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((Short description|Simplifying assumptions about the behavior of the stress–energy tensor in general relativity))
((Multiple issues|
((original research|date=October 2012))
((more footnotes needed|date=October 2012))
))

In [[theory of relativity|relativistic]] [[classical field theory|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 ((Lang|la|per se)), but are rather mathematically imposed boundary conditions that attempt to capture a belief that "energy should be positive".<ref name="ARX-2014">((cite news |last=Curiel |first=E. |title=A Primer on Energy Conditions |url=https://archive.org/details/arxiv-1405.0403 |arxiv=1405.0403 |year=2014))</ref> Many energy conditions are known to not correspond to [[Reality|physical reality]]—for example, the observable effects of [[dark energy]] are well known to violate the strong energy condition.<ref name="ARX-2018">((cite journal |last=Farnes |first=J.S. |title=A Unifying Theory of Dark Energy and Dark Matter: Negative Masses and Matter Creation within a Modified ΛCDM Framework |journal=Astronomy & Astrophysics |volume=620 |pages=A92 |arxiv=1712.07962 |year=2018 |doi=10.1051/0004-6361/201832898 |bibcode=2018A&A...620A..92F |s2cid=53600834 ))</ref><ref name=Visser>((Cite book |arxiv = gr-qc/0001099|doi = 10.1142/9789812792129_0014|chapter = Energy Conditions and Their Cosmological Implications|title = Cosmo-99|pages = 98–112|year = 2000|last1 = Visser|first1 = Matt|last2 = Barceló|first2 = Carlos|isbn = 978-981-02-4456-9|s2cid = 119446302))</ref>

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 [[black hole thermodynamics|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'') <math>T^{ab}</math>. 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 [[eigenvalue]]s and [[eigenvector]]s of the matter tensor. A more subtle but no less important feature is that they are imposed ''eventwise'', at the level of [[tangent space]]s. Therefore, they have no hope of ruling out objectionable [[global spacetime structure|global features]], such as [[closed timelike curve]]s.

==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 vector|timelike]] or [[null vector]]s and the matter tensor.

First, a unit timelike vector field <math>\vec{X}</math> can be [[Congruence (general relativity)|interpreted]] as defining the world lines of some family of (possibly noninertial) ideal observers. Then the [[scalar field]]

:<math> \rho = T_{ab}   X^a X^b </math>

can be interpreted as the total [[mass–energy equivalence|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 <math>-{T^a}_b X^b</math> represents (after a projection) the [[momentum]] measured by our observers.

Second, given an arbitrary null vector field <math>\vec{k},</math> the scalar field

:<math> \nu = T_{ab}   k^a k^b </math>

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 <math>\vec{X}</math>, again interpreted as describing the motion of a family of ideal observers, the ''Raychaudhuri scalar'' is the scalar field obtained by taking the [[trace (linear algebra)|trace]] of the [[tidal tensor]] corresponding to those observers at each event:

:<math> {E[\vec{X}]^m}_m = R_{ab}   X^a X^b </math>

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

:<math> \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}   T g_{ab} \right)   X^a X^b,</math>

where <math>T = {T^m}_m</math> is the trace of the matter tensor.

==Mathematical statement==<!-- This section is linked from [[Kip Thorne]] -->
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'' <math>\vec{k}</math>,

:<math>\nu = T_{ab} k^a k^b \ge 0.</math>

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) <math>C</math> of the null vector field <math>\vec{k},</math> we must have

:<math> \int_C T_{ab}   k^a k^b d\lambda \ge 0.</math>

===Weak energy condition===
The '''weak energy condition''' stipulates that for every ''timelike vector field'' <math>\vec{X},</math> the matter density observed by the corresponding observers is always non-negative:

:<math>\rho = T_{ab} X^a X^b \ge 0.</math>

===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) <math>\vec{Y},</math> the vector field <math>-{T^a}_b Y^b</math> 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'' <math>\vec{X}</math>, the trace of the tidal tensor measured by the corresponding observers is always non-negative:

:<math>\left( T_{ab} - \frac{1}{2} T g_{ab} \right)   X^a X^b \ge 0</math>

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).<ref name=Visser/>

==Perfect fluids==
[[File:EnergyConditions.svg|200px|thumb|right|Implications among some energy conditions, in the case of a perfect fluid]]

[[Fluid solution|Perfect fluids]] possess a matter tensor of form

:<math> T^{ab} = \rho u^a u^b + p h^{ab},</math>

where <math>\vec{u}</math> is the [[four-velocity]] of the matter particles and where <math>h^{ab}\equiv g^{ab} + u^{a}u^{b}</math> 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 fields in general relativity|frame]] aligned with the motion of the matter particles, the components of the matter tensor take the diagonal form

:<math> 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}.</math>

Here, <math>\rho</math> is the energy [[density]] and <math>p</math> is the [[pressure]].

The energy conditions can then be reformulated in terms of these eigenvalues:
* The null energy condition stipulates that <math>\rho + p \ge 0.</math>
* The weak energy condition stipulates that <math>\rho \ge 0, \; \; \rho + p \ge 0.</math>
* The dominant energy condition stipulates that <math>\rho \ge |p|.</math>
* The strong energy condition stipulates that <math>\rho + p \ge 0, \; \; \rho + 3 p \ge 0.</math>

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

:<math> \varepsilon = \frac{-\pi^2}{720}   \frac{\hbar}{d^4} </math>

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

:<math>p = w\rho,</math>

where <math> \rho </math> is the matter energy density, <math>p</math> is the matter pressure, and <math>w</math> is a constant. Then the strong energy condition requires <math>w \ge -1/3</math>; but for the state known as a false vacuum, we have <math>w = -1</math>.<ref>((Cite book|title=Relativistic Cosmology|author1= G.F.R. Ellis |author2=R. Maartens |author3=M.A.H. MacCallum|chapter= Section 6.1|publisher= Cambridge University Press|year= 2012))</ref>

==See also ==
* [[Congruence (general relativity)]]
* [[Exact solutions in general relativity]]
* [[Frame fields in general relativity]]
* [[Positive energy theorem]]
* [[Quantum inequalities]]

==Notes==
((Reflist))

==References==
*((cite book|title-link=The Large Scale Structure of Space-Time|author1=Hawking, Stephen|author2=Ellis, G. F. R.|title=The Large Scale Structure of Space-Time|location=Cambridge|publisher=[[Cambridge University Press]]|year=1973|isbn=0-521-09906-4)) The energy conditions are discussed in §4.3.
*((cite book | author=Poisson, Eric | title=A Relativist's Toolkit: The Mathematics of Black Hole Mechanics | location=Cambridge | publisher=Cambridge University Press | year=2004 | isbn=0-521-83091-5| bibcode=2004rtmb.book.....P )) Various energy conditions (including all of those mentioned above) are discussed in ''Section 2.1''.
*((cite book | author=Carroll, Sean M. | title=Spacetime and Geometry: An Introduction to General Relativity | location=San Francisco | publisher=[[Addison-Wesley]] | year = 2004 | isbn=0-8053-8732-3)) Various energy conditions are discussed in ''Section 4.6''.
*((cite book|title-link=General Relativity (book)|author=Wald, Robert M.|title=General Relativity|location=Chicago|publisher=[[University of Chicago Press]]|year=1984|isbn=0-226-87033-2)) Common energy conditions are discussed in ''Section 9.2''.
*((cite book |author1=Ellis, G. F. R. |author2=Maartens, R.  |author3=MacCallum, M.A.H. | title=Relativistic Cosmology| location=Cambridge | publisher=Cambridge University Press | year = 2012 | isbn=978-0-521-38115-4)) Violations of the strong energy condition is discussed in ''Section 6.1''.

[[Category:Mathematical methods in general relativity]]

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