Taming Nonlocality in Theories with Planck-Scale Deformed Lorentz Symmetry

We report a general analysis of worldlines for theories with deformed relativistic symmetries and momentum dependence of the speed of photons. Our formalization is faithful to Einstein’s program, with spacetime points viewed as an abstraction of physical events. The emerging picture imposes the renunciation of the idealization of absolutely coincident events, but is free from some pathologies which had been previously conjectured.

Figure 1

In the argument of Ref. 11 a key role is played by the assumption that a photon which Alice sees emitted in $P_A$ (from a source at a distance $L$ from Alice) with speed $v$ (and momentum $p$) should be seen by boosted Bob as a photon emitted from $P_B$, obtained by classical, undeformed boost of $P_A$, with speed obtained from the speed $v$ with a deformed boost. In this figure we expose the logical inconsistency of such criteria of “selective applicability” of deformed boosts by allowing for a second photon which according to Alice also has speed $v$ and is emitted from a point $P_{A^{\prime }}$ such that the two photons share the same worldline: a single worldline would be mapped by a relativistic boost into two wildly different worldlines.

Figure 2

We here show a hard-photon worldline as seen by Alice (solid blue line), by DSR-boosted Bob (dashed blue line) and by classically boosted Bob (dashed black line). In spite of assuming (for visibility) the unrealistically huge $\Pi =0.05/\ell$, $\xi =0.15$, the difference between DSR boosts and undeformed boosts is minute near the origin. But according to Bob’s coordinates the emission of the hard particle appears to occur slightly off the (thick) worldline of the source.

Figure 3

A case with two hard (violet lines) worldlines, with momentum ${\Pi }_v=0.13/\ell$, a “semihard” (blue line) worldline with momentum ${\Pi }_b={\Pi }_v/2$, and a ultrasoft worldline (red line, with ${\Pi }_r\ll 1/\ell$). According to Alice (whose lines are solid, while boosted Bob has dashed lines) three of the worldlines give a distant coincidence of events, while two of the worldlines cross in the origin.