Null energy condition and superluminal propagation

TL;DR

The paper investigates whether violating the null energy condition (NEC) necessarily leads to instabilities, using an EFT of derivatively coupled scalars around coordinate-dependent backgrounds. It proves NEC violation implies instabilities in broad classes of isotropic solids and fluids, but also constructs explicit, stable NEC-violating examples that rely on anisotropy and superluminal modes, highlighting a nuanced, frame-dependent stability landscape. The work connects NEC considerations to relativistic solids/fluids, Kelvin-like hydrodynamics, and ghost condensate constructions, and discusses implications for massive gravity and cosmological models aiming for $w<-1$ without pathologies. Overall, the study clarifies when NEC violation signals genuine pathologies and when stable NEC-violating backgrounds can exist, guiding the search for viable modified gravity scenarios.

Abstract

We study whether a violation of the null energy condition necessarily implies the presence of instabilities. We prove that this is the case in a large class of situations, including isotropic solids and fluids relevant for cosmology. On the other hand we present several counter-examples of consistent effective field theories possessing a stable background where the null energy condition is violated. Two necessary features of these counter-examples are the lack of isotropy of the background and the presence of superluminal modes. We argue that many of the properties of massive gravity can be understood by associating it to a solid at the edge of violating the null energy condition. We briefly analyze the difficulties of mimicking $\dot H>0$ in scalar tensor theories of gravity.

Figures (6)

• Figure 1: The dispersion relations (bold lines) of a subluminal scalar in the original frame in which eq. (\ref{['subluminal']}) holds (left), and in a highly boosted frame (right).
• Figure 2: The dispersion relations (bold lines) of a superluminal scalar in the original frame (left) and in a highly boosted frame (right).
• Figure 3: The creation of a negative energy particle and two gravitons out of vacuum in the boosted frame (right). The bold arrows are the particles momenta, which sum up to zero. The same process appears in the original frame as the decay of a superluminal particle into two gravitons (left).
• Figure 4: The bold lines depict the past causal cones in the original reference frame (left) and in a highly boosted one (right). Left: The two distinct vacuum choices require no incoming wave in the light shaded region ($t<0$) and in the dark shaded region ($t'<0$), respectively. As clear from the picture these choices are inequivalent: an incoming particle moving along the left branch of the past causal cone is discarded in the former choice while it is accepted (as outgoing) in the latter.
• Figure 5: The eigenvalues $\lambda_I$ of $\tilde{L}_{IJ}$ as functions of $\nu$. The configuration shown corresponds to a stable system: all the positive derivative roots $\nu_{I+}$ are on the right of the negative derivative roots $\nu_{I-}$ (taking into account also the roots $\pm1$ corresponding to the dispersion relation of the graviton). As clear from the picture, this property is equivalent to conditions (i), (ii).