Gravitational energy

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

In classical mechanics, two or more masses always have a gravitational potential. Conservation of energy requires that this gravitational field energy is always negative.^[2] The gravitational potential energy is the potential energy an object has because it is within a gravitational field.

The force one point mass ${\displaystyle M}$ exerts onto another point mass ${\displaystyle m}$ is given by Newton's law of gravitation: ${\displaystyle F=G{\frac {mM}{r^{2}}}}$

To get the total work done by the gravitational force from infinity to the final distance $R$ (for example the radius of Earth) of the two mass points, the force is integrated with respect to displacement:

${\displaystyle W=\int _{\infty }^{R}G{\frac {mM}{r^{2}}}dr=}$ ${\displaystyle -G\left.{mM \over r}\right\vert _{\infty }^{R}}$

Because ${\displaystyle \lim _{r\rightarrow \infty }{\frac {1}{r}}=0}$, the total work done on the object can be written as:^[3]

Gravitational Potential Energy ${\displaystyle E=-G{\frac {mM}{R}}}$

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^[4] which allows for 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

• Gravitational binding energy
• Gravitational potential
• Standard gravitational parameter
• Gravitational wave

References

1. Lambourne, Robert J. A. (2010). Relativity, Gravitation and Cosmology (Illustrated ed.). Cambridge University Press. p. 222. ISBN 0521131383. Retrieved 2012-11-20.
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. "Is this reasoning for the integral correct?". math.stackexchange.com. Retrieved 2016-10-20.
4. Lev Davidovich Landau & Evgeny Mikhailovich Lifshitz, The Classical Theory of Fields, (1951), Pergamon Press, ISBN 7-5062-4256-7