References - Bogoliubov Laboratory of Theoretical Physics - JINR

Schwinger gauge. Probably for the first time introduced independently by Schwinger
[5] and Dirac [6] (and widely used in many works, including [7] and our current study),
this choice demands that the tetrad matrix e α i and its inverse ei α satisfy e�0 b =0,e 0 � b =0.
Landau-Lifshitz gauge (see, e.g., Ref. [8]) fixes the tetrad so that e �a 0 =0,ea �0 =0.
Symmetric gauge. Using the Minkowski flat metric gαβ =diag(c 2 , −1, −1, −1), we
. Now assume that the
can move the anholonomic index down and define eαi := gαβe β
i
resulting matrix is invariant under the transposition operation that can be symbolically
written as eαi = eiα. Such a tetrad was used by Pomeransky and Khriplovich [9] and
Dvornikov [10].
We choose the Schwinger gauge by specifying the coframe as (5). The other tetrads
are obtained from our e α i with the help of the Lorentz transformation e′α i =Λα βe β
i ,where
Λ α β =
� λ λqb/c
λcq a
δ a b +(λ − 1)qa qb/q 2
�
, q a = ξ WKa
, λ =
V
1
� . (10)
1 − q2 The constant ξ conveniently parametrizes different choices of tetrads. Namely, for ξ =1/2
the Lorentz matrix (10) transforms our tetrad to that of Pomeransky and Khriplovich,
and for ξ = 1 we obtain the tetrad of Landau and Lifshitz.
The spin precession in the gravitational field of rotating object is given by
where we denote
ρ =
2γ +1
γ +1
GM γ
r +
r3 γ +1
Ω = G
c 2 r 3
�
3r(r · J)
r2 �
− J + ρ × v
c2 , (11)
3G
c2r3 �
r
2ξ
· v)
(r · (J × v)) − J × v +(2ξ− 1)(r
r2 3 r2 �
J × r .
(12)
Putting ξ = 0, thus specifying to the Schwinger tetrad, we find that the classical formula
(11) perfectly reproduces the quantum result (9). If we choose another tetrad by putting
ξ =1/2 in (12), the equation (11) yields the result by Pomeransky and Khriplovich [9]
and Dvornikov [10] which differs from Eq. (9).
For particles in a rotating frame, the angular velocity of spin precession is given by
Ω = −ω + ξγ v × (v × ω)
γ +1 c2 . (13)
The correct result for the Schwinger gauge (ξ = 0) was first obtained in [11]. An
explanation of the dependence of the angular velocity of the spin precession on the gauge
of a tetrad was presented recently [12] on the basis of the Thomas precession.
The Lense-Thirring effect or frame dragging is one of the most impressive predictions
of the general relativity. This effect is currently analyzed in the Gravity Probe B experiment
[13]. However, relativistic corrections to the LT effect are not observable in this
experiment as well as in other experiments inside the solar system [14]. Nevertheless,
it is necessary to take the relativistic corrections to the LT precession into account for
the investigation of physical phenomena in the binary stars such as pulsar systems. In
this case, both components of a system undergo a mutual LT precession about the total
angular momentum J. Since the spin precession effects are well observable [15], the use of
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