Energy condition
In relativistic classical field theories of gravitation, particularly general relativity, an energy condition is a generalization of the statement "the energy density of a region of space cannot be negative" in a relativistically phrased mathematical formulation.
There are multiple possible alternative ways to express such a condition such that can be applied to the matter content of the theory.
This is both a strength, since a good general theory of gravitation should be maximally independent of any assumptions concerning non-gravitational physics, and a weakness, because without some further criterion the Einstein field equation admits putative solutions with properties most physicists regard as unphysical, i.e. too weird to resemble anything in the real universe even approximately.
Roughly speaking, they crudely describe properties common to all (or almost all) states of matter and all non-gravitational fields that are well-established in physics while being sufficiently strong to rule out many unphysical "solutions" of the Einstein field equation.
Mathematically speaking, the most apparent distinguishing feature of the energy conditions is that they are essentially restrictions on the eigenvalues and eigenvectors of the matter tensor.
A more subtle but no less important feature is that they are imposed eventwise, at the level of tangent spaces.
Therefore, they have no hope of ruling out objectionable global features, such as closed timelike curves.
In order to understand the statements of the various energy conditions, one must be familiar with the physical interpretation of some scalar and vector quantities constructed from arbitrary timelike or null vectors and the matter tensor.
can be interpreted as defining the world lines of some family of (possibly noninertial) ideal observers.
, again interpreted as describing the motion of a family of ideal observers, the Raychaudhuri scalar is the scalar field obtained by taking the trace of the tidal tensor corresponding to those observers at each event: This quantity plays a crucial role in Raychaudhuri's equation.
For example, the averaged null energy condition states that for every flowline (integral curve)
the matter density observed by the corresponding observers is always non-negative: The dominant energy condition stipulates that, in addition to the weak energy condition holding true, for every future-pointing causal vector field (either timelike or null)
The strong energy condition stipulates that for every timelike vector field
, the trace of the tidal tensor measured by the corresponding observers is always non-negative: There are many classical matter configurations which violate the strong energy condition, at least from a mathematical perspective.
Moreover, observations of dark energy/cosmological constant show that the strong energy condition fails to describe our universe, even when averaged across cosmological scales.
Furthermore, it is strongly violated in any cosmological inflationary process (even one not driven by a scalar field).
is the projection tensor onto the spatial hyperplane elements orthogonal to the four-velocity, at each event.
(Notice that these hyperplane elements will not form a spatial hyperslice unless the velocity is vorticity-free, that is, irrotational.)
Finally, there are proposals for extension of the energy conditions to spacetimes containing non-perfect fluids, where the second law of thermodynamics provides a natural Lyapunov function to probe both stability and causality, where the physical origin of the connection between stability and causality lies in the relationship between entropy and information.
[4] These attempts generalize the Hawking-Ellis vacuum conservation theorem (according to which, if energy can enter an empty region faster than the speed of light, then the dominant energy condition is violated, and the energy density may become negative in some reference frame[5]) to spacetimes containing out-of-equilibrium matter at finite temperature and chemical potential.
Indeed, the idea that there is a connection between causality violation and fluid instabilities has a long history.
For example, in the words of W. Israel: “If the source of an effect can be delayed, it should be possible for a system to borrow energy from its ground state, and this implies instability”.
[6] It is possible to show that this is a restatement of the Hawking-Ellis vacuum conservation theorem at finite temperature and chemical potential.
[4][5] While the intent of the energy conditions is to provide simple criteria that rule out many unphysical situations while admitting any physically reasonable situation, in fact, at least when one introduces an effective field modeling of some quantum mechanical effects, some possible matter tensors which are known to be physically reasonable and even realistic because they have been experimentally verified, actually fail various energy conditions.
Being negative for parallel plates, the vacuum energy is positive for a conducting sphere.)
However, various quantum inequalities suggest that a suitable averaged energy condition may be satisfied in such cases.
In particular, the averaged null energy condition is satisfied in the Casimir effect.
The strong energy condition is obeyed by all normal/Newtonian matter, but a false vacuum can violate it.
