Gravitational energyorgravitational potential energy is the potential energyamassive object has due to its position in a gravitational field. It is the mechanical work done by the gravitational force to bring the 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 the kinetic energies of the objects as they fall 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.
For two pairwise interacting point particles, the gravitational potential energy 
where 
is the displacement vector between the two particles and 
denotes the scalar product. Since the gravtitational force is always parallel to the axis joining the particles, this simplifies to:
where 
Close to the Earth's surface, the gravitational field is approximately constant, and the gravitational potential energy of an object reduces to
where is the object's mass, 
is the gravity of Earth, and 
is the height of the object's center of mass above a chosen reference level.[1]
Inclassical 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 magnitude of the force between a point mass,
To get the total work done by the gravitational force in bringing point mass 
Because 
In the common situation where a much smaller mass
If
is small, as it must be close to the surface where 
is constant, then this expression can be simplified using the binomial approximation
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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5118b8bff7cdd16760ce444539901d5f161d5144)
As the gravitational field is 
, this reduces to
Taking 
at the surface (instead of at infinity), the familiar expression for gravitational potential energy emerges:[5]
Ingeneral 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.[citation needed]