### Weak energy condition

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In relativistic classical field theories of gravitation, particularly general relativity, an **energy condition** is one of various alternative conditions which can be applied to the matter content of the theory, when it is either not possible or desirable to specify this content explicitly. 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.

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.

## Contents

## 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*) $T^\{ab\}$. 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 $\backslash vec\{X\}$ can be interpreted as defining the world lines of some family of (possibly noninertial) ideal observers. Then the scalar field

- $\backslash rho\; =\; T\_\{ab\}\; \backslash ,\; X^a\; \backslash ,\; X^b$

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 $-\{T^a\}\_b\; \backslash ,\; X^b$ represents (after a projection) the momentum measured by our observers.

Second, given an arbitrary null vector field $\backslash vec\{k\}$, the scalar field

- $\backslash nu\; =\; T\_\{ab\}\; \backslash ,\; k^a\; \backslash ,\; k^b$

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 $\backslash vec\{X\}$, 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:

- $\{E[\backslash vec\{X\}]^m\}\_m\; =\; R\_\{ab\}\; \backslash ,\; X^a\; \backslash ,\; X^b$

This quantity plays a crucial role in Raychaudhuri's equation. Then from Einstein field equation we immediately obtain

- $\backslash frac\{1\}\{8\; \backslash pi\}\; \backslash ;\; \{E[\backslash vec\{X\}]^m\}\_m\; =\; \backslash frac\{1\}\{8\; \backslash pi\}\; R\_\{ab\}\; \backslash ,\; X^a\; \backslash ,\; X^b\; =\; \backslash left(\; T\_\{ab\}\; -\; \backslash frac\{1\}\{2\}\; \backslash ,\; T\; \backslash ,\; g\_\{ab\}\; \backslash right)\; \backslash ,\; X^a\; \backslash ,\; X^b,$

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* $\backslash vec\{k\}$,

- $\backslash rho\; =\; T\_\{ab\}\; \backslash ,\; k^a\; \backslash ,\; k^b\; \backslash ge\; 0.$

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 $\backslash vec\{k\}$, we must have

- $\backslash int\_C\; T\_\{ab\}\; \backslash ,\; k^a\; \backslash ,\; k^b\; \backslash ,\; d\backslash lambda\; \backslash ge\; 0.$

### Weak energy condition

The **weak energy condition** stipulates that for every *timelike vector field* $\backslash vec\{X\}$, the matter density observed by the corresponding observers is always non-negative:

- $\backslash rho\; =\; T\_\{ab\}\; \backslash ,\; X^a\; \backslash ,\; X^b\; \backslash ge\; 0.$

### 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) $\backslash vec\{Y\}$, the vector field $-\{T^a\}\_b\; \backslash ,\; Y^b$ 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* $\backslash vec\{X\}$, the trace of the tidal tensor measured by the corresponding observers is always non-negative:

- $\backslash left(\; T\_\{ab\}\; -\; \backslash frac\{1\}\{2\}\; \backslash ,\; T\; \backslash ,\; g\_\{ab\}\; \backslash right)\; \backslash ,\; X^a\; \backslash ,\; X^b\; \backslash ge\; 0$

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.^{[1]} However, it is clear that such a violation would violate the classical regime of general relativity, and one would be required to use an alternative theory of gravity. ^{[2]}

## Perfect fluids

Perfect fluids possess a matter tensor of form

- $T^\{ab\}\; =\; \backslash rho\; \backslash ,\; u^a\; \backslash ,\; u^b\; +\; p\; \backslash ,\; h^\{ab\},$

where $\backslash vec\{u\}$ is the four-velocity of the matter particles and where $h^\{ab\}\backslash equiv\; g^\{ab\}\; +\; u^\{a\}u^\{b\}$ 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

- $T^\{\backslash hat\{a\}\; \backslash hat\{b\}\}\; =\; \backslash left[\; \backslash begin\{matrix\}$

\rho& 0 & 0 & 0 \\ 0 & p & 0 & 0 \\ 0 & 0 & p & 0 \\ 0 & 0 & 0 & p \end{matrix} \right] . Here, $\backslash rho$ is the energy density and $p$ is the pressure.

The energy conditions can then be reformulated in terms of these eigenvalues:

- The weak energy condition stipulates that $\backslash rho\; \backslash ge\; 0,\; \backslash ;\; \backslash ;\; \backslash rho\; +\; p\; \backslash ge\; 0\; .$
- The null energy condition stipulates that $\backslash rho\; +\; p\; \backslash ge\; 0\; .$
- The strong energy condition stipulates that $\backslash rho\; +\; p\; \backslash ge\; 0,\; \backslash ;\; \backslash ;\; \backslash rho\; +\; 3\; p\; \backslash ge\; 0\; .$
- The dominant energy condition stipulates that $\backslash rho\; \backslash ge\; |p|\; .$

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

- $\backslash varepsilon\; =\; \backslash frac\{-\backslash pi^2\}\{720\}\; \backslash ,\; \backslash frac\{\backslash hbar\}\{d^4\}$

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.) Fortunately, 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

- $p\; =\; w\backslash rho$,

where $\backslash rho$ is the matter energy density, $p$ is the matter pressure, and $w$ is a constant. In the case where $w\; =\; -1\; \backslash Rightarrow\; \backslash rho\; =\; const$, which is known as the false vacuum, the strong energy condition is violated. However, this equation of state only becomes relevant above temperatures of at least $10^\{12\}$ Kelvin. This would only happen under highly extreme conditions when quantum effects become dominant, and where classical general relativity would not apply. The main reason for wanting to falsify the strong energy condition is to avoid the initial singularity in the universe. The aforementioned suggests that a non-singular universe model would not obey the laws of classical physics at all times in the past. ^{[3]}

## See also

- Congruence (general relativity)
- Exact solutions in general relativity
- Frame fields in general relativity

## References

**The energy conditions are discussed in ยง4.3.**

**Various energy conditions (including all of those mentioned above) are discussed in***Section 2.1*.

**Various energy conditions are discussed in***Section 4.6*.

**Common energy conditions are discussed in***Section 9.2*.

**Violations of the strong energy condition is discussed in***Section 6.1*.