Cosmological gravitomagnetism and Mach’s principle

Christoph Schmid
Phys. Rev. D 74, 044031 – Published 28 August 2006

Abstract

The spin axes of gyroscopes experimentally define local nonrotating frames, i.e. the time evolution of axes of inertial frames. But what physical cause governs the time evolution of gyroscope axes? We consider linear perturbations of Friedmann-Robertson-Walker (FRW) cosmologies with k=0, i.e. spatially flat. We ask the following question: Will cosmological vector perturbations (i.e. vorticity or rotational perturbations) exactly drag the spin axes of gyroscopes relative to the directions of geodesics to quasars in the asymptotic unperturbed FRW space? Using Cartan’s formalism with local orthonormal bases, we cast the laws of linear cosmological gravitomagnetism into a form showing the close correspondence with the laws of ordinary magnetism. Our results, valid for any equation of state and any form of the energy-momentum tensor for cosmological matter, are as follows: (1) the dragging of a gyroscope axis by rotational perturbations of matter beyond the H-dot radius from the gyroscope is exponentially suppressed, where H is the Hubble rate, and the dot is the derivative with respect to cosmic time. (2) If the perturbation of matter is a homogeneous rotation inside some radius around a gyroscope, then exact dragging of the gyroscope axis by the rotational perturbation is reached exponentially fast as the rotation radius gets larger than the H-dot radius. (3) For the most general linear cosmological perturbations, the time evolution of all gyroscope spin axes and the axis directions of all local inertial frames exactly follow a weighted average of the rotational motion of cosmological matter, i.e. there is exact frame-dragging everywhere. The weight function is the density of measured angular momentum of matter times (1/r) times the Yukawa force (d/dr)[(1/r)exp(μr)], where r is the geodesic distance from the source to the gyroscope. The exponential cutoff is given by μ2=4(dH/dt). Except for the Yukawa cutoff the weight function is the same as in the integrated form of Ampère’s law. Our results demonstrate (in first-order perturbations of any type for FRW cosmologies with k=0) the validity of Mach’s hypothesis that axes of local nonrotating frames precisely follow an average of the motion of cosmic matter.

  • Figure
  • Received 4 November 2005

DOI:https://doi.org/10.1103/PhysRevD.74.044031

©2006 American Physical Society

Authors & Affiliations

Christoph Schmid*

  • Institut für Theoretische Physik, ETH-Hönggerberg, CH-8093 Zürich

  • *Electronic address: chschmid@itp.phys.ethz.ch

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Issue

Vol. 74, Iss. 4 — 15 August 2006

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