Energy condition

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) T a b . However, the Einstein field equation is not very choosy about what kinds of states of matter or nongravitational 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 nongravitational 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 nongravitational fields which are well-established in physics, while being sufficiently strong to rule out many unphysical "solutions" of the Einstein field equation. (It does not hold for matter described by a super-field, i.e., the Dirac field.)

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 X → 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 nongravitational fields) measured by the observer from our family (at each event on his world line). Similarly, the vector field with components − T a b X b represents (after a projection) the momentum measured by our observers.

Second, given an arbitrary null vector field k → , 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 X → , 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 T = T m m 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 k → ,

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) C of the null vector field k → , we must have

Weak energy condition

The weak energy condition stipulates that for every timelike vector field X → , 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) Y → , the vector field − T a b Y b 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 future-pointing timelike vector field X → , the trace of the tidal tensor measured by the corresponding observers is always non-negative:

There are many matter configurations which violate the strong energy condition, at least from a mathematical perspective. It is not clear whether these violations are physically possible in a classical regime. For instance, a scalar field with a positive potential can violate this condition. Moreover, it is violated in any cosmological inflationary process.

Perfect fluids

Perfect fluids possess a matter tensor of form

where u → is the four-velocity of the matter particles and where h a b ≡ g a b + u a u b 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, ρ is the energy density and p 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 strong energy condition

While the intent of the energy conditions is to provide simple criteria which rule out many unphysical situations while admitting any physically reasonable situation, in fact, at least when one introduces an effective field modeling some quantum mechanical effects, some possible matter tensors which are known to be physically reasonable and even realistic because they have been experimentally verified to 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 ρ is the matter energy density, p is the matter pressure, and w is a constant. In the case where w = − 1 ⇒ ρ = c o n s t , which is known as the false vacuum, the strong energy condition is violated.