In general relativity, any spacetime geometry is possible, as long as one has the necessary type and distribution of matter. However, this means that some geometries require exotic kinds of matter that do not exist in reality. To avoid this, one can impose some conditions on the energetic behavior of matter to force it to be physical. For example, one can require that energy is always positive and that all energy fluxes are causal. But which conditions do real matter, and in particular quantum matter, actually satisfy?
Classically, the energy conditions would state ideas such that at every point in spacetime any observer sees positive energy densities (weak energy condition), or in addition to that all observers only see causal fluxes of energy (dominant energy condition). These conditions are very interesting from a phenomenological point of view and also lead to a myriad of results in classical general relativity. For example, Hawking’s area theorem, the positive mass theorems of Schoen–Yau and Witten, among other results.
Unfortunately, quantum effects violate all of the classical energy conditions imposed on classical matter. One common example of this is the Casimir system. If one brings two parallel conducting plates together, a net force will pull them together. This force originates from the quantum fields around the plates, and in between the plates, the energy density in the quantum field is negative enough to violate all classical energy conditions.
Despite this, some ideas reminisce. Recently, particular attention has been devoted to the so-called averaged null energy condition (ANEC) and quantum null energy condition (QNEC). These conditions are believed to hold in quantum field theory and, hence, they specify what sorts of matter are physical within the realm of quantum mechanics. Essentially, they allow the existence of negative energy densities as long as they satisfy some remaining conditions. The ANEC, for example, demands that there is sufficient positive energy somewhere else to balance out the negative energy.
These ideas can be useful to generalize classical theorems of general relativity to scenarios in which quantum mechanics is relevant. Furthermore, the validity of energy conditions allows us to understand further why observed macroscopic masses are always positive, even though quantum mechanics allows negative energies.
The following references discuss energy conditions in general relativity and quantum field theory.