DSR-relativistic spacetime picture and the phenomenology of Planck-scale-modified time dilation

TL;DR

The paper addresses whether Planck-scale modified dispersion relations affect time dilation within LIV and DSR frameworks. It formulates the most studied DSR model, a 1+1D $\kappa$-Poincaré (bicrossproduct) theory, derives finite boosts, worldlines, and a covariant spacetime metric, and computes an all-orders time-dilation factor $\gamma_{\text{dsr}}$. It shows that LIV leaves time dilation unmodified, while DSR yields a deformed but well-behaved time-dilation factor that remains close to the standard SR result at sub-Planckian energies and saturates for $\ell>0$, challenging naïve extrapolations from Finsler or heuristic arguments. The work clarifies the phenomenology of Planck-scale physics for time dilation and provides a consistent framework for analyzing DSR-time dilation across related models, though experimental tests at current energies are unlikely to be feasible.

Abstract

The most active area of research in quantum-gravity phenomenology investigates the possibility of Planck-scale-modified dispersion relations, focusing mainly on two alternative scenarios: the "LIV" scenario, characterized by a specific mechanism of breakdown of relativistic symmetries, and the "DSR" scenario, which preserves overall relativistic invariance but with deformed laws of relativistic transformation. Two recent studies of modified dispersion relations, one relying on Finsler geometry and one based on heuristic reasoning, raised the possibility of potentially observable effects for time dilation and argued that this might apply also to the LIV and DSR scenarios. We observe that the description of Lorentz transformations in the LIV scenario is such that time dilation cannot be modified. The DSR scenario allows for modifications of time dilation, and establishing their magnitude required us to obtain novel results on the effects of finite DSR boosts in the spacetime sector, with results showing in particular that the modification of time dilation is too small for experimental testing.

Figures (2)

• Figure 1: Time dilation factors as a function of energy for high-energy muons ($\epsilon \gg M \simeq 100\,\text{MeV}$). The deformation scale is taken to be Planckian ($\ell^{-1}\simeq10^{19}\,\text{GeV}$). The heuristic LIV prediction, Eq. \ref{['GAC2']}, and the Finsler prediction, Eq. \ref{['GAC2']} with opposite sign, are compared to our DSR prediction in Eq. \ref{['gamma dsr energy']} and to the standard special relativity (SR) result, Eq. \ref{['eq:gammaSR']}. Both the LIV and Finsler curves deviate significantly from the special-relativistic curve already at relatively low energies around $10^6\,\text{GeV}$. In contrast, the DSR curve essentially overlaps with the special-relativistic one throughout this low-energy range.
• Figure 2: Time dilation factors as a function of energy for high-energy muons ($\epsilon \gg M \simeq 100\,\text{MeV}$). As in Fig. \ref{['fig:plot Finsler/Auger vs DSR']}, we take $\ell^{-1}\simeq10^{19}\,\text{GeV}$. The DSR prediction in Eq. \ref{['gamma dsr energy']} starts to deviate significantly from the special-relativistic (SR) one only at high energies around $10^{18}\,\text{GeV}$.