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Gravitational energy

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Type of potential energy

Image depictingEarth'sgravitational field. Objects accelerate towards the Earth, thus losing their gravitational energy and transforming it intokinetic energy.

Gravitational energy orgravitational potential energy is thepotential energy an object withmass has due to thegravitational potential of its position in agravitational field. Mathematically, it is the minimummechanical work that has to be done against the gravitational force to bring a mass from a chosen reference point (often an "infinite distance" from the mass generating the field) to some other point in the field, which is equal to the change in thekinetic energies of the objects as theyfall towards each other. Gravitational potential energy increases when two objects are brought further apart and is converted to kinetic energy as they are allowed to fall towards each other.

Formulation

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For two pairwise interacting point particles, the gravitational potential energy $U {\displaystyle U}$ is the work that an outside agent must do in order to quasi-statically bring the masses together (which is therefore, exactly opposite the work done by the gravitational field on the masses): $U=-W_{g}=-\int {\vec {F}}_{g}\cdot d{\vec {r}}$ where ${\textstyle d{\vec {r}}}$ is thedisplacement vector of the mass, ${\vec {F_{g}}}$ is gravitational force acting on it and ${\textstyle \cdot }$ denotesscalar product.

Newtonian mechanics

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Inclassical mechanics, two or moremasses always have agravitational potential.Conservation of energy requires that this gravitational field energy is alwaysnegative, so that it is zero when the objects are infinitely far apart.^[1] The gravitational potential energy is the potential energy an object has because it is within a gravitational field.

The magnitude & direction of gravitational force experienced by a point mass $m {\displaystyle m}$ , due to the presence of another point mass $M {\displaystyle M}$ at a distance $r {\displaystyle r}$ , is given byNewton's law of gravitation.^[2]Taking origin to be at the position of $M {\displaystyle M}$ , ${\vec {F_{g}}}=-{\frac {GMm}{r^{2}}}{\hat {r}}$ To get the total work done by the gravitational force in bringing point mass $m {\displaystyle m}$ from infinity to final distance $R {\displaystyle R}$ (for example, the radius of Earth) from point mass ${\textstyle M}$ , the force is integrated with respect to displacement:

$W_{g}=\int {\vec {F}}_{g}\cdot d{\vec {r}}=-\int _{\infty }^{R}{\frac {GMm}{r^{2}}}dr=\left.{\frac {GMm}{r}}\right|_{\infty }^{R}={\frac {GMm}{R}}$ Gravitational potential energy being the minimum (quasi-static) work that needs to be done against gravitational force in this procedure,

Gravitational Potential Energy

$U=-{\frac {GMm}{R}}$

Simplified version for Earth's surface

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In the common situation where a much smaller mass $m {\displaystyle m}$ is moving near the surface of a much larger object with mass $M {\displaystyle M}$ , the gravitational field is nearly constant and so the expression for gravitational energy can be considerably simplified. The change in potential energy moving from the surface (a distance $R {\displaystyle R}$ from the center) to a height $h {\displaystyle h}$ above the surface is ${\begin{aligned}\Delta U&={\frac {GMm}{R}}-{\frac {GMm}{R+h}}\\&={\frac {GMm}{R}}\left(1-{\frac {1}{1+h/R}}\right)\end{aligned}}$ If $h/R$ is small, as it must be close to the surface where $g {\displaystyle g}$ is constant, then this expression can be simplified using thebinomial approximation ${\frac {1}{1+h/R}}\approx 1-{\frac {h}{R}}$ to ${\begin{aligned}\Delta U&\approx {\frac {GMm}{R}}\left[1-\left(1-{\frac {h}{R}}\right)\right]\\\Delta U&\approx {\frac {GMmh}{R^{2}}}\\\Delta U&\approx m\left({\frac {GM}{R^{2}}}\right)h\end{aligned}}$ As the gravitational field is $g=GM/R^{2}$ , this reduces to $\Delta U\approx mgh$ Note, this is the change of energy in gaining some height $h {\displaystyle h}$ from the surface.

General relativity

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Main article:Mass in general relativity

A 2 dimensional depiction of curvedgeodesics ("world lines"). According togeneral relativity, mass distortsspacetime and gravity is a natural consequence of Newton's First Law. Mass tells spacetime how to bend, and spacetime tells mass how to move.

Ingeneral relativity gravitational energy is extremely complex, and there is no single agreed upon definition of the concept. It is sometimes modelled via theLandau–Lifshitz pseudotensor^[3] that allows retention for the energy–momentum conservation laws ofclassical mechanics. Addition of the matterstress–energy tensor to the Landau–Lifshitz pseudotensor results in a combined matter plus gravitational energy pseudotensor that has a vanishing4-divergence in all frames—ensuring the conservation law. Some people object to this derivation on the grounds thatpseudotensors are inappropriate in general relativity, but the divergence of the combined matter plus gravitational energy pseudotensor is atensor.^{[citation needed]}

References

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^For a demonstration of the negativity of gravitational energy, seeAlan Guth, The Inflationary Universe: The Quest for a New Theory of Cosmic Origins (Random House, 1997),ISBN 0-224-04448-6, Appendix A—Gravitational Energy.
^MacDougal, Douglas W. (2012).Newton's Gravity: An Introductory Guide to the Mechanics of the Universe (illustrated ed.). Springer Science & Business Media. p. 10.ISBN 978-1-4614-5444-1.Extract of page 10
^Lev Davidovich Landau &Evgeny Mikhailovich Lifshitz,The Classical Theory of Fields, (1951), Pergamon Press,ISBN 7-5062-4256-7

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