Induced Holonomy Groups for Thomas Precession on a Sphere

​
γ
1.5
ϕ
-0.5
Δϕ
0.5
θ
1
Δθ
-0.2
This Demonstration calculates the Thomas precession induced when a point in constant-speed motion is transported around a spherical triangle on the sphere of constant
γ
, where
γ=
-1/2
(1-
2
v
/
2
c
)
is the Lorentz factor. The sphere represents the directions of velocity vectors at constant speed (constant
γ
). The spherical triangle is defined by the north pole and two other points with colatitude and longitude coordinates
(θ-Δθ,ϕ-Δϕ)
and
(θ+Δθ,ϕ+Δϕ)
. This represents a cycle of motion at constant speed. You can change
θ
,
Δθ
,
ϕ
,
Δϕ
to move the spherical triangle, which is plotted on the surface of the constant-speed sphere. The Thomas precession of the coordinate frame induced by transport around the loop is then shown as green, blue, and purple coordinate frames.

Details

Snapshot 1 shows the Thomas precession around a loop with
γ=3
(i.e. when the speed is
v=2
2
c/3
). In this case, the holonomy group induced by Thomas precession is trivial, just as it is when
γ=1
(nonrelativistic motion) and Thomas precession induced by torque-free motion state transport on the sphere's surface is path independent.
Snapshot 2 shows the case
γ=2
(i.e. when the speed is
v=
3
c/2
), when the holonomy group collapses to the group
SO(2)≅U(1)
of rotations about the lone axis defined by the initial velocity direction. The case
γ=1
is, of course, still trivial. At all other values of
γ
(i.e.
γ∉{1,2,3}
), the holonomy group is the whole of
SO(3)
.
Snapshot 3 shows the
γ=4
case, when there is a large rotation of the coordinate frame induced by Thomas precession arising from motion state transport around the spherical triangle.

References

[1] C. W. Misner, K. S. Thorne, and J. A. Wheeler, Gravitation, San Francisco: W. H. Freeman, 1973 Section 6.6.
[2] A. Dragan and T. Odrzygóźdź, "A Half-Page Derivation of the Thomas Precession," American Journal of Physics, 81(631), 2013 p. 631–632. doi:10.1119/1.4807564.

External Links

Geodesic Precession on a Timelike Circular Orbit around a Schwarzschild Black Hole
Thomas Precession in Accelerated Planar Motion

Permanent Citation

Selene Rodd-Routley
​
​"Induced Holonomy Groups for Thomas Precession on a Sphere"​
​http://demonstrations.wolfram.com/InducedHolonomyGroupsForThomasPrecessionOnASphere/​
​Wolfram Demonstrations Project​
​Published: October 26, 2015