# 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 body 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 only one body is accelerating measurably, and its acceleration is constant (for example, dropping a ball on Earth). For such scenarios the Newtonian formula can – for the potential energy of the accelerating body with respect to the stationary – be reduced to:

${\displaystyle U=mgh}$

where ${\displaystyle U}$ is the gravitational potential energy, ${\displaystyle m}$ is the mass of the object accelerating, ${\displaystyle g}$ is the acceleration of the object, and ${\displaystyle h}$ is the distance between the bodies.[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.