Energy Condition - Perfect Fluids

Perfect Fluids

Perfect fluids possess a matter tensor of form

where is the four-velocity of the matter particles and where 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} \rho& 0 & 0 & 0 \\ 0 & p & 0 & 0 \\ 0 & 0 & p & 0 \\ 0 & 0 & 0 & p \end{matrix} \right] .

Here, is the energy density and is the pressure.


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

  • The weak energy condition stipulates that
  • The null energy condition stipulates that
  • The strong energy condition stipulates that
  • The dominant energy condition stipulates that

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.

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