Gravitational energy or gravitational potential energy is the potential energy a massive object has in relation to another massive object due to gravity. It is the potential energy associated with the gravitational field, which is released (converted into kinetic energy) when the objects fall towards each other. Gravitational potential energy increases when two objects are brought further apart.
For two pairwise interacting point particles, the gravitational potential energy is given by
To get the total work done by an external force to bring point mass from infinity to the final distance (for example the radius of Earth) of the two mass points, the force is integrated with respect to displacement:
Because , the total work done on the object can be written as:
Gravitational Potential Energy
In the common situation where a much smaller mass is moving near the surface of a much larger object with mass , 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 from the center) to a height above the surface is
If is small, as it must be close to the surface where is constant, then this expression can be simplified using the binomial approximation
As the gravitational field is , this reduces to
Taking at the surface (instead of at infinity), the familiar expression for gravitational potential energy emerges:
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 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.
^For a demonstration of the negativity of gravitational energy, see Alan Guth, The Inflationary Universe: The Quest for a New Theory of Cosmic Origins (Random House, 1997), ISBN0-224-04448-6, Appendix A—Gravitational Energy.