Relativistic extended uncertainty principle from spacetime curvature

Fabian Wagner

Phys. Rev. D 105, 025005 – Published 6 January 2022

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 that 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 assumption 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, there are 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.

Received 6 December 2021
Accepted 21 December 2021

DOI:https://doi.org/10.1103/PhysRevD.105.025005

Physics Subject Headings (PhySH)

Quantum cosmology Quantum fields in curved spacetime Quantum gravity

Particles & FieldsGravitation, Cosmology & Astrophysics

Authors & Affiliations

Fabian Wagner ^*

Institute of Physics, University of Szczecin, Wielkopolska 15, 70-451 Szczecin, Poland

^*fabian.wagner@usz.edu.pl

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Issue

Vol. 105, Iss. 2 — 15 January 2022

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Images

Figure 1
Schematic visualization of the squared absolute value of two wave functions (eigenfunctions of the Laplacian) color coded from violet (vanishing) to red on a flat two-dimensional background, here as a plane embedded in three-dimensional space, and confined to a disk whose boundary is displayed in black.
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Figure 2
Three geodesic balls with equal geodesic radius $ρ = 0.4 R_{S}$ but different distances from the center of symmetry $r_{0} = 3.5 R_{S}$ (left), $r_{0} = 2.5 R_{S}$ (mid), and $r_{0} = 1.5 R_{S}$ (right) on a spatial section of the Schwarzschild static patch characterized by the Schwarzschild radius $R_{S}$ and described in terms of Schwarzschild coordinates. Surfaces of geodesic balls are colored blue while black hole horizons are indicated in black.
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Figure 3
Schematic visualization of the squared absolute value of two wave functions (eigenfunctions of the Laplacian) color coded from violet (vanishing) to red on a sphere, imbedded in three-dimensional Euclidean space, and confined to a disk whose boundary is indicated in black.
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Figure 4
The left plot shows the trajectory followed by a massive particle in the equatorial ( $x − y$ ) plane of a fast black hole rotating as $a_{J} / G M = 0.5$ and with an outer horizon of radius $r_{+}$ symbolized by the black disk in the center. Its starting point lies on the $x$ axis at a distance $100 r_{+}$ from the source with initial velocity $u (τ = 0) ≃ (1.010, - 0.0010, 0.0000, 0.0001)$ in Boyer-Lindquist coordinates. The color, ranging from violet to red, indicates an increase in the affine parameter $τ$ . Inset in the top left corner is a plot displaying all the corrections to the uncertainty relation in units of $ℏ$ logarithmically experienced by a particle of mass $m = ℏ / r_{+}$ with position uncertainty $ρ = 10^{- 1} r_{+}$ along this orbit as a function of proper time. On the right-hand side, the fully relativistic (blue curve and nonrelativistic (orange, dotted curve) corrections are compared allowing a closer look at the first peak of the uncertainty relation.
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Figure 5
A geodesic ball on a flat background, deformed by Lorentz transformations. The velocity parameter corresponding to the transformation, in units of the speed of light, is color-coded, increasing in steps of 0.1 from red (0) to violet (0.8).
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Figure 6
Lower bounds on the relativistic momentum uncertainties of all linear combinations of two eigenstates of the Laplacian (22) with principal quantum numbers $(n, n^{'}) \leq (10, 11)$ to lowest nonvanishing order as functions of the parameter $a \in [0, 1]$ characterizing the relative weight and in units of $ℏ / ρ$ . Black curves correspond to linear combinations including the ground state (24), while the color measures the sum of the principal quantum numbers $n$ and $n^{'}$ for the others. The eigenvalue of the ground state is represented by the violet line.
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