# Gravitational energy

Image depicting Earth's gravitational field. Objects accelerate towards the Earth, thus losing their gravitational energy and transforming it into kinetic energy.

Gravitational energy is the potential energy a physical object with mass has in relation to another massive object due to gravity. It is potential energy associated with the gravitational field. Gravitational energy is dependent on the masses of two bodies, their distance apart and the gravitational constant (G).[1]

In everyday cases (i.e. close to the Earth's surface), the gravitational field is considered to be constant. For such scenarios the Newtonian formula for potential energy can be reduced to:

${\displaystyle U=mgh}$

where ${\displaystyle U}$ is the gravitational potential energy, ${\displaystyle m}$ is the mass, ${\displaystyle g}$ is the gravitational field, and ${\displaystyle h}$ is the height.[1] This formula treats the potential energy as a positive quantity.

## 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 an external force to bring point mass ${\displaystyle m}$  from infinity to the final distance ${\displaystyle R}$  (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

${\displaystyle U=-G{\frac {mM}{R}}}$

## General relativity

A depiction of curved geodesics ("world lines"). According to general relativity, mass distorts spacetime and gravity is a natural consequence of Newton's First Law.

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] that allows retention for the energy-momentum conservation laws of classical mechanics. Addition of the matter stress–energy–momentum tensor to the Landau–Lifshitz pseudotensor results in a combined matter plus gravitational energy pseudotensor that has a vanishing 4-divergence in all frames - ensuring 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.

## References

1. ^ a b "Gravitational Potential Energy". hyperphysics.phy-astr.gsu.edu. Retrieved 10 January 2017.
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. ^ Tsokos, K. A. (2010). Physics for the IB Diploma Full Colour (revised ed.). Cambridge University Press. p. 143. ISBN 978-0-521-13821-5. Extract of page 143
4. ^ Lev Davidovich Landau & Evgeny Mikhailovich Lifshitz, The Classical Theory of Fields, (1951), Pergamon Press, ISBN 7-5062-4256-7