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Quantum astrometric observables I: time delay in classical and quantum gravity

Igor Khavkine

TL;DR

The paper tackles the problem of defining and computing diffeomorphism-invariant observables in gravity by introducing the time delay δτ(s) = s - τ(s) as a concrete, operational astrometric observable. It develops a classical framework in which the observable is gauge-invariant and subject to causal inequalities derived from Lorentzian geometry, including maximal light-speed properties and geodesic extremality. It then sketches a quantum treatment using linearized gravity, modeling measurements with a dynamical apparatus, and shows that in Minkowski space the emission-time statistics are Gaussian with a mean set by the classical result, while the variance encodes quantum fluctuations of the graviton vacuum; regularization via detector smearing is essential. Overall, the work proposes a tractable, physically interpretable benchmark (astrometric observables) to study causality and quantum gravitational effects, and discusses extensions to more general backgrounds and nonperturbative regimes as well as implications for comparing different quantum gravity theories.

Abstract

A class of diffeomorphism invariant, physical observables, so-called astrometric observables, is introduced. A particularly simple example, the time delay, which expresses the difference between two initially synchronized proper time clocks in relative inertial motion, is analyzed in detail. It is found to satisfy some interesting inequalities related to the causal structure of classical Lorentzian spacetimes. Thus it can serve as a probe of causal structure and in particular of violations of causality. A quantum model of this observable as well as the calculation of its variance due to vacuum fluctuations in quantum linearized gravity are sketched. The question of whether the causal inequalities are still satisfied by quantized gravity, which is pertinent to the nature of causality in quantum gravity, is raised, but it is shown that perturbative calculations cannot provide a definite answer. Some potential applications of astrometric observables in quantum gravity are discussed.

Quantum astrometric observables I: time delay in classical and quantum gravity

TL;DR

The paper tackles the problem of defining and computing diffeomorphism-invariant observables in gravity by introducing the time delay δτ(s) = s - τ(s) as a concrete, operational astrometric observable. It develops a classical framework in which the observable is gauge-invariant and subject to causal inequalities derived from Lorentzian geometry, including maximal light-speed properties and geodesic extremality. It then sketches a quantum treatment using linearized gravity, modeling measurements with a dynamical apparatus, and shows that in Minkowski space the emission-time statistics are Gaussian with a mean set by the classical result, while the variance encodes quantum fluctuations of the graviton vacuum; regularization via detector smearing is essential. Overall, the work proposes a tractable, physically interpretable benchmark (astrometric observables) to study causality and quantum gravitational effects, and discusses extensions to more general backgrounds and nonperturbative regimes as well as implications for comparing different quantum gravity theories.

Abstract

A class of diffeomorphism invariant, physical observables, so-called astrometric observables, is introduced. A particularly simple example, the time delay, which expresses the difference between two initially synchronized proper time clocks in relative inertial motion, is analyzed in detail. It is found to satisfy some interesting inequalities related to the causal structure of classical Lorentzian spacetimes. Thus it can serve as a probe of causal structure and in particular of violations of causality. A quantum model of this observable as well as the calculation of its variance due to vacuum fluctuations in quantum linearized gravity are sketched. The question of whether the causal inequalities are still satisfied by quantized gravity, which is pertinent to the nature of causality in quantum gravity, is raised, but it is shown that perturbative calculations cannot provide a definite answer. Some potential applications of astrometric observables in quantum gravity are discussed.

Paper Structure

This paper contains 25 sections, 3 theorems, 57 equations, 7 figures, 1 table.

Table of Contents

  1. Introduction
  2. Operational description and gauge invariance
  3. Classical mathematical model
  4. Causal inequalities
  5. Maximality of light speed
  6. Geodesic extremality
  7. Explicit calculation in classical linearized gravity
  8. Minkowski space
  9. Approximately Minkowski space
  10. Tetrad formalism
  11. Geodesic triangle construction
  12. Time delay and gauge invariance
  13. Sketch of quantum mathematical model
  14. Quantum linearized gravity
  15. Time in quantum mechanics
  16. ...and 10 more sections

Key Result

Theorem 1

Given two gauge-equivalent lab-equipped spacetimes $(M,g,O,e^a_i)$ and $(M',g',O',e^{\prime a}_i)$, the corresponding time delays (keeping $s$ and $v^i$ fixed) are equal:

Figures (7)

  • Figure 1: Geometry of the experimental protocol defining the reception time $s$, emission time $\tau(s)$, and the time delay $\delta\tau(s)=s-\tau(s)$. The synchronization/ejection point is $O$. The signal emission point is $P$ and the signal reception point is $Q$.
  • Figure 2: Illustration of the conclusion of Theorem \ref{['light-bound']}. Successive emission times $\tau < \tau$ imply successive reception times $s < s'$. The dashed line represents a case, ruled out by the theorem, where the signal might appear superluminal.
  • Figure 3: Illustration of the proof of Theorem \ref{['twin-bound']}. The auxiliary dashed curve interpolates between $OQ$ and $OP$, as $\lambda$ varies from $0$ to $1$. As it does so, its proper time length $T(\lambda)$ is shown to decrease monotonically, thus implying $\tau(s) < s$.
  • Figure 4: Schematic structure of the $H$-term in $r$, Eq. \ref{['H-struct']}. Notation follows Eqs. \ref{['H-struct']}--\ref{['darr-struct']}.
  • Figure 5: Schematic structure of the $J$-term in $r$, Eq. \ref{['J-struct']}. Notation follows Eqs. \ref{['H-struct']}--\ref{['darr-struct']}.
  • ...and 2 more figures

Theorems & Definitions (9)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Theorem 1
  • Theorem 2
  • proof
  • Theorem 3
  • proof