nLab energy condition

Idea

The theory of classical gravity (general relativity) a priori and straightforwardly admits the coupling to gravity of (matter-)fields with any covariant local Lagrangian. The resulting Einstein equations will always exist and will equate the Einstein tensor with the given stress-energy tensor of the matter fields.

While that hence gives consistent theories in themselves – a set of differential equations –, not all choices of matter Lagrangians are physically reasonable. One consistency condition for physical reasonable choices is for instance that from far away any non-vanishing matter distribution should behave as a single ball of positive mass. An energy condition is an extra condition imposed on a stress-energy tensor that is meant to reflect physical plausibility conditions of this kind.

There are various conditions of various strengths considered in the literature. One is called the dominant energy condition and that indeed implies the above asymptotic positivity of mass distributions, this is the content of the positive energy theorem.

Related concepts

• phantom dark energy

References

Review with focus on the situation in quantum gravity:

• Jackson R. Fliss: Modave lectures on energy conditions in quantum field theory and semi-classical gravity [arXiv:2605.18964]

See also:

• Wikipedia, Energy condition

On the averaged null energy condition:

• Felipe Rosso, Global aspects of conformal symmetry and the ANEC in dS and AdS [arXiv:1912.08897]

  (for conformal field theory on de Sitter spacetime and anti de Sitter spacetime)

• Hidde Stoffels: Positive thinking: A search for energy conditions [arXiv:2411.11050]

On the quantum null energy condition:

• Taha A Malik, Rafael Lopez-Mobilia, Proof of the quantum null energy condition for free fermionic field theories [arXiv:1910.07594]

• Philipp Stanzer, Numerical Relativity, Holography and the Quantum Null Energy Condition [arXiv:2009.07040]

• Stefan Hollands, Roberto Longo:

  A New Proof of the QNEC [arXiv:2503.04651]

• Tanay Kibe, Pratik Roy: Quantum null energy condition in quenched 2d CFTs [arXiv:2503.17448]