Modified Dispersion Relation for Massive Particles

• Modified dispersion relations for massive particles are deformations of the standard relativistic energy-momentum relation, introducing nonlinear corrections from quantum gravity, medium-induced phenomena, or extended spacetime geometry.
• These modifications offer key insights into high-energy astrophysics, gravitational physics, and cosmology by predicting observable deviations in particle transport, black hole thermodynamics, and cosmological redshifts.
• They enable precision tests of fundamental principles like the weak equivalence principle and provide frameworks through Hamilton and Finsler geometry to explore quantum gravity phenomenology.

A modified dispersion relation (MDR) for massive particles is a deformation of the standard relativistic relationship between energy, momentum, and rest mass, induced by new physics such as quantum gravity effects, medium-induced phenomena, or extended spacetime geometry. MDRs have substantial implications across high-energy particle transport, gravitational physics, cosmology, astrophysics, and the foundations of relativity, and are characterized by additional terms in the on-shell relation that typically depend nonlinearly on momentum, energy, or background fields.

1. General Forms and Theoretical Motivation

The standard special relativistic dispersion relation is $E^2 = p^2 + m^2$ (setting $c = 1$). MDRs introduce corrections:

• In effective quantum gravity models and various phenomenological frameworks, MDRs can be parameterized generically as

$H(x, p) = g^{ab}(x)p_a p_b + \Xi\, h(x, p) = M^2$

where $g^{ab}(x)$ is a Lorentzian metric, $h(x, p)$ is a higher-rank polynomial or nonlinear function in momenta, and $\Xi$ is a coupling constant possibly dependent on mass or model parameters (Hohmann et al., 29 Apr 2024).

Prominent MDRs include:

• Polynomial and non-polynomial deformations (e.g., cubic or quartic in momentum) as in the $q$-de Sitter or $\kappa$-Poincaré frameworks (Barcaroli et al., 2018, Barcaroli et al., 2016).
• “Thermal mass” or effective mass terms arising from medium effects, e.g., interactions with a dark medium or plasma (Masina et al., 2011).
• Planck-scale or curvature-induced modifications where effective mass receives geometric contributions from phase-space curvature or extra dimensions (Horváth et al., 18 Oct 2025).
• Generalized uncertainty principle (GUP)-motivated MDRs with terms $\propto p^3$, $p^4$, or higher, reflecting minimal length and/or maximal momentum (Majhi et al., 2013, Mukherjee et al., 2019).

2. Specific Examples of Modified Dispersion Relations

Hamiltonian/Phase Space Formulations

• General static spherically symmetric MDR:

$H(r, E, P, L) = -\frac{m^2}{2}$

with $E$, $P$, $L$ the conserved energy, radial momentum, and angular momentum, respectively; $H$ may include nonlinear or non-metric terms (Hohmann, 2023).

• $\kappa$-Poincaré (bicrossproduct basis):

$$H = -\frac{2}{\ell^2} \sinh^2{\left[\frac{\ell}{2}(-cE + dP)\right]} + \frac{1}{2} e^{\ell(-cE + dP)} \left( (-a + c^2)E^2 - 2cd\, EP + (b + d^2)P^2 + \frac{L^2}{r^2} \right)$$

where $\ell$ is the deformation scale (often Planck length), and metric coefficients are chosen to recover standard Schwarzschild in the $\ell \rightarrow 0$ limit (Hohmann, 2023, Läänemets et al., 2022, Barcaroli et al., 2016).

Curvature-Induced and Strong Gravity Modifications

• In Kaluza-Klein and strong gravity regimes, the effective mass is

$$m_{\rm eff} = \sqrt{ m^2 + \frac{\hbar^2}{6} \mathcal{R} }$$

so that the MDR is

$E^2 = p^2 + m^2 + \frac{\hbar^2}{6} \mathcal{R}$

where $\mathcal{R}$ is the Ricci scalar of the phase-space (“kinetic”) geometry. For sufficiently negative $\mathcal{R}$, $m_{\rm eff}$ becomes imaginary, triggering a tachyonic instability or decay (Horváth et al., 18 Oct 2025).

Planck-Scale Deformations in Cosmological Context

• Homogeneous and isotropic (FLRW) MDRs:

$$H_{q\mathrm{FLRW}} = -\frac{4}{\ell^2} \sinh^2{\left(\frac{\ell p_t}{2}\right)} + a(t)^{-2} e^{\ell p_t} w^2$$

where $p_t$ is conjugate to FLRW time, $a(t)$ the scale factor, $w$ the FLRW spatial invariant, and $\ell$ the deformation parameter (Barcaroli et al., 2016).

• General perturbation:

$$H(t, p_t, w) = -p_t^2 + a(t)^{-2} w^2 + \epsilon h(t, p_t, w)$$

with $h$ modeling quantum gravity-induced corrections (Pfeifer, 2018).

• Cosmological models: Modified dispersion relations of the type

$\omega^2 = c_s^2 k^2 + c_s^2 k_c^{-2} k^4$

as in Dirac–Born–Infeld-inspired nonminimal kinetic coupling (DINKIC) inflation (Qiu et al., 2022).

3. Physical Consequences and Observables

Equivalence Principle Violation

For general MDRs, the acceleration of a free-falling (classical or quantum) massive particle depends on its mass unless the MDR is two-homogeneous or differs from the homogeneous case by a total derivative. The weak equivalence principle (WEP) is satisfied only if the coupling constant and structure of the modification scale with mass to appropriately cancel mass dependence:

• For modifications $h(x, \lambda p) = \lambda^n h(x, p)$, $n$-homogeneous, WEP is preserved if $\Xi(M) = M^{2-n} \hat{\Xi}$.
• If not, the Eötvös parameter

$\eta = \frac{2|a_1 - a_2|}{|a_1 + a_2|}$

is nonzero, and experimental limits (e.g., from MICROSCOPE) provide constraints on model parameters, e.g., $\hat{\Xi}^{-1} \gtrsim 10^{15}~\mathrm{GeV}/c^2$ for deformations in the $\kappa$-Poincaré family (Hohmann et al., 29 Apr 2024).

Black Hole and Cosmological Phenomenology

• MDRs modify Hawking temperature, entropy, and evaporation rates of black holes by introducing corrections $\propto M_{\rm Pl}^{-2}$ or higher, where $M_{\rm Pl}$ is the Planck mass, in the emission spectrum. For massive particles, these corrections scale with both the mass of the particle and thermodynamic parameters (Tao et al., 2015, Kamali et al., 2016, Feng et al., 2018).
• In strong gravity or Kaluza-Klein scenarios, effective mass may become imaginary, implying gravitationally-triggered decay of particles in deep gravity wells—an effect unaccounted for in standard GR (Horváth et al., 18 Oct 2025).
• MDRs affect cosmological redshift and lateshift: energy-dependent corrections to the standard expansion-induced redshift appear, and for both massless and massive particles, arrival times acquire energy/momentum dependence (Barcaroli et al., 2016, Pfeifer, 2018).

Kinetic Theory and Collective Modes

• For relativistic transport, the dispersion relations of sound, heat, and shear modes for massive particles become mass dependent via scaled mass $z = m/T$ in kinetic coefficients (Bessel and Bickley functions). Massive systems exhibit sound–heat channel coupling, and critical wavenumbers for collective mode existence depend non-trivially on $z$ (Lin et al., 7 May 2025).
• Landau damping (collisionless dissipation) is modified: in massive systems, the analytic structure changes from two discrete branch points in $c$-plane (massless) to an entire continuous branch cut $c \in [-1,1]$. This fundamentally alters the long-time, low-frequency response and convergence properties of hydrodynamic expansions (Lin et al., 7 May 2025).

Matter Density Profiles in Strong-Field Gravity

• In collisionless kinetic gases, the $\kappa$-Poincaré MDR shifts the density of bound orbiting particles toward smaller radii and reduces the density of radially infalling particles near the event horizon, reflecting higher coordinate velocities and the impact of non-standard geometry (Hohmann, 2023).

4. Mathematical and Geometric Structure

MDRs are naturally formulated in the language of Hamilton geometry:

• The Hamilton function $H(x,p)$ defines both the dynamics and the “metric” structure on phase space.
• Non-quadratic corrections, e.g., $$H_{q\mathrm{DS}} = g^{ab}p_a p_b + \ell G^{abc}p_a p_b p_c$$, lead to nonstandard motion:
  • Momentum space and spacetime acquire curvature dependent on both position and momentum (Barcaroli et al., 2018).
  • Hamilton's equations include additional source terms (force-like contributions) beyond autoparallel motion.
• The passage from the Hamiltonian to the (parametrization-invariant) test particle action reveals a link between MDRs and Finsler geometry, providing a geometric criterion for WEP violation or satisfaction (Hohmann et al., 29 Apr 2024).

5. Experimental and Observational Implications

• Precision tests of free-fall (MICROSCOPE, satellite, and laboratory experiments) constrain MDR parameter space via the Eötvös parameter.
• Observables such as black hole shadow size, time delays (Shapiro delay), gravitational lensing deflection, and multi-frequency discrepancies in astrophysical signals (e.g., GRBs, neutrino bursts) can be systematically compared to MDR predictions to constrain or detect Planck-scale or curvature-driven MDR effects (Läänemets et al., 2022, Horváth et al., 18 Oct 2025).
• In cosmology, step-like features in primordial spectra, altered complexity dynamics, and energy-dependent lateshifts serve as potential observational diagnostics for MDRs (Qiu et al., 2022, Li et al., 2023).

6. Interpretational Frameworks

A comprehensive theory of MDRs for massive particles requires:

• Specification of the modification function $h(x,p)$ and its symmetry properties.
• Consideration of both quantum gravity and medium-induced scenarios.
• Geometrization via Finsler or Hamilton geometry for rigorous analysis of dynamics, trajectory dependence, and connection to classical tests (WEP, redshift).
• Assessment of the analytic structure of response functions (e.g., presence of continuous or discrete branch cuts) in transport and kinetic theory contexts.

The explicit form and physical consequences of a modified dispersion relation depend sensitively on the underlying symmetry principles, coupling to background fields or media, and the precise nature of corrections—whether Planck-suppressed, curvature-induced, or medium-dependent. MDRs stand as a central probe in the search for quantum gravity phenomenology, strong-field gravity effects, and possible violations or refinements of the pillars of general relativity and quantum field theory.