[Paper Review] Lorentz transformations in de Sitter relativity

R. Aldrovandi, Juan P. Beltrán Almeida+2 more|ArXiv.org|Sep 25, 2007

Noncommutative and Quantum Gravity Theories5 references3 citations

TL;DR

This paper demonstrates that Lorentz transformations in de Sitter relativity preserve both the speed of light and the de Sitter length scale $ l = \sqrt{\Lambda/3} $, showing that an invariant length parameter—relevant to quantum gravity—need not break Lorentz symmetry. The result implies that de Sitter relativity provides a consistent framework for high-energy physics where local spacetime geometry becomes de Sitter-like near the Planck scale, preserving Lorentz invariance through conformal structure in stereographic coordinates.

ABSTRACT

The properties of Lorentz transformations in de Sitter relativity are studied. It is shown that, in addition to leaving invariant the velocity of light, they also leave invariant the length-scale related to the curvature of the de Sitter spacetime. The basic conclusion is that it is possible to have an invariant length parameter without breaking the Lorentz symmetry. This result may have important implications for the study of quantum kinematics, and in particular for quantum gravity.

Motivation & Objective

• To investigate whether Lorentz symmetry can coexist with an invariant length scale, such as the Planck length, in a high-energy regime.
• To explore the kinematic structure of de Sitter spacetime as a natural extension of special relativity when the cosmological constant $ \Lambda $ is non-zero.
• To show that the de Sitter length parameter $ l $, related to spacetime curvature, remains invariant under Lorentz transformations.
• To establish a geometric and algebraic framework for Lorentz transformations in de Sitter spacetime using stereographic projection and conformal mapping.
• To assess the implications for quantum gravity, particularly in scenarios involving local spacetime fluctuations near the Planck scale.

Proposed method

• The de Sitter spacetime $ dS(4,1) $ is embedded in a five-dimensional pseudo-Euclidean space $ \mathbf{E}^{4,1} $, defined by the constraint $ \eta_{AB}\chi^A\chi^B = -l^2 $.
• Infinitesimal de Sitter transformations are expressed as $ \delta\chi^A = -\mathcal{E}^A{}_B \chi^B $, with parameters $ \mathcal{E}^{AB} $, and mapped to four-dimensional Minkowski spacetime via stereographic projection.
• Stereographic coordinates $ x^a $ are defined via $ \chi^a = \Omega(x) x^a $ and $ \chi^4 = -l\Omega(x)(1 + \sigma^2/(4l^2)) $, with $ \Omega(x) = 1/(1 - \sigma^2/(4l^2)) $, yielding a conformally flat metric $ g_{ab} = \Omega^2 \eta_{ab} $.
• Lorentz transformations in the projected spacetime are derived as $ \delta x^a = -\epsilon^a{}_b x^b - \epsilon^b \left[ \delta^a_b - \frac{1}{4l^2}(2\eta_{bc}x^a x^c - \sigma^2 \delta^a_b) \right] $, showing linearity in coordinates and momenta.
• The Lorentz generators $ L_{ab} $ and de Sitter translation generators $ \Pi_b $ are defined, with $ \Pi_b = P_b - \frac{1}{4l^2}K_b $, ensuring the full de Sitter algebra is realized.
• Invariance of $ l $ under Lorentz transformations is proven via the conformal factor $ \Omega $, which is Lorentz-invariant, thus preserving both $ c $ and $ l $.

Experimental results

Research questions

• RQ1 Can Lorentz symmetry be preserved in the presence of a fundamental length scale such as the Planck length?
• RQ2 How do Lorentz transformations behave in de Sitter spacetime, and do they preserve the de Sitter length parameter $ l $?
• RQ3 Is it possible to have a non-zero cosmological constant $ \Lambda $ without breaking Lorentz invariance in the high-energy regime?
• RQ4 What is the role of the conformal factor $ \Omega $ in preserving invariants under Lorentz transformations in stereographic coordinates?
• RQ5 Can local spacetime fluctuations near the Planck scale be consistently described within a de Sitter relativistic framework?

Key findings

• Lorentz transformations in de Sitter spacetime leave both the speed of light $ c $ and the de Sitter length parameter $ l = \sqrt{\Lambda/3} $ invariant, demonstrating that Lorentz symmetry is compatible with a fundamental length scale.
• The conformal factor $ \Omega(x) $, which defines the stereographic projection, is Lorentz-invariant, ensuring that the de Sitter metric $ g_{ab} = \Omega^2 \eta_{ab} $ transforms covariantly under Lorentz transformations.
• The stereographic coordinate system provides a linear realization of Lorentz boosts in both position and momentum space, preserving the algebraic structure of the Lorentz group.
• The de Sitter length parameter $ l $ is invariant under Lorentz transformations because it is geometrically tied to the curvature of spacetime and not subject to Lorentz contraction.
• The invariance of $ l $ implies that the cosmological constant $ \Lambda $, related by $ l = \sqrt{\Lambda/3} $, is also preserved under Lorentz transformations, avoiding the need to break Lorentz symmetry to maintain a fundamental scale.
• This framework suggests that high-energy phenomena could locally induce a de Sitter geometry with $ l \sim l_{\text{Pl}} $, and observers in such regions would all measure the same $ l $, preserving relativistic invariance.