Gravitational length stretching: Curvature-induced modulation of quantum probability densities

Abstract We introduce gravitational length stretching (GLS) as a novel quantum-gravitational phenomenon that distorts the spatial probability distribution of quantum particles in curved spacetime. Through quantum field theory in curved spacetime (QFT-CS), we demonstrate that GLS originates from the amplitude modulation function <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mrow> <m:mi>B</m:mi> </m:mrow> <m:mrow> <m:mi>k</m:mi> </m:mrow> </m:msub> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>t</m:mi> <m:mo>,</m:mo> <m:mover accent="true"> <m:mrow> <m:mi>x</m:mi> </m:mrow> <m:mrow> <m:mo>→</m:mo> </m:mrow> </m:mover> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> {B}_{k}\left(t,\overrightarrow{x}) in WKB solutions, governed by the covariant continuity equation <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mrow> <m:mrow> <m:mo>∇</m:mo> </m:mrow> </m:mrow> <m:mrow> <m:mi>μ</m:mi> </m:mrow> </m:msub> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:msubsup> <m:mrow> <m:mi>B</m:mi> </m:mrow> <m:mrow> <m:mi>k</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msubsup> <m:msup> <m:mrow> <m:mi>k</m:mi> </m:mrow> <m:mrow> <m:mi>μ</m:mi> </m:mrow> </m:msup> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo>=</m:mo> <m:mn>0</m:mn> </m:math> {\nabla }_{\mu }\left({B}_{k}^{2}{k}^{\mu })=0 . This effect causes quantum objects to stretch in regions of stronger curvature, increasing their proper spatial extent by <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mo>∼</m:mo> <m:mn>20</m:mn> <m:mo>%</m:mo> </m:math> \sim 20 \% over millimeter scales for ultracold neutrons (UCNs) in Earth’s gravity. Our derivation of the Schrödinger equation in weak gravitational fields from the Klein–Gordon equation in Cartesian Schwarzschild coordinates reveals curvature-induced corrections absent in semi-classical approaches. Crucially, QFT-CS yields a covariant quantization condition <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo>∫</m:mo> </m:mrow> <m:msub> <m:mrow> <m:mi>k</m:mi> </m:mrow> <m:mrow> <m:mi>μ</m:mi> </m:mrow> </m:msub> <m:mi mathvariant="normal">d</m:mi> <m:msup> <m:mrow> <m:mi>x</m:mi> </m:mrow> <m:mrow> <m:mi>μ</m:mi> </m:mrow> </m:msup> <m:mo>=</m:mo> <m:mi>n</m:mi> <m:mi>π</m:mi> </m:math> \int {k}_{\mu }{\rm{d}}{x}^{\mu }=n\pi , resolving ambiguities in the noncovariant semi-classical formalism. We predict observable energy level shifts ( <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi mathvariant="normal">Δ</m:mi> <m:msub> <m:mrow> <m:mi>E</m:mi> </m:mrow> <m:mrow> <m:mi>n</m:mi> </m:mrow> </m:msub> <m:mo>∕</m:mo> <m:msub> <m:mrow> <m:mi>E</m:mi> </m:mrow> <m:mrow> <m:mi>n</m:mi> </m:mrow> </m:msub> <m:mspace width="0.33em"/> <m:mo>∼</m:mo> <m:mspace width="0.33em"/> <m:mn>1</m:mn> <m:mo>%</m:mo> </m:math> \Delta {E}_{n}/{E}_{n}\hspace{0.33em} \sim \hspace{0.33em}1 \% ) and probability density gradients ( <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mrow> <m:mo>∂</m:mo> </m:mrow> <m:mrow> <m:mi>z</m:mi> </m:mrow> </m:msub> <m:msubsup> <m:mrow> <m:mi>B</m:mi> </m:mrow> <m:mrow> <m:mi>k</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msubsup> <m:mo>≈</m:mo> <m:mn>202</m:mn> <m:mspace width="0.33em"/> <m:msup> <m:mrow> <m:mi mathvariant="normal">m</m:mi> </m:mrow> <m:mrow> <m:mo>−</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msup> </m:math> {\partial