# Law of gravity: Wikis

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# Encyclopedia

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Gravitation keeps the planets in orbit around the Sun. (Not to scale)

Gravitation, or gravity, is one of the four fundamental interactions of nature, along with strong interaction, electromagnetic force and weak interaction. It is the means by which objects with mass attract one another.[1] In everyday life, gravitation is most familiar as the agent that lends weight to objects with mass and causes them to fall to the ground when dropped. Gravitation causes dispersed matter to coalesce, thus accounting for the existence of the Earth, the Sun, and most of the macroscopic objects in the universe. 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 fluid flow occurs under the influence of a density gradient and gravity; for heating the interiors of forming stars and planets to very high temperatures; and for various other phenomena observed on Earth.

Modern physics describes gravitation using the general theory of relativity, in which gravitation is a consequence of the curvature of spacetime which governs the motion of inertial objects. The simpler Newton's law of universal gravitation provides an accurate approximation for most calculations.

## History of gravitational theory

### Scientific revolution

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 apocryphal)[2] 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.[3] 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.”[4]

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 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 General Theory of Relativity, which accounted for the small 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.

### Equivalence principle

The equivalence principle, explored by a succession of researchers including Galileo, Loránd Eötvös, and Einstein, expresses the idea that all objects fall in the same way. The simplest way to test the weak equivalence principle is to drop two objects of different masses or compositions in a vacuum, and see if they hit the ground at the same time. These experiments demonstrate that all objects fall at the same rate with negligible friction (including air resistance). More sophisticated tests use a torsion balance of a type invented by Loránd Eötvös. Satellite experiments are planned for more accurate experiments in space.[5]

Formulations of the equivalence principle include:

• The weak equivalence principle: The trajectory of a point mass in a gravitational field depends only on its initial position and velocity, and is independent of its composition.[6]
• The Einsteinian equivalence principle: The outcome of any local non-gravitational experiment in a freely falling laboratory is independent of the velocity of the laboratory and its location in spacetime.[7]
• The strong equivalence principle requiring both of the above.

The equivalence principle can be used to make physical deductions about the gravitational constant, the geometrical nature of gravity, the possibility of a fifth force, and the validity of concepts such as general relativity and Brans-Dicke theory.

### General relativity

General relativity
$G_{\mu \nu} + \Lambda g_{\mu \nu}= {8\pi G\over c^4} T_{\mu \nu}$
Einstein field equations
Introduction
Mathematical formulation
Resources

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, and describes free-falling inertial objects as being accelerated relative to non-inertial observers on the ground.[8][9] In Newtonian physics, however, no such acceleration can occur unless at least one of the objects is being operated on by a force.

Einstein proposed that spacetime is curved by matter, and that free-falling objects are moving along locally straight paths in curved spacetime. These straight lines are called geodesics. Like Newton's First Law, Einstein's theory stated that if there is a force applied to an object, it would deviate from the geodesics in spacetime.[10] For example, we are no longer following the geodesics while standing because the mechanical resistance of the Earth exerts an upward force on us. Thus, we are non-inertial on the ground. This explains why moving along the geodesics in spacetime is considered inertial.

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:

The tests of general relativity included:[11]

‹The template Fn is being considered for deletion.›  2

• 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 Stanley Eddington in 1919.[12][13] The Newtonian corpuscular theory also predicted a lesser deflection of light, but Eddington found that the results of the expedition confirmed the predictions of general relativity over those of the Newtonian theory. However this interpretation of the results was later disputed.[14] More recent tests using radio interferometric measurements of quasars passing behind the Sun have more accurately and consistently confirmed the deflection of light to the degree predicted by general relativity.[15] See also gravitational lens.
• The time delay of light passing close to a massive object was first identified by Irwin I. Shapiro in 1964 in interplanetary spacecraft signals.
• Gravitational radiation has been indirectly confirmed through studies of binary pulsars.
• Alexander Friedmann in 1922 found that Einstein equations have non-stationary solutions (even in the presence of the cosmological constant). In 1927 Georges Lemaître showed that static solutions of the Einstein equations, which are possible in the presence of the cosmological constant, are unstable, and therefore the static universe envisioned by Einstein could not exist. Later, in 1931, Einstein himself agreed with the results of Friedmann and Lemaître. Thus general relativity predicted that the Universe had to be non-static—it had to either expand or contract. The expansion of the universe discovered by Edwin Hubble in 1929 confirmed this prediction.[16]

### Gravity and quantum mechanics

Several decades after the discovery of general relativity it was realized that general relativity is incompatible with quantum mechanics.[17] It is possible to describe gravity in the framework of quantum field theory like the other fundamental forces, such that 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.[18][19] This reproduces general relativity in the classical limit. However, this approach fails at short distances of the order of the Planck length,[17] 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,[20] or more currently M-theory, and, on the other hand, it may be a background independent theory such as loop quantum gravity or causal dynamical triangulation.

## 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 expressed below as the standard average.

g = 9.81 m/s2 = 32.2 ft/s2

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 (32.2 ft/s) after one second, 19.6 m/s (64.4 ft/s) after two seconds, and so on, adding 9.81 m/s (32.2 ft/s) to each resulting velocity. Also, again ignoring air resistance, any and all objects, when dropped from the same height, will hit the ground at the same time.

According to Newton's 3rd Law, the Earth itself experiences an equal [in force] and opposite [in direction] force to that acting on the falling object, meaning that the Earth also accelerates towards the object (until the object hits the earth, then the Law of Conservation of Energy states that it will move back with the same acceleration with which it initially moved forward, canceling out the two forces of gravity.). 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 near the surface of the Earth

Ball falling freely under gravity. See text for description.

Under an assumption of constant gravity, Newton's law of universal 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 $h = \tfrac{v^2}{2g}$ for the maximum height reached by a vertically projected body with velocity v is useful for small heights and small initial velocities only.

### 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 B1913+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.

## Anomalies and discrepancies

There are some observations that are not adequately accounted for, which may point to the need for better theories of gravity or perhaps be explained in other ways.

• Pioneer anomaly: The two Pioneer spacecraft seem to be slowing down in a way which has yet to be explained.[21]
• Flyby anomaly: Various spacecraft have experienced greater accelerations during slingshot maneuvers than expected.
• Accelerating expansion: The metric expansion of space seems to be speeding up. Dark energy has been proposed to explain this. A recent alternative explanation is that the geometry of space is not homogeneous (due to clusters of galaxies) and that when the data are reinterpreted to take this into account, the expansion is not speeding up after all[22], however this conclusion is disputed[23].
• Anomalous increase of the AU: Recent measurements indicate that planetary orbits are widening faster than if this was solely through the sun losing mass by radiating energy.
• Extra energetic photons: Photons travelling through galaxy clusters should gain energy and then lose it again on the way out. The accelerating expansion of the universe should stop the photons returning all the energy, but even taking this into account photons from the cosmic microwave background radiation gain twice as much energy as expected. This may indicate that gravity falls off faster than inverse-squared at certain distance scales[24].
• Dark flow: Surveys of galaxy motions have detected a mystery dark flow towards an unseen mass. Such a large mass is too large to have accumulated since the Big Bang using current models and may indicate that gravity falls off slower than inverse-squared at certain distance scales[24].
• Extra massive hydrogen clouds: The spectral lines of the Lyman-alpha forest suggest that hydrogen clouds are more clumped together at certain scales than expected and, like dark flow, may indicate that gravity falls off slower than inverse-squared at certain distance scales[24].

## Notes

• ^ 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
• ^ 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.)

## Footnotes

1. ^ Does Gravity Travel at the Speed of Light?, UCR Mathematics. 1998. Retrieved 3 July 2008
2. ^ Ball, Phil (06 2005). "Tall Tales". Nature News. doi:10.1038/news050613-10.
3. ^ Galileo (1638), Two New Sciences, First Day Salviati speaks: "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."
4. ^ *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.
5. ^
6. ^ Paul S Wesson (2006). Five-dimensional Physics. World Scientific. p. 82. ISBN 9812566619.
7. ^ Haugen, Mark P.; C. Lämmerzahl (2001). Principles of Equivalence: Their Role in Gravitation Physics and Experiments that Test Them. Springer. ISBN 978-3-540-41236-6.
8. ^ http://www.black-holes.org/relativity6.html
9. ^ http://laser.phys.ualberta.ca/~egerton/genrel.htm
10. ^ Law of Geodesic Motion http://blog.sauliaus.info/temp/gravity.pdf
11. ^ Pauli, Wolfgang Ernst (1958). "Part IV. General Theory of Relativity". Theory of Relativity. Courier Dover Publications. ISBN 9780486641522.
12. ^ Dyson, F.W.; Eddington, A.S.; Davidson, C.R. (1920). "A Determination of the Deflection of Light by the Sun's Gravitational Field, from Observations Made at the Total Eclipse of May 29, 1919". Phil. Trans. Roy. Soc. A 220: 291–333. doi:10.1098/rsta.1920.0009. . Quote, p. 332: "Thus the results of the expeditions to Sobral and Principe can leave little doubt that a deflection of light takes place in the neighbourhood of the sun and that it is of the amount demanded by Einstein's generalised theory of relativity, as attributable to the sun's gravitational field."
13. ^ Weinberg, Steven (1972). Gravitation and cosmology. John Wiley & Sons. . Quote, p. 192: "About a dozen stars in all were studied, and yielded values 1.98 ± 0.11" and 1.61 ± 0.31", in substantial agreement with Einstein's prediction θʘ = 1.75"."
14. ^ Earman, John; Glymour, Clark (1980). "Relativity and Eclipses: The British eclipse expeditions of 1919 and their predecessors". Historical Studies in the Physical Sciences 11: 49–85.
15. ^ Weinberg, Steven (1972). Gravitation and cosmology. John Wiley & Sons. p. 194. .
16. ^ See W.Pauli, 1958, pp.219–220
17. ^ a b Randall, Lisa (2005). Warped Passages: Unraveling the Universe's Hidden Dimensions. Ecco. ISBN 0-06-053108-8.
18. ^ Feynman, R. P.; Morinigo, F. B., Wagner, W. G., & Hatfield, B. (1995). Feynman lectures on gravitation. Addison-Wesley. ISBN 0201627345.
19. ^ Zee, A. (2003). Quantum Field Theory in a Nutshell. Princeton University Press. ISBN 0-691-01019-6.
20. ^ Greene, Brian (2000). The elegant universe: superstrings, hidden dimensions, and the quest for the ultimate theory. New York: Vintage Books. ISBN 0375708111.
21. ^ Wanted: Einstein Jr, The Economist, 6th March 2008.
22. ^ Dark energy may just be a cosmic illusion, New Scientist, issue 2646, 7th March 2008.
23. ^ Swiss-cheese model of the cosmos is full of holes, New Scientist, issue 2678, 18th October 2008.
24. ^ a b c "Where Matter Fears to Tread", New Scientist issue 2669, 14 March 2009

## References

• Halliday, David; Robert Resnick; Kenneth S. Krane (2001). Physics v. 1. New York: John Wiley & Sons. ISBN 0-471-32057-9.
• Serway, Raymond A.; Jewett, John W. (2004). Physics for Scientists and Engineers (6th ed.). Brooks/Cole. ISBN 0-534-40842-7.
• Tipler, Paul (2004). Physics for Scientists and Engineers: Mechanics, Oscillations and Waves, Thermodynamics (5th ed.). W. H. Freeman. ISBN 0-7167-0809-4.

• Thorne, Kip S.; Misner, Charles W.; Wheeler, John Archibald (1973). Gravitation. W.H. Freeman. ISBN 0-7167-0344-0.

"Law of Gravity"
CSI: Crime Scene Investigation episode
Episode no. Season 7
Episode 15
Written by Richard Catalani and Carol Mendelsohn
Directed by Richard J. Lewis
Guest stars Liev Schreiber
Original airdate February 8, 2007
Season 7 episodes
List of CSI: Crime Scene Investigation episodes
For the natural phenomenon by which all objects attract each other, see gravitation.

"Law of Gravity" is the fifteenth episode of the seventh season of the television series CSI: Crime Scene Investigation.

## Plot

Before the start of his shift, Keppler eats dinner at a diner where he runs into an older man named Frank McCarty. Frank, who appears to know Keppler from New Jersey, asks Keppler for a favor, but Keppler quickly says no before Frank can explain.

The CSIs are called to a hotel room where a man and a woman have been shot and killed; the woman's left hand has been cut off. Brass informs them the man is Dennis Graves, a cop from Trenton, New Jersey, and that Frank found the bodies. Keppler denies knowing either Graves or McCarty, despite also being from Trenton.

Brass questions McCarty, who says Graves picked up a prostitute that night. Nick finds long blonde hairs and shoe impressions in the bathroom; neither matches either victim. The CSIs realize a witness may have survived.

During the autopsy, Doc Robbins removes butt implants from the dead woman. The serial numbers on the implants lead to Courtney Evans, and Warrick goes to her house to process it. Grissom, who has returned from his sabbatical, joins him at the house and asks Warrick how the lab was during his absence. Warrick tells Grissom about the friction in the lab caused by Keppler's use of reverse forensics. Nick learns Courtney's implants were paid for by a pimp named Ricky.

Keppler meets Frank at a bar. Frank tells Keppler he cut off Courtney's hand because she scratched him; he had to make sure the CSIs could not get his DNA from her nails. He also says he killed Graves because Graves was about to be indicted, and he would have told the grand jury about Frank going vigilante and killing a drug dealer. He thanks Keppler for killing the man who raped his daughter, Amy, but then threatens to expose him as the murderer if Keppler does not clear him of the Graves/Evans murder. Frank tells Keppler he still has the gun with the dead man's blood and Keppler's fingerprints on it. If Keppler were exposed, not only would he be arrested, but every case he ever worked would be tainted and the criminals set free.

Fingerprints at Courtney's house lead to a homeless man, who says he found her purse in a dumpster and went to her house to return it to her. Keppler calls Catherine, who tells him about the dumpster; Keppler says he is nearby and offers to process. At the dumpster, Keppler finds Courtney's missing hand. However, when Catherine asks, Keppler says he found nothing there.

Nick and Archie find elevator video footage of Frank and a blonde. Because Frank had previously denied any knowledge of a blonde woman, Nick begins to suspect Frank is a dirty cop. Nick, Keppler, and Sofia go to Frank's hotel room and find it has been cleaned out. The only thing in the room is a paper bag with a bloody gun in it. Nick is confused, but excited the killer has given them such damning evidence, but Keppler worries the gun is the gun he himself used to kill Amy's rapist and not the one Frank used to kill Graves.

Grissom and Sara reunite at the lab, but Sara backs away from Grissom, saying she has been processing a trash dump site and smells bad.

Archie finds a voice mail from Suzy, the blonde, on Courtney's cell phone. In it, Suzy tells her friend Frank kept calling her Amy. Keppler is stunned to hear this. He leaves the lab. Through flashbacks, he realizes Frank raped Amy and allowed him to kill an innocent man.

Nick runs the fingerprints from the gun and find they lead to Keppler. Nick takes this evidence to Grissom, Catherine, and Warrick. They discuss the possibility of him being set up, but also speculate he may be dirty. Grissom orders his team to treat Keppler like a suspect.

Archie and Greg track Keppler to a deserted field in Henderson using his car's GPS, but Keppler disables it. They begin to track his cell phone.

Brass calls Trenton PD and learns about the history between Frank and Keppler. He calls Catherine to tell her his suspicions that both Frank and Keppler, along with Graves, are dirty. Catherine and Sara learn Suzy works out of a motel called the Blue Siren.

Keppler finds Ricky and demands to know how to find Suzy. He then calls Frank and tells him to meet him at the Blue Siren. Frank comes to the motel and enters a room where Suzy is waiting. Frank points a gun at Suzy, but Keppler enters and points a gun at Frank. Keppler confronts Frank and accuses him of raping his own daughter. Suzy runs out of the room, but when Frank fires at her Keppler takes the bullet. Catherine then bursts into the room, where Frank proceeds to point his gun at her, but Keppler quickly fires at Frank six times, killing him. The police arrive and Keppler is rushed to an ambulance, where he dies. Catherine bursts into tears and is led away by Grissom.

Nick and Warrick head out to the deserted field in Henderson, where they find a cooler with the hand and Keppler's cell phone inside. Keppler had left a note telling the CSIs to check under the nails for Frank's DNA.

## Awards

Liev Schreiber submitted this episode for consideration in the category of "Outstanding Guest Actor in a Drama Series" on his behalf for the 2007 Emmy Awards. [1]

## References

1. "2007 Emmys CONFIRMED Episode Submissions". The Envelope Forum, Los Angeles Times. Retrieved on 2007-06-18.