Gravitational energy

Gravitational energy is the gravitational potential energy a body with mass has in relation to another massive object. 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 cases where the acceleration is fairly constant, such as dropping a ball on Earth, the Newtonian formula – for the gravitational energy of one of the bodies relative to the other – can be reduced to:

E=mgh

where $E$ is the gravitational energy (in Joules), $m$ is the mass of the object accelerating (in kilograms), $g$ is the acceleration of the object (in meters per second per second) and $h$ is the distance between the bodies (in meters).^[1] Note that this formula gives the potential energy and treats it as a positive number.

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

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:

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

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

Gravitational Potential Energy

$E=-G{\frac {mM}{R}}$

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.

References

^ ^a ^b "Gravitational Potential Energy". hyperphysics.phy-astr.gsu.edu. Retrieved 10 January 2017.
^ 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.
^ 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
^ Lev Davidovich Landau & Evgeny Mikhailovich Lifshitz, The Classical Theory of Fields, (1951), Pergamon Press, ISBN 7-5062-4256-7

[gpe-1] "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

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Newtonian mechanics

General relativity

See also

References