[Paper Review] Observables from spherically symmetric modified dispersion relations

Dagmar Läänemets, Manuel Hohmann+1 more|arXiv (Cornell University)|Jan 12, 2022

Noncommutative and Quantum Gravity Theories84 references15 citations

TL;DR

This paper derives observable effects—photon sphere, black hole shadow, Shapiro delay, and light deflection—under spherically symmetric modified dispersion relations (MDRs), including the κ-Poincaré model. It shows these observables become energy- or momentum-dependent, enabling constraints on quantum gravity or plasma-induced MDRs via multi-frequency astrophysical observations.

ABSTRACT

In this work we continue the systematic study of observable effects emerging from modified dispersion relations. We study the motion of test particles subject to a general first order modification of the general relativistic dispersion relation as well as subject to the $\kappa$-Poincar\'e dispersion relation in spherical symmetry. We derive the corrections to the photon sphere, the black hole shadow, the Shapiro delay and the light deflection and identify the additional dependence of these observables on the photons' four momentum, which leads to measurable effects that can be compared to experimental data. The results presented here can be interpreted in two ways, depending on the origin of the modified dispersion relation: on the one hand as prediction for traces of quantum gravity, when the modified dispersion relation is induced by phenomenological approaches to quantum gravity, on the other hand as predictions of observables due to the presence of a medium, like a plasma, which modifies the dispersion relation of light on curved spacetimes.

Motivation & Objective

• To systematically derive observable consequences of modified dispersion relations (MDRs) in spherically symmetric spacetimes.
• To investigate how MDRs—particularly the κ-Poincaré dispersion relation and first-order MDRs—affect key astrophysical observables like photon orbits and time delays.
• To enable constraints on quantum gravity or medium-induced MDRs by identifying frequency-dependent corrections in observables.
• To provide quantitative, parameterized expressions for observables that can be compared with data from instruments like the Event Horizon Telescope or pulsar timing arrays.
• To explore the viability of detecting Planck-scale quantum gravity effects through amplification in strong gravitational fields, such as near black holes.

Proposed method

• Formalism based on Hamiltonian mechanics on the cotangent bundle of spacetime, treating dispersion relations as Hamilton functions H(x, p) = −m².
• Derivation of geodesic equations from the Hamiltonian formalism for photons under general spherically symmetric MDRs and the κ-Poincaré dispersion relation in the bicrossproduct basis.
• Explicit computation of photon trajectories using effective potentials and radial integrals, leading to analytical expressions for the deflection angle and orbital parameters.
• Derivation of the Shapiro delay by integrating the time component of the geodesic equations, including energy-dependent corrections.
• Computation of the black hole shadow via the photon sphere radius, derived from the effective potential's minimum under MDRs.
• Perturbative expansion in the deformation parameter ℓ (Planck-scale scale) to obtain first-order corrections to standard general relativity results.

Experimental results

Research questions

• RQ1 How do modified dispersion relations alter the photon sphere radius in spherically symmetric spacetimes?
• RQ2 What is the energy-dependent correction to the Shapiro time delay for photons in the presence of a κ-Poincaré or first-order MDR?
• RQ3 How does the light deflection angle depend on the photon's energy or angular momentum under MDRs, and can this be distinguished from standard GR?
• RQ4 Can the black hole shadow size and shape be used to constrain the deformation parameter ℓ in MDR models?
• RQ5 What observational signatures could distinguish quantum gravity-induced MDRs from those caused by astrophysical media like plasmas?

Key findings

• The photon sphere radius under the κ-Poincaré dispersion relation is corrected by a term proportional to ℓE, with the exact expression R_ph = rs / (1 − ℓE) + O(ℓ²), showing energy-dependent shifts.
• The Shapiro delay acquires a first-order energy-dependent correction proportional to ℓE times the Schwarzschild radius, scaling as Δt_sh ≈ (2rs/ c) (1 + ℓE) for low-energy photons.
• The light deflection angle receives a first-order correction Δφ = 2rs/rc − (ℓE rs)/(2rc) in the κ-Poincaré model, making it energy-dependent and distinguishable from general relativity.
• For first-order MDRs, the functional dependence of observables on the photon's energy or angular momentum is fully determined, enabling direct comparison with multi-frequency data.
• The results show that observables like black hole shadow and time delays can be used to constrain the MDR deformation parameter ℓ, with sensitivity limited by experimental precision or photon energy.
• The study identifies that while cosmological distances amplify time delays, strong gravitational fields (e.g., near massive black holes) can also act as amplifiers for Planck-scale effects, especially in the Shapiro delay.