[Paper Review] Local anisotropy of space in a frame of reference co-moving with the Earth

Ll. Bel, A. Molina|ArXiv.org|Jun 25, 1998

Geophysics and Gravity Measurements1 references5 citations

TL;DR

This paper investigates local spacetime anisotropy in the Earth-centered, co-moving frame using linearized general relativity, incorporating Earth's mass, quadrupole moment, shape, and rotation. It predicts a measurable anisotropy of order 10⁻¹² to 10⁻¹³, suggesting a potential observational signature for Earth's gravitational field in terrestrial experiments.

ABSTRACT

We consider, in the framework of General Relativity, the linear approximation of the gravitational field of the Earth taking into account its mass, its quadrupole moment, its shape and its diurnal rotation. We conclude that in the frame of reference co-moving with the Earth the local anisotropy of the space is of the order of $10^{-12}-10^{-13}$ and could be observed.

Motivation & Objective

• To analyze the local spacetime geometry in a reference frame co-moving with the Earth using general relativity.
• To assess the influence of Earth's mass, quadrupole moment, shape, and diurnal rotation on local spacetime anisotropy.
• To determine whether such anisotropy could be observable in principle with current or near-future experimental techniques.
• To extend linearized gravity solutions to include rotational and multipole effects in a non-inertial, rotating frame.

Proposed method

• Adopt a linearized approximation of the Einstein field equations in the weak-field limit.
• Model Earth's gravitational field using its monopole (mass), quadrupole moment, and rotational effects.
• Use a non-inertial, rotating reference frame co-moving with Earth's surface to compute local metric deviations.
• Compute the spatial metric components and analyze their anisotropy in the co-moving frame.
• Apply perturbation theory to include the effects of Earth's rotation and non-spherical mass distribution.
• Derive the effective spatial metric tensor and evaluate its anisotropy via the ratio of principal spatial components.

Experimental results

Research questions

• RQ1 What is the magnitude of local spacetime anisotropy in a frame co-moving with the Earth?
• RQ2 How do Earth's quadrupole moment and rotation contribute to spacetime anisotropy in the co-moving frame?
• RQ3 Can the predicted anisotropy be detected with current or foreseeable experimental techniques?
• RQ4 How does the inclusion of rotational and multipole effects modify the spatial geometry in the Earth's vicinity?

Key findings

• The local anisotropy of space in the Earth-centered, co-moving frame is predicted to be on the order of 10⁻¹² to 10⁻¹³.
• This anisotropy arises primarily from the combined effects of Earth's quadrupole moment and diurnal rotation.
• The spatial metric components deviate from isotropy due to the non-spherical mass distribution and rotation in the rotating frame.
• The predicted anisotropy is within the range of sensitivity for high-precision interferometric experiments such as those using laser ranging or atomic clocks.
• The result suggests that spacetime anisotropy in the co-moving frame is a measurable effect in principle, though extremely small.