Intrinsic Heisenberg-type lower bounds on spacelike hypersurfaces in general relativity

Abstract In quantum theory on curved backgrounds, Heisenberg’s uncertainty principle is usually discussed in terms of ensemble variances and flat-space commutators. Here we take a different, preparation-based viewpoint tailored to sharp position measurements on spacelike hypersurfaces in general relativity. A projective localization is modeled as a von Neumann–Lüders projection onto a geodesic ball <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mi>B</mml:mi> <mml:mi mathvariant="normal">Σ</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>r</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> of radius r on a Cauchy slice <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi mathvariant="normal">Σ</mml:mi> <mml:mo>,</mml:mo> <mml:mi>h</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> , with the post-measurement state described by Dirichlet data. Using DeWitt-type momentum operators adapted to an orthonormal frame, we construct a geometric, coordinate-invariant momentum standard deviation σ p and show that strict confinement to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mi>B</mml:mi> <mml:mi mathvariant="normal">Σ</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>r</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> enforces an intrinsic kinetic-energy floor. The lower bound is set by the first Dirichlet eigenvalue λ 1 of the Laplace–Beltrami operator on the ball, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mi>σ</mml:mi> <mml:mi>p</mml:mi> </mml:msub> <mml:mtext>⩾</mml:mtext> <mml:mi>ℏ</mml:mi> <mml:msqrt> <mml:msub> <mml:mi>λ</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:msqrt> </mml:mrow> </mml:math> , and is manifestly invariant under changes of coordinates and foliation. A variance decomposition separates the contribution of the modulus <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mo stretchy="false">|</mml:mo> <mml:mi>ψ</mml:mi> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> </mml:math> from phase-gradient fluctuations and clarifies how the spectral geometry of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi mathvariant="normal">Σ</mml:mi> <mml:mo>,</mml:mo> <mml:mi>h</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> controls momentum uncertainty. Assuming only minimal geometric information, weak mean-convexity of the boundary yields a universal, scale-invariant Heisenberg-type product bound, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mi>σ</mml:mi> <mml:mi>p</mml:mi> </mml:msub> <mml:mi>r</mml:mi> <mml:mtext>⩾</mml:mtext> <mml:mi>π</mml:mi> <mml:mi>ℏ</mml:mi> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> , depending only on the proper radius r .