# Fermi-Walker transport and gyroscopes

In GR, the geodesic equation for a test particle can be seen in 2 ways.

(1) It is a fundamental postulate consistent with the EP that is experimentally verified.

(2) It is derived from more fundamental postulates such as the Hilbert action with minimally coupled matter as an approximation for very small extended bodies.

Can Fermi-Walker transport of a gyroscope along a worldline be seen in the same 2 ways, as either a fundamental postulate consistent with the EP, or derived as the small body approximation for a rotating extended body? In the second case, googling seems to indicate that the equations for the extended body are the Mathisson-Papapetrou equations. Is that right, or are there other derivations?

I can't see F-W transport as coming from a postulate in the sense of the geodesic principle. With the geodesic principle we are postulating that freely falling test particles/observers have geodesic worldlines that is that their 4-velocity ##\xi^{a}## automatically satisfies ##\xi^{b}\nabla_{b}\xi^{a} = 0## by hypothesis. Of course there are treatments where the geodesic principle is instead a theorem resulting from some other assumptions (you mentioned one derivation; Geroch and Jang have another way of deriving it: http://jmp.aip.org/resource/1/jmapaq/v16/i1/p65_s1?isAuthorized=no [Broken])

F-W transport on the other hand isn't saying that a certain class of observers automatically Fermi transports the spatial basis vectors (the gyroscopes) of an initial Lorentz frame by hypothesis. It just tells us what an arbitrary observer has to do in order for the gyroscopes to remain stabilized throughout i.e. for there to be no local rotation of the spatial basis vectors.

Last edited by a moderator:
I vote for the second alternative. Fermi-Walker transport is a consequence of the equations of motion for an extended body in the small body limit.

An observer can always construct a set of Fermi-Walker transported axes, but the statement that a small body with no external torques follows these axes is nontrivial.

1 person
Is Fermi-Walker transport relevant to quantum entanglement? Two particles are entangled, then particle A remains in place while particle B is transported to a distant location. When the spin of particle A is measured to be spin up, the spin of particle B must be spin down - but along which axis? Does the axis depend on the path we used to transport particle B to its new location?

1 person
Bill_K, thanks for the viewpoint in post #3. I've just come across an article by Pirani, first published in 1956, and recently republished in 2009 as a "golden oldie". In that article, he is advocating the use of tetrads, and justifies the physical significance of Fermi-Walker transport by starting from the Mathisson-Papapetrou model for a spinning top, and showing that if internal angular momentum is conserved, the angular momentum is Fermi-Walker transported. Apparently Pirani wrote a children's book that was condemned for inciting alcoholism and violence, and another book for young people in favour of nuclear power.

Is Fermi-Walker transport relevant to quantum entanglement? Two particles are entangled, then particle A remains in place while particle B is transported to a distant location. When the spin of particle A is measured to be spin up, the spin of particle B must be spin down - but along which axis? Does the axis depend on the path we used to transport particle B to its new location?

Something like http://physics.aps.org/synopsis-for/10.1103/PhysRevLett.108.230404, but with 2 particles http://arxiv.org/abs/0705.0173? The paper for the gravitational Aharonov-Bohm proposal is free at http://arxiv.org/abs/1109.4887.

Edit:These seem closer:

http://arxiv.org/abs/quant-ph/0307114
Einstein-Podolsky-Rosen correlation in gravitational field
Hiroaki Terashima, Masahito Ueda
Phys.Rev. A69 (2004) 032113. DOI: 10.1103/PhysRevA.69.032113

http://arxiv.org/abs/0902.2366
Relativistic Einstein-Podolsky-Rosen Correlations in curved spacetime via Fermi-Walker Transport
Knut Bakke, Alexandre M. de M. Carvalho, Claudio Furtado
Int. J. Quantum Inform. 08, 1277 (2010). DOI: 10.1142/S0219749910006952

http://arxiv.org/abs/1108.3896
Localized qubits in curved spacetimes
Matthew C. Palmer, Maki Takahashi, Hans F. Westman
Annals of Physics 327 (2012) pp. 1078-1131. DOI: 10.1016/j.aop.2011.10.009

Last edited:
I can't see F-W transport as coming from a postulate in the sense of the geodesic principle. With the geodesic principle we are postulating that freely falling test particles/observers have geodesic worldlines that is that their 4-velocity ##\xi^{a}## automatically satisfies ##\xi^{b}\nabla_{b}\xi^{a} = 0## by hypothesis. Of course there are treatments where the geodesic principle is instead a theorem resulting from some other assumptions (you mentioned one derivation; Geroch and Jang have another way of deriving it: http://jmp.aip.org/resource/1/jmapaq/v16/i1/p65_s1?isAuthorized=no [Broken])

F-W transport on the other hand isn't saying that a certain class of observers automatically Fermi transports the spatial basis vectors (the gyroscopes) of an initial Lorentz frame by hypothesis. It just tells us what an arbitrary observer has to do in order for the gyroscopes to remain stabilized throughout i.e. for there to be no local rotation of the spatial basis vectors.

I vote for the second alternative. Fermi-Walker transport is a consequence of the equations of motion for an extended body in the small body limit.

An observer can always construct a set of Fermi-Walker transported axes, but the statement that a small body with no external torques follows these axes is nontrivial.

Thanks, that seems provably right. Since a classical spin couples to curvature, it seems only an approximation to say that a free falling observer can use gyroscopes to align his spatial basis vectors so that they are Fermi-Walker transported - one can't have a free falling spin, just as one can't have a free falling charge.

For example, Obukhov, Silenko and Teryaev write: "In the present work, we do not consider a relatively weak influence of the spin on particle’s trajectory produced by the Mathisson-Papapetrou force which results in a weak violation of the equivalence principle by the curvature-dependent terms."

Incidentally, Dixon wrote a somewhat recent and really interesting review of Mathisson's work and the developments that followed: http://www.actaphys.uj.edu.pl/sup1/pdf/s1p0027.pdf [Broken]. It was part of a conference about Mathisson: http://www.actaphys.uj.edu.pl/supplement.htm [Broken].

Last edited by a moderator: