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. The hope is then that any reasonable matter theory will satisfy this condition or at least will preserve the condition if it is satisfied by the starting conditions. Energy conditions are not physical constraints , but are rather mathematically  imposed boundary conditions that attempt to capture a belief that "energy should be positive". Many energy conditions are known to not correspond to physical reality—for example, the observable effects of dark energy are well-known to violate the strong energy condition. In general relativity, energy conditions are often used (and required) in proofs of various important theorems about black holes, such as the no hair theorem or the laws of black hole thermodynamics. Motivation. In general relativity and allied theories, the distribution of the mass, momentum, and stress due to matter and to any non-gravitational fields is described  by the energy–momentum tensor (or "matter tensor") formula_1. However, the Einstein field equation in itself does not specify what kinds of states of matter or non-gravitational fields are admissible in a spacetime model. 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. The energy conditions represent such criteria. 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. Some observable quantities. 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. First, a unit timelike vector field formula_2 can be interpreted as defining the world lines of some family of (possibly noninertial) ideal observers. Then the scalar field can be interpreted as the total mass-energy density (matter plus field energy of any non-gravitational fields) measured by the observer from our family (at each event on his world line). Similarly , the vector field with components formula_4 represents (after a projection) the momentum measured by our observers. Second, given an arbitrary null vector field formula_5 the scalar field can be considered a kind of limiting case of the mass-energy density. Third, in the case of general relativity, given an arbitrary timelike vector field formula_2, 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. Then  from Einstein field equation we immediately obtain where formula_10 is the trace of the matter tensor. Mathematical statement. There are several alternative energy conditions in common use: Null energy condition. The null energy condition stipulates that for every future-pointing "null vector field" formula_11, Each of these has an "averaged" version, in which the properties noted above are to hold only "on average" along the flowlines of the appropriate vector fields. Otherwise, the Casimir effect leads to exceptions. For example, the averaged null energy condition states that for every flowline (integral curve) formula_13 of the null vector field formula_5 we  must have Weak energy condition. The weak energy condition stipulates that for every "timelike vector field" formula_16 the matter density observed by the corresponding observers is always non-negative: Dominant energy condition. 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) formula_18 the vector field formula_19 must be a future-pointing causal vector. That is, mass–energy can never be observed to be flowing faster than light. Strong energy condition. The strong energy condition stipulates that for every "timelike vector field" formula_2, 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. For instance, a scalar field with a positive potential can violate this condition. 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). Perfect fluids. Perfect fluids possess a matter tensor of form where formula_23 is  the four-velocity of the matter particles and where formula_24 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".) With respect to a frame aligned with the motion of the matter particles, the components of the matter tensor take the diagonal form Here, formula_26 is the energy density and formula_27 is the pressure. The energy conditions can then be reformulated in terms of these eigenvalues: The implications among these conditions are indicated  in the figure at right. Note that some of these conditions allow "negative" pressure. Also, note that despite the names the strong energy condition does not imply the weak energy condition "even in the context of perfect fluids". Attempts at falsifying the energy conditions. 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. In particular, in the Casimir effect, in the region between two conducting plates held parallel at a very small separation "d", there is a "negative" energy density between the plates. (Be mindful, though, that the Casimir effect is topological, in that the sign of the vacuum energy depends on both the geometry and topology of the configuration. 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. Indeed, for energy–momentum tensors arising from effective field theories on Minkowski spacetime, the averaged null energy condition holds for everyday quantum fields. Extending these results is an open problem. The strong energy condition is obeyed by all normal/Newtonian matter, but a false vacuum can violate it. Consider the linear barotropic equation state where formula_34 is the matter energy density, formula_27 is the matter pressure, and formula_36 is a constant. Then the strong energy condition requires formula_37; but for the state known as a false vacuum, we have formula_38