Gravitational energy

For two pairwise interacting point particles, the gravitational potential energy ${\displaystyle U}$ is given by

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

where ${\displaystyle M}$ and ${\displaystyle m}$ are the masses of the two particles, ${\displaystyle R}$ is the distance between them, and ${\displaystyle G}$ is the gravitational constant.[1]

Close to the Earth's surface, the gravitational field is approximately constant, and the gravitational potential energy of an object reduces to

$${\displaystyle U=mgh}$$

where ${\displaystyle m}$ is the object's mass, ${\textstyle g={GM_{\oplus }}/{R_{\oplus }^{2}}}$ is the gravity of Earth, and ${\displaystyle h}$ is the height of the object's center of mass above a chosen reference level.[1]

In classical mechanics, two or more masses always have a gravitational potential. Conservation of energy requires that this gravitational field energy is always negative, so that it is zero when the objects are infinitely far apart.[2] The gravitational potential energy is the potential energy an object has because it is within a gravitational field.

The force between a point mass, ${\displaystyle M}$, and another point mass, ${\displaystyle m}$, is given by Newton's law of gravitation:[3]

$${\displaystyle F={\frac {GMm}{r^{2}}}}$$

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:

$${\displaystyle W=\int _{\infty }^{R}{\frac {GMm}{r^{2}}}dr=-\left.{\frac {GMm}{r}}\right|_{\infty }^{R}}$$

Because ${\textstyle \lim _{r\to \infty }{\frac {1}{r}}=0}$, the total work done on the object can be written as:[4]

In the common situation where a much smaller mass ${\displaystyle m}$ is moving near the surface of a much larger object with mass ${\displaystyle M}$, 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 ${\displaystyle R}$ from the center) to a height ${\displaystyle h}$ above the surface is

$${\displaystyle {\begin{aligned}\Delta U&={\frac {GMm}{R}}-{\frac {GMm}{R+h}}\\&={\frac {GMm}{R}}\left(1-{\frac {1}{1+h/R}}\right).\end{aligned}}}$$

If ${\displaystyle h/R}$ is small, as it must be close to the surface where ${\displaystyle g}$ is constant, then this expression can be simplified using the binomial approximation

$${\displaystyle {\frac {1}{1+h/R}}\approx 1-{\frac {h}{R}}}$$

to

$${\displaystyle {\begin{aligned}\Delta U&\approx {\frac {GMm}{R}}\left[1-\left(1-{\frac {h}{R}}\right)\right]\\\Delta U&\approx {\frac {GMmh}{R^{2}}}\\\Delta U&\approx m\left({\frac {GM}{R^{2}}}\right)h.\end{aligned}}}$$

As the gravitational field is ${\displaystyle g=GM/R^{2}}$, this reduces to

$${\displaystyle \Delta U\approx mgh.}$$

Taking ${\displaystyle U=0}$ at the surface (instead of at infinity), the familiar expression for gravitational potential energy emerges:[5]

$${\displaystyle U=mgh.}$$

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

In general relativity gravitational energy is extremely complex, and there is no single agreed upon definition of the concept. It is sometimes modelled via the Landau–Lifshitz pseudotensor[6] that allows retention for the energy–momentum conservation laws of classical mechanics. Addition of the matter stress–energy 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.

• Gravitational binding energy
• Gravitational potential
• Gravitational potential energy storage
• Positive energy theorem