The null energy condition and instability

We extend previous work showing that violation of the null energy condition implies instability in a broad class of models, including gauge theories with scalar and fermionic matter as well as any perfect fluid. Simple examples are given to illustrate these results. The role of causality in our results is discussed. Finally, we extend the fluid results to more general systems in thermal equilibrium. When applied to the dark energy, our results imply that $w=p/\rho$ is unlikely to be less than $-1$.

Figure 1

In a model that violates the NEC, an initially smooth field configuration (top) is unstable. The system can lower its total energy by developing gradients (bottom), thereby decreasing its kinetic energy, while leaving its potential energy unchanged.

Figure 2

A flow chart of the general version of the proof that for models described by the action (32), only solutions satisfying the NEC can be stable.

Figure 3

Representation of simple Lorentz invariants that the Lagrangian may depend upon. Each dot represents a Lorentz index, and a line connecting them denotes contraction using the metric. Rectangles (with two indices) are field strengths, small circles covariant derivatives of scalar fields, and large circles epsilon tensors. All Lorentz indices are ultimately contracted, and we suppress color indices for simplicity.

Figure 4

Representation of the most general Lagrangian of the type considered in this paper. The notation is the same as in Fig. 3. We focus on the elements shaded black. In graphical terms, $M$ for this term in the Lagrangian can be obtained from the figure by simply removing these elements. $Z$ represents the remainder of the Lagrangian.

Figure 5

The NEC, causality (here $C$), and quantum mechanical stability (the absence of ghosts, here $\mathrm{G̸}$) are all determined by the sign-definiteness of matrices $A$ and $B$. The NEC holds for $A>0$; the model is causal for $A^{-1}B>0$ and ghost-free for $B>0$. A model is ghost-free and causal only if it obeys the NEC.