Palais–Smale compactness condition: Difference between revisions

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'''Gravitational energy''' is [[potential energy]] associated with the [[gravitational field]].  This phrase is found frequently in scientific writings about [[quasar]]s (quasi-stellar objects) and other [[active galaxies]].  Quasars generate and emit their energy from a very small region.  The emission of large amounts of power from a small region requires a power source far more efficient than the nuclear fusion that powers stars. The release of gravitational energy<ref name="Lambourne">{{Cite book |title=Relativity, Gravitation and Cosmology |author=Lambourne, Robert J. A. |url=http://books.google.com/books?id=GUySYQaDM1cC&pg=PA222&dq=the+release+of+gravitational+energy+by+matter&hl=en&sa=X&ei=JvirUKTBMobo8gTrxYHgDQ&ved=0CC0Q6AEwAA#v=onepage&q=the%20release%20of%20gravitational%20energy%20by%20matter&f=false |edition=Illustrated |year=2010 |publisher=Cambridge University Press |isbn=0521131383 |page=222 |accessdate=2012-11-20}}</ref> by matter falling towards a massive black hole is the only process known that can produce such high power continuously. Stellar explosions – [[supernova]]s and [[gamma-ray burst]]s – can do so, but only for a few weeks.<ref name="Lambourne" />
 
== Newtonian mechanics ==
 
According to [[classical mechanics]], between two or more [[mass]]es (or other forms of [[energy-momentum 4-vector|energy-momentum]]) a gravitational potential energy exists, from which the gravitational field [[energy density]] can be calculated. [[Conservation of energy]] requires that this gravitational field energy is always negative.<ref>[[Alan Guth]]'' The Inflationary Universe: The Quest for a New Theory of Cosmic Origins'' (1997), Random House , ISBN 0-224-04448-6 Appendix A: ''Gravitational Energy'' demonstrates the negativity of gravitational energy.</ref>
 
The gravitational energy density is:
:<math> u = \frac{ -|\mathbf{g}|^2}{8 \pi G} \, ,</math><ref>[http://www.grc.nasa.gov/WWW/k-12/Numbers/Math/Mathematical_Thinking/possible_scalar_terms.htm NASA site] Gravitational energy density by analogy with EM</ref>
 
where ''G'' is Newton's [[gravitational constant]], and '''g''' is the [[gravitational field]] vector.
 
== General relativity ==
 
{{Main|Mass in general relativity}}
 
In [[general relativity]] gravitational energy is extremely complex, and there is no single agreed upon definition of the concept. It is sometimes modeled via the [[Landau-Lifshitz pseudotensor]]<ref>[[Lev Davidovich Landau]] & [[Evgeny Mikhailovich Lifshitz]], ''The Classical Theory of Fields'', (1951), Pergamon Press, ISBN 7-5062-4256-7</ref> which allows the energy-momentum conservation laws of [[classical mechanics]] to be retained. Addition of the matter [[stress-energy-momentum tensor]] to the Landau-Lifshitz pseudotensor results in a combined matter plus gravitational energy pseudotensor which has a vanishing 4-[[divergence]] in all frames; the vanishing divergence ensures the conservation law. Some people object to this derivation on the grounds that [[pseudotensor]]s are inappropriate in general relativity, but the divergence of the combined matter plus gravitational energy pseudotensor is a [[tensor]].
 
== See also ==
 
* [[Gravitational binding energy]]
* [[Gravitational potential]]
* [[Standard gravitational parameter]]
* [[Gravitational wave]]
 
== References ==
 
{{Reflist}}
 
<!--Categories-->
[[Category:Forms of energy]]
[[Category:Gravitation]]
[[Category:Conservation laws]]
[[Category:Tensors in general relativity]]
 
 
{{classicalmechanics-stub}}
{{relativity-stub}}

Latest revision as of 20:20, 20 April 2013

Gravitational energy is potential energy associated with the gravitational field. This phrase is found frequently in scientific writings about quasars (quasi-stellar objects) and other active galaxies. Quasars generate and emit their energy from a very small region. The emission of large amounts of power from a small region requires a power source far more efficient than the nuclear fusion that powers stars. The release of gravitational energy[1] by matter falling towards a massive black hole is the only process known that can produce such high power continuously. Stellar explosions – supernovas and gamma-ray bursts – can do so, but only for a few weeks.[1]

Newtonian mechanics

According to classical mechanics, between two or more masses (or other forms of energy-momentum) a gravitational potential energy exists, from which the gravitational field energy density can be calculated. Conservation of energy requires that this gravitational field energy is always negative.[2]

The gravitational energy density is:

u=−|g|28πG,[3]

where G is Newton's gravitational constant, and g is the gravitational field vector.

General relativity

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In general relativity gravitational energy is extremely complex, and there is no single agreed upon definition of the concept. It is sometimes modeled via the Landau-Lifshitz pseudotensor[4] which allows the energy-momentum conservation laws of classical mechanics to be retained. Addition of the matter stress-energy-momentum tensor to the Landau-Lifshitz pseudotensor results in a combined matter plus gravitational energy pseudotensor which has a vanishing 4-divergence in all frames; the vanishing divergence ensures the conservation law. Some people object to this derivation on the grounds that pseudotensors are inappropriate in general relativity, but the divergence of the combined matter plus gravitational energy pseudotensor is a tensor.

See also

References

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  1. ↑ 1.0 1.1 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

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  2. ↑ Alan Guth The Inflationary Universe: The Quest for a New Theory of Cosmic Origins (1997), Random House , ISBN 0-224-04448-6 Appendix A: Gravitational Energy demonstrates the negativity of gravitational energy.
  3. ↑ NASA site Gravitational energy density by analogy with EM
  4. ↑ Lev Davidovich Landau & Evgeny Mikhailovich Lifshitz, The Classical Theory of Fields, (1951), Pergamon Press, ISBN 7-5062-4256-7
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