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^{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.

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 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 or Null Vector s and the matter tensor.

First, an arbitrary timelike vector field

ec{X}

can be Interpreted as defining the world lines of some family of (possibly noninertial) ideal observers. Then the Scalar Field
:

\mu = T_{ab} \, X^a \, 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 \, X^b

represents (after a projection) the Momentum measured by our observers.

Second, given an arbitrary null vector field

ec{k}

, the scalar field
:

u = 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

ec{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

{Link without Title} ^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
:

rac{1}{8 \pi} \; {E

{Link without Title} ^m}_m = rac{1}{8 \pi} R_{ab} \, X^a \, X^b = \left( T_{ab} - rac{1}{2} \, T \, g_{ab} ight) \, X^a \, X^b
where

T = {T^m}_m

is the trace of the matter tensor.

MATHEMATICAL STATEMENT

There are currently approximately half a dozen alternative energy conditions in common use:

The weak energy condition stipulates that for every future-pointing ''timelike vector field'' $ec{X}$ , the matter density observed by the corresponding observers is always non-negative:

: $\mu = T_{ab} \, X^a \, X^b \ge 0$

The null energy condition stipulates that for every future-pointing ''null vector field'' $ec{k}$ ,

: $\mu = T_{ab} \, k^a \, k^b \ge 0$

The strong energy condition stipulates that for every future-pointing ''timelike vector field'' $ec{X}$ , the trace of the tidal tensor measured by the corresponding observers is always non-negative:

: $\left( T_{ab} - rac{1}{2} \, T \, g_{ab}
ight) \, X^a \, X^b \ge 0$

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) $ec{Y}$ , 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.

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. For example, the averaged null energy condition states that for every
flowline (integral curve)

C

of the null vector field

ec{k}

, we must have
:

\int_C T_{ab} \, k^a \, k^b \, d\lambda \ge 0

PERFECT FLUIDS

Perfect Fluids possess a matter tensor of form
:

T^{ab} = \mu \, u^a \, u^b + p \, h^{ab}

where

ec{u}

is the Four-velocity of the matter particles and where

h^{ab}

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^{\hat{a} \hat{b}} = \left[ \begin{matrix} 
\mu & 0 & 0 & 0 \
0   & p & 0 & 0 \
0   & 0 & p & 0 \
0   & 0 & 0 & p \end{matrix} 
ight]

Here,

\mu

is of course the mass Density and

p

is the Pressure .

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

the weak energy condition stipulates that $\mu \ge 0, \; \; \mu + p \ge 0$

the null energy condition stipulates that $\mu + p \ge 0$

the strong energy condition stipulates that $\mu + p \ge 0, \; \; \mu + 3 p \ge 0$

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Energy Condition