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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
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, 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 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.