Causal loops in the theory of relative locality

We find that relative locality, a recently proposed Planck-scale deformation of special relativity, suffers from the existence of causal loops. A simple and general construction of such on-shell loop processes is studied. We then show that even in one of the weakest deformations of the Poincaré group in relative locality, causality can be violated.

Figure 1

A causal loop in the two-particle phase space.

Figure 2

A two-event causal loop in relative locality.

Figure 3

The region of $(p_0^1,p_1^1)$ that possibly leads to a causal loop by events A and B when $p_0^0=p_0^1$, $p_1^0=-p_1^1$, $r_0=1.1k_0=0.88\kappa$, $k_1=-r_1=0.1\kappa$.