Gravitation

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Gravitation is a natural phenomenon by which all objects with mass attract each other, and is one of the fundamental forces of physics. In everyday life, gravitation is most commonly thought of as the agency that gives objects weight. It is responsible for keeping the Earth and the other planets in their orbits around the Sun; for keeping the Moon in its orbit around the Earth, for the formation of tides; for convection (by which hot fluids rise); for heating the interiors of forming stars and planets to very high temperatures; and for various other phenomena that we observe. Gravitation is also the reason for the very existence of the Earth, the Sun, and most macroscopic objects in the universe; without it, matter would not have coalesced into these large masses and life, as we know it, would not exist.

Modern physics describes gravitation using the general theory of relativity, but the much simpler Newton's law of universal gravitation provides an excellent approximation in most cases.

The terms gravitation and gravity are mostly interchangeable in everyday use, but in scientific usage a distinction may be made. "Gravitation" is a general term describing the attractive influence that all objects with mass exert on each other, while "gravity" specifically refers to a force that is supposed in some theories (such as Newton's) to be the cause of this attraction. By contrast, in general relativity gravitation is due to spacetime curvatures that cause inertially moving objects to accelerate towards each other.

File:Solar sys8.jpg
Gravitation keeps the planets in orbit about the Sun. (Not to scale)

History of gravitational theory

Early history

Efforts to understand gravity began in ancient times. Philosophers in ancient India explained the phenomenon from the 8th century BC.[1] According to Kanada, founder of the Vaisheshika school, "Weight causes falling; it is imperceptible and known by inference."[2]

In the 4th century BC, the Greek philosopher Aristotle believed that there was no effect without a cause, and therefore no motion without a force. He hypothesized that everything tried to move towards its proper place in the crystalline spheres of the heavens, and that physical bodies fell toward the center of the Earth in proportion to their weight.

Brahmagupta, in the Brahmasphuta Siddhanta (628 AD), responded to critics of the heliocentric system of Aryabhata (476–550 AD) stating that "all heavy things are attracted towards the center of the earth" and that "all heavy things fall down to the earth by a law of nature, for it is the nature of the earth to attract and to keep things, as it is the nature of water to flow, that of fire to burn, and that of wind to set in motion... The earth is the only low thing, and seeds always return to it, in whatever direction you may throw them away, and never rise upwards from the earth."[3][4]

In the 9th century, the eldest Banū Mūsā brother, Muhammad ibn Musa, in his Astral Motion and The Force of Attraction, hypothesized that there was a force of attraction between heavenly bodies,[5] foreshadowing Newton's law of universal gravitation.[6] In the 1000s, the Persian scientist Ibn al-Haytham (Alhacen), in the Mizan al-Hikmah, discussed the theory of attraction between masses, and it seems that he was aware of the magnitude of acceleration due to gravity.[7] In 1121, Al-Khazini, in The Book of the Balance of Wisdom, differentiated between force, mass, and weight,[8] and discovered that gravity varies with the distance from the centre of the Earth,[9] though he believed that the weight of heavy bodies increase as they are farther from the centre of the Earth.[10] All these early attempts at trying to explain the force of gravity were philosophical in nature and it would be Isaac Newton that gave the first correct desciption of gravity.

Scientific revolution

Modern work on gravitational theory began with the work of Galileo Galilei in the late 16th century and early 17th century. In his famous (though probably apocryphal) experiment dropping balls from the Tower of Pisa, and later with careful measurements of balls rolling down inclines, Galileo showed that gravitation accelerates all objects at the same rate. This was a major departure from Aristotle's belief that heavier objects are accelerated faster. (Galileo correctly 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.

Newton's theory of gravitation

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 nearly.”

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 be accounted by the actions of the other planets. Calculations by John Couch Adams and Urbain Le Verrier both predicted the general position of the planet, and Le Verrier's calculations are what led Johann Gottfried Galle to the discovery of Neptune.

Ironically, it was another discrepancy in a planet's orbit that helped to point out flaws in Newton's theory. By the end of the 19th century, it was known that the orbit of Mercury 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 General Theory of Relativity, which accounted for the discrepancy in Mercury's orbit.

Although Newton's theory has been superseded, most modern non-relativistic gravitational calculations are still made using Newton's theory because it is a much simpler theory to work with than General Relativity, and gives sufficiently accurate results for most applications.

General relativity

In general relativity, the effects of gravitation are ascribed to spacetime curvature instead of 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 discovered the field equations of general relativity, which relate the presence of matter and the curvature of spacetime and are named after him. The Einstein field equations are a set of 10 simultaneous, non-linear, differential equations. The solutions of the field equations are the components of the metric tensor of spacetime. A metric tensor describes a geometry of spacetime. The geodesic paths for a spacetime are calculated from the metric tensor.

Notable solutions of the Einstein field equations include:

General relativity has enjoyed much success because of how its predictions of phenomena which are not called for by the theory of gravity have been regularly confirmed. For example:

Gravity and quantum mechanics

Several decades after the discovery of general relativity it was realized that general relativity is incompatible with quantum mechanics.[11] It is possible to describe gravity in the framework of quantum field theory like the other fundamental forces, with the attractive force of gravity arises due to exchange of virtual gravitons, in the same way as the electromagnetic force arises from exchange of virtual photons.[12][13] This reproduces general relativity in the classical limit. However, this approach fails at short distances of the order of the Planck length,[14] where a more complete theory of quantum gravity (or a new approach to quantum mechanics) is required. Many believe the complete theory to be string theory,[15] or more currently M Theory.

Specifics

Earth's gravity

Every planetary body (including the Earth) is surrounded by its own gravitational field, which exerts an attractive force on all objects. Assuming a spherically symmetrical planet (a reasonable approximation), the strength of this field at any given point is proportional to the planetary body's mass and inversely proportional to the square of the distance from the center of the body.

The strength of the gravitational field is numerically equal to the acceleration of objects under its influence, and its value at the Earth's surface, denoted g, is approximately 9.81 m/s² (32.2 ft/s²) as the standard average. This means that, ignoring air resistance, an object falling freely near the earth's surface increases its velocity with 9.81 m/s (32.2 ft/s or 22 mph) for each second of its descent. Thus, an object starting from rest will attain a velocity of 9.81 m/s (32 ft/s) after one second, 19.6 m/s (64 ft/s) after two seconds, and so on, adding 9.8 m/s to each resulting velocity. According to Newton's 3rd Law, the Earth itself experiences an equal and opposite force to that acting on the falling object, meaning that the Earth also accelerates towards the object. However, because the mass of the Earth is huge, the acceleration of the Earth by this same force is negligible, when measured relative to the system's center of mass.

Equations for a falling body

File:Falling ball.jpg
Ball falling freely under gravity. See text for description.

The kinematical and dynamical equations describing the trajectories of falling bodies are considerably simpler if the gravitational force is assumed constant. This assumption is reasonable for objects falling to Earth over the relatively short vertical distances of our everyday experience, but does not hold over larger distances, such as spacecraft trajectories, since the acceleration due to Earth's gravity is much smaller at large distances.

Under an assumption of constant gravity, Newton’s law of gravitation simplifies to F = mg, where m is the mass of the body and g is a constant vector with an average magnitude of 9.81 m/s². The acceleration due to gravity is equal to this g. An initially-stationary object which is allowed to fall freely under gravity drops a distance which is proportional to the square of the elapsed time. The image on the right, spanning half a second, was captured with a stroboscopic flash at 20 flashes per second. During the first 1/20th of a second the ball drops one unit of distance (here, a unit is about 12 mm); by 2/20ths it has dropped at total of 4 units; by 3/20ths, 9 units and so on.

Under the same constant gravity assumptions, the potential energy, Ep, of a body at height h is given by Ep = mgh (or Ep = Wh, with W meaning weight). This expression is valid only over small distances h from the surface of the Earth. Similarly, the expression $h = v^2/2g$ for the maximum height reached by a vertically projected body with velocity v is useful for small heights and small initial velocities only. In case of large initial velocities we have to use the principle of conservation of energy to find the maximum height reached. This same expression can be solved for v to determine the velocity of an object dropped from a height h immediately before hitting the ground, $v=\sqrt{2gh}$, assuming negligible air resistance.

Gravity and astronomy

The discovery and application of Newton's law of gravity accounts for the detailed information we have about the planets in our solar system, the mass of the Sun, the distance to stars, quasars and even the theory of dark matter. Although we have not traveled to all the planets nor to the Sun, we know their masses. These masses are obtained by applying the laws of gravity to the measured characteristics of the orbit. In space an object maintains its orbit because of the force of gravity acting upon it. Planets orbit stars, stars orbit galactic centers, galaxies orbit a center of mass in clusters, and clusters orbit in superclusters. The force of gravity is proportional to the mass of an object and inversely proportional to the square of the distance between the objects.

In general relativity, gravitational radiation is generated in situations where the curvature of spacetime is oscillating, such as is the case with co-orbiting objects. The gravitational radiation emitted by the solar system is far too small to measure. However, gravitational radiation has been indirectly observed as an energy loss over time in binary pulsar systems such as PSR 1913+16. It is believed that neutron star mergers and black hole formation may create detectable amounts of gravitational radiation. Gravitational radiation observatories such as LIGO have been created to study the problem. No confirmed detections have been made of this hypothetical radiation, but as the science behind LIGO is refined and as the instruments themselves are endowed with greater sensitivity over the next decade, this may change.

Notes

• Template:Fnb Proposition 75, Theorem 35: p.956 - I.Bernard Cohen and Anne Whitman, translators: Isaac Newton, The Principia: Mathematical Principles of Natural Philosophy. Preceded by A Guide to Newton's Principia, by I. Bernard Cohen. University of California Press 1999 ISBN 0-520-08816-6 ISBN 0-520-08817-4
• Template:Fnb Max Born (1924), Einstein's Theory of Relativity (The 1962 Dover edition, page 348 lists a table documenting the observed and calculated values for the precession of the perihelion of Mercury, Venus, and Earth.)

References

1. Dick Teresi (2002), Lost Discoveries: The Ancient Roots of Modern Science - from the Babylonians to the Maya, Simon & Schuster, New York, ISBN 0-684-83718-8:

"Two hundred years before Pythagoras, philosophers in northern India had understood that gravitation held the solar system together, and that therefore the sun, the most massive object, had to be at its centre."

2. Brahmagupta (628 AD). Brahmasphuta Siddhanta ("The Opening of the Universe").
3. Al-Biruni (1030). Ta'rikh al-Hind (Indica).
4. K. A. Waheed (1978). Islam and The Origins of Modern Science, p. 27. Islamic Publication Ltd., Lahore.
5. Robert Briffault (1938). The Making of Humanity, p. 191.
6. Dr. Nader El-Bizri, "Ibn al-Haytham or Alhazen", in Josef W. Meri (2006), Medieval Islamic Civilization: An Encyclopaedia, Vol. II, p. 343-345, Routledge, New York, London.
7. Donald Routledge Hill (1993), Islamic Science and Engineering, p. 61, Edinburgh University Press. (cf. Salah Zaimeche PhD (2005), Merv, p. 5, Foundation for Science Technology and Civilization.)
8. Professor Mohammed Abattouy (2002). "The Arabic Science of weights: A Report on an Ongoing Research Project", The Bulletin of the Royal Institute for Inter-Faith Studies 4, p. 109-130.
9. N. Khanikoff, ed. and trans. (1858-1860), "Analysis and Extracts of ... Book of the Balance of Wisdom, An Arabic Work on the Water-Balance, Written by 'Al-Khâzinî in the Twelfth Century", chap. 5, sect. 3.1, Journal of the American Oriental Society 6, p. 36.
10. Randall, Lisa (2005). Warped Passages: Unraveling the Universe's Hidden Dimensions. Ecco. ISBN 0-06-053108-8.
11. Feynman, R. P. (1995). Feynman lectures on gravitation. Addison-Wesley. ISBN 0201627345. Unknown parameter |coauthors= ignored (help)
12. Zee, A. (2003). Quantum Field Theory in a Nutshell. Princeton University Press. ISBN 0-691-01019-6.
13. Randall, Lisa (2005). Warped Passages: Unraveling the Universe's Hidden Dimensions. Ecco. ISBN 0-06-053108-8.
14. Greene, Brian (2000). The elegant universe: superstrings, hidden dimensions, and the quest for the ultimate theory. New York: Vintage Books. ISBN 0375708111.
• Halliday, David (2001). Physics v. 1. New York: John Wiley & Sons. ISBN 0-471-32057-9. Unknown parameter |coauthors= ignored (help)
• Serway, Raymond A. (2004). Physics for Scientists and Engineers (6th ed. ed.). Brooks/Cole. ISBN 0-534-40842-7. Unknown parameter |coauthors= ignored (help)
• Tipler, Paul (2004). Physics for Scientists and Engineers: Mechanics, Oscillations and Waves, Thermodynamics (5th ed. ed.). W. H. Freeman. ISBN 0-7167-0809-4.