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*To*: <escepticos en ccdis.dis.ulpgc.es>*Subject*: [escepticos] Inercia*From*: "J.S." <j.susaeta@...>*Date*: Mon, 24 Dec 2001 12:54:54 +0100*References*: <F124p7mvjkO22ZhCl8a0000a910 en hotmail.com> <3C24F15B.70405 en teleline.es> <200112222122.fBMLME179051 en borja.sarenet.es> <00c001c18b34$577db160$21fda8c0 en User> <002601c18b37$cd0985a0$0100a8c0 en uno>*Reply-to*: escepticos en ccdis.dis.ulpgc.es*Sender*: owner-escepticos en dis.ulpgc.es

Hola. A continuación mando el capítulo 1 del libro que mencionó Carlos Ungil hace pocos días (Ciufolini & Wheeler, 'Gravitation and Inertia', Princeton University Press, 1995). Creo que es de interés para el tema de la 'inercia extrínseca' que venimos debatiendo. He eliminado las abundantes remisiones bibliográficas, y también parte de las remisiones a otros capítulos de la obra. Un comentario: lo de los 15 segundos de arco de desviación observados por Gauss me suenan raros. estaba -Gauss- midiendo un triángulo plano, entre tres cumbres. La masa de la Tierra es, creo, insuficiente como para generar esa curvatura. Lo curioso es que el texto dice que eso era prueba de la curvatura de la Tierra, pero Gauss no estaba -sus visuales eran rectas- midiendo un triángulo esférico... Creo que a los autores se les han 'cruzado los cables' en este comentario. Saludos Javier ***************************** A First Tour In this chapter we introduce some basic concepts and ideas of Einstein's General Theory of Relativity which are developed in the book. 1,1 SPACETIME CURVATURE, GRAVITATION, AND INERTIA "Gravity is not a foreign and physical force transmitted through space and time. It is a manifestation of the curvature of spacetime." That, in a nutshell, is Einstein's theory. What this theory is and what it means, we grasp more fully by looking at its intellectual antecedents. First, there was the idea of Riemann that space, telling mass how to move, must itself-by the principle of action and reaction-be af- fected by mass. It cannot be an ideal Euclidean perfection, standing in high mightiness above the battles of matter and energy. Space geometry must be a participant in the world of physics. Second, there was the contention of Ernst Mach that the "acceleration relative to absolute space" of Newton is only prop- erly understood when it is viewed as acceleration relative to the sole significant mass there really is, the distant stars. According to this "Mach principle," in- ertia here arises from mass there. Third was that great insight of Einstein that we summarize in the phrase "free fall is free float": the equivalence princi- ple, one of the best-tested principles in physics, from the inclined tables of Galilei and the pendulum experiments of Galilei, Huygens, and Newton to the highly accurate torsion balance measurements of the twentieth century, and the Lunar Laser Ranging experiment. With those three clues vibrating in his head, the magic of the mind opened to Einstein what remains one of mankind's most precious insights: gravity is manifestation of spacetime curvature. Euclid's (active around 300 B.C.) fifth postulate states that, given any straight line and any point not on it, we can draw through that point one and only one straight line parallel to the given line, that is, a line that will never meet the given one (this altemative formulation of the fifth postulate is essentially due to Proclos). This is the parallel postulate. In the early 1800s the discussion grew lively about whether the properties of parallel lines as presupposed in Euclidean geometry could be derived from the other postulates and axioms, or whether the parallel postulate had to be assumed independently. More than two thousand years after Euclid, Karl Friedrich Gauss, János Bolyai, and Niko- lai Ivanovich Lobachevskiy discovered pencil-and-paper geometric systems that satisfy all the axioms and postulates of Euclidean geometry except the parallel postulate. These geometries showed not only that the parallel postulate must be assumed in order to obtain Euclidean geometry but, more important, that non-Euclidean geometries as mathematical abstractions can and do exist. Consider the two-dimensional surface of a sphere, itself embedded in the three-dimensional space geometry of everyday existence. Euclid's system ac- curately describes the geometry of ordinary three-dimensional space, but not the geometry on the surface of a sphere. Let us consider two lines locally parallel on the surface of a sphere. They propagate on the surface as straight as any lines could possibly be, they bend in their courses one whit neither to left or fight. Yet they meet and cross. Clearly, geodesic lines (on a surface, a geodesic is the shortest line between two nearby points) on the curved surface of a sphere do not obey Euclid's parallel postulate. The thoughts of the great mathematician Karl Friedrich Gauss about curva- ture stemmed not from theoretical spheres drawn on paper but from concrete, down-to-Earth measurements. Commissioned by the govemment in 1827 to make a survey map of the region for miles around G6ttingen, he found that the sum of the angles in his largest survey triangle was different from 180°. The deviation from 180° observed by Gauss-almost 15 seconds of arc -was both inescapable evidence for and a measure of the curvature of the surface of Earth. To recognize that straight and initially parallel lines on the surface of a sphere can meet was the first step in exploring the idea of a curved space. Second came the discovery of Gauss that we do not need to consider a sphere or other two- dimensional surface to be embedded in a three-dimensional space to define its geometry. It is enough to consider measurements made entirely within that two- dimensional geometry, such as, would be made by an ant forever restricted to live on that surface. The ant would know that the surface is curved by measuring that the sum of the internal angles of a large dangle differs from 180°, or by measuring that the ratio between a large circumference and its radius R differs from 2pi. Gauss did not limit himself to thinking of a curved two-dimensional surface floating in a flat three-dimensional universe. In an 1824 letter to Ferdinand Karl Schweikart, he dared to conceive that space itself is curved: "Indeed I have therefore from time to time in jest expressed the desire that Euclidean geometry would not be correct." He also wrote: "Although geometers have given much attention to general investigations of curved surfaces and their results cover a significant portion of the domain of higher geometry, this subject is still so far from being exhausted, that it can well be said that, up to this time, but a small portion of an exceedingly fruitful field has been cultivated" (Royal Society of G6ttingen, 8 October 1827). The inspiration of these thoughts, dreams, and hopes passed from Gauss to his student, Bernhard Riemann. Bemhard Riemann went on to generalize the ideas of Gauss so that they could be used to describe curved spaces of three or more dimensions. Gauss had found that the curvature in the neighborhood of a given point of a specified two-dimensional space geometry is given by a single number: the Gaussian curvature. Riemann found that six numbers are needed to describe the curvature of a three-dimensional space at a given point, and that 20 numbers at each point are required for a four-dimensional geometry: the 20 independent components of the so-called Riemann curvature tensor. In a famous lecture he gave 10 June 1854, entitled *On the Hypotheses That Lie at the Foundations of Geometry*, Riemann emphasized that the truth about space is to be discovered not from perusal of the 2000-year-old books of Euclid but from physical experience. He pointed out that space could be highly irregular at very small distances and yet appear smooth at everyday distances. At very great di stances, he also noted, large-scale curvature of space might show up, perhaps even bending the universe into a closed system like a gigantic ball. He wrote: "Space [in the large] if one ascribes to it a constant curvature, is nec- essarily finite, provided only that this curvature has a positive value, however small. . .. It is quite conceivable that the geometry of space in the very small does not satisfy the axioms of [Euclidean] geometry. . . . The curvature in the three directions can have arbitrary values if only the entire curvature for every sizeable region of space does not differ greatly from zero. . . . The properties which distinguish space from other conceivable triply-extended magnitudes are only to be deduced from experience." But as Einstein was later to remark, "Physicists were still far removed from such a way of thinking: space was still, for them, a rigid, homogeneous some- thing, susceptible of no change or conditions. Only the genius of Riemann, solitary and uncomprehended, had already won its way by the middle of the last century to a new conception of space, in which space was deprived of its rigidity, and in which its power to take part in physical events was recognized as possible." Even as the 39-year-old Riemann lay dying of tuberculosis at Selasca on Lake Maggiore in the summer of 1866, having already achieved his great mathemat- ical description of space curvature, he was working on a unified description of electromagnetism and gravitation. Why then did he not, half a century before Einstein, arrive at a geometric account of gravity? No obstacle in his way was greater than this: he thought only of space and the curvature of space, whereas Einstein discovered that he had to deal with spacetime and the spacetime cur- vature. Einstein could not thank Riemann, who ought to have been still alive. A letter of warm thanks he did, however, write to Mach. In it he explained how mass there does indeed influence inertia here, through its influence on the enveloping spacetime geometry. Einstein's geometrodynamics had trans- muted Mach's bit of philosophy into a bit of physics, susceptible to calculation, prediction, and test. Let us bring out the main idea in what we may call poor man's language. Inertia here, in the sense of local inertial frames, that is the grip of spacetime here on mass here, is fully defined by the geometry, the curvature, the structure of spacetime here. The geometry here, however, has to fit smoothly to the geometry of the immediate surroundings; those domains, onto their surroundings; and so on, all the way around the great curve of space. Moreover, the geometry in each local region responds in its curvature to the mass in that region. Therefore every bit of momentum-energy, wherever located, makes its influence felt on the geometry of space throughout the whole universe-and felt, thus, on inertia right here. The bumpy surface of a potato is easy to picture. It is the two-dimensional analogue of a bumpy three-sphere, the space geometry of a universe loaded irregularly here and there with concentrations and distributions of momentum- energy. If the spacetime has a Cauchy surface, that three-geometry once known-mathematical solution as it is of the so-called initial-value problem of geometrodynamics -the future evolution of that geometry follows straightforwardly and deterministically. In other words, inertia (local inertial frames) everywhere and at all times is totally fixed, specified, determined, by the initial distribution of momentum- energy, of mass and mass-in-motion. The mathematics cries out with all the force at its command that mass there does determine inertia here. We will enter into the mathematics of this initial-value problem-so thor- oughly investigated in our day- in chapter 5. Of all the contributions made to inertia here by all the mass and mass-in-motion in the whole universe, the fractional contribution made by a particular mass at a particular distance is of the order of magnitude [fractional contribution by a given mass, *there* to the determination of the direction of axes of the local gyroscopes, the compass of inertia, *here*] is of order of [mass, *there*]/[distance, *there* to *here*] In this rough measure of the voting power, the "inertia-contributing power" of any object or any concentration of energy, its mass is understood to be expressed in the same geometric units as the distance. Does this whole idea of voting fights and inertia-contributing power make sense? It surely does so if the total voting power of all the mass there is in the whole universe adds up to 100%. But does it? Let us run a check on the closed Friedmann model universe (chap. 4)? There the total amount of mass is of the order of ~ 6 x 10^56 g (see § 4.2). This amount, translated into geometric units by way of the conversion factor 0,742 x 10^-28 cm/g, is ~ 4,5 x 10^28 cm of mass. It is much harder to assign an effective distance at which that mass lies from us, and for two reasons. First, distances are chang- ing with time. So at what time is it that we think of the distance as being measured? If we got into all the subtleties of that question we would be fight back at the full mathematics of the so-called initial-value problem (see chap. 5), mathematics sufficiently complicated to have persuaded us to short circuit it in this introduction by our quick and rough poor man's account of inertia. This problem of "when" to measure distances let us therefore resolve by taking them at the instant of maximum expansion, when they are not changing. What is the other problem about defining an effective distance to all the mass in the universe? Simple! Some lies at one distance, some at another. What separation does it make sense to adopt as a suitable average measure of the distance between inertia here and mass there? It is surely too much to accept as an average the mileage all the way around to the opposite side of the three-sphere, the most remote bit of mass in the whole universe ! As a start let us nevertheless look at that maximum figure a minute. It is half of the circumference of the three-sphere. In other words, it is pi times the radius of the three-sphere at maximum expansion, ~ 10 to 20 * 10^9 yr * 3 x 10^7 s/yr * 3 x 10^10 cm/s ~~ 2 x 10^28 cm, or ~ 6 x 10^28 cm. It is too big, we know, but surely not tenfold too big, not fivefold too big, conceivably not even twice as great as a reasonable figure for an effective average distance from here to mass there. Then, we say, why not adopt for that fuzzy average distance figure the very amount of mass figure that we have for the three-sphere model universe, ~ 4,5 x 10^28 cm? This figure is surely of the right order of magnitude, and in our rough and ready way of figuring inertia, gives the satisfying 100% signal of "all votes in" when all the mass is counted. This is the poor man's version of the origin of inertia! Now for inertia-determination in action! Mount a gyroscope on frictionless gimbals or, better, float it weightless in space to eliminate the gravity force that here on Earth grinds surface against surface. Picture our ideal gyro as sitting on a platform at the North Pole with the weather so cloudy that it has not one peek at the distant stars. Pointing initially to the flag and flagpole at a comer of the support platform, will the gyro continue to point that way as a 24 hour candle gradually bums away? No. The clouds do not deceive it. It does not see the star to which its spin axis points, but to that star it nevertheless continues to point as the day wears on. Earth turns beneath the heedless gyro, one rotation and many another as the days go by. That is the inertia-determining power of the mass spread throughout space, as that voting power is seen in its action on the gyro. Wait! As conscientious workers against election fraud, let us inspect the ballot that each mass casts. Where is the one signed "Earth"? Surely it was entitled to vote on what would be the free-float frame at the location of the gyro. The vote tabulator sheepishly confesses. "I know that Earth voted for having the frwne of reference turn with it. However, its vote had so little inertia-determining power compared to all the other masses in the universe that I left it out." We inspectors reply, "you did wrong. Let us see how much damage you have donel' Yes, the voting power of Earth at the location of the gyro is small, but it is not zero. It is of the order of magnitude of [mass of Earth/radius of Earth]~= 0,44 cm/6,4*10^8 cm~= 0,69*10^-9 roughly only one-billionth as much influence as all the rest of the universe together. "It was not only that its voice was so weak that I threw out its vote," the vote tabulator interrupts to say. "The free-float frame of reference that Earth wanted the gyro axis to adhere to was so little different from the frame demanded for the gyro by the faraway stars that I could not believe that Earth really took its own wishes seriously. It wants the gyroscope axis to creep slowly around in a twenty-four hour day rather than keep pointing at one star? Who cares about that peanut difference?" "Again you are wrong," we tell him. "You do not know our friends the astronomers. For a long time there was so little that they could measure about a faraway star except its direction that they developed the art of measuring angles to a fantastic discrimination. With Very Long Baseline Interferometry one can measure the angular separation of the distant quasars down to less than a tenth of millisecond of arc and the rate of turning of Earth down to less than I milliarcsec per year. Content to wait a day to see the effect of Earth on the direction of the spin axis of the gyro? No. Figure our friends as watching it for a whole 365-day year. Do you know how many milliseconds of arc the axis of the gyro would tum through in the course of a whole year, relative to the distant stars, if it followed totally and exclusively the urging of Earth? Four-hundred and seventy-three billion milliarcseconds! And that is the tuming effect which you considered so miniscule that you threw away Earth's ballot! We are here to see justice done." The corrected turning rate of the gyro, relative to the distant stars, is of the order of magnitude of [voting power of Earth]*[rate of turn desired by Earth]=[0,698 billionth of total voting power]*[473 billion milliarcsec per year]=[330 milliarcsec per year] Rough and ready though the reasoning that leads to this estimate, it gives the right order of magnitude. However, nobody has figured out how to operate on Earth's surface a gyroscope sufficiently close to friction-free that it can detect the predicted effect. What to call this still undetected phenomenon? "Dragging," Einstein called it in his 1913 letter of thanks to Mach, already two years before he arrived at the final formulation of his geometric theory of gravity; and *frame-dragging* the effect is often still called today following the lead of Jeffrey Cohen (see below, ref. 24). Others often call it the *Lense-Thirring effect*, after the 1918 paper of Hans Thirring and J. Lense, that gave a quantitative analysis of it (see also W. de Sitter 1916 and §§ 3.4.3 and 6.10 below). Today still other words are used for that force at work on the gyro which causes the slow tuming of its axis: *gyrogravitation* or gravitomagnetism (see fig. 1.2). Such a new name is justified for a new force. This force differs as much from everyday gravity as a magnetic force differs from an electric force. Magnetism known since Greek times was analyzed (William Gilbert, *De Magnete*, 1600) long after electricity. It took even longer to recognize that an electric charge going round and round in a circuit produces magnetism (H.C. Oersted, 1820). Gravitomagnetism or gyrogravitation, predicted in 1896-1916 but still not brought to the light of the day, is produced by the motion of mass around and around in a circle. How amazing that a new force of nature should be enveloping us all and yet sitll stand there undetected! Active work is in progress to detect gravitomagnetism. The detecting gyroscopes will be otrbiting Earth. The one gyroscope is a quartz spehre smaller than a hand, carried along with an electrostatic suspension and a sensitive readout devices, the whole under development by the Stanford group of C. W. F. Everitt and his associates for more than 20 years. The other gyroscope is huge, roughly four Earth radii in diameter. It consists of a sphere of 30 cm of radius covered with reflecting mirrors, going round and round in Earth orbit every 3,758 hours. An axis perpendicular to that or- bit we identify with the gyroscope axis. Such a satellite, LAGEOS, already exists. It was launched in 1976. Its position is regularly read with a precision better than that of any other object in the sky, or an un- certainty of about a centimeter. For it the predicted gravitomagnetic precession is 31 milliarcsec per year, quite enough to be detected if there were not enor- mously larger precessions at work that drown it out, about 126° per year. This devastating competition arises from the nonsphericity of the Earth. Fortunately the possibility exists to launch another LAGEOS satellite going around at an- other inclination, so chosen that the anomalies in Earth's shape give a reversed orbit tuming but the Earth's angular momentum gives the same Lense-Thirring frame-dragging (Ciufolini 1984). In other words the two LAGEOS satel- lites, with their supplementary inclinations, will define an enormous gyroscope unaffected by the multipole mass moments of the Earth but only a%ected by the Lense-Thirring drag. If this is done the small "difference" between the "large" tuming rates will yield a direct measure of a force of nature new to man. Still more important, we will have direct evidence that mass there govems inertia here! 1,2 RELATION OF GRAVITY TO THE OTHER FORCES OF NATURE In this book we will trace out the meaning and consequences of Einstein's great 1915 battle-tested and still standard picture of gravity as manifestation of the curvature of four-dimensional spacetime. I weigh all that is. Nowhere in the universe Is there anything over which I do not have dominion. As spacetime, As curved all-pervading spacetime, I reach everywhere. My name is gravity. What of the other forces of nature? Every other force-the electric force that rules the motion of the atomic elec- tron, the weak nuclear force that govems the emission of electrons and neutrinos from radioactive nuclei, and the strong nuclear force that holds together the con- stituents of particles heavier than the electron-Remands for its understanding, today's researches argue, a geometry of more than four dimensions, perhaps as many as ten. The extra six dimensions are envisaged as curled up into an ultrasmall cavity, with one such cavity located at each point in spacetime. This tiny world admits of many a different organ-pipe resonance, many a different vibration frequency, each with its own characteristic quantum energy. Each of these modes of vibration of the cavity shows itself to us of the larger world as a particle of a particular mass, with its own special properties. Such ideas are pursued by dozens of able investigators in many leading cen- ters today, under the name of "grand unified field theories (GUTS)" or "string theories." They open up exciting prospects for deeper understanding. How- ever, this whole domain of research is in such a turmoil that no one can report with any confidence any overarching final view, least of all any final conclusion as to how gravity fits into the grand pattem. The theories of the unification of forces with greatest promise today all have this striking feature that they, like the battle-tested, but simpler and older Ein- stein gravitation theory, build themselves on the boundary of a boundary principle, though in a higher dimensional version; for example, "the eight- dimensional boundary of the nine-dimensional boundary of a ten-dimensional region is automatically zero." Hidden to the uninstructed student of modern field theory is this vital organizing power of the boundary principle (see § 2.8). Hidden to the casual observer of a beautiful modern building is the steel frame- work that alone supports it; and hidden in Einstein's great account of geometric gravity, hidden until Élie Cartan's penetrating insight brought it to light, is the unfolding of it all, from the grip of spacetime on mass to the grip of mass on spacetime, and from the automatic conservation of momentum-energy- without benefit of gears and pinions or any sophisticated manual to instruct nature what to dw-to all the rich garden of manifestations of gravity in na- ture, the unfolding of all this from "the one-dimensional boundary of the two-dimensional boundary of a three-dimensional region is zero" and "the two- dimensional boundary of the three-dimensional boundary of a four-dimensional region is zero." Nowhere more decisively than in the fantastic austerity of this organizing idea-the boundary principle-does nature instruct man that it is at heart utterly simple-and that some day we will see it so. *********************

**Follow-Ups**:**Re: [escepticos] Inercia**- From: Carlos Ungil <Carlos.Ungil@...>

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