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Gravitational potential energy
The positive gravitational attraction between two massive objects is accompanied by a negative amount of gravitational potential energy in the field which attracts them. As the distance between them approaches infinity, the gravitational attraction approaches zero from the positive side of the real number line and the gravitational potential energy approaches zero from the negative side. Therefore, as two massive objects move towards each other, the motion accelerates under gravity causing an increase in the (positive) kinetic energy of the system and an increase of the same amount in the (negative) gravitational potential energy. This is because the law of conservation of energy requires that the net energy of the system will not change. Gravitational binding energy is a kind of potential energy.[clarification needed] 
A universe in which positive energy dominates will eventually collapse in a "Big Crunch", while an "open" universe in which negative energy dominates will either expand indefinitely or eventually disintegrate in a "big rip". In the zero-energy universe model ("flat" or "Euclidean"), the total amount of energy in the universe is exactly zero: its amount of positive energy in the form of matter is exactly cancelled out by its negative energy in the form of gravity. (It is unclear which of these models accurately describes the real universe.)
Quantum field effects
In quantum theory, the uncertainty principle allows the vacuum of space to be filled with virtual particle-antiparticle pairs which appear spontaneously and exist for only a short time before, typically, annihilating themselves again. Some of these virtual particles can have negative energy. Their behaviour plays a role in several important phenomena, as described below.
In the Casimir effect, two flat plates placed very close together restrict the wavelengths of quanta which can exist between them. This in turn restricts the types and hence number and density of virtual particle pairs which can form in the intervening vacuum and can result in a negative energy density. This causes an attractive force between the plates, which has been measured.
It is possible to arrange multiple beams of laser light such that destructive quantum interference suppresses the vacuum fluctuations. Such a squeezed vacuum state involves negative energy. The repetitive waveform of light leads to alternating regions of positive and negative energy.
According to the theory of the Dirac sea, developed by Paul Dirac in 1930, the vacuum of space is full of negative energy. This theory was developed to explain the anomaly of negative-energy quantum states predicted by the Dirac equation.
The Dirac sea theory correctly predicted the existence of antimatter two years prior to the discovery of the positron in 1932 by Carl Anderson. However, the Dirac sea theory treats antimatter as a hole where there is the absence of a particle rather than as a real particle. Quantum field theory (QFT), developed in the 1930s, deals with antimatter in a way that treats antimatter as made of real particles rather than the absence of particles, and treats a vacuum as being empty of particles rather than full of negative-energy particles like in the Dirac sea theory.
Quantum field theory has displaced the Dirac sea theory as a more popular explanation of these aspects of physics. Both the Dirac sea theory and quantum field theory are equivalent by means of a Bogoliubov transformation, so the Dirac sea can be viewed as an alternative formulation of quantum field theory, and is thus consistent with it.
Negative energy appears in the speculative theory of wormholes, where it is needed to keep the wormhole open. A wormhole directly connects two locations which may be separated arbitrarily far apart in both space and time, and in principle allows near-instantaneous travel between them.
A theoretical principle for a faster-than-light (FTL) warp drive for spaceships has been suggested, involving negative energy. The Alcubierre drive comprises a solution to Einstein's equations of general relativity, in which a bubble of spacetime is moved rapidly by expanding space behind it and shrinking space in front of it.
- Alan Guth The Inflationary Universe: The Quest for a New Theory of Cosmic Origins (1997), Random House, ISBN 0-224-04448-6 Appendix A: Gravitational Energy demonstrates the negativity of gravitational energy.
- Stephen Hawking; The Grand Design, 2010, Page 180.
- Everett, Allen; Roman, Thomas (2012). Time Travel and Warp Drives. University of Chicago Press. p. 167. ISBN 978-0-226-22498-5.
- Ford and Roman 2000
- Stephen Hawking; A Brief History of Time, Bantam 1988, Pages 105-107. ISBN 0-593-01518-5
- López de Recalde, Andrea (2017). The Standard Electro-Weak Theory, 2nd Edition. Morrisville, North Carolina: LuLu Press, Inc. p. 65. ISBN 978-1-365-65887-7.
- Lawrence H. Ford and Thomas A. Roman; "Negative energy, wormholes and warp drive", Scientific American January 2000, 282, Pages 46–53.