Relativistic Extended Uncertainty Principle from Spacetime Curvature

Abstract

The investigations presented in this study are directed at relativistic modifications of the uncertainty relation derived from the curvature of the background spacetime. These findings generalize previous work which is recovered in the nonrelativistic limit. Applying the 3+1-splitting in accordance with the ADM-formalism, we find the relativistic physical momentum operator and compute its standard deviation for wave functions confined to a geodesic ball on a spacelike hypersurface. Its radius can then be understood as a measure of position uncertainty. Under the assumtion of small position uncertainties in comparison to background curvature length scales, we obtain the corresponding corrections to the uncertainty relation in flat space. Those depend on the Ricci scalar of the effective spatial metric, the particle is moving on, and, if there are nonvanishing time-space components of the spacetime metric, gradients of the shift vector and the lapse function. Interestingly, this result is applicable not only to massive but also to massless particles. Over all, this is not a covariant, yet a consistently general relativistic approach. We further speculate on a possible covariant extension.

arXiv comment22 pages, 5 figures

References in corpus (56)

• The Dynamics of General Relativity
• Hilbert Space Representation of the Minimal Length Uncertainty Relation
• Quantum gravity and minimum length
• A Generalized Uncertainty Principle in Quantum Gravity
• Generalized Uncertainty Principle in Quantum Gravity from Micro-Black Hole Gedanken Experiment
• Minimal Length Scale Scenarios for Quantum Gravity
• Entropic Uncertainty Relations and their Applications
• Discreteness of Space from the Generalized Uncertainty Principle
• The algebraic structure of the generalized uncertainty principle
• On Gravity and the Uncertainty Principle
• Nonpointlike Particles in Harmonic Oscillators
• D-branes and Short Distances in String Theory
• The principle of relative locality
• Stronger uncertainty relations for the sum of variances
• Short Distance vs. Long Distance Physics: The Classical Limit of the Minimal Length Uncertainty Relation
• The Generalized Uncertainty Principle in (A)dS Space and the Modification of Hawking Temperature from the Minimal Length
• (Anti-)de Sitter Black Hole Thermodynamics and the Generalized Uncertainty Principle
• Natural extension of the Generalised Uncertainty Principle
• Duality and zero-point length of spacetime
• Minimum Length from Quantum Mechanics and Classical General Relativity
• Extended uncertainty principle and the geometry of (anti)-de Sitter space
• Limits on the Measurability of Space-time Distances in (the Semi-classical Approximation of) Quantum Gravity
• Generalized Uncertainty Principle, Classical Mechanics, and General Relativity
• Born Reciprocity in String Theory and the Nature of Spacetime
• Quantum gravitational decoherence from fluctuating minimal length and deformation parameter at the Planck scale
• Lorentz violation and generalized uncertainty principle
• Horizon wave-function for single localized particles: GUP and quantum black hole decay
• Quantum Gravity, Dynamical Phase Space and String Theory
• Relativistic Generalized Uncertainty Principle
• Minimal Length and Small Scale Structure of Spacetime
• Challenge to Macroscopic Probes of Quantum Spacetime Based on Noncommutative Geometry
• Generalised uncertainty relations from superpositions of geometries
• A Unique Connection for Born Geometry
• Relativistic deformed kinematics from momentum space geometry
• Phenomenology of GUP stars
• A violation of the uncertainty principle implies a violation of the second law of thermodynamics
• On the Minimal Length Uncertainty Relation and the Foundations of String Theory
• Spacetime with zero point length is two-dimensional at the Planck scale
• A closer look at the uncertainty relation of position and momentum
• Generalized uncertainty principle or curved momentum space?
• Quantum mechanics in de Sitter space
• Gauge Symmetry in Phase Space, Consequences for Physics and Spacetime
• Phenomenological consequences of a geometry in the cotangent bundle
• Gravitationally induced uncertainty relations in curved backgrounds
• Generalized Uncertainty Principle from Quantum Geometry
• Path Integral for non-relativistic Generalized Uncertainty Principle corrected Hamiltonian
• Planck Scale Effect in the Entropic Force Law
• Uncertainty principle on 3-dimensional manifolds of constant curvature
• Quantum Gravity, Dynamical Energy-Momentum Space and Vacuum Energy
• How the Geometric Calculus Resolves the Ordering Ambiguity of Quantum Theory in Curved Space
• Horizon of quantum black holes in various dimensions
• On deformations of classical mechanics due to Planck-scale physics
• Gravitational effects on the Heisenberg Uncertainty Principle: a geometric approach
• On the Physics of the Minimal Length: The Question of Gauge Invariance
• A New Approach to Generalised Uncertainty Relations
• Covariant bandlimitation from Generalized Uncertainty Principles

Cited by in corpus (8)

• Modified theories of Gravity: Why, How and What?
• 30 years in: Quo vadis generalized uncertainty principle?
• Geometrizing the Klein-Gordon and Dirac equations in Doubly Special Relativity
• Reinterpreting deformed Heisenberg algebras
• Effect of minimal length uncertainty on neutrino oscillation
• Covariant Space Time Line Elements in the Friedmann Lemaitre Robertson Walker Geometry
• Towards quantum Mechanics on the curved cotangent bundle
• Probing the interior of the Schwarzschild black hole using congruences: LQG vs. GUP