Energy condition

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) ${\displaystyle T^{ab}}$. 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.

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 ${\displaystyle {\vec {X}}}$ can be interpreted as defining the world lines of some family of (possibly noninertial) ideal observers. Then the scalar field

${\displaystyle \rho =T_{ab}X^{a}X^{b}}$

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 ${\displaystyle -{T^{a}}_{b}X^{b}}$ represents (after a projection) the momentum measured by our observers.

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

${\displaystyle \nu =T_{ab}k^{a}k^{b}}$

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

${\displaystyle {E[{\vec {X}}]^{m}}_{m}=R_{ab}X^{a}X^{b}}$

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

${\displaystyle {\frac {1}{8\pi }}{E[{\vec {X}}]^{m}}_{m}={\frac {1}{8\pi }}R_{ab}X^{a}X^{b}=\left(T_{ab}-{\frac {1}{2}}Tg_{ab}\right)X^{a}X^{b},}$

where ${\displaystyle T={T^{m}}_{m}}$ is the trace of the matter tensor.

There are several alternative energy conditions in common use:

Implications among some energy conditions, in the case of a perfect fluid.

Perfect fluids possess a matter tensor of form

${\displaystyle T^{ab}=\rho u^{a}u^{b}+ph^{ab},}$

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

${\displaystyle T^{{\hat {a}}{\hat {b}}}={\begin{bmatrix}\rho &0&0&0\\0&p&0&0\\0&0&p&0\\0&0&0&p\end{bmatrix}}.}$

Here, ${\displaystyle \rho }$ is the energy density and ${\displaystyle p}$ is the pressure.

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

• The null energy condition stipulates that ${\displaystyle \rho +p\geq 0.}$
• The weak energy condition stipulates that ${\displaystyle \rho \geq 0,\;\;\rho +p\geq 0.}$
• The dominant energy condition stipulates that ${\displaystyle \rho \geq |p|.}$
• The strong energy condition stipulates that ${\displaystyle \rho +p\geq 0,\;\;\rho +3p\geq 0.}$

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.

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

${\displaystyle \varepsilon ={\frac {-\pi ^{2}}{720}}{\frac {\hbar }{d^{4}}}}$

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.) 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

${\displaystyle p=w\rho ,}$

where ${\displaystyle \rho }$ is the matter energy density, ${\displaystyle p}$ is the matter pressure, and ${\displaystyle w}$ is a constant. Then the strong energy condition requires ${\displaystyle w\geq -1/3}$; but for the state known as a false vacuum, we have ${\displaystyle w=-1}$.[4]

• Congruence (general relativity)
• Exact solutions in general relativity
• Frame fields in general relativity
• Positive energy theorem

• Hawking, Stephen; Ellis, G. F. R. (1973). The Large Scale Structure of Space-Time. Cambridge: Cambridge University Press. ISBN 0-521-09906-4. The energy conditions are discussed in §4.3.
• Poisson, Eric (2004). A Relativist's Toolkit: The Mathematics of Black Hole Mechanics. Cambridge: Cambridge University Press. Bibcode:2004rtmb.book.....P. ISBN 0-521-83091-5. Various energy conditions (including all of those mentioned above) are discussed in Section 2.1.
• Carroll, Sean M. (2004). Spacetime and Geometry: An Introduction to General Relativity. San Francisco: Addison-Wesley. ISBN 0-8053-8732-3. Various energy conditions are discussed in Section 4.6.
• Wald, Robert M. (1984). General Relativity. Chicago: University of Chicago Press. ISBN 0-226-87033-2. Common energy conditions are discussed in Section 9.2.
• Ellis, G. F. R.; Maartens, R.; MacCallum, M.A.H. (2012). Relativistic Cosmology. Cambridge: Cambridge University Press. ISBN 978-0-521-38115-4. Violations of the strong energy condition is discussed in Section 6.1.