Gravito-magnetic gyroscope precession in Palatini f(R) gravity
aa r X i v : . [ g r- q c ] M a r Gravito-magnetic gyroscope precession in Palatini f ( R ) gravity Matteo Luca Ruggiero ∗ Dipartimento di Fisica, Politecnico di Torino, Corso Duca degli Abruzzi 23, Torino, ItalyINFN, Sezione di Torino, Via Pietro Giuria 1, Torino, Italy (Dated: November 29, 2018)We study gravito-magnetic effects in the Palatini formalism of f ( R ) gravity. On using the Kerr-deSitter metric, which is a solution of f ( R ) field equations, we calculate the impact of f ( R ) gravityon the gravito-magnetic precession of an orbiting gyroscope. We show that, even though an f ( R )contribution is present in principle, its magnitude is negligibly small and far to be detectable in thepresent (like GP-B) and foreseeable space missions or observational tests around the Earth. I. INTRODUCTION
Among the theories that have been proposed to explainthe present acceleration of the Universe [1, 2, 3] withoutrequiring the existence of a dark energy, f ( R ) theories ofgravity received much attention in recent years. In thesetheories the gravitational Lagrangian depends on an ar-bitrary function f of the scalar curvature R ; they arealso referred to as “extended theories of gravity”, sincethey naturally generalize, on a geometric ground, GeneralRelativity (GR): namely, when f ( R ) = R the action re-duces to the usual Einstein-Hilbert action, and Einstein’stheory is obtained. Extended theories of gravity canbe studied in different (non equivalent) formalisms (seeCapozziello and Francaviglia [4] and references therein):in order to obtain the field equations in the metric for-malism the action is varied with respect to metric tensoronly; in the Palatini formalism the action is varied withrespect to the metric and the affine connection, whichare supposed to be independent from one another. Ac-tually, f ( R ) provide cosmologically viable models, whereboth the inflation phase and the accelerated expansionare reproduced (see Nojiri and Odintsov [5] and refer-ences therein) and, furthermore, they have been used toreproduce the rotation curves of galaxies without needfor dark matter [6, 7]. It is worthwhile noticing that, be-cause of the excellent agreement of GR with Solar Systemand binary pulsar observations, every modified theoryof gravity should have the correct Newtonian and post-Newtonian limits, in order to agree with GR tests (seee.g. Will [8]), and this is an important issue for f ( R )gravity too (for a comprehensive review of f ( R ) theories,where this and other issues are dealt with in details, werefer to the recent paper by Sotiriou and Faraoni [9]).In this paper we are concerned with gravito-magnetic(GM) effects in f ( R ) theories of gravity. GM effects arepost-Newtonian effects originated by the rotation of thesources of the gravitational field: this gives raise to thepresence of off-diagonal terms in the metric tensor, whichare responsible for the dragging of the inertial frames.These effects are expected in GR, but are generally very ∗ Electronic address: [email protected] small and, hence, very difficult to detect [10, 11, 12]. Inrecent years, there have been some attempts to measurethese effects (see e.g. Ciufolini and Pavlis [13], Iorio [14]and references therein); in April 2004 Gravity Probe Bwas launched to accurately measure the frame dragging(and the geodetic precession) of an orbiting gyroscope:the final results are going to be published [15].Here, working on an exact solution of the field equa-tions in the Palatini formalism (GM effects and otherPost-Newtonian effects were obtained in metric f ( R )gravity by Clifton [16]), we want to evaluate the impactof f ( R ) theories on GM effects, in order to see if there arecorrections to the GR predictions that can be detected(at least in principle) by GP-B or other foreseeable ex-periments around the Earth. II. VACUUM FIELD EQUATIONS OFPALATINI f ( R ) GRAVITY
The equations of motion of f ( R ) extended theories ofgravity can be obtained starting from the action: A = A grav + A mat = Z [ √ gf ( R ) + 2 χL mat ( ψ, ∇ ψ )] d x. (1)The gravitational part of the Lagrangian is representedby a function f ( R ) of the scalar curvature R . The totalLagrangian contains also a first order matter part L mat functionally depending on matter fields Ψ, together withtheir first derivatives, equipped with a gravitational cou-pling constant χ = πGc . In the Palatini formalism themetric g and the affine connection Γ are supposed to beindependent, so that the scalar curvature R has to beintended as R ≡ R ( g, Γ) = g αβ R αβ (Γ), where R µν (Γ) isthe Ricci-like tensor of the connection Γ.By independent variations with respect to the metric g and the connection Γ, we obtain the following equationsof motion: f ′ ( R ) R ( µν ) (Γ) − f ( R ) g µν = χT µν , (2) ∇ Γ α [ √ gf ′ ( R ) g µν ] = 0 , (3)where f ′ ( R ) = df ( R ) /dR , T µν is the matter source stress-energy tensor and ∇ Γ means covariant derivative withrespect to the connection Γ [4, 9].The equation of motion (2) can be supplemented bythe scalar-valued equation obtained by taking the con-traction of (2) with the metric tensor: f ′ ( R ) R − f ( R ) = χT, (4)where T is the trace of the energy-momentum tensor.Equation (4) is an algebraic equation for the scalarcurvature R : it is called the structural equation and itcontrols the solutions of equation (2).We are interested into solutions of the field equation invacuum, in particular outside a rotating source of matter:so the field equations become f ′ ( R ) R ( µν ) (Γ) − f ( R ) g µν = 0 , (5) ∇ Γ α [ √ gf ′ ( R ) g µν ] = 0 , (6)and they are, again, supplemented by the scalar equation f ′ ( R ) R − f ( R ) = 0 . (7)The trace equation (7) is an algebraic equation for R which admits constant solutions R = c i . Then, it is pos-sible to show that, under suitable conditions (see Ferrariset al. [17],Allemandi et al. [18]), the field equations (5,6)reduce to R µν ( g ) = kg µν , (8) with k = c i /
4, which are identical to GR equations witha cosmological constant Λ: in practice, it is Λ = k in ournotation. Indeed, we may say that k is a measure of thenon-linearity of the theory (if f ( R ) = R , eq. (7) has onlythe solution R = 0 → k = 0).Consequently, the vacuum solutions of GR with acosmological constant can be used in Palatini f ( R )gravity: the role of the f ( R ) function is determining thesolutions of the structural equation (7). It is useful topoint out that, for a given f ( R ) function, in vacuumcase the solutions of the field equations of Palatini f ( R )gravity are a subset of the solutions of the field equationsof metric f ( R ) gravity (see e.g. Magnano [19]). Soevery solution of eqs. (8) is also a solution of the fieldequations of metric f ( R ) gravity with constant scalarcurvature R .The Kerr-de Sitter metric, which is an exact solutionof the field equations in the form (8), describes a rotatingblack-hole in a space-time with a cosmological constant[20, 21, 22] and can be used to investigate GM effects inextended theories of gravity.The Kerr-de Sitter metric in the standard Boyer-Lindquist coordinates x µ = ( t, r, θ, φ ) has the form The space-time metric has signature ( − , , , G = c = 1, greek indices run from0 to 3, and latin ones run from 1 to 3, boldface letters like x refers to three-vectors. ds = − (cid:20) − M r Σ − k r + a sin θ ) (cid:21) dt − a (cid:20) M r
Σ + k r + a ) (cid:21) sin θdtdφ + Σ∆ dr + Σ χ dθ + (cid:20) M r Σ a sin θ + (1 + k a )( r + a ) (cid:21) sin θdφ , (9)whereΣ = r + a cos θ , χ = 1 + k a cos θ , (10)∆ = r − M r + a − k r ( r + a ) . (11)The mass of the source is M , while J = M a is itsangular momentum (which is perpendicular to the θ = π/ k = 0 the Kerr-de Sitter metric (9)reduces to the Kerr metric. Other limiting cases canbe checked: for instance, when a = 0, we obtain theSchwarzschild-de Sitter solution, and when M = a = 0we have the de Sitter space-time.It is interesting to consider the “weak-field” approxi-mation of the metric (9): in other words, we expand itup to linear terms in M/r, M a/r , kr , kar, kM r . Whatwe get is ds = − (cid:18) − Mr − k r (cid:19) dt + (cid:18) Mr − k r (cid:19) dr + r dθ + r sin θ dφ − a (cid:18) Mr + k r (cid:19) sin θdφdt. (12)We notice that on using Boyer-Lindquist coordinates theweak field metric (12) does not contain terms in the form kM r . However, these coordinates are not directly relatedto physical lengths. We can express the metric (12) in themore familiar isotropic coordinates, which is necessary toproperly deal with gravito-magnetic effects [12]. To thisend we introduce a new radial coordinate ρ ρ = r (cid:18) − Mr − kr (cid:19) , (13)and, up to the required approximation order, the metricturns out to be: ds = − (cid:18) − Mρ − k ρ (cid:19) dt + (cid:18) Mρ − k ρ (cid:19) (cid:0) dρ + ρ dθ + ρ sin θdφ (cid:1) + − a (cid:18) Mρ + k ρ + 56 M kρ (cid:19) sin θdφdt. (14)On using these coordinates we see that terms in the form kM ρ are present: as a consequence, their absence in themetric (14) was due to the use of Boyer-Lindquist coor-dinates.By inspection of the metric (14) we see that for a = 0,we obtain the weak field limit of the Schwarzschild-deSitter solution, for k = 0 we obtain the weak field limitof the Kerr solution. III. GRAVITO-MAGNETIC FIELD EFFECTS INPALATINI f ( R ) GRAVITY
According to what we have seen, the terms containing k in the metric (9) are due to the non linearity of thegravity Lagrangian in the framework of Palatini f ( R )gravity, or to the presence of a cosmological constant inthe framework of GR. Some effects of these terms havebeen already investigated in the literature. For instanceKerr et al. [20], Sereno and Jetzer [23] showed that, due tothe cosmological constant, the mean motion for circulargeodesics is modified according to ω k = Mρ − k θ = π/ φ k = πkA M p − e , (16)where A is the semi-major axis of the (unperturbed) or-bit, and e is its eccentricity. The effects of the cosmologi-cal constant on gravitational lensing was recently studied by Rindler and Ishak [24],Sereno [25] and by Ruggiero[26] in connection with Palatini f ( R ) gravity. Actually,the effects (15,16) of the k -term (whose expected orderof magnitude is comparable to the cosmological constant k ≃ Λ ≃ − m − ), though present in principle, aretoo small to be detected.Here, we would like to study the GM precession of anorbiting gyroscope, focusing on the corrections due to the k -term.We remember that such an orbiting gyroscope under-goes also a geodetic precession with an angular frequency(averaged over a revolution) that can be written in theform Ω S − O = Ω MS − O + Ω kS − O (17)where Ω MS − O = 32 MA L (1 − e ) / (18)is the classical GR geodetic (or de Sitter) precession,where L is the specific orbit angular momentum of thegyroscope, moving along the orbit with semi-major axis A and eccentricity e , and Ω kS − O = − k L (19)is the correction due to the k -term (see e.g. Sereno andJetzer [23]).The angular velocity Ω S − O can be referred to as aspin-orbit term, because it is due to the coupling of thegyroscope’s spin S with the orbit angular momentum L .As for the GM precession, it can be calculated, usingthe standard approach (see e.g. Ciufolini and Wheeler[28]), starting from the GM potential g φ = − M aρ sin θ − ka ρ sin θ − M akρ sin θ (20)of the metric (14). The first term in eq. (20) is the“classical” GM potential arising in GR, while the othertwo terms are proportional to k . As a consequence, wecan write the angular frequency of GM precession in theform Ω S − S = Ω JS − S + Ω kaS − S + Ω JkS − S (21)where Ω JS − S = − (cid:20) J | ρ | − J · ρ ) ρ | ρ | (cid:21) , (22)is the GR gravito-magnetic precession (see e.g. Ciufoliniand Wheeler [28]), and Ω kaS − S = ka J , (23) Ω JkS − S = 5 Jk ρ h ˆ J + (cid:16) ˆ J · ˆ ρ (cid:17) ˆ ρ i . (24)are new contributions due to the presence of the k -term.The angular velocity Ω S − S can be referred to as aspin-spin term, because it is due to the coupling of thegyroscope’s spin S with the spin angular momentum J of the source of the gravitational field. We point out thatall contributions in (21) do not depend on the velocity ofthe gyroscope; furthermore, the term Ω kaS − S is a constantcontribution over the whole orbit.It is interesting now evaluate the magnitude of the newcontributions (23,24). To this end, it is useful to remem-ber that the GP-B mission is expected to measure theprecession (geodetic plus frame-dragging) of the orbitinggyroscope with an accuracy of 0.1 milliarcseconds/year,which is a very hard task, as the long story of this mis-sion teaches [15]. Taking into account that the geodeticprecession has a magnitude of about 6.6 arcseconds/year,and the frame-dragging effect of 0.041 arcseconds/year,to give an idea of the magnitude of the effects of the nonlinearity of the gravity Lagrangian, we may calculate theratio between the two angular frequencies (23) and (24)and the GR one (22), at a distance R = | x | = ρ from thesource Ω kaS − S Ω JS − S ≃ kR M , (25)Ω
JkS − S Ω JS − S ≃ kR . (26) Using k = 10 − m − , i.e. the current estimate of thecosmological constant, M equal to the Earth mass, R ≃ Km , i.e. order of magnitude of the GP-B orbit, weget Ω kaS − S Ω JS − S ≃ − . (27)Ω JkS − S Ω JS − S ≃ − . (28)Accordingly, we may conclude that the impact of f ( R )gravity on GM gyroscope precession is very small, andcompletely negligible for a mission like GP-B. For sim-ilar reasons, we can say that f ( R ) gravity is not rele-vant for other experiments aimed at the measurementof the gravito-magnetic field of the Earth, such as thoseperformed in the past with LAGEOS satellites (see Ciu-folini and Pavlis [13]), or those that are planned in thenext months such as LARES [29].On the other hand , if GP-B will confirm the GR pre-dictions for the orbiting gyroscope precession and no ad-ditional terms will be seen, from the expected accuracyof 0.1 milliarcseconds/year, we might deduce an estimatefor the k -term in (23): | k | ≤ − m − , which is con-siderably greater than the current best estimates of thecosmological constant. IV. CONCLUSIONS
We have studied gravito-magnetic effects in the frame-work of f ( R ) gravity. Namely, thanks to the analogy, inthe Palatini formalism, between general relativistic vac-uum field equations with cosmological constant and vac-uum f ( R ) field equations, we have considered the Kerr-de Sitter metric as a solution of f ( R ) field equations.Since this metric describes a rotating black-hole, it issuitable to evaluate the gravito-magnetic effects. In par-ticular, we have considered the weak-field approximationof the Kerr-de Sitter metric (which, as far as we know,has never been studied before) and then we have calcu-lated the contribution to the gravito-magnetic precessionof an orbiting gyroscope due to the non linearity of thegravity Lagrangian. We have shown that, though presentin principle, this contribution is very small, and far to bedetectable by a mission like GP-B and, probably, also byother foreseeable tests around the Earth. This confirmsthat the non-linearities appearing in f ( R ) become impor-tant on length scales much larger than the Solar System(e.g. on the cosmological scale) and their effects on localphysics are probably negligible. ACKNOWLEDGMENTS
The author would like to thank Prof. B. Mashhoonand Dr. M. Sereno for useful discussions. The authoracknowledges financial support from the Italian Ministryof University and Research (MIUR) under the nationalprogram “Cofin 2005” -
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