Mach's principle: Exact framedragging via gravitomagnetism in perturbed FriedmannRobertsonWalker universes with K=({+}1,0)
Abstract
We show that there is exact dragging of the axis directions of local inertial frames by a weighted average of the cosmological energy currents via gravitomagnetism for all linear perturbations of all FriedmannRobertsonWalker (FRW) universes and of Einstein's static closed universe, and for all energymomentumstress tensors and in the presence of a cosmological constant. This includes FRW universes arbitrarily close to the Milne Universe and the de Sitter universe. Hence the postulate formulated by Ernst Mach about the physical cause for the timeevolution of inertial axes is shown to hold in general relativity for linear perturbations of FRW universes.  The timeevolution of local inertial axes (relative to given local fiducial axes) is given experimentally by the precession angular velocity {omega}vector{sub gyro} of local gyroscopes, which in turn gives the operational definition of the gravitomagnetic field: Bvector{sub g}{identical_to}2{omega}vector{sub gyro}. The gravitomagnetic field is caused by energy currents Jvector{sub {epsilon}} via the momentum constraint, Einstein's G{sup 0}circumflex{sub icircumflex} equation, ({delta}+{mu}{sup 2})Avector{sub g}=16{pi}G{sub N}Jvector{sub {epsilon}} with Bvector{sub g}=curl Avector{sub g}. This equation is analogous to Ampere's law, but it holds for all timedependent situations. {delta} is the de RhamHodge Laplacian, and {delta}=curl curl for the vorticity sector in Riemannian 3space.  Inmore »
 Authors:

 ETH Zurich, Institute for Theoretical Physics, 8093 Zurich (Switzerland)
 Publication Date:
 OSTI Identifier:
 21266378
 Resource Type:
 Journal Article
 Journal Name:
 Physical Review. D, Particles Fields
 Additional Journal Information:
 Journal Volume: 79; Journal Issue: 6; Other Information: DOI: 10.1103/PhysRevD.79.064007; (c) 2009 The American Physical Society; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 05562821
 Country of Publication:
 United States
 Language:
 English
 Subject:
 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; ANGULAR VELOCITY; BESSEL FUNCTIONS; COSMOLOGICAL CONSTANT; DE SITTER GROUP; DECOMPOSITION; EUCLIDEAN SPACE; FIELD THEORIES; GENERAL RELATIVITY THEORY; GRAVITATION; GYROSCOPES; LAPLACIAN; MATHEMATICAL SOLUTIONS; PERTURBATION THEORY; PRECESSION; SPHERICAL CONFIGURATION; SPHERICAL HARMONICS; TIME DEPENDENCE; UNIVERSE; VECTOR FIELDS
Citation Formats
Schmid, Christoph. Mach's principle: Exact framedragging via gravitomagnetism in perturbed FriedmannRobertsonWalker universes with K=({+}1,0). United States: N. p., 2009.
Web. doi:10.1103/PHYSREVD.79.064007.
Schmid, Christoph. Mach's principle: Exact framedragging via gravitomagnetism in perturbed FriedmannRobertsonWalker universes with K=({+}1,0). United States. doi:10.1103/PHYSREVD.79.064007.
Schmid, Christoph. Sun .
"Mach's principle: Exact framedragging via gravitomagnetism in perturbed FriedmannRobertsonWalker universes with K=({+}1,0)". United States. doi:10.1103/PHYSREVD.79.064007.
@article{osti_21266378,
title = {Mach's principle: Exact framedragging via gravitomagnetism in perturbed FriedmannRobertsonWalker universes with K=({+}1,0)},
author = {Schmid, Christoph},
abstractNote = {We show that there is exact dragging of the axis directions of local inertial frames by a weighted average of the cosmological energy currents via gravitomagnetism for all linear perturbations of all FriedmannRobertsonWalker (FRW) universes and of Einstein's static closed universe, and for all energymomentumstress tensors and in the presence of a cosmological constant. This includes FRW universes arbitrarily close to the Milne Universe and the de Sitter universe. Hence the postulate formulated by Ernst Mach about the physical cause for the timeevolution of inertial axes is shown to hold in general relativity for linear perturbations of FRW universes.  The timeevolution of local inertial axes (relative to given local fiducial axes) is given experimentally by the precession angular velocity {omega}vector{sub gyro} of local gyroscopes, which in turn gives the operational definition of the gravitomagnetic field: Bvector{sub g}{identical_to}2{omega}vector{sub gyro}. The gravitomagnetic field is caused by energy currents Jvector{sub {epsilon}} via the momentum constraint, Einstein's G{sup 0}circumflex{sub icircumflex} equation, ({delta}+{mu}{sup 2})Avector{sub g}=16{pi}G{sub N}Jvector{sub {epsilon}} with Bvector{sub g}=curl Avector{sub g}. This equation is analogous to Ampere's law, but it holds for all timedependent situations. {delta} is the de RhamHodge Laplacian, and {delta}=curl curl for the vorticity sector in Riemannian 3space.  In the solution for an open universe the 1/r{sup 2}force of Ampere is replaced by a Yukawa force Y{sub {mu}}(r)=(d/dr)[(1/R)exp({mu}r)], formidentical for FRW backgrounds with K=(1,0). Here r is the measured geodesic distance from the gyroscope to the cosmological source, and 2{pi}R is the measured circumference of the sphere centered at the gyroscope and going through the source point. The scale of the exponential cutoff is the Hdot radius, where H is the Hubble rate, dot is the derivative with respect to cosmic time, and {mu}{sup 2}=4(dH/dt). Analogous results hold in closed FRW universes and in Einstein's closed static universe.We list six fundamental tests for the principle formulated by Mach: all of them are explicitly fulfilled by our solutions.We show that only energy currents in the toroidal vorticity sector with l=1 can affect the precession of gyroscopes. We show that the harmonic decomposition of toroidal vorticity fields in terms of vector spherical harmonics Xvector{sub lm}{sup } has radial functions which are formidentical for the 3sphere, the hyperbolic 3space, and Euclidean 3space, and are formidentical with the spherical Bessel, Neumann, and Hankel functions.  The Appendix gives the de RhamHodge Laplacian on vorticity fields in Riemannian 3spaces by equations connecting the calculus of differential forms with the curl notation. We also give the derivation the Weitzenboeck formula for the difference between the de RhamHodge Laplacian {delta} and the ''rough'' Laplacian {nabla}{sup 2} on vector fields.},
doi = {10.1103/PHYSREVD.79.064007},
journal = {Physical Review. D, Particles Fields},
issn = {05562821},
number = 6,
volume = 79,
place = {United States},
year = {2009},
month = {3}
}