What forces act in relativistic gyroscope precession?

What forces act in relativistic gyroscope precession? Oldˇrich Semer´ak∗ Department of Theoretical Physics, Faculty of Mathematics and Physics, Charles University, V Holeˇsoviˇck´ ach 2, 180 00 Praha 8, Czech Republic

Abstract The translation of the relativistic motion into the language of forces, proposed in [27], is employed to interpret the gyroscope precession in general relativity. The precession is referred to the comoving Frenet triad built up along the projection of the gyroscope’s trajectory onto the 3-space of the local hypersurface-orthogonal observer. The contributions of the centrifugal, the gravitational, and the dragging+Coriolis forces are identified respectively with the Thomas, the geodetic, and the gravitomagnetic components of precession. Explicit expressions are given for several simple types of motion in the Kerr (or simpler) field in order to show that the general formulas obtained are not only very simple, but also yield clear results in accord with intuition in concrete situations.

PACS numbers: 04.20.-q, 95.30.Sf, 97.60.Lf

∗

Electronic address: [email protected]

1

1

Introduction

The study of the gyroscopic precession in general relativity has a long history and represents many different approaches. There are three specifically relativistic contributions to this phenomenon: • The special relativistic Thomas precession [1], arising along a curved (accelerated) trajectory as an additional rotation accompanying the composition of not-aligned Lorentz boosts. • The de Sitter [2], Fokker [3], “geodetic” (“geodesic”), or “gravitoelectric” (GE) [4] precession, associated with the gyro’s transport in curved spacetime (in the static – “GE” – field produced by the source’s mass density). • The Lense-Thirring [5], Schiff [6], “dragging”, “motional”, “hyperfine”, or “gravitomagnetic” (GM) [4] precession. This is caused by deformation (differential dragging) of the inertial space by mass currents within the source and is often described as a consequence of the existence of the “GM” component of the gravitational field. All of these effects have been studied in the post-Newtonian limit ([7], §40.7) as well as in the full theory [8], as referred to various frames (asymptotic inertial frame [9], orthonormal tetrad of some local observer [10], or the Frenet-Serret frame attached to the gyro’s trajectory or to its projection to some 3-space [11]), and have been given different interpretations (cf. e.g. [12, 13, 14, 15]). Recently the interest in relativistic gyroscope precession has been maintained by the achieved observational evidence for the geodetic precession [16] and by the hope for a soon direct detection of the dragging effect in the GM field of the Earth [17, 18, 19, 20] (cf. [14]).1 In particular, the question has been studied [22] what information (about his orbit) can the observer infer from the behaviour of his local inertial compass, and the interpretation of gyroscope precession has been proposed in classical-like terms of spatial fields and forces introduced after the Lorentz force (“gravitoelectromagnetism”, GEM; see mainly [23, 24]) or after the Newtonian inertial forces [25, 26]. In [27] I suggested the language of forces which is general and covariant and verified that for many simple types of test-particle motion it yields clear description in accord with intuition. In the present note I show that this approach is successful also in interpretation of the relativistic gyroscope precession. The calculations will be performed in geometrized units (c = G = 1); Greek indices go from 0 while Latin from 1 to 3.

2

General situation

In a general spacetime (gµν , −+++), consider a congruence of hypersurface-orthogonal observers (HOOs),2 denoting τHOO their proper time, uµHOO their 4-velocity, aµHOO = DuµHOO /dτHOO their 4-acceleration and t the time coordinate such that the space-like slices locally orthogonal to uµHOO correspond to t = const (then uHOO i = 0). Suppose each HOO carries with himself a local orthonormal frame (HOF) {eµαˆ } (eµtˆ = uµHOO ) whose transport is regulated by ([7], eq. (13.60)) ~ × ~eαˆ )µ . (1) DF eµαˆ /dτHOO = (Ω HOF 1

A very-high-accuracy gyroscope test is proposed [21] also to decide between the PPN theories of gravity. The HOOs has been referred to [28] as a generalization of the Newtonian rest observers and proved [29] to exist in every spacetime indeed. 2

2

Here

DF eµαˆ /dτHOO = Deµαˆ /dτHOO + aµHOO uHOO ν eναˆ − uµHOO aHOO ν eναˆ

(2)

is the Fermi derivative of eµαˆ along uµHOO , the vector product in the HOO’s 3-space acts according to ~ (3) (Ω × ~eαˆ )µ = uHOO ν ǫνµρσ ΩρHOF eσαˆ HOF

and ΩµHOF represents the angular velocity of the HOF’s spatial vectors relative to comoving Fermi-Walker transported vectors (“gyroscopes”). The arrows denote the spatial parts of the corresponding (contravariant) 4-vectors.3 Then have a test inertial gyroscope with proper time τ , 4-velocity uµ and 4-acceleration µ a = Duµ /dτ . At each instant τ define the unit tangent, inward normal and binormal to the projection of the gyroscope’s trajectory onto the 3-space of the local HOO. The tangent (the relative velocity of the gyro with respect to the local HOO) vˆµ is given by the unique decomposition (4) uµ = γˆ (uµHOO + vˆµ ), where γˆ = (1 − vˆ2 )−1/2 = −uν uνHOO = ut /utHOO and vˆ2 = vˆν vˆν = ~vˆ · ~vˆ. The inward normal nµ and binormal bµ read nµ = vˆ0ν ∇ν vˆ0µ , 4 bµ = (~vˆ × ~n)µ , (5)

where vˆ0µ = vˆµ /ˆ v . At each point of the gyro’s trajectory the unit vectors ~vˆ0 , ~n0 , ~b0 form the orthonormal Frenet basis obeying the relations ~vˆ = ~n × ~b , 0 0 0

~n0 = ~b0 × ~vˆ0 ,

~b = ~vˆ × ~n . 0 0 0

(6)

The change of vˆ0µ along uµ is again described by the transport law DF vˆ0µ /dτ = Dˆ v0µ /dτ + aµ uν vˆ0ν − uµ aν vˆ0ν = uν ǫνµρσ Ωρ vˆ0σ = γˆ uHOO ν ǫνµρσ Ωρ vˆ0σ , i.e., in a 3-vector notation,

(7)

~ × ~vˆ , DF ~vˆ0 /dτ = γˆ Ω 0

(8)

~ = ~vˆ × D ~vˆ /dτ . γˆ Ω 0 F 0

(9)

~ is the angular velocity of rotation of the direction ~vˆ (and thus of the Frenet triad) where Ω 0 ~ (that normal with relative to comoving gyroscopes. Accordingly, the relevant component of Ω ~ respect to vˆ0 ) can be found from

In order to translate the relativistic test motion in classical-like language, I suggested [27] to split the corresponding 4-acceleration into 5 parts, defined in terms of quantities measured by the local HOO and interpreted respectively as contributions from the “gravitational” (“GE”), the “dragging” (“GM”), the “Coriolis” and the “centrifugal” forces and of the body’s “tangent intrinsic inertial resistance”: aµ = aµg + aµd + aµC + aµcf + aµti , 3

(10)

This notation is most appropriate in case of 4-vectors purely spatial with respect to the HOO, i.e., whose t-components are zero. The latter applies to aµHOO , eˆµı , ΩµHOF , and also to vˆµ , nµ , bµ , ΩµG (= −Ωµ ), aµg and aµd below. 4 Eq. (4) determines vˆµ only along the worldline of the gyroscope, so one must either extend the vˆµ to a ˆ vector field before applying the gradient, or note that vˆν ∇ν just stands for the spatial absolute derivative (D) with respect to the sequence of proper times (ˆ τ ) of the HOOs whose worldlines are crossed by that of the gyro: ˆ ˆ τ (cf. [24], Sect. V.). vˆν ∇ν = γ ˆ −1 (δνµ + uµHOO uHOO ν ) uν ∇µ = γˆ −1 D/dτ = D/dˆ

3

where aµg = γˆ 2 aµHOO ,

(11)

aµd = γˆ 2 vˆν ∇ν uµHOO ,

(12)

~ˆ)µ ] , ~ = γˆ 2 [uµHOO aHOO ν vˆν + (Ω HOF × v

aµC µ

µ

acf

(13)

µ

= γˆ vˆ Dˆ v0 /dτ − aC ,

µ

µ

3

ati

(14)

µ

= γˆ (ˆ v0 + vˆuHOO ) Dˆ v /dτ .

(15)

By introducing this decomposition into (7), one obtains from (9) that the gyroscope precesses relative to the comoving Frenet triad with the angular velocity ~ = −Ω ~ =Ω ~ [Thom] + Ω ~ [geod] + Ω ~ [drag] , Ω G G G G

(16)

~ [Thom] = −ˆ Ω v −2~vˆ × ~acf , G ~ [geod] = −~vˆ × ~ag , Ω G ~ ΩG [drag] = −~vˆ × ~ad − vˆ−2~vˆ × ~aC .

(17) (18)

~ˆ · Ω ~ˆ ]. ~ ~ ~ ~k ~ ⊥ = γˆ 2 (Ω ˆ 2 [Ω vˆ−2~vˆ × ~aC = γˆ2 Ω 0 0 HOF − (v HOF ) v HOF − ΩHOF ) = γ HOF

(20)

where I denoted

(19)

Note that

The Coriolis term (13) and the Coriolis-like “corrections” in the centrifugal term (14) arise due to the fact that the HOF (in general) rotates relative to comoving gyroscopes. If the HOF is locked to the symmetries of spacetime (when the latter has any),5 this rotation demonstrates the presence of dragging. Then it is natural to include the Coriolis-type terms into the dragging part. Just this I did in (19). The rest of this note will show how clear interpretation of the gyroscope precession the decomposition (16)-(19) yields in several “standard” situations in the Kerr or simpler field.

3

In the Kerr spacetime

There HOO=ZAMO (zero-angular-momentum observer) and HOF=LNRF (locally non-rotating frame). Using the results of Section 4. of [27], the explicit forms of (17)-(19) will be given in Boyer-Lindquist coordinates (t, r, θ, φ). Let me remind several quantities which will be relevant below: √ A∆−1 Σ−1 (1, 0, 0, ωK ), (21) uµZAMO = aµZAMO

= M Σ−2 A−1





0, Σ(r 4 − a4 ) + 2∆(ra sin θ)2 , − (r 2 + a2 ) ra2 sin 2θ, 0 , (22)

ΩµLNRF = −M aΣ−2 A−1 sin θ





0, ra2 ∆ sin 2θ, 2r 2 (r 2 + a2 ) + Σ(r 2 − a2 ), 0 ,

Γrφφ = ∆Σ−3 sin2 θ [M (2r 2 − Σ)a2 sin2 θ − rΣ2 ] ,

Γθφφ

= −Σ

−3

2

2

2 2

sin θ cos θ [∆Σ + 2M r(r + a ) ] ,

(23) (24) (25)

where M and a are parameters of the Kerr solution, and ∆ = r 2 − 2M r + a2 , Σ = r 2 + a2 cos2 θ, A = (r 2 + a2 )2 − ∆a2 sin2 θ, ωK = 2M ar/A. 5

... as e.g. in the Kerr case – see below...

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One also observes that ~ˆ)µ =⇒ ~vˆ × ~a = ~vˆ × (±~a ) , ~ vˆν ∇ν uµZAMO = ± (Ω LNRF × v C d

(26)

with the plus signs holding for r- and θ-components and the minus signs for the φ-component. ~ [drag] is expressed in terms of Ω ~ Thus the Ω : the r- and θ- components as G LNRF ˆ

and the φ-component as

~ ⊥ )r,θ − 2ˆ Ωr,θ [drag] = −(Ω γ 2 (ˆ v φ )2 Ωr,θ G LNRF LNRF

(27)

~k ΩφG [drag] = γˆ 2 (1 + vˆ2 )(Ω )φ ; LNRF

(28)

ˆ vˆφ = A/Σ sin θ vˆφ stands for the azimuthal LNRF-component of ~vˆ. To specify the sense of a given component of precession [imagining its projection onto the (z = r cos θ)-plane], I will use the words “prograde”/“retrograde” when referring to the rotation (spin) of the Kerr source, whereas the words “forward”/“backward” when referring to the gyroscope’s (azimuthal) orbital rotation (to its orbital axial angular momentum). The orbit will be called co-rotating/counter-rotating if its orbital angular velocity with respect to infinity, ω = dφ/dt, is greater/smaller than ωK (⇔ its axial angular momentum is > 0/ < 0).

p

3.1

Gyroscope in (any) purely radial motion

For a gyroscope having ~vˆ = (ˆ v r , 0, 0) we obtain ~ Ω G

= −(0, ΩθLNRF , γˆ 2 vˆrˆ∆A−3/2 a2 cos θ), ~ [Thom] = −ˆ Ω γ 2~b = −ˆ γ 2~b0 vˆ/R = −ˆ γ 2 vˆrˆΣ−1 A−1/2 a2 cos θ (0, 0, 1), G ˆ ~b , ~ [geod] = −ˆ Ω γ 2 vˆaθ G

ZAMO 0

~ [drag] = −Ω ~⊥ = −(0, ΩθLNRF , 0), Ω G LNRF

(29) (30) (31) (32)

where R = 1/n is the (first) curvature of the projection of the gyroscope’s trajectory onto the p instantaneous 3-space of the local ZAMO, vˆrˆ = Σ/∆ vˆr is the radial LNRF-component of √ ~vˆ, and aθˆ = Σ aθZAMO is the latitudinal LNRF-component of ~aZAMO . ZAMO The effect of (30)-(32) is exactly as one expected:6 pointing momentarily in the direction of its ~vˆ, the radially moving gyroscope is deflected (away from the ~vˆ) towards the symmetry axis by the Thomas term and away from the axis by the geodetic term; the dragging term, describing the influence of the shear of the reference ZAMO-congruence, deflects such gyroscope in the retrograde sense (with respect to the source’s rotation) along the (θ = const)-surface. Whereas the Thomas and geodetic precessions vanish when vˆ = 0, the effect of dragging is independent of velocity. In the special case of purely radial motion in the equatorial plane the Thomas and geodetic ~ is given simply by −Ω ~ terms vanish and Ω and has just the θ-component. The gyroscope G LNRF moving along the axis does not precess at all with respect to the comoving Frenet triad: ~ × ~vˆ = ~0. Ω G 6 Imagine the Boyer-Lindquist mesh depicted in the Kerr-Schild coordinates (e.g. [30]) and realize that the meridional section of the (θ = const)-surface, along which the gyro moves, is a hyperbola which is bent away from the axis.

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3.2

Gyroscope in (any) purely azimuthal motion

When ~vˆ = (0, 0, vˆφ ) we obtain ~ [Thom] = −ˆ Ω γ 2~b = −ˆ γ 2 (ω − ωK ) sin−1 θ (−Γθφφ , Γrφφ /∆, 0), G ~ [geod] = −ˆ Ω γ 2 (ω − ωK )Σ−1 A sin θ (−aθZAMO , arZAMO /∆, 0), G ~ [drag] = −ˆ ~ Ω γ 2 (1 + vˆ2 ) Ω . G

LNRF

(33) (34) (35)

Again in accord with intuition, outside the horizon the Thomas contribution deflects the gyroscope (consider that momentarily aligned with its ~vˆ again) into the region of greater r and smaller |90◦ − θ|, i.e., it acts “away from the symmetry axis” and “backward” (with respect to the sense of the orbit). The geodetic part pulls the gyroscope’s axis into the region of smaller r and smaller |90◦ − θ| – it is also “away from the axis”, but always “forward”. The effect of dragging is, in the same language, “towards the axis” and retrograde (against the ~ [Thom] and Ω ~ [geod] vanish source’s spin independently of the orbital sense). Again the Ω G G ~ when vˆ = 0 (ω = ωK ), in contrast to ΩG [drag]. Realizing that close to the horizon the range of permitted values of angular velocity shrinks according to √ | ω − ωK | ≤ | ωmin,max − ωK | = ∆ ΣA−1 sin−1 θ, (36) one finds that when approaching the black hole (∆ → 0+ ), the Thomas precession fades away as ∼ ∆1/2 , the dragging effect (at least its θ-component) remains finite, while the geodetic term (its θ-component) diverges as ∼ ∆−1/2 . This fits in the interpretation of relativistic particle dynamics – and particularly of its peculiar features occurring in the extreme fields in the vicinity of black holes – suggested by [31] and generalized in [32, 27]: at the horizon the (repulsive) centrifugal force vanishes as ∼ ∆, the effects of dragging as ∼ ∆1/2 , and the (attractive) gravitational force predominates (namely remains finite).7 Consequently, sufficiently close to the horizon the gyroscope precesses – ~ is dominated by the (“forward”) geodetic against the intuition – forward [9, 34], because its Ω G term, given exactly by the gravitational force. For purely azimuthal motion in the equatorial plane (33)-(35) point just in the θ-direction, ΩθG [Thom] = γˆ2 r −4 (r 3 − M a2 ) (ω − ωK ), θ

2

ΩG [geod] = −ˆ γ Mr θ

2

−4 2

∆

−1

2

(37)

2 2

2

(38)

.

(39)

[(r + a ) − 4M a r] (ω − ωK ),

ΩG [drag] = γˆ (1 + vˆ ) M ar

−2

2

2

(3r + a )A

−1

~ ) appears especially clearly since the Here the role of single forces (of the components of Ω G precession occurs just within the equatorial plane: Thomas contribution is backward, the geodetic term forward, and the gravitomagnetic precession is always retrograde. The total angular velocity comes out surprisingly simple for the equatorial circular geodesics, i.e., for p ω = (a ± r 3 /M )−1 (for co-/counter- rotating orbits): √

q

gθθ ΩθG = ± M/r 3

(40)

(this is consistent with the famous Schiff’s result [6] as was shown in [9]; note that the exact formula for the gyroscope precession along circular geodesics in the Kerr geometry was derived already by [35]). 7

See [33] and references therein for an alternative interpretation involving the reversal of direction of the centrifugal effect.

6

As compared with the equatorial plane, the situation is quite opposite in the innermost region of the (r > 0)-part of Kerr spacetime, where the gravitational field is “repulsive” (namely arZAMO < 0) and the dragging (its angular velocity ωK ) strengthens outwards (namely ∂ωK /∂r > 0 ⇔ ΩθLNRF > 0), which makes the geodetic precession backward and the gravitomagnetic precession prograde (the Thomas contribution keeps its usual backward sense). In particular, inside the ring singularity (i.e., at r = 0), the θ-components of (33)-(35) read ΩθG [Thom, geod, drag] = γˆ 2 M a−3 sin θ cos−4 θ [aω sin2 θ, aω, −1 − a2 ω 2 sin2 θ].

4

(41)

In the Schwarzschild spacetime

For a = 0, the ZAMO and his LNRF (22)-(23) go over to the static observer (SO) and his frame (SF), and the Boyer-Lindquist to Schwarzschild coordinates. The familiar absence of ~ = ~0 which leads, thanks to dragging in the Schwarzschild field appears in the fact that Ω SF eqs. (27) and (28), to the absence also of the corresponding component of precession.

4.1

Gyroscope in radial motion

Here (30) and (31) yield zeros, too: namely the projection of the radial trajectory onto the local SO’s 3-space is not curved, and its tangent ~vˆ is just parallel to the gravitational field (to −~aSO ).

4.2

Gyroscope in azimuthal motion

From (33) and (34) we find that ~ [Thom] = γˆ2 ω (− cos θ, r −1 sin θ, 0), Ω G   2 −1 −1 ~ [geod] = −ˆ Ω γ ω 0, M r (r − 2M ) sin θ, 0 . G

(42) (43)

Their θ-components add together in ΩθG = γˆ2

ω r − 3M r − 3M ω sin θ = sin θ. r r − 2M r r − 2M − r 3 ω 2

(44)

This is a welknown result: the position of the gyroscope with respect to the circular photon orbit at r = 3M decides about the sense of its precession [9]. Below r = 3M the geodesic precession becomes so large that, independently of ω, the gyroscope precesses forward even in the co-rotating Frenet triad. Eq. (44) shows that this result is also independent of θ (and not precisely correlated with the occurrence of the rotosphere – the region below r = 3M sin2 θ, where the 4-acceleration of a stationary observer depends on his ω in a counter-intuitive manner [36]).

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In the flat spacetime

Setting also M = 0, the reference congruence becomes inertial, the coordinates spherical, and just the Thomas term survives. For the circular motion, eq. (42) gives (when choosing θ = 90◦ ) (45) ΩθG = ΩθG [Thom] = γ 2 ω/r = (1 − ω 2 r 2 )−1 ω/r

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(the hats need not be used since in flat spacetime the velocity with respect to the HOO is simply the coordinate velocity, vˆi = v i ). The Thomas precession is, however, usually understood to be the effect referred to the global inertial frame. Our Frenet basis rotates with proper angular velocity γω relative to this frame, so the expression (45) – and already the general one (17) – in fact comprises also a “classical” contribution (backward with this angular velocity). Thus the magnitude (with sign) of the angular velocity of the “net” Thomas effect is √ (46) gθθ ΩθG − γω = γ(γ − 1)ω. This is a proper precession frequency (with respect to “distant stars”) as seen by the corotating observer. The inertial observer sees just (γ − 1)ω, which agrees with [7], eq. (6.29).

Acknowledgements I am grateful for support from the grant GACR-202/96/0206 of the Grant Agency of the C.R. and from the grant GAUK-318 of the Charles University.

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