauss's Theorema Egregium (Latin for "Remarkable Theorem") is a foundational result in differential geometry proved by Carl Friedrich Gauss that concerns the curvature of surfaces. The theorem says that the Gaussian curvature of a surface can be determined entirely by measuring angles, distances and their rates on the surface itself, without further reference to the particular way in which the surface is embedded in the ambient 3-dimensional Euclidean space. Thus the Gaussian curvature is an intrinsic invariant of a surface.' back |
Torah - Wikipedia, Torah - Wikipedia, the free encyclopedia, 'Torah (. . . "Instruction, Teaching"), or the Pentateuch . . . , is the central reference of the religious Judaic tradition. It has a range of meanings. It can most specifically mean the first five books of the twenty-four books of the Tanakh, and it usually includes the rabbinic commentaries. The term Torah means instruction and offers a way of life for those who follow it; it can mean the continued narrative from Genesis to the end of the Tanakh, and it can even mean the totality of Jewish teaching, culture and practice.' back |
Trapped surface - Wikipedia, Trapped surface - Wikipedia, the free encyclopedia, ' Closed trapped surfaces are a concept used in black hole solutions of general relativity which describe the inner region of an event horizon. Roger Penrose defined the notion of closed trapped surfaces in 1965. A trapped surface is one where light is not moving away from the black hole. The boundary of the union of all trapped surfaces around a black hole is called an apparent horizon. back |