Gravitational Fields

7 Key excerpts on "Gravitational Fields"

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  Intermediate Dynamics

  • Patrick Hamill(Author)
  • 2022(Publication Date)
  • Cambridge University Press

    (Publisher)

  The field extends throughout all space. It is not easy to grasp the field concept; it is particularly difficult to describe exactly what is filling all of space. We know that if we place a test mass at any point in the field it will feel a P(r) = field point r – r' r' r M = source point Figure 9.2 Particle M at the source point r  generates a gravitational field everywhere. The gravitational field at point P(r) (the “field point”) is given by Equation (9.3). 5 Sometimes, I like to change the definition of field slightly and state, “A field is a region of space in which some physical quantity is defined at every point.” This focuses the mind on the space rather than on the physical quantity. It is not, however, the standard definition of field. 9.2 The Gravitational Field 215 force, so we think of the mass M as producing something which exerts a force on any other mass in this space. The field is everywhere, but it exerts a force only when a material body is placed in the field. At the surface of the Earth the gravitational field is approximately given by g = − 9.8 m/s 2 ˆ k. Here ˆ k is a unit vector pointing upward at the surface. For points far above the surface, the field is approximately g(r) = −(GM E /r 2 ) ˆ r, where the origin is at the center of the Earth and M E is the mass of the Earth. (Do not confuse the field vector g(r) with the quantity g which is an abbreviation for the numerical value 9.8 m/s 2 .) The field concept is quite different from action at a distance. In action at a distance we consider the force one body exerts directly on another. When using the field concept, we think of the interaction of bodies M and m as a two-step process in which mass M generates a field and then mass m interacts with the field, rather than interacting with M directly. Thus the field approach decouples the sources from the test body used to determine the field.

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  Relativistic Astrophysics

  • Marek Demiański, D. Ter Haar(Authors)
  • 2013(Publication Date)
  • Pergamon

    (Publisher)

  CHAPTER 1 GRAVITATIONAL FIELD 1.1. Newton's Theory of Gravitation Classical celestial mechanics and models of stars are based on the assumption that bodies attract each other according to Newton's theory of gravitation. The fundamental quantity describing the gravitational field in this theory is the gravitational potential φ. The distribution of mass is represented by the density of matter p as a function of position; the gravitational potential φ is then determined from the Poisson equation Αφ=4πβρ, (1.1) where (?=6.67·10~ 8 cm 3 g~ 1 s~ 2 is the gravitational constant. For the potential φ to be determined uniquely, boundary conditions must be given. For a bounded distribution of mass it is usually assumed that far from the sources, at infinity, the gravitational potential vanishes as 1/r. The Poisson equation can then be solved uniquely and φ can be written in the form *(r,0=-GJ^dV. (1.2) Far from the sources, for |f|=r>Ä, where R characterizes the size of the region filled with matter, : 7 . can be represented by the series |r-r'| r rj ta 2 rj tab The small Latin indices a,b,c, ... run over the values 1, 2, 3, and repeated indices denote summation, e.g. x a p a =x i p l +x 2 p 2 +x 3 P3· A comma denotes partial differentiation, e.g. dr r a =-— . Substituting this expansion into (1.2) gives • ox * ( r f 0 -~ + c ( i ) i x > ( ^ 0 d V -y G ( i ) J x ; x ; p ( * ' , 0 d V + ..., (1.4) where M = Jpd 3 jc. In the centre-of-mass system this reads *(r.O-— -J G ß -f i ) , (1.5) r 6 V r J.ab 1 (1.3) 2 GRAVITATIONAL FIELD where g<*= J p (jc', 0(3x'V 6 -x /c x;5 eb )dV (1.6) denotes the quadrupole moment of the mass distribution. The equations of motion for a test particle can be found from Newton's second law, which takes the form mx f l =-m^ t a . (1.7) Owing to the equality of gravitational and inertial mass, an experimentally confirmed fact, gravitational forces are locally indistinguishable from inertial forces.

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  The Big Picture

  The Universe in Five S.T.E.P.S.

  • John Beaver(Author)
  • 2022(Publication Date)
  • Springer

    (Publisher)

  But this is only because we happen to be relatively close to an enormous quantity of mass—the 5:972 ˆ 10 24 kg of mostly rock and metal that makes up Earth. For astronomers, it is gravity that holds it all together and has the greatest effect on the motions of large bodies in the universe, even though it is the intrinsically weakest of the physical forces, all else being equal. The reason is that all else is not equal; gravitation is the only one of the fundamental forces that is both long-range and only attractive. The much-stronger electrical force, for example, can either attract or repel, and so it tends to cancel out at large distances. 11.1. NEWTON’S GRAVITY 147 11.1.1 THE CLASSICAL GRAVITATIONAL FIELD Equation (11.1), taken at face value, asserts that one object exerts an instantaneous influence (a gravitational force) upon another, even if the two are separated by a vast expanse of the vacuum of space. But it is possible to look at Newton’s gravity in a different way; we replace the action at a distance with the local action of the gravitational field. Let us return to our example of two point masses, m 1 and m 2 , placed a distance d from each other. From the point of view of Newton, we say that m 1 exerts a gravitational force, at a distance, on m 2 . But from the perspective of the gravitational field, we say something subtly different: the presence of m 1 causes there to be a gravitational field, E g, that extends throughout space. We can think of this field as a property of space itself. And so we have the gravitational field, E g, due to the presence of m 1 . Our field theory must then tell us how to calculate this field. For this simple case of the gravitational field due to a single point mass, the answer is very simple. The field points everywhere toward m 1 , and it has this magnitude, for a point in space located a distance d from m 1 : g D G m 1 d 2 : (11.4) But this is only half the story; we must also say what the field does.

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  Geophysical Field Theory and Method, Part A

  Gravitational, Electric, and Magnetic Fields

  • Alexander A. Kaufman(Author)
  • 1992(Publication Date)
  • Academic Press

    (Publisher)

  11.2 Determination of the Gravitational Field 165 caused by different masses within the body, and their contributions essen-tially depend on the location and distance of these masses from the observation point. In particular, masses located closer give a larger contri-bution, while remote parts of the body produce smaller effects. It is obvious that there are always such masses within the body that their contribution to the total field is so small that with the given accuracy of measuring it cannot be detected. For instance, we can imagine such changes of a shape, dimensions, location of the body, as well as the density of some of its parts, that the measured field would remain the same. In other words, due to errors in determination of the secondary field there can be an unlimited number of different distributions of masses that generate practically the same field at observation points. Inasmuch as the secondary field is caused by all masses within the body -that is, an integrated effect is measured-some changes of masses in relatively remote parts of the body can be significant; but their contribu-tion to the field would still remain small. At the same time similar changes in those parts of the body closer to observation points will result in much larger changes of the field. For this reason, in performing an interpreta-tion it is appropriate to distinguish at least two groups of parameters describing the distribution of masses, namely; 1. Parameters that have a sufficiently strong effect on the field; that is, relatively small changes of their values produce a change of the field that can be detected. 2. Parameters that have a noticeable influence on the field only if their values are significantly changed. It simply means that they cannot be defined from a field measured with some error.

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  Principles of the Gravitational Method

  • Alex Kaufman, Richard O. Hansen, Alex A. Kaufman(Authors)
  • 2007(Publication Date)
  • Elsevier Science

    (Publisher)

  Gravitational Field of the Earth 63 Now we demonstrate one interesting feature of the magnitude of the surface force as a function of the latitude. First, assuming that the fluid Earth is almost a sphere with a radius a , we can represent the distance r as r ¼ a cos l . Then, Equa-tion (2.18) has the form g s ¼ ð g a o 2 a cos 2 l Þ or g s ¼ ½ð g a o 2 a Þ þ o 2 a sin 2 l Š ð 2 : 22 Þ At points of the equator we have g e s ¼ g a o 2 a ð 2 : 23 Þ and, therefore, we obtain g s ð l Þ ¼ g e s ð 1 þ q sin 2 l Þ ð 2 : 24 Þ Here the parameter q ¼ o 2 a g e s ð 2 : 25 Þ is very small. In fact, assuming that g s ¼ 9 : 8 m = s 2 ; a ¼ 6 : 4 10 6 m ; and o ¼ 7 : 3 10 5 s 1 we have q ¼ 1 289 ð 2 : 26 Þ This parameter can be interpreted in the following way. At the pole Equation (2.24) gives g p s ¼ g e s ð 1 þ q Þ or q ¼ g p s g e s g e s ð 2 : 27 Þ Equations (2.24 and 2.27) look like as Clairaut’s formulas which will be derived later. However, this similarity is superficial, since the former do not contain the flattening of the earth. A variation of the field magnitude, g s with latitude, Equation (2.24), is caused by only a change of the centripetal acceleration on the spherical surface. 2.1.3. The gravitational field g In order to study the attraction of masses of the earth which moves around the axis of rotation, it seems appropriate to use the field g s , which depends on the distribution of masses and the angular velocity, as well as coordinates of the point. Besides, it has a physical meaning of the reaction force per unit mass. However, it has one very serious shortcoming, namely, unlike the attraction force it is directed outward. In other words, it differs strongly from the attraction field, in spite of the fact that the contribution of rotation is extremely small. To overcome this problem we introduce the gravitational field g which differs from the reaction field in direction only: g ð p Þ ¼ g s ð p Þ ð 2 : 28 Þ Methods in Geochemistry and Geophysics 64

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  Effects of Hypergravity and Microgravity on Biomedical Experiments, The

  • Thais Russomano, Gustavo Dalmarco, Felipe Prehn Falcao(Authors)
  • 2022(Publication Date)
  • Springer

    (Publisher)

  Compared to the other three fundamental interactions in nature, gravitation is the weakest one. It, however, acts over great distances and is always present. Gravitation is interpreted differently by classic mechanics, relativity, and quantum physics. In classic mechanics, gravitation arises out of the force of gravity. In general relativity, it is the mass that curves space time, not a force. In quantum gravity theories, the gravitation is the postulated carrier of the gravitational force, time space itself is envisioned as discrete in nature, or both. The gravitational attraction of the Earth endows objects with weight that causes them to fall to the ground when dropped (the Earth also moves toward the object but only by an infinitesimal amount). FIGURE 1.1: The gravitational force keeps the planets in orbit about the sun (http://en.wikipedia .org/wiki/Gravity). GENERAL CONCEPTS IN PHYSICS—DEFINITION OF PHYSICAL TERMS 3 The Universal Law of Gravitation was postulated by the English physicist and mathemati- cian Sir Isaac Newton (1642–1727). There is a popular story that the origin of this theory happened when Newton was sitting under a tree and an apple fell on his head. This is almost certainly not an exact truth, with embellishment of details, as what happens in most legends. Probably, the more correct version of the story is that Newton, upon observing (or imagining!) an apple fall from a tree, began to think that the apple is accelerated because its velocity changes from zero as it is hanging on the tree then moves toward the ground. He then considered that the same force that pulled the apple toward the Earth is the same one that makes the Moon to orbit our planet. Newton’s theory of how a celestial body can orbit another celestial body can be illustrated by his well-known example shown in Figure 1.2. Suppose that a cannon ball is fired horizontally from a high mountain on top of the Earth.

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  The Curious History of Relativity

  How Einstein's Theory of Gravity Was Lost and Found Again

  • Jean Eisenstaedt(Author)
  • 2018(Publication Date)
  • Princeton University Press

    (Publisher)

  C H A P T E R T H R E E Toward a New Theory of Gravitation Before getting down to the details of general relativity and an- alyzing each of the theory’s guiding principles, let us try to un- derstand toward which new horizons Einstein is going to take us. What is general relativity? It is a theory of gravitation that re- places Newton’s and which predicts the behavior of material and light particles subject to a gravitational field. General rela- tivity disregards all other physical phenomena, such as quan- tum and electromagnetic ones, to focus only on a classical (i.e., nonquantum) physics, a kind of particle ballistics. A gravitational field is simply a concept, a theoretical tool that expresses the existence of gravitation at each point of space- time, a field that we will be able to construct thanks to general relativity. Einstein’s theory of gravitation first tells us the nature of that field, in which space it exists, what its equations are and how to write them, and finally how material and light particles behave in it. To better understand the changes (the numerous changes, in fact) that will take place, let us go back for a moment to Newton’s theory of gravitation. As a consequence of its absolute space and absolute time, his theory operates in an absolute space-time that we shall call Newtonian. The position of each particle, material or luminous, is defined by its four coordinates, three spatial and one temporal; they determine where and when. Given a distri- bution of matter of mass M, the Sun, for instance, we know how to calculate the gravitational field created by this distribution everywhere in the universe. It is essentially an M / r 2 force field. Any test particle, that is, a particle small enough so that we may ignore its own gravitational field, of inertial mass m, will be sub- ject to that force field. But this particle is also affected by its own 58

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