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

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.

💡 Research Summary

The paper investigates whether Planck‑scale modifications of the dispersion relation can lead to observable changes in time dilation, focusing on two leading quantum‑gravity phenomenology frameworks: Lorentz‑invariance‑violation (LIV) and doubly‑special relativity (DSR). The authors begin by recalling the commonly studied modified dispersion relation
(M^{2}\simeq \epsilon^{2}-p^{2}-\ell,\epsilon,p^{2})
with (\ell) of order the Planck length. In the LIV scenario the Lorentz transformation laws remain exactly those of special relativity, so any modification of the dispersion relation cannot affect the standard time‑dilation factor (\gamma=\epsilon/M). They explicitly refute recent heuristic proposals that insert a correction term proportional to (\ell\epsilon^{3}/M^{2}) into the gamma factor, showing that such a term would contradict the defining property of LIV (unchanged boost laws). Consequently, LIV predicts no measurable deviation in time dilation at any energy.

The bulk of the work is devoted to the DSR case, where the Lorentz group is deformed in a way that preserves two invariant scales (the speed of light and a Planck‑scale energy). The authors adopt the most studied DSR model based on the κ‑Poincaré Hopf algebra in the bicross‑product basis. In 1+1 dimensions the algebra reads (eqs. 3‑4) and the Casimir yields the same modified dispersion relation (1) at first order in (\ell). Crucially, they keep the canonical Poisson brackets ({x^{\mu},p_{\nu}}=-\delta^{\mu}_{\nu}) so that the phase‑space structure is standard.

A key technical achievement is the derivation of the finite‑boost transformation of all phase‑space variables. By integrating the infinitesimal Poisson‑bracket equations (10) they obtain explicit expressions for the boosted energy and momentum (11) and for the boosted coordinates (12). These formulas reduce smoothly to the ordinary Lorentz transformation when (\ell\to0) and, unlike previous literature, provide the missing spacetime‑coordinate transformation in this DSR model.

They then construct the boost matrix (\Lambda(\epsilon,p;\eta)) that depends on the particle’s energy and momentum, and prove that the inverse boost corresponds to (\eta’=-\eta), preserving the relativity principle. Using the Hamiltonian flow generated by the Casimir, they derive the world‑line equation (x(t)=V(\epsilon,p),t+x_{0}) with a non‑linear velocity function (22). They verify that world‑lines are covariant under the finite DSR boosts, and they identify the deformed velocity‑addition law (27).

Finally, they compute the DSR time‑dilation factor. In the ultra‑relativistic limit (\epsilon\gg M) the factor takes the form
(\gamma_{\text{DSR}}\simeq \frac{\epsilon}{M}\bigl