References - Bogoliubov Laboratory of Theoretical Physics - JINR

REMARK ON SPIN PRECESSION FORMULAE
Lewis Ryder
School of Physical Sciences, University of Kent, Canterbury CT2 8EN, UK
Abstract
It is common to represent spin as a 4-vector in relativistic formulations, but
from a fundamental point this is incorrect : spin should be represented by a rank
2 antisymmetric tensor. Such a relativistic tensor is displayed, which for Dirac
particles turns out to be the Foldy-Wouthuysen mean spin operator. The precession
formula for such a rank 2 tensor is shown to be the same as that for a 4-vector.
As is well known from undergraduate physics, a vector a, fixed in a rotating body,
when viewed from an inertial frame has a time dependence
Da
Dt
= da
dt
+ Ω ∧ a.
If spin is described by a 3-vector S, the same formula holds for its rate of change in a
rotating frame:
DS dS
= + Ω ∧ S. (1)
Dt dt
This is the non-relativistic formula for spin precession. To obtain a relativistic formula
the usual procedure (see for example Weinberg [1]) is to replace the 3-vector S by a 4vector
Sμ =(S0, S). With this formulation, the precession rate of a spinning object in a
satellite in orbit round the Earth is given by
Ω = ΩdeSitter + ΩLense−Thirring = 3GM
2c2 GI
r × v +
r3 c2 · r)r
{3(ω
r3 r2 − ω}.
and this formula for Lense-Thirring and de Sitter precession agrees well with observations.
It is useful to rewrite the formulae above in coordinate notation. If a body rotates
about its z axis with angular velocity ω the coordinates proper to the body are
x ′ =cosωt − y sin ωt, y ′ = x sin ωt + y cos ωt, z ′ = z, t ′ = t,
so the invariant interval in Minkowski spacetime is (with ρ 2 = x 2 + y 2 )
ds 2 =
�
1 − ω2ρ2 c2 �
c 2 dt 2 − dx 2 − dy 2 − dz 2 +2ω(ydxdt− xdydt).
Equivalently, with ds 2 = gμνdx μ dx ν , the metric tensor is
gμν =
⎛
⎜
⎝
1 − ω2 ρ 2
c 2 ωy −ωx 0
ωy −1 0 0
−ωx 0 −1 0
0 0 0 −1
451
⎞
⎟
⎠ .