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| Dominant energy condition | |
|---|---|
| Name | Dominant energy condition |
| Field | General relativity |
| Introduced | 1960s |
| Major contributors | Roger Penrose, Stephen Hawking, Robert Geroch, Hermann Bondi, Richard Feynman |
| Related concepts | Weak energy condition, Null energy condition, Strong energy condition, Energy–momentum tensor |
Dominant energy condition The dominant energy condition (DEC) is an energy condition used in general relativity and mathematical physics to constrain the allowed forms of the energy–momentum tensor in spacetime models. It is employed in proofs in causal structure and singularity theorems and interacts with results attributed to Roger Penrose, Stephen Hawking, Robert Geroch, Gerard 't Hooft, and Jacob Bekenstein. The DEC links notions of energy density measured by observers associated with timelike vectors and the causal propagation of energy and momentum, and it appears in discussions involving the Einstein field equations, black hole thermodynamics, and cosmological models related to Friedmann–Lemaître–Robertson–Walker metric.
The DEC is defined as a condition on the energy–momentum tensor T_{ab} introduced in the context of classical Einstein field equations research by contributors including Hermann Bondi and formalized in work by Roger Penrose and Robert Geroch. It requires that for any future-directed timelike vector field associated with observers familiar from Kepler problem analogies or references in the literature of Richard Feynman and John Wheeler, the vector -T^a{}_b v^b is future-directed causal. This requirement was used in developments by Stephen Hawking and in classical treatments by John Archibald Wheeler collaborators such as Kip Thorne.
Physically, the DEC asserts that local energy density is nonnegative for every timelike observer affiliated with frames studied by Arthur Eddington and that energy flux does not propagate faster than light as in analyses by Lev Landau and Evgeny Lifshitz. It is invoked in arguments about energy conditions in connection with the Hawking area theorem of Stephen Hawking and the cosmic censorship conjectures discussed by Roger Penrose and Robert Wald. The DEC ensures compatibility with causal propagation principles also considered in investigations by John von Neumann and Paul Dirac into relativistic field behavior.
Mathematically, the DEC can be expressed using tensor notation developed in textbooks by Misner, Thorne and Wheeler and formal treatments by Wald, Robert M.: for every future-directed timelike vector v^a, the vector -T^a{}_b v^b is future-directed timelike or null. Equivalently, the scalar T_{ab} v^a v^b is nonnegative and T_{ab} v^b is causal for each v^a, an approach that appears in proofs by Robert Geroch and Gary Gibbons. This formulation is employed in theorems by Stephen Hawking and Roger Penrose and in examinations by Hawking and Ellis.
Classic examples satisfying the DEC include the perfect fluid models used by Alexander Friedmann and Georges Lemaître in Friedmann cosmology when pressure obeys certain inequalities, and electromagnetic fields as treated by James Clerk Maxwell and relativistic extensions in work influenced by Oliver Heaviside. The stress–energy of dust, radiation under standard equations of state considered by Albert Einstein collaborators, and many classical scalar field configurations used by Andre Linde and Alan Guth in inflationary models meet or fail the DEC depending on parameters. Counterexamples include exotic matter models introduced in wormhole studies by Morris and Thorne (1988), Casimir effect scenarios analyzed by Hendrik Casimir and quantum corrections explored by Gerard 't Hooft and Stephen Fulling, and certain scalar fields in models discussed by Robert Brandenberger and Viatcheslav Mukhanov.
When the DEC holds, it implies the weak energy condition and constrains the causal character of energy flow, contributing to results like the area nondecrease theorem associated with Stephen Hawking and the singularity theorems by Roger Penrose and Stephen Hawking. It is related to, but logically distinct from, the null energy condition used in analyses by Alan Guth and the strong energy condition appearing in classical cosmology by Georges Lemaître and Alexander Friedmann. The DEC plays a role in the proof strategies of uniqueness theorems for black holes developed by Stephen Hawking, Jerzy Lewandowski, and Piotr Chrusciel.
Quantum field theory in curved spacetime, as developed by Stephen Fulling, Paul Davies, and Nicolas Birrell, exhibits violations of the DEC in contexts such as the Casimir effect, Hawking radiation around Schwarzschild black hole solutions studied by Stephen Hawking, and squeezed vacuum states referenced in work by Roy Glauber. Semiclassical analyses by James Hartle, Gary Gibbons, and Jerzy Lewandowski show that quantum inequalities and averaged energy conditions proposed by Lawrence Ford and Thomas Roman can partially replace the DEC in some proofs. Violations are central to theoretical constructs involving traversable wormholes by Morris and Thorne and warp-drive metrics popularized by Miguel Alcubierre.
The DEC is applied in proofs of classical results in black hole physics attributed to Stephen Hawking and in global structure theorems refined by Robert Geroch and Roger Penrose. It is used in constructing physically reasonable models of cosmology explored in the literature by Alan Guth, Andrei Linde, Sean Carroll, and Vera Rubin-adjacent observational studies. The condition informs modeling of compact objects in work by Subrahmanyan Chandrasekhar, Kip Thorne, and John Wheeler and guides constraints in numerical relativity codes developed by teams at Max Planck Institute for Gravitational Physics, Caltech, and MIT.