Man has long been aware of the concept of up and down, and that different objects have different weights. Ancient Greek philosopher Aristotle believed that objects move downward toward the center of the universe, which was their natural place, and he was convinced that heavier objects must fall faster than lighter objects. These ideas were held as true for almost 2000 more years until Galileo proposed that a falling body would fall with a uniform acceleration. Newton brought in additional enlightenment to gravitational thought by developing the first convincing mathematical theory of gravity — two masses are attracted toward each other by a force whose effect decreases according to the inverse square of the distance between them. From this point, gravity is no longer the earthbound force, but a relational force between any objects. Newton’s law of universal gravitation, in turn, was replaced by Einstein’s General theory of relativity, which describes gravity as a consequence of the curvature of spacetime rather than a force. A complete theory of quantum gravity is required when trying to describe gravity according to the principles of quantum mechanics. Human minds have been freed from the inevitable fall to the force of geocentrism, and the idea of gravity is becoming more relational and fluid. The idea of gravity has lost its fateful weight.
Gravity 370BC, 1585, 1687, 1915, ?
Gravity 370BC, 1585, 1687, 1915, ?
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Contents
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Aristotelian Physics
6
Wikipedia
Aristotelian Physics
7
Aristotle
Physics Book IV Place, Void And Time
21
Scientific Revolution
22
Wikipedia
Gravity
23
Galileo Galilei
Dialogues Concerning Two New Sciences
45
Newton's Law of Universal Gravitation
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Wikipedia
Gravity
47
Isaac Newton
Mathematical Principles of Natural Philosophy
65
General Relativity
66
Wikipedia
Gravity
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Albert Einstein
The Field Equations of Gravitation
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Quantum Gravity
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Wikipedia
History of Gravitational Theory
370BC
Aristotelian physics
In Aristotle’s system heavy bodies are not attracted to the earth by an external force of gravity, but tend toward the center of the universe because of an inner gravitas or heaviness.
Wikipedia
Aristotelian Physics
NATURAL PLACE The Aristotelian explanation of gravity is that all bodies move toward their natural place. For the elements earth and water, that place is the center of the (geocentric) universe;1 the natural place of water is a concentric shell around the earth because earth is heavier; it sinks in water. The natural place of air is likewise a concentric shell surrounding that of water; bubbles rise in water. Finally, the natural place of fire is higher than that of air but below the innermost celestial sphere (carrying the Moon). In Book Delta of his Physics (IV.5), Aristotle defines topos (place) in terms of two bodies, one of which contains the other: a “place� is where the inner surface of the former (the containing body) touches the outer surface of the other (the contained body). This definition remained dominant until the beginning of the 17th century, even though it had been questioned and debated by
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Aristotle
Physics Book IV Place, Void And Time
LECTURE 5 Necessary previous notions for the definition of place. CHAPTER 4 What then after all is place? The answer to this question may be elucidated as follows. Let us take for granted about it the various characteristics which are supposed correctly to belong to it essentially. We assume then: 1 Place is what contains that of which it is the place. 2 Place is no part of the thing. 3 The immediate place of a thing is neither less nor greater than the thing. 4 Place can be left behind by the thing and is separable. In addition: 5 All place admits of the distinction of up and down, and each of the bodies is naturally carried to its appropriate place and rests there, and this makes the place either up or down. Having laid these foundations, we
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Aristotelian Physics
philosophers since antiquity.2 The most significant early critique was made in terms of geometry by the 11th-century Arab polymath al-Hasan Ibn al-Haytham (Alhazen) in his Discourse on Place.3
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NATURAL MOTION Terrestrial objects rise or fall, to a greater or lesser extent, according to the ratio of the four elements of which they are composed. For example, earth, the heaviest element, and water, fall toward the center of the cosmos; hence the Earth and for the most part its oceans, will have already come to rest there. At the opposite extreme, the lightest elements, air and especially fire, rise up and away from the center.4 The elements are not proper substances in Aristotelian theory (or the modern sense of the word). Instead, they are abstractions used to explain the varying natures and behaviors of actual materials in terms of ratios between them.
Aristotle
Physics Book IV Place, Void And Time
must complete the theory. We ought to try to make our investigation such as will render an account of place, and will not only solve the difficulties connected with it, but will also show that the attributes supposed to belong to it do really belong to it, and further will make clear the cause of the trouble and of the difficulties about it. Such is the most satisfactory kind of exposition. First then we must understand that place would not have been thought of, if there had not been a special kind of motion, namely that with respect to place. It is chiefly for this reason that we suppose the heaven also to be in place, because it is in constant movement. Of this kind of change there are two species—locomotion on the one hand and, on the other, increase and diminution. For these too involve variation of place: what was then in this place has now in turn changed to what is larger or smaller. Again, when we say a thing is
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Wikipedia
Aristotelian Physics
Motion and change are closely related in Aristotelian physics. Motion, according to Aristotle, involved a change from potentiality to actuality.5 He gave example of four types of change. Aristotle proposed that the speed at which two identically shaped objects sink or fall is directly proportional to their weights and inversely proportional to the density of the medium through which they move.6 While describing their terminal velocity, Aristotle must stipulate that there would be no limit at which to compare the speed of atoms falling through a vacuum, (they could move indefinitely fast because there would be no particular place for them to come to rest in the void). Now however it is understood that at any time prior to achieving terminal velocity in a relatively resistance-free medium like air, two such objects are expected to have nearly identical speeds because both are experiencing a force of gravity proportional to their
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Aristotle
Physics Book IV Place, Void And Time
‘moved’, the predicate either 1 belongs to it actually, in virtue of its own nature, or 2 in virtue of something conjoined with it. In the latter case it may be either a) something which by its own nature is capable of being moved, e.g. the parts of the body or the nail in the ship, or b) something which is not in itself capable of being moved, but is always moved through its conjunction with something else, as ‘whiteness’ or ‘science’. These have changed their place only because the subjects to which they belong do so. We say that a thing is in the world, in the sense of in place, because it is in the air, and the air is in the world; and when we say it is in the air, we do not mean it is in every part of the air, but that it is in the air because of the outer surface of the air which surrounds it; for if all the air were its place, the place of a thing would not be equal to the
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Aristotelian Physics
masses and have thus been accelerating at nearly the same rate. This became especially apparent from the eighteenth century when partial vacuum experiments began to be made, but some two hundred years earlier Galileo had already demonstrated that objects of different weights reach the ground in similar times.7
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UNNATURAL MOTION Apart from the natural tendency of terrestrial exhalations to rise and objects to fall, unnatural or forced motion from side to side results from the turbulent collision and sliding of the objects as well as transmutation between the elements (On Generation and Corruption). CHANCE In his Physics Aristotle examines accidents (συμβεβηκός, sumbebekos) that have no cause but chance. “Nor is there Vany definite cause for an accident, but only chance (τύχη, tukhe),
Aristotle
Physics Book IV Place, Void And Time
thing—which it is supposed to be, and which the primary place in which a thing is actually is. When what surrounds, then, is not separate from the thing, but is in continuity with it, the thing is said to be in what surrounds it, not in the sense of in place, but as a part in a whole. But when the thing is separate and in contact, it is immediately ‘in’ the inner surface of the surrounding body, and this surface is neither a part of what is in it nor yet greater than its extension, but equal to it; for the extremities of things which touch are coincident. Further, if one body is in continuity with another, it is not moved in that but with that. On the other hand it is moved in that if it is separate. It makes no difference whether what contains is moved or not. Again, when it is not separate it is described as a part in a whole, as the pupil in the eye or the hand in the body: when it is separate, as the water in the cask or the wine in the jar. For the hand
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namely an indefinite (ἀόριστον) cause” (Metaphysics V, 1025a25). CONTINUUM AND VACUUM Aristotletes against the indivisibles of Democritus (which differ considerably from the historical and the modern use of the term “atom”). As a place without anything existing at or within it, Aristotle argued against the possibility of a vacuum or void. Because he believed that the speed of an object’s motion is proportional to the force being applied (or, in the case of natural motion, the object’s weight) and inversely proportional to the viscosity of the medium, he reasoned that objects moving in a void would move indefinitely fast — and thus any and all objects surrounding the void would immediately fill it. The void, therefore, could never form.8 The “voids” of modern-day astronomy (such as the Local Void adjacent to our own galaxy) have the opposite
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Physics Book IV Place, Void And Time
is moved with the body and the water in the cask.r
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Aristotelian Physics
effect: ultimately, bodies off-center are ejected from the void due to the gravity of the material outside.9 q
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Aristotle
Physics Book IV Place, Void And Time
17
Notes 1
De Caelo II. 13-14.
2
For instance, by Simplicius in his Corollaries on Place.
3
El-Bizri, Nader (2007). "In Defence of the Sovereignty of Philosophy: al-Baghdadi's Critique of Ibn al-Haytham's Geometrisation of Place". Arabic Sciences and Philosophy (Cambridge University Press) 17: 57–80. doi:10.1017/ s0957423907000367.
4
Tim Maudlin (2012-07-22). Philosophy of Physics: Space and Time: Space and Time (Princeton Foundations of Contemporary Philosophy) (p. 2). Princeton University Press. Kindle Edition.
5
Bodnar, Istvan, "Aristotle's Natural Philosophy" in The Stanford Encyclopedia of Philosophy (Spring 2012 Edition, ed. Edward N. Zalta)
6
Gindikin, S.G. (1988). Tales of Physicists and Mathematicians. Birkh. p. 29. ISBN 9780817633172. LCCN 87024971.
7
Lindberg, D. (2008), The beginnings of western science: The European scientific tradition in philosophical, religious, and
institutional context, prehistory to AD 1450 (2nd ed.), University of Chicago Press. 8
Land, Helen, The Order of Nature in Aristotle's Physics: Place and the Elements (1998).
9
Tully; Shaya; Karachentsev; Courtois; Kocevski; Rizzi; Peel (2008). "Our Peculiar Motion Away From the Local Void". The Astrophysical Journal 676 (1): 184. arXiv:0705.4139. Bibcode:2008ApJ...676..184T. doi:10.1086/527428.
Works Cited Aristotelian physics. (2015, October 31). In Wikipedia, The Free Encyclopedia. Retrieved 01:02, December 8, 2015, from https://en.wikipedia.org/w/index. php?title=Aristotelian_physics&oldid=688301041 Aquinas, Thomas. (n.d.). Commentary on Aristotle's Physics. Retrieved December 8, 2015, from http://dhspriory.org/thomas/Physics.htm
1585
Scientific revolution
Galileo found that, counter to Aristotle’s teachings, all objects accelerated equally when falling.
Wikipedia
Gravity
Modern work on gravitational theory began with the work of Galileo Galilei in the late 16th and early 17th centuries. In his famous (though possibly apocryphal10) experiment dropping balls from the Tower of Pisa, and later with careful measurements of balls rolling down inclines, Galileo showed that gravity accelerates all objects at the same rate. This was a major departure from Aristotle’s belief that heavier objects accelerate faster.11 Galileo postulated air resistance as the reason that lighter objects may fall more slowly in an atmosphere. Galileo’s work set the stage for the formulation of Newton’s theory of gravity. q
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Galileo Galilei
Dialogues Concerning Two New Sciences
FIRST DAY Interlocutors: Salviati, Sagredo And Simplicio Salv.The argument is, as you see, ad hominem, that is, it is directed against those who thought the vacuum a prerequisite for motion. Now if I admit the argument to be conclusive and concede also that motion cannot take place in a vacuum, the assumption of a vacuum considered absolutely and not with reference to motion, is not thereby invalidated. But to tell you what the ancients might possibly have replied and in order to better understand just how conclusive Aristotle’s demonstration is, we may, in my opinion, deny both of his assumptions. And as to the first, I greatly doubt that Aristotle ever tested by experiment whether it be true that two stones, one weighing ten times as much as the other, if allowed to fall, at the same instant, from a height of, say, 100
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Dialogues Concerning Two New Sciences
cubits, would so differ in speed that when the heavier had reached the ground, the other would not have fallen more than 10 cubits. Simp. His language would seem to indicate that he had tried the experiment, because he says: We see the heavier; now the word see shows that he had made the experiment. Sagr. But I, Simplicio, who have made the test can assure you that a cannon ball weighing one or two hundred pounds, or even more, will not reach the ground by as much as a span ahead of a musket ball weighing only half a pound, provided both are dropped from a height of 200 cubits. Salv. But, even without further experiment, it is possible to prove clearly, by means of a short and conclusive argument, that a heavier body does not move more rapidly than a lighter one provided both bodies
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Dialogues Concerning Two New Sciences
are of the same material and in short such as those mentioned by Aristotle. But tell me, Simplicio, whether you admit that each falling body acquires a definite speed fixed by nature, a velocity which cannot be increased or diminished except by the use of force (violenza) or resistance. Simp. There can be no doubt but that one and the same body moving in a single medium has a fixed velocity which is determined by nature and which cannot be increased except by the addition of momentum (impeto) or diminished except by some resistance which retards it. Salv. If then we take two bodies whose natural speeds are different, it is clear that on uniting the two, the more rapid one will be partly retarded by the slower, and the slower will be somewhat hastened by the swifter. Do you not agree
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Dialogues Concerning Two New Sciences
with me in this opinion? Simp. You are unquestionably right. Salv. But if this is true, and if a large stone moves with a speed of, say, eight while a smaller moves with a speed of four, then when they are united, the system will move with a speed less than eight; but the two stones when tied together make a stone larger than that which before moved with a speed of eight. Hence the heavier body moves with less speed than the lighter; an effect which is contrary to your supposition. Thus you see how, from your assumption that the heavier body moves more rapidly than the lighter one, I infer that the heavier body moves more slowly. Simp. I am all at sea because it appears to me that the smaller stone when added to the larger increases its weight and by adding weight I do not see how it can fail to increase its speed or, at least, not to diminish it.
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Galileo Galilei
Dialogues Concerning Two New Sciences
Salv. Here again you are in error, Simplicio, because it is not true that the smaller stone adds weight to the larger. Simp. This is, indeed, quite beyond my comprehension. Salv. It will not be beyond you when I have once shown you the mistake under which you are laboring. Note that it is necessary to distinguish between heavy bodies in motion and the same bodies at rest. A large stone placed in a balance not only acquires additional weight by having another stone placed upon it, but even by the addition of a handful of hemp its weight is augmented six to ten ounces according to the quantity of hemp. But if you tie the hemp to the stone and allow them to fall freely from some height, do you believe that the hemp will press down upon the stone and thus accelerate its motion or do you think the motion will be retarded
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Dialogues Concerning Two New Sciences
by a partial upward pressure? One always feels the pressure upon his shoulders when he prevents the motion of a load resting upon him; but if one descends just as rapidly as the load would fall how can it gravitate or press upon him? Do you not see that this would be the same as trying to strike a man with a lance when he is running away from you with a speed which is equal to, or even greater, than that with which you are following him? You must therefore conclude that, during free and natural fall, the small stone does not press upon the larger and consequently does not increase its weight as it does when at rest. Simp. But what if we should place the larger stone upon the smaller? Salv. Its weight would be increased if the larger stone moved more rapidly; but we have already concluded that when the small stone moves
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Dialogues Concerning Two New Sciences
more slowly it retards to some extent the speed of the larger, so that the combination of the two, which is a heavier body than the larger of the two stones, would move less rapidly, a conclusion which is contrary to your hypothesis. We infer therefore that large and small bodies move with the same speed provided they are of the same specific gravity. Simp. Your discussion is really admirable; yet I do not find it easy to believe that a bird-shot falls as swiftly as a cannon ball. Salv. Why not say a grain of sand as rapidly as a grindstone? But, Simplicio, I trust you will not follow the example of many others who divert the discussion from its main intent and fasten upon some statement of mine which lacks a hair’s-breadth of the truth and, under this hair, hide the fault of another which is as big as a ship’s
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cable. Aristotle says that “an iron ball of one hundred pounds falling from a height of one hundred cubits reaches the ground before a one-pound ball has fallen a single cubit.� I say that they arrive at the same time. You find, on making the experiment, that the larger outstrips the smaller by two finger-breadths, that is, when the larger has reached the ground, the other is short of it by two finger-breadths; now you would not hide behind these two fingers the ninety-nine cubits of Aristotle, nor would you mention my small error and at the same time pass over in silence his very large one. Aristotle declares that bodies of different weights, in the same medium, travel (in so far as their motion depends upon gravity) with speeds which are proportional to their weights; this he illustrates by use of bodies in which it is possible to perceive the
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Dialogues Concerning Two New Sciences
pure and unadulterated effect of gravity, eliminating other considerations, for example, figure as being of small importance (minimi momenti), influences which are greatly dependent upon the medium which modifies the single effect of gravity alone. Thus we observe that gold, the densest of all substances, when beaten out into a very thin leaf, goes floating through the air; the same thing happens with stone when ground into a very fine powder. But if you wish to maintain the general proposition you will have to show that the same ratio of speeds is preserved in the case of all heavy bodies, and that a stone of twenty pounds moves ten times as rapidly as one of two; but I claim that this is false and that, if they fall from a height of fifty or a hundred cubits, they will reach the earth at the same moment.
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Simp. Perhaps the result would be different if the fall took place not from a few cubits but from some thousands of cubits. Salv. If this were what Aristotle meant you would burden him with another error which would amount to a falsehood; because, since there is no such sheer height available on earth, it is clear that Aristotle could not have made the experiment; yet he wishes to give us the impression of his having performed it when he speaks of such an effect as one which we see. Simp. In fact, Aristotle does not employ this principle, but uses the other one which is not, I believe, subject to these same difficulties. r
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Notes 10
Ball, Phil (June 2005). "Tall Tales". Nature News. doi:10.1038/news050613-10.
11
Galileo (1638), Two New Sciences.
Works Cited Gravity. (2015, November 28). In Wikipedia, The Free Encyclopedia. Retrieved 01:01, December 8, 2015, from https://en.wikipedia.org/w/index. php?title=Gravity&oldid=692857802 Galileo Galilei, Dialogues Concerning Two New Sciences by Galileo Galilei. Translated from the Italian and Latin into English by Henry Crew and Alfonso de Salvio. With an Introduction by Antonio Favaro (New York: Macmillan, 1914). Retrieved December 7, 2015 from the World Wide Web: http://oll.libertyfund.org/titles/753
1687
Newton’s Law of Universal Gravitation
Every point mass attracts every single other point mass by a force pointing along the line intersecting both points. The force is proportional to the product of the two masses and inversely proportional to the square of the distance between them.
Wikipedia
Gravity
In 1687, English mathematician Sir Isaac Newton published Principia, which hypothesizes the inverse-square law of universal gravitation. In his own words, “I deduced that the forces which keep the planets in their orbs must (be) reciprocally as the squares of their distances from the centers about which they revolve: and thereby compared the force requisite to keep the Moon in her Orb with the force of gravity at the surface of the Earth; and found them answer pretty 12 nearly.� The equation is the following: m1 m2 F = G _____ r2 Where F is the force, m1 and m2 are the masses of the objects interacting, r is the distance between the centers of the masses and G is the gravitational constant. Newton’s theory enjoyed its greatest success when it was used to predict the existence of Neptune based on motions of Uranus that could not
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Isaac Newton
Mathematical Principles of Natural Philosophy
DEFINITIONS DEFINITION I. The quantity of matter is the measure of the same, arising from its density and bulk conjunctly. Thus air of a double density, in a double space, is quadruple in quantity; in a triple space, sextuple in quantity. The same thing is to be understood of snow, and fine dust or powders, that are condensed by compression or liquefaction and of all bodies that are by any causes whatever differently condensed. I have no regard in this place to a medium, if any such there is, that freely pervades the interstices between the parts of bodies. It is this quantity that I mean hereafter everywhere under the name of body or mass. And the same is known by the weight of each body; for it is proportional to the weight, as I have found by experiments on pendulums, very accurately made, which shall be shewn hereafter.
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Gravity
be accounted for by the actions of the other planets. Calculations by both John Couch Adams and Urbain Le Verrier predicted the general position of the planet, and Le Verrier’s calculations are what led Johann Gottfried Galle to the discovery of Neptune. A discrepancy in Mercury’s orbit pointed out flaws in Newton’s theory. By the end of the 19th century, it was known that its orbit showed slight perturbations that could not be accounted for entirely under Newton’s theory, but all searches for another perturbing body (such as a planet orbiting the Sun even closer than Mercury) had been fruitless. The issue was resolved in 1915 by Albert Einstein’s new theory of general relativity, which accounted for the small discrepancy in Mercury’s orbit. Although Newton’s theory has been superseded by the Einstein’s general relativity, most modern non-relativistic gravitational calculations are
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Isaac Newton
Mathematical Principles of Natural Philosophy
DEFINITION II. The quantity of motion is the measure of the same, arising from the velocity and quantity of matter conjunctly. The motion of the whole is the sum of the motions of all the parts; and therefore in a body double in quantity, with equal velocity, the motion is double; with twice the velocity, it is quadruple. DEFINITION III. The vis insita, or innate force of matter, is a power of resisting, by which every body, as much as in it lies, endeavours to persevere in its present stale, whether it be of rest, or of moving uniformly forward in a right line. This force is ever proportional to the body whose force it is ; and differs nothing from the inactivity of the mass, but in our manner of conceiving it. A body, from the inactivity of matter, is not without difficulty put out of its state of rest or motion. Upon which account, this vis insita, may, by a most significant name,
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Gravity
still made using the Newton’s theory because it is simpler to work with and it gives sufficiently accurate results for most applications involving sufficiently small masses, speeds and energies. q
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Isaac Newton
Mathematical Principles of Natural Philosophy
be called vis inertia, or force of inactivity. But a body exerts this force only, when another force, impressed upon it, endeavours to change its condition; and the exercise of this force may be considered both as resistance and impulse; it is resistance, in so far as the body, for maintaining its present state, withstands the force impressed; it is impulse, in so far as the body, by not easily giving way to the impressed force of another, endeavours to change the state of that other. Resistance is usually ascribed to bodies at rest, and impulse to those in motion; but motion and rest, as commonly conceived, are only relatively distinguished; nor are those bodies always truly at rest, which commonly are taken to be so.
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DEFINITION IV. An impressed force is an action exerted upon a body, in order to change its state, either of rest, or of moving uniformly forward in a right line.
Isaac Newton
Mathematical Principles of Natural Philosophy
This force consists in the action only; and remains no longer in the body, when the action is over. For a body maintains every new state it acquires, by its vis inertice only. Impressed forces are of differe.it origins as from percussion, from pressure, from centripetal force. DEFINITION V. A centripetal force is that by which bodies are drawn or impelled, or any way tend, towards a point as to a centre. Of this sort is gravity, by which bodies tend to the centre of the earth magnetism, by which iron tends to the loadstone; and that force, what ever it is, by which the planets are perpetually drawn aside from the rectilinear motions, which otherwise they would pursue, and made to revolve in curvilinear orbits. A stone, whirled about in a sling, endeavours to recede from the hand that turns it; and by that endeavour, distends the sling, and that with so much the greater force, as it is revolved with the greater
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Mathematical Principles of Natural Philosophy
velocity, and as soon as ever it is let go, flies away. That force which opposes itself to this endeavour, and by which the sling perpetually draws back the stone towards the hand, and retains it in its orbit, because it is directed to the hand as the centre of the orbit, I call the centripetal force. And the same thing is to be understood of all bodies, revolved in any orbits. They all endeavour to recede from the centres of their orbits; and wore it not for the opposition of a contrary force which restrains them to, and detains them in their orbits, which I therefore call centripetal, would fly off in right lines, with an uniform motion. A projectile, if it was not for the force of gravity, would not deviate towards the earth, but would go off from it in a right line, and that with an uniform motion, if the resistance of the air was taken away. It is by its gravity that it is drawn aside perpetually from its rectilinear course, and made to deviate towards the earth,
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Mathematical Principles of Natural Philosophy
more or less, according to the force of its gravity, and the velocity of its motion. The less its gravity is, for the quantity of its matter, or the greater the velocity with which it is projected, the less will it deviate from a rectilinear course, and the farther it will go. If a leaden ball, projected from the top of a mountain by the force of gunpowder with a given velocity, and in a direction parallel to the horizon, is carried in a curve line to the distance of two miles before it falls to the ground; the same, if the resistance of the air were taken away, with a double or decuple velocity, would fly twice or ten times as far. And by increasing the velocity, we may at pleasure increase the distance to which it might be projected, and diminish the curvature of the line, which it might describe, till at last it should fall at the distance of 10, 30, or 90 degrees, or even might go quite round the whole earth before it falls; or lastly, so that it might never fall to the earth, but go
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Mathematical Principles of Natural Philosophy
forward into the celestial spaces, and proceed in its motion in infinitum. And after the same manner that a projectile, by the force of gravity, may be made to revolve in an orbit, and go round the whole earth, the moon also, either by the force of gravity, if it is endued with gravity, or by any other force, that impels it towards the earth, may be perpetually drawn aside towards the earth, out of the rectilinear way, which by its innate force it would pursue; and would be made to revolve in the orbit which it now describes; nor could the moon with out some such force, be retained in its orbit. If this force was too small, it would not sufficiently turn the moon out of a rectilinear course: if it was too great, it would turn it too much, arid draw down the moon from its orbit towards the earth. It is necessary, that the force be of a just quantity, and it belongs to the mathematicians to find the force, that may serve exactly to retain a body in a given orbit, with a given velocity; and
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Isaac Newton
Mathematical Principles of Natural Philosophy
vice versa, to determine the curvilinear way, into which a body projected from a given place, with a given velocity, may be made to deviate from its natural rectilinear way, by means of a given force. The quantity of any centripetal force may be considered as of three kinds; absolute, accelerative, and motive. DEFINITION VI. The absolute quantity of a centripetal force is the measure of the same proportional to the efficacy of the cause that propagates it from the centre, through the spaces round about. Thus the magnetic force is greater in one load-stone and less in another according to their sizes and strength of intensity.
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DEFINITION VII. The accelerative quantity of a centripetal force is the measure, of the same, proportional to the velocity which it generates in a given time.
Mathematical Principles of Natural Philosophy
Isaac Newton
Thus the force of the same loadstone is greater at a less distance, and less at a greater: also the force of gravity is greater in valleys, less on tops of exceeding high mountains; and yet less (as shall hereafter be shown), at greater distances from the body of the earth; but at equal distances, it is the same everywhere; because (taking away, or allowing for, the resistance of the air), it equally accelerates all falling bodies, whether heavy or light, great or small.
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DEFINITION VIII. The motive quantity of a centripetal force, is the measure of the same proportional to the motion which it generates in a given time. Thus the weight is greater in a greater body, less in a less body; and, in the same body, it is greater near to the earth, and less at remoter distances. This sort of quantity is the centripetency, or propension of the whole body towards the centre, or, as I may say, its weight;
Isaac Newton
Mathematical Principles of Natural Philosophy
and it is always known by the quantity of an equal and contrary force just sufficient to hinder the descent of the body. These quantities of forces, we may, for brevity s sake, call by the names of motive, accelerative, and absolute forces; and, for distinction s sake, consider them, with respect to the bodies that tend to the centre; to the places of those bodies; and to the centre of force towards which they tend; that is to say, I refer the motive force to the body as an endeavour a n d p ro p e n s i t y o f the whole towards a centre, arising from the propensities of the several parts taken together ; the accelerative force to the place of the body, as a certain power or energy diffused from the centre to all places around to move the bodies that are in them: and the absolute force to the centre, as endued with some cause, without which those motive forces would not be propagated through the spaces round about; whether that cause be some central body (such as is the load-stone,
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Isaac Newton
Mathematical Principles of Natural Philosophy
in the centre of the magnetic force, or the earth in the centre of the gravitating force), or anything else that does not yet appear. For I here design only to give a mathematical notion of those forces, without considering their physical causes and seats. Wherefore the accelerative force will stand in the same relation to the motive, as celerity does to motion. For the quantity of motion arises from the celerity drawn into the quantity of matter: and the motive force arises from the accelerative force drawn into the same quantity of matter. For the sum of the actions of the accelerative force, upon the several ; articles of the body, is the motive force of the whole. Hence it is, that near the surface of the earth, where the accelerative gravity, or force productive of gravity, in all bodies is the same, the motive gravity or the weight is as the body: but if we should ascend to higher regions, where the accelerative gravity is less, the weight would be
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Isaac Newton
Mathematical Principles of Natural Philosophy
equally diminished, and would always be as the product of the body, by the accelerative gravity. So in those regions, where the accelerative gravity is diminished into one half, the weight of a body two or three times less, will be four or six times less. I likewise call attractions and impulses, in the same sense, accelerative, and motive; and use the words attraction, impulse or propensity of any sort towards a centre, promiscuously, and indifferently, one for another; considering those forces not physically, but mathematically: wherefore, the reader is not to imagine, that by those words, I anywhere take upon me to define the kind, or the manner of any action, the causes or the physical reason thereof, or that I attribute forces, in a true and physical sense, to certain centres (which are only mathematical points); when at any time I happen to speak of centres as attracting, or as endued with attractive powers. r
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Isaac Newton
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Mathematical Principles of Natural Philosophy
Notes 12
Chandrasekhar, Subrahmanyan (2003). Newton's Principia for the common reader. Oxford: Oxford University Press. (pp.1–2). The quotation comes from a memorandum thought to have been written about 1714. As early as 1645 Ismaël Bullialdus had argued that any force exerted by the Sun on distant objects would have to follow an inverse-square law. However, he also dismissed the idea that any such force did exist. See, for example, Linton, Christopher M. (2004). From Eudoxus to Einstein—A History of Mathematical Astronomy. Cambridge: Cambridge University Press. p. 225. ISBN 978-0-521-82750-8.
Works Cited Gravity. (2015, November 28). In Wikipedia, The Free Encyclopedia. Retrieved 01:01, December 8, 2015, from https://en.wikipedia.org/w/index. php?title=Gravity&oldid=692857802 Newton, Isaac. (1846). The Mathematical Principles of Natural Philosophy. Retrieved December 8, 2015, from https://archive.org/ details/newtonspmathema00newtrich
1915
General Relativity
In general relativity, the effects of gravitation are ascribed to spacetime curvature instead of to a force.
Wikipedia
History of Gravitational Theory
In general relativity, the effects of gravitation are ascribed to spacetime curvature instead of to a force. The starting point for general relativity is the equivalence principle, which equates free fall with inertial motion. The issue that this creates is that free-falling objects can accelerate with respect to each other. In Newtonian physics, no such acceleration can occur unless at least one of the objects is being operated on by a force (and therefore is not moving inertially). To deal with this difficulty, Einstein proposed that spacetime is curved by matter, and that free-falling objects are moving along locally straight paths in curved spacetime. (This type of path is called a geodesic). More specifically, Einstein and Hilbert discovered the field equations of general relativity, which relate the presence of matter and the curvature of spacetime and are named after Einstein. The Einstein field equations are a set of 10 simultaneous,
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Albert Einstein
The Field Equations of Gravitation
I have shown in two recently published reports,13 how one can arrive at field equations of gravitation, that are in agreement with the postulate of general relativity, i.e. which in their general form are covariant in respect to arbitrary substitutions of spacetime variables. The line of development was as follows. At first I found equations, that contain Newton's theory as approximation and that are covariant in respect to arbitrary substitutions of the determinant 1. Afterwards I found, that those equations in general correspond to covariant ones, if the scalar of the energy tensor of "matter" vanishes. The coordinate system had to be specialized in accordance with the simple rule, that √−g is made to 1, whereby the equations of the theory experience an eminent simplification. In the course of this, however, one had to introduce the hypothesis, that the scalar of the energy tensor
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Wikipedia
History of Gravitational Theory
non-linear, differential equations. The solutions of the field equations are the components of the metric tensor of spacetime. A metric tensor describes the geometry of spacetime. The geodesic paths for a spacetime are calculated from the metric tensor. Notable solutions of the Einstein field equations include: • The Schwarzschild solution, which describes spacetime surrounding a spherically symmetric non-rotating uncharged massive object. For compact enough objects, this solution generated a black hole with a central singularity. For radial distances from the center which are much greater than the Schwarzschild radius, the accelerations predicted by the Schwarzschild solution are practically identical to those predicted by Newton’s theory of gravity. • The Reissner– Nordström solution,
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Albert Einstein
The Field Equations of Gravitation
of matter vanishes. Recently I find now, that one is able to dispense with hypothesis concerning the energy tensor of matter, if one fills in the energy tensor of matter into the field equations in a somehow different way than it was done in my two earlier reports. The field equations for vacuum, upon which I based the explanation of the perihelion motion of mercury, remain untouched by this modification. I give the complete consideration again at this place, so that the reader is not forced to uninterruptedly consultate the earlier reports. From the well known Riemannian covariant of fourth rank, the following covariant of second rank is derived: Gim=Rim+Sim (1)
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(1a) (1b) We obtain the ten general covariant equations of the gravitational field
Wikipedia
History of Gravitational Theory
in which the central object has an electrical charge. For charges with a geometrized length which are less than the geometrized length of the mass of the object, this solution produces black holes with an event horizon surrounding a Cauchy horizon. • The Kerr solution for rotating massive objects. This solution also produces black holes with multiple horizons. • The cosmological Robertson– Walker solution, which predicts the expansion of the universe. General relativity has enjoyed much success because of the way its predictions of phenomena which are not called for by the older theory of gravity have been regularly confirmed. For example: • General relativity accounts for the anomalous perihelion precession of the planet Mercury.
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Albert Einstein
The Field Equations of Gravitation
in spaces, in which "matter" is absent, by putting (2) These equations can be formed in a simpler way, when one choses the reference system so that √−g=1. Then Sim vanishes due to (1b), so that one obtains instead of (2) (3) √-g=1
(3a)
Here we put (4) which magnitudes we will denote as the "components" of the gravitational field. If "matter" exists in the considered space, then its energy tensor appears on the right hand side of (2) or (3). We put (2a) where we put (5)
Wikipedia
History of Gravitational Theory
• The prediction that time runs slower at lower potentials has been confirmed by the Pound– Rebka experiment, the Hafele– Keating experiment, and the GPS. • The prediction of the deflection of light was first confirmed by Arthur Eddington in 1919, and has more recently been strongly confirmed through the use of a quasar which passes behind the Sun as seen from the Earth. See also gravitational lensing. • The time delay of light passing close to a massive object was first identified by Irwin Shapiro in 1964 in interplanetary spacecraft signals. • Gravitational radiation has been indirectly confirmed through studies of binary pulsars. • The expansion of the universe (predicted by the Robertson–Walker metric) was confirmed by Edwin Hubble in 1929. q
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Albert Einstein
The Field Equations of Gravitation
T is the scalar of the energy tensor of "matter", the right hand side of (2a) is a tensor. If we specialize the coordinate system in the ordinary way again, then we obtain instead of (2a) the equivalent equations
(6) √−g=1 (3a) Like always we assume, that the divergence of the energy tensor of matter vanishes in the sense of the general differential calculus (MomentumEnergy theorem). When specializing the coordinate choice in accordance with (3a), it follows from it, that the Tim shall fulfill the conditions (7) or (7a) If one multiplies (6) by and sums over i and m, then one obtains14 in respect to (7) and in respect to the
Albert Einstein
The Field Equations of Gravitation
relation following from (3a) (8) the conservation law for matter and the gravitational field together in the form (8a) The reasons that drove me to the introduction of the second member on the right-hand side of (2a) and (6), become clear from the following considerations, that are completely analogous to those given at the place just mentioned (p. 785). If we multiply (6) by gim and sum over the indices i and m, then we obtain after simple calculation (9) where corresponding to (5) it is put for abbreviation (8b) Note, that it follows from the additional term, that in (9) the energy
Albert Einstein
The Field Equations of Gravitation
tensor of the gravitational field occurs besides that of matter in the same way, which is not the case in equations (21) l.c.. Furthermore one derives instead of equation (22) l.c., in the way as it is given there by the aid of the energy equation, the relations: (10) From our additional term it follows, that these equations contain no new condition in respect to (9), so that concerning the energy tensor of matter, no other presupposition has to be made than the one, that it has to be in agreement with the momentumenergy theorem. By that, the general theory of relativity as a logical building is eventually finished. The relativity postulate in its general form that makes the space-time coordinates to physically meaningless parameters, is directed with stringent necessity to a very specific theory of gravitation that explains
Albert Einstein
The Field Equations of Gravitation
the perihelion motion of mercury. However, the general relativity postulate offers nothing new about the essence of the other natural processes, which wasn't already taught by the special theory of relativity. My opinion regarding this issue, recently expressed at this place, was erroneous. Any physical theory equivalent to the special theory of relativity, can be included in the general theory of relativity by means of the absolute differential calculus, without that the latter gives any criterion for the admissibility of that theory. r
Albert Einstein
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The Field Equations of Gravitation
Notes 13
14
Sitzungsber. XLIV, p. 778 and XLVI, p. 799, 1915 Concerning the derivation see Sitzungsber. XLIV, 1915, p. 784/785. For the following, I request the reader to use the derivations given on p. 785 for comparison.
Works Cited History of gravitational theory. (2015, November 27). In Wikipedia, The Free Encyclopedia. Retrieved 01:15, December 8, 2015, from https:// en.wikipedia.org/w/index.php?title=History_of_ gravitational_theory&oldid=692663911 Translation:The Field Equations of Gravitation. (2013, August 18). In Wikisource . Retrieved 00:59, December 8, 2015, from https://en.wikisource.org/w/index.php?title=Translation:The_ Field_Equations_of_Gravitation&oldid=4554953
?
Quantum Gravity
Nevertheless, a number of open questions remain, the most fundamental of which is how general relativity can be reconciled with the laws of quantum physics to produce a complete and selfconsistent theory of quantum gravity.
Wikipedia
Gravity
In the decades after the disc o very of general relativity, it was realized that general relativity is inc o mpatible with quantum mechanics.15 It is p o ssible t o describe gravity in the framew o rk of quantum field the o ry like the o ther fundamental f o rces, such that the attractive f o rce of gravity arises due t o exchange o f virtual gravit o ns, in the same way as the electr o magnetic f o rce arises fr o m exchange o f virtual ph o t o ns.16 17 This repr o duces general relativity in the classical limit. H o wever, this appr o ach fails at sh o rt distances o f the o rder o f the Planck length,18 where a m o re c o mplete the o ry o f quantum gravity ( o r a new appr o ach t o quantum mechanics) is required. q
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Notes 15
Randall, Lisa (2005). Warped Passages: Unraveling the Universe's Hidden Dimensions. Ecco.
16
Feynman, R. P.; Morinigo, F. B.; Wagner, W. G.; Hatfield, B. (1995). Feynman lectures on gravitation. Addison-Wesley. ISBNÂ 0-201-62734-5.
17
Zee, A. (2003). Quantum Field Theory in a Nutshell. Princeton University Press.
18
Randall, Lisa (2005). Warped Passages: Unraveling the Universe's Hidden Dimensions. Ecco. ISBNÂ 0-06-053108-8.
Works Cited Gravity. (2015, November 28). In Wikipedia, The Free Encyclopedia. Retrieved 01:01, December 8, 2015, from https://en.wikipedia.org/w/index. php?title=Gravity&oldid=692857802 History of gravitational theory. (2015, November 27). In Wikipedia, The Free Encyclopedia. Retrieved 01:15, December 8, 2015, from https:// en.wikipedia.org/w/index.php?title=History_of_ gravitational_theory&oldid=692663911