De Felice F., Bini D. Classical Measurements in Curved Space-Times

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10.1 Behavior of spin in general space-times 233

so that, acting on both sides with P (u), we ﬁnd, from (10.1),

(a(U) · S)γν =

(fw,U,u)

dτ

Σ(U, u) −(ν(U, u) ·Σ(U, u))F

(G)

(fw,U,u)

, (10.9)

recalling the deﬁnition (3.156) of the Fermi-Walker gravitational force

dτ

= −F

(G)

(fw,U,u)

. (10.10)

Let us evaluate the term (a(U ) · S). We have

a(U) · S = a(U) · Σ(U, u)+(ν(U, u) · Σ(U, u))u · a(U)

= γF(U, u) · Σ(U, u)

+(ν(U, u) ·Σ(U, u))u ·∇

U. (10.11)

But

u ·∇

U = −

dγ

dτ

+ γν ·F

(G)

(fw,U,u)

= −γF(U, u) · ν, (10.12)

wherewehaveusedEq.(6.85), that is,

dγ

dτ

= γ[F (U, u)+F

(G)

(fw,U,u)

] ·ν(U, u). (10.13)

Hence

a(U) · S = γ [F (U, u) ·Σ(U, u) − (ν(U, u) · Σ(U, u))F (U, u) ·ν] . (10.14)

Using this in (10.9) and setting ν(U, u)=ν and F (U, u)=F to simplify notation,

we obtain

(fw,U,u)

dτ

Σ(U, u)=(ν ·Σ(U, u))F

(G)

(fw,U,u)

+(a(U) · S)γν

=(ν ·Σ(U, u))F

(G)

(fw,U,u)

+(F · Σ(U, u) − (ν · Σ(U, u))F · ν)γ

ν. (10.15)

Let us now consider the evolution of the magnitude of Σ(U, u), that is ||Σ(U, u)||,

and of its unit direction

Σ(U, u) ≡

Σ separately. Using (10.7) we ﬁnd

dτ

ln ||Σ(U, u)|| =(ν ·

Σ)



χ + F

(G)

(fw,U,u)



, (10.16)

where

χ = F ·

Σ −(ν ·

Σ)(F · ν). (10.17)

Using the relative standard time parameterization, we deduce

(fw,U,u)

dτ

(U,u)

Σ=γχ



Σ ×

(ν ×

Σ)



(

Σ ·ν)



Σ ×

(G)

(fw,U,u)

Σ)



. (10.18)

234 Measurements of spinning bodies

We can also cast (10.18)intheform

(fw,U,u)

dτ

(U,u)

Σ=ζ

(project)

(fw)

Σ, (10.19)

where

(project)

(fw)

= γχ(

Σ ×

ν)+

(ν ·

Σ)



Σ ×

(G)

(fw,U,u)



Σ ×



γχν +

(ν ·

Σ)F

(G)

(fw,U,u)



(10.20)

is the precession rate of the gyroscope as it would be locally measured by u.

Boosted spin vector

Let us examine now the complementary problem of describing the precession of

a gyroscope as it would be measured by the observer U who carries it, relative to

a frame, say {e(u)

}, adapted to the observers of the congruence C

. According

to U, the axes e(u)

are moving axes; hence their orientation in LRS

is deﬁned

by boosting them onto LRS

itself, that is

E(U)

= B

(lrs)

(U, u)e(u)

. (10.21)

Our aim here is to study the precession rate of the spin vector S with respect to

the axes E(U)

. However, since the boost is an isometry, the precession of S with

respect to the axes E(U)

is the same as the precession of the boosted vector

S(U, u)=B

(lrs)

(u, U)S, (10.22)

momentarily at rest with respect to u, as measured with respect to the axes

e(u)

. Using the representation (3.143) of the boost, we have

S(U, u)=Σ(U, u) −

1+γ

(ν(U, u) · Σ(U, u))ν(U, u). (10.23)

The evolution of S(U, u) along U is a consequence of the similar result for Σ(U, u)

and can be cast in the form

(fw,U,u)

dτ

(U,u)

S(U, u)=ζ

(boost)

(fw)

S(U, u), (10.24)

where ζ

(boost)

(fw)

is made up of two terms:

(boost)

(fw)

= ζ

(Thomas)

+ ζ

(geo)

, (10.25)

where

(Thomas)

= −

1+γ

ν(U, u) ×

F (U, u), (10.26)

10.1 Behavior of spin in general space-times 235

and

(geo)

1+γ

ν(U, u) ×

(G)

(fw,U,u)

. (10.27)

The Thomas precession arises from the acceleration of the world line of the

observer who carries the gyroscope and is due to the non-gravitational force

F (U, u) deﬁned in (6.75). The geodesic precession is due to the gravitational

force F

(G)

(fw,U,u)

only.

Let us derive (10.24). Start from (10.23) and replace Σ(U, u) using (10.6)to

obtain

S(U, u)=S −

[ν(U, u) ·Σ(U, u)]

1+γ

(U + u)

= S −

γ[ν(U, u) ·S(U, u)]

1+γ

(U + u), (10.28)

because

ν(U, u) · Σ(U, u)=γ[ν(U, u) ·S(U, u)]. (10.29)

Let f denote the quantity

f =

γ[ν(U, u) ·S(U, u)]

1+γ

. (10.30)

Diﬀerentiate both sides of (10.28) along U:

dτ

S(U, u)=

dτ

− (u + U)

dτ

−



a(U)+

dτ



=(a(U ) ·S)U − (u + U)

dτ

−



a(U) − F

(G)

(fw,U,u)



f, (10.31)

where we have used the properties of S of being orthogonal to U and of being

Fermi-Walker transported along it, that is

(fw,U )

dτ

S =

dτ

S − (a(U) · S)U =0. (10.32)

From (10.28)wehave

a(U) · S = a(U) ·S(U, u)+(u · a(U))f. (10.33)

But

u ·a(U)=u ·∇

U = −

dγ

dτ

+ γν(U, u) ·F

(G)

(fw,U,u)

, (10.34)

so that Eq. (10.13) gives

u ·a(U)=−γν(U, u) ·F (U, u). (10.35)

236 Measurements of spinning bodies

Therefore

dτ

S(U, u)=[a(U) ·S(U, u) − fγν(U, u) · F (U, u)] U

−

dτ

(u + U) −



a(U) − F

(G)

(fw,U,u)



= γ [F (U, u) ·S(U, u) −fν(U, u) ·F (U, u)] U

−(u + U )

dτ

−



a(U) − F

(G)

(fw,U,u)



f. (10.36)

First project both sides of (10.36) orthogonally to u, obtaining

(fw,U,u)

dτ

S(U, u)=γ

[F (U, u) ·S(U, u)

−fν(U, u) ·F (U, u)] ν(U, u)

−γν

dτ



−γF + F

(G)

(fw,U,u)



f. (10.37)

Then take the scalar product of both sides of (10.36) with u, to give

u ·

dτ

S(U, u)=−γ

(F (U, u) ·S(U, u) −fν(U, u) · F (U, u))

dτ

(γ +1)− (u · a(U)) f, (10.38)

that is, since u ·S(U, u)=0,

(G)

(fw,U,u)

·S(U, u)=−γ

(F (U, u) ·S(U, u)

−fν(U, u) ·F (U, u))

dτ

(γ +1)

+ fγν(U, u) · F (U, u)

= −γ

F (U, u) ·S(U, u)

+ γ(fν(U, u) · F (U, u))(1 + γ)

dτ

(γ +1). (10.39)

From this it follows that

dτ

=(1+γ)

−1



(G)

(fw,U,u)

+ γ

F (U, u)



·S(U, u)

−γfν(U, u) ·F(U, u). (10.40)

Substituting this expression into (10.37), we obtain the ﬁnal result,

(fw,U,u)

dτ

S(U, u)=

1+γ

[ν(U, u) ⊗S(U, u) −ν(U, u) ·S(U, u)P (u)]



γF(U, u) −F

(G)

(fw,U,u)



, (10.41)

which is equivalent to (10.24).

10.2 Motion of a test gyroscope in a weak gravitational ﬁeld 237

If the frame {e(u)

} is orthonormal, then we can further manipulate (10.24).

In this case we have

(fw,U,u)

dτ

(U,u)

S(U, u)

ˆa

dS(U, u)

ˆa

dτ

(U,u)



(fw)

ˆa

ˆc

+Γ

ˆa

ˆc

ν(U, u)



S(U, u)

ˆc

. (10.42)

Introducing the quantities

ˆa

(fw)

= −

η(u)

ˆa

bˆc

(fw)

bˆc

(10.43)

and the spatial curvature angular velocity

ˆa

(sc)

= −

η(u)

ˆa

bˆc

[ˆc|

ν(U, u)

, (10.44)

already deﬁned in (4.18) and (4.20) respectively, we have

(fw,U,u)

dτ

(U,u)

S(U, u)

ˆa

dS(U, u)

dτ

(U,u)

+[(ζ

(fw)

+ ζ

(sc)

) ×

S(U, u)]

ˆa

, (10.45)

and hence

dS(U, u)

ˆa

dτ

(U,u)



(boost)

(fw)

− ζ

(fw)

− ζ

(sc)



S(U, u)



ˆa

≡



(tot)

S(U, u)



ˆa

, (10.46)

or equivalently

dS(U, u)

ˆa

dτ

(U,u)

= η(u)

ˆa

bˆc

(tot)

S(U, u)

ˆc

. (10.47)

Note that a change of parameter along U implies a rescaling of the precession

angular velocity. For instance, the above formula can also be written as

dS(U, u)

ˆa

dτ

= η(u)

ˆa

bˆc

[γζ

(tot)

]

S(U, u)

ˆc

, (10.48)

when the world line U is parameterized by the proper time τ

.Inthiscasethe

eﬀective total angular velocity is

(tot)

= γζ

(tot)

. (10.49)

10.2 Motion of a test gyroscope in a weak gravitational ﬁeld

Let us consider the post-Newtonian (PN) treatment of a weak gravitational ﬁeld

within general relativity and evaluate the orders of magnitude of the quantities we

shall meet in our treatment in terms of powers of 1/c, c being the velocity of light

in vacuum. In a PN coordinate system, say (t, x

)(a =1, 2, 3), the space-time

metric can be written as

238 Measurements of spinning bodies

= −(1 − 2Φ)dt

+2Φ

dtdx

+ (1 + 2Φ)δ

+ O(4), (10.50)

where Φ = O(2) and Φ

= O(3). We use the three-dimensional notation



Φ=Φ

∂

+Φ

∂

+Φ

∂

(10.51)

and the ordinary Euclidean space operations for grad, curl, div, and vector prod-

uct, unless otherwise speciﬁed. Let u be the family of observers at rest with

respect to the coordinate grid

u =

√

−g

∂

=(1+Φ)∂

+ O(4), (10.52)

so that



=(−1+Φ)dt +Φ

+ O(4). (10.53)

Let us ﬁx an orthonormal frame adapted to u as follows:

ˆa

=Φ

∂

+(1− Φ)∂

,ω

ˆa

=(1+Φ)δ

ˆa

. (10.54)

The observers u have acceleration

a(u)=−grad

Φ+O(4) ; (10.55)

they also have vorticity

ω(u)=

curl



Φ+O(4) (10.56)

and expansion

θ(u)

= ∂

Φ δ

+ O(4). (10.57)

A direct calculation shows that

∇

ˆa

= −∂

Φ u + ω(u) ×

ˆa

+ O(4)

= a(u)

u + ω(u) ×

ˆa

+ O(4), (10.58)

and hence

(fw)

ˆa

= ω(u) ×

ˆa

, (10.59)

which implies

ω(u)=ζ

(fw)

. (10.60)

Similarly, the non-vanishing spatial Ricci rotation coeﬃcients

ˆc



∇

ˆa



=Γ

ˆc

ˆa

(10.61)

We note that for these three ﬁelds, that is acceleration, vorticity, and expansion, the

coordinate and frame components coincide.

10.3 Motion of a test gyroscope in Schwarzschild space-time 239

are as follows:

ˆx

ˆyˆy

=Γ

ˆx

ˆzˆz

= −Γ

ˆy

ˆxˆy

= −Γ

ˆz

ˆxˆz

= −∂

Φ,

ˆy

ˆxˆx

=Γ

ˆy

ˆzˆz

= −Γ

ˆx

ˆyˆx

= −Γ

ˆz

ˆyˆz

= −∂

Φ,

ˆz

ˆxˆx

=Γ

ˆz

ˆyˆy

= −Γ

ˆx

ˆzˆx

= −Γ

ˆy

ˆz ˆy

= −∂

Φ. (10.62)

If U = γ[u + ν(U, u)

ˆa

] is the 4-velocity of the observer carrying the gyroscope

and we assume that it satisﬁes the slow motion condition ||ν(U, u)||  1, then

we can evaluate the spatial curvature angular velocity

ˆc

ˆaˆc

= −

ˆa

(sc)

, (10.63)

and ﬁnd that

(sc)

= −ν ×

grad

Φ. (10.64)

Finally, we have

(boost)

(fw)

= −

ν ×

F −

ν ×

a(u)=−

ν ×

F +

ν ×

grad

Φ, (10.65)

so that the precession angular velocity of (10.46) becomes

(tot)

= −

ν ×

F +

ν ×

grad

+ ν ×

grad

Φ −

curl



= −

ν ×

F +

ν ×

grad

−

curl



Φ. (10.66)

When a gyroscope moves along a geodesic (F = 0), Eq. (10.66) is known as the

Schiﬀ formula;thatis,

(tot)

ν ×

grad

Φ −

curl



Φ. (10.67)

The part of ζ

(tot)

which is contributed by the intrinsic spin of the gravity source

is responsible for a relativistic eﬀect implied by the Lense-Thirring metric, and

known as the Lense-Thirring eﬀect.

10.3 Motion of a test gyroscope in Schwarzschild space-time

We shall now study the motion of a test gyroscope in Schwarzschild space-time.

Our aim here is to deduce the precession of a gyroscope carried by an observer U,

but with respect to a frame e(m)

given by (8.18), adapted to the static observers.

In this way we can specialize the general formulas given above.

The evolution of the boosted spin vector S(U, m) along U canbecastinthe

form

(fw,U,m)

dτ

(U,m)

S(U, m)=ζ

(boost)

(fw)

(U, m) ×

S(U, m), (10.68)

240 Measurements of spinning bodies

where

(boost)

(fw)

= ζ

(Thomas)

+ ζ

(geo)

1+γ

ν(U, m) ×



−F (U, m)+γ

−1

(G)

(fw,U,m)



1+γ

ν(U, m) ×

[−F (U, m) −a(m)] . (10.69)

(boost)

(fw)

has a component only along the latitudinal frame direction

θ,

(boost)

(fw)

1+γ

ν(U, m)



−F (U, m)

ˆr

− a(m)

ˆr



, (10.70)

with

F (U, m)

ˆr

=Γ



− rζ





− rζ



a(m)

ˆr



1 −



−1/2

, (10.71)

with

N =



1 −



1/2

. (10.72)

We then have

(boost)

(fw)

1+γ

ν(U, m)



−



− rζ



−



= −γ

ν(U, m)

rγ

1+γ

ν(U, m)

. (10.73)

Recalling that

ν(U, m)

rζ

, (10.74)

we have

(boost)

(fw)

= −γ

Mζ

1+γ

. (10.75)

Moreover,

1+γ

||ν(U, m)||

1+γ

− 1

1+γ

= γ −1. (10.76)

Therefore

(boost)

(fw)

= −γ

Mζ

+ ζ(γ − 1) = ζγ



−



− ζ

ζγ



1 −



− ζ. (10.77)

10.4 Motion of a spinning body in Schwarzschild space-time 241

Taking into account now that ζ

(fw)

= 0 and that, from (8.48) with θ = π/2,

(sc)

= −ζe

, (10.78)

we have the Shiﬀ formula in Schwarzschild space-time,

(tot)

= ζ

(boost)

(fw)

− ζ

(sc)

ζγ



1 −



. (10.79)

Finally, from (8.69), we see that

(tot)

(U)

. (10.80)

As a convention the physical (orthonormal) component along −∂

, perpendicular

to the equatorial plane, is taken to be along the positive z-axis, denoted by ˆz.

Therefore,

(tot) ˆz

= −

(U)

. (10.81)

This result agrees with the expression for

ζ given in Section 9.2. In fact, assuming

U is a geodesic parameterized by the proper time and taking into account the

discussion at the end of Section 10.1 we have

ζ ≡ ζ

(tot)

ˆz

= −τ

(U)=Γ



− 1



, (10.82)

where the expression (8.69)forτ

(U) has been used. Finally, introducing the

proper frequency

ζ =Γζ yields

ζ =Γ



− 1



. (10.83)

10.4 Motion of a spinning body in Schwarzschild space-time

The equations of motion are given by (10.2) and (10.3) coupled to the supple-

mentary condition (10.4), where, as stated, P

is the total 4-momentum of the

particle and S

μν

is its (antisymmetric) spin tensor. In those equations, U is the

time-like unit tangent vector of the center-of-mass line used to make the multipole

reduction. Equation (10.2) is also referred to as the Riemann force equation.

Let the Schwarzschild metric be written in spherical-type coordinates (8.1),

and introduce an orthonormal frame adapted to the static observers:

=(1− 2M/r)

−1/2

∂

ˆr

=(1− 2M/r)

1/2

∂

r sin θ

∂

, (10.84)

242 Measurements of spinning bodies

with dual

=(1− 2M/r)

1/2

dt, ω

ˆr

=(1− 2M/r)

−1/2

dr,

= rdθ, ω

= r sin θdφ. (10.85)

Let us assume that U is tangent to a (time-like) spatially circular orbit, which

we here denote as a U-orbit, with

U =Γ[∂

+ ζ∂

]=γ[e

+ νe

],γ=(1− ν

)

−1/2

. (10.86)

ζ is the angular velocity of the orbital revolution as it would be measured at

inﬁnity, ν is the magnitude of the local proper linear velocity measured in the

frame (10.84), and Γ is a normalization factor given by

Γ=



−g

− ζ

φφ



−1/2

(10.87)

in order to ensure that U · U = −1. The angular velocity ζ is related to ν by

ζ =



−

φφ

ν. (10.88)

Here ζ and therefore also ν are assumed to be constant along the U-orbit. We

limit our analysis to the equatorial plane (θ = π/2) of the Schwarzschild solution;

again, as a convention, the physical (orthonormal) component along −∂

, per-

pendicular to the equatorial plane, will be taken to be along the positive z-axis

and will be indicated by ˆz, as necessary.

Among the circular orbits analyzed in detail in Section 8.2, particular attention

is devoted to the time-like spatially circular geodesics which corotate (ζ

)and

counter-rotate (ζ

−

) relative to a pre-assigned positive sense of variation of the

azimuthal coordinate φ. They have respective angular velocities ζ

≡±ζ

±(M/r

)

1/2

, so that

=Γ

[∂

+ ζ

∂

]=γ

± ν

], (10.89)

where



r − 2M



1/2

,γ



r − 2M

r − 3M



1/2

, (10.90)

and with the time-like condition ν

< 1 satisﬁed if r>3M. Here γ

is the local

(geodesic) Lorentz factor relative to the frame (10.84).

Let us now introduce the Lie relative curvature (8.66)ofeachU-orbit,

(lie)

≡ k

(lie)

ˆr

= −∂

ˆr

√

φφ

= −



1 −



1/2

= −

, (10.91)

as well as a Frenet-Serret intrinsic frame along U, deﬁned by

= U, E

ˆr

= e

ˆr

ˆz

= e

ˆz

= γ[νe

+ e

], (10.92)