The effects of strong gravity on the dispersion relation of massive particles in the Kaluza-Klein theory

TL;DR

This work investigates how strong gravitational fields in a five-dimensional Kaluza-Klein framework modify the uncertainty principle and the dispersion relation for massive particles. By combining KK gravity with general relativity in both Jordan and Einstein frames, it derives a curvature-driven modification to the dispersion relation, yielding an effective mass $m_{\mathrm{eff}}=\sqrt{m^2+\frac{\hbar^2\mathcal{R}}{6}}$ where $\mathcal{R}$ is the phase-space Ricci scalar; the four-dimensional KK solution reduces to a generalized Schwarzschild metric with a scalar field, subject to a constraint $a^2=b^2+3d^2$. The paper shows that $\mathcal{R}$ is generally nonzero and can diverge near horizons, allowing $m_{\mathrm{eff}}$ to become imaginary and potentially signaling curvature-induced decay, with the parameter space (governed by $(a,b,d)$) determining proximity to GR and the magnitude of corrections. The results illuminate geometry-induced mass corrections and their potential astrophysical and cosmological implications, and they discuss observational constraints and the significance of frame choice for interpreting the physics.

Abstract

We derive a modified dispersion relation for massive particles within the frameworks of five-dimensional Kaluza-Klein theory and general relativity, taking into account strong gravitational effects. The resulting effective mass depends on the curvature of the underlying phase space. Notably, in regions with strong gravitational fields, the effective mass may become imaginary, implying the possibility of particle decay induced by spacetime curvature.

Figures (7)

• Figure 1: The effects of Kaluza -- Klein theory and a strong gravitational field on the uncertainty relation of massive particles.
• Figure 2: The ratio of the $g_{00}$ terms of the KK and the GR Schwarzschild solutions' metric in the Jordan frame as a function of the radial coordinate $r$, in units of $a$.
• Figure 3: Phase space ' cat'. Left: Jordan frame, right: Einstein frame. Phase space curvature ratio of the Kaluza -- Klein theory and usual general relativity. Different colors correspond to different $r$ radial coordinate values, while the vertical dashed lines denote special values of $d$ (at $d=\pm \sqrt{(a-4/9a)/3}$ for the Einstein ' cat'). The ' nose' indicates the (0,0) coordinate.
• Figure 4: Phase space curvature ratio of the Kaluza -- Klein and the general relativistic Schwarzschild metric calculated from $G_\mathrm{ij}$ for different values of parameter $d$ (connected to the scalar field) as a function of the radial coordinate $r$, given in the Jordan frame. The larger plot is a magnified part of the smaller one indicated by a dotted rectangle.
• Figure 5: Properties of the Kaluza -- Klein theory in the Jordan frame, shown for the parameter $d$, which is related to the scalar field, $\sigma$. Green region shows where the model is closest to general relativity, $b$ is set to be positive.