Gravitomagnetic effects in conformal gravity Jackson Levi Said∗,1 Joseph Sultana†,2 and Kristian Zarb Adami‡1,3 1Physics Department, University of Malta, Msida, MSD 2080, Malta 2Mathematics Department, University of Malta, Msida, MSD 2080, Malta 3Physics Department, University of Oxford, Oxford, OX1 3RH, United Kingdom (Dated: 14 January 2014) Gravitomagnetic effects are characterized by two phenomena: first, the geodetic effect which describes the precession of the spin of a gyroscope in a free orbit around a massive object, second, 4 theLense-Thirringeffectwhichdescribestheprecessionoftheorbitalplaneaboutarotatingsource 1 mass. We calculate both these effects in the fourth-order theory of conformal Weyl gravity for the 0 test case of circular orbits. We show that for the geodetic effect a linear term arises which may 2 be interesting for high radial orbits, whereas for the Lense-Thirring effect the additional term has a diminishing effect for most orbits. Circular orbits are also considered in general leading up to a n generalization of Kepler’s third law. a J PACSnumbers: 04.20.-q,04.50.Gh 0 1 I. INTRODUCTION sponding field equations are then given by [1] ] c q √ gg g δIW = 2αW - The validity of any alternative theory to Ein- − µα νβδg − µν r αβ g stein’s general relativity depends on how well it = 1T , (3) [ agrees with his theory in the weak field limit as −2 µν 1 well as observational tests. One of the possi- where Tµν is the stress-energy tensor, and ble alternatives to Einstein’s second order the- v 8 orywhich has been proposedduring the lasttwo Wµν =2Cαµνβ;βα+CαµνβRαβ. (4) 9 decades is conformal Weyl gravity [1–3]. Instead 8 of choosing the gravitationalaction by requiring Itcanbeseenfrom(3)thatanyvacuumsolution 2 of Einstein’s field equations is also a vacuum so- thatthetheorybenohigherthansecondorderas 1. in the case of the Einstein-Hilbert action, Weyl lutionofWeylgravity;withtheconversenotnec- 0 essarily being true. Despite being a fourth-order gravity employs the principle of local conformal 4 theory with highly nonlinear field equations, a invariance of spacetime to fix the gravitational 1 number of exact solutions [4–9] have been found : action, meaning that the theory is invariant un- v which generalize the well-known Kerr-Newmann der local conformal stretching of the type i and cylindrical solutions of general relativity. X The exact static and spherically symmetric vac- r g (x) Ω2(x)g (x), (1) a µν µν uum solution for conformal gravity is given, up → to a conformal factor, by the metric [1] whereΩ(x)isasmoothstrictlypositivefunction. dr2 Thisleadstoaconformallyinvariantfourthorder ds2 = B(r)dt2+ +r2(dθ2+sin2θdφ2), − B(r) theory with a unique action given by (5) where IW = α d4x( g)1/2CλµνκCλµνκ β(2 3βγ) − − B(r)=1 − 3βγ+γr kr2, (6) Z − r − − = 2α d4x( g)1/2[R Rµκ (Rν)2/3] − − µκ − ν and β, γ, and k are integration constants. Z This solution includes as special cases the +a total derivative, (2) Schwarzschild solution (γ = 0 = k) and the Schwarzschild-deSitter (γ =0) solution; the lat- where C is the conformalWeyl tensorand α λµνκ ter requiring the presence of a cosmologicalcon- is a purely dimensionless coefficient. The corre- stant in Einstein gravity. Moreover the constant γ has dimensions of acceleration, and so the so- lutionprovidesacharacteristic,constantacceler- ation without having introduced one at the La- ∗[email protected] †[email protected] grangian(such as in the relativistic implementa- ‡[email protected] tion of MOND with TeVeS [10]). 2 The magnitude and the origin (i.e. whether it chosen coordinate system [17], geodetic preces- is system dependent like β or cosmological like sioncanbeconsideredasduetoaLense-Thirring k) of the integration constant γ in (6) remains drag. A combined observable phenomenon of unknown. Whenit is associatedwith the inverse these two effects [18] occurs in the Earth-Moon Hubble length, i.e. γ 1/R , the effects of the systemaroundtheSunthroughtheprecessionof H ≃ accelerationdue to the γr terminthe metric are theMoon’sperigee,whichisdetectedbymeasur- comparablewiththoseduetotheNewtonianpo- ing the lunar orbit using laser ranging between tentialterm2β/r r /r(r istheSchwarzschild stations on Earth and reflectors on the Moon’s s s ≡ radius), on length scales given by surface [19–21]. A spaceexperimentto test these twopredictions r /r2 γ 1/R or r (r R )1/2. (7) s ≃ ≃ H ≃ s H of Einstein’s theory is Gravity Probe B (GP-B) [22]whichwaslaunchedon20April2004ina642 As noted in [1], for a galaxy of mass M 1011 M with r 1016 cm and R 1028 cm≃, km polar Earth orbit with data collection last- ⊙ s H this scale is r ≃1022 cm, i.e. rough≃ly the size ing almost a year. Measurement of the geodetic ∼ and frame-dragging effect was done by means of of the galaxy, a fact that prompted Mannheim, cryogenic gyroscopes with one or more of these O’BrienandKazanastoproducefitstothegalac- referenced to a remote star by means of an on- tic rotation curves using the metric of Eq. (6) board telescope. For the chosen orbit the two above, without the need to invoke the presence effects result in a precession along two perpen- of dark matter as in standard Einstein’s theory dicularplanessothatGP-Bcouldmeasurethese (see Refs. [11, 12] for recent work on this issue). independently. Analysisofthedatafromthefour However, an issue arises in that in addition to onboard gyroscopes showed a geodetic drift rate having flat rotation curves in the region of in- of 6601.8 18.3 mas/yr and a frame dragging terest for galaxies, we must also have stability − ± drift rate of 37.2 7.2 mas/yr, which are in for the orbits under consideration [37]. Circular − ± accordance with the general relativistic predic- orbits offer a good toy model for investigating tions of 6606.1 mas/yr and 39.2 mas/yr re- different gravitationaleffects in alternativetheo- − − spectively. riesofgravity. Duetotheirimportancewebegin The classical tests of general relativity including with a short investigationof circular orbits lead- the bending of light [23–26], time delay [27] and ing up to a statement of Kepler’s third law. perihelionprecession[28]havealreadybeenstud- The geodetic precession, also known as de Sitter ied in conformal gravity. In this paper we inves- precession is a general relativistic phenomenon tigatecircularorbitsinSec. II,derivingKepler’s discovered in 1916 by de Sitter [14], who found third law. In Sec. III the geodetic precession ef- that the spin of a gyroscope precesses with re- fect is determined for circular orbits. In Sec. IV spect to a distant inertial frame when it makes theLense-Thirringeffectiscalculatedintermsof free (or “geodetic”) orbits around a nonrotating the precessionvelocity. Finally in Sec. V we end massive object. For a circular orbit the amount with a summary of conclusions and a discussion. of geodetic precession per orbit is given by 1/2 3GM ∆φgeodetic = 2π"1−(cid:18)1− c2R (cid:19) # II. CIRCULARTOHRIRBDITLSAAWND KEPLER’S 3πGM GM , for <<1,(8) ≈ c2R c2R We investigate circular orbits in conformal where M is the mass of the massive object and gravityusingaLagrangianapproachforthemet- R is the radius of the orbit. This effect is also ric in Eq.(5). In this case the Lagrangian takes observed in flat spacetime, where in this case the form it has a purely kinematical origin and is known 1 ∂xµ∂xν as Thomas precession [15]. Another closely re- = gµν L 2 ∂τ ∂τ latedrelativisticprecessionisthe Lense-Thirring 1 r˙2 effect [16] (or frame dragging effect) discovered = Bt˙2+ +r2θ˙2+r2sin2θφ˙2 , (9) 2 − B by Lense and Thirring in 1918, and refers to the (cid:20) (cid:21) precessionofthegyroscopeduetotherotationof where dots denote differentiationwith respectto the massive object which produces a draggingof proper time, τ. Using the four-velocity normal- nearby inertial frames. Although these are two ization condition u uµ = 1, where u dxµ, µ − µ ≡ dτ independenteffectsinthesensethatthelatterre- it follows that = 1. It must be noted at L −2 quires rotation of the source of the gravitational this point that this Lagrangian and the associ- field, it can be shown that in an appropriately ated timelike geodesics representing trajectories 3 of free massive particles are not conformally in- For k <<γ <<β this can be written as variant, unlike the theory of conformal gravity. Moreoverconformalinvarianceimpliesthatthere is no prescribed scale in the theory, which has a β γ dimensionless coupling constant α as shown in ω2 + , (14) Eq. (2). However recently Edery et al. in Ref. ∼ R3 2R [29] (see also Refs. [30, 31]) considered spon- taneous symmetry breaking in Weyl gravity and showedthatthiscanprovideamechanismtogen- which is similar to Kepler’s third law, with the erate a scale in the theory. second term on the right-hand-side representing Takingadvantageofthe independence ofthe La- the correction from conformal gravity. grangianonthet andφ coordinates,andusing − − θ = π/2 as the plane of the circular orbits with- out loss of generality due to sphericalsymmetry, theEuler-Lagrangeequationsofmotionaregiven by Bt˙=E, (10) III. THE GEODETIC EFFECT r2φ˙ =L, (11) where E and L are the energy and angular mo- The precession is calculated for orbits in the mentum respectively. The third Euler-Lagrange equatorial plane θ = π2 since by symmetry con- equation is given by siderationsarotationcanalwaysbemadetothis plane. Rotating coordinates are first introduced (B−12r˙)˙ [ (β(2−3βγ) +γ 2kr)t˙2 through the transformation − − r2 − ∂ + (B−1)r˙2+2rφ˙2]=0. (12) ∂r φ φ ωt, (15) For circular orbits r = R, r˙ 0 and therefore → − ≡ the above equation yields 2 dφ 1 β(2 3βγ) where ω is the coordinate angular frequency of ω2 = = − +γ 2kR . dt 2R R2 − the rotation. This transforms the metric in (cid:18) (cid:19) (cid:18) (cid:19) (13) Eq.(6) to 2 β(2 3βγ) r2ω ds2 = 1 − 3βγ+γr kr2 r2ω2 dt dφ −(cid:18) − r − − − (cid:19)" − 1− β(2−r3βγ) −3βγ+γr−kr2−r2ω2 # dr2 1 β(2−3βγ) 3βγ+γr kr2 + +r2 − r − − dφ2. 1 β(2−3βγ) 3βγ+γr kr2 1 β(2−3βγ) 3βγ+γr kr2 r2ω2 − r − − − r − − − (16) Comparing this with Rindler’s canonical form r2ω w = [34] 3 1 β(2−3βγ) 3βγ+γr kr2 r2ω2 − r − − − (19) ds2 = e2Φ dt w dxi 2+k dxidxj, (17) − − i ij k11 =1 β(2−3βγ) 3βγ+γr kr2 (20) (cid:0) (cid:1) − r − − the following nonvanishing components arise 1 β(2−3βγ) 3βγ+γr r2 k+ω2 e2Φ =1 β(2−3βγ) 3βγ+γr kr2 r2ω2 k33 = − r − − , − r − − − r2(1−3βγ)−βr(2−3βγ)+r(cid:0)3γ−kr(cid:1)4 (18) (21) 4 whereLatinindicesrefertospacelikecoordinates with respect to the inertial frame, where the re- only. lation uφ =ω = 2π was used. ∆t We considerfreecircularorbits forwhichthe ac- celeration [35] vanishes IV. LENSE-THIRRING EFFECT a= kijΦ Φ 1/2 =0, (22) ,i ,j whichimplies tha(cid:0)tΦ,r =0.(cid:1)Hence arelationcan The Lense-Thirringeffect describes the correc- be established between the angular velocity and tion of a rotating spacetime on the precession the metric parameters such that of orbits. The effect can be quantified through a consideration of the Sagnac effect as in [36– β(2 3βγ) γ ω2 = − + k. (23) 38],whichrepresentsthedifferenceintraveltime 2r3 2r − or phase shift of corotating and counterrotating Substituting this in Eqs.(18, 19, 21) gives light waves in the field of a central massive and spinning object. In order to calculate this in 3β(2 3βγ) γr e2Φ =1 − 3βγ+ (24) conformal gravity we must consider the rotating − 2r − 2 metric found in [4], namely r2 β(2−3βγ) + γ k w = 2r3 2r − (25) dr2 dy2 3 1 q3β(2−3βγ) 3βγ+ γr ds2 =(bf ce) + − 2r − 2 − a d (cid:18) (cid:19) 1 3β 2−3βγ 3βγ+ rγ 1 k33 = − 2r − 2 . + d(bdφ cdt)2 a(edφ fdt)2 , r2(1 3βγ) βr(2 3βγ)+r3γ kr4 bf ce − − − − − − − (26) − h (31i) whichis the canonicalCarterformofthe metric, The proper rotation rate of a gyrocompass with where respect to the rotating frame (15) is given in terms of the canonical form (17) by [35] a=j2+ur+pr2+vr3 kr4, (32) − Ω= 1 eΦ kikkjl(w w )(w w ) 1/2 d=1+r′y py2+sy3 j2ky4, (33) 2√2 i,j − j,i k,l− l,k − − b=j2+r2, (34) eΦ (cid:2) 1/2 (cid:3) = k11k33w 2 . (27) e=j 1 y2 , (35) 2 3,1 − h i c=j, (36) Substituting Eqs. (24 - 26) in this expressionwe (cid:0) (cid:1) getΩ=ω,sothatthecoordinaterateω isreally f =1, (37) the orbital rotational frequency. and u, p, v, k, r′ and s are constants satisfying Now for the Mannheim-Kazanas metric (6) the the constraint uv r′s = 0. The angular mo- relationbetweenproperandcoordinatetimesfor − mentum of the source is represented by j. The a circular orbit (r = R, r˙ = 0) with coordinate general relativistic result representing the Kerr angular velocity ω, is given by solution is recoveredwhen 3β(2 3βγ) γR ∆τ = 1 − 3βγ+ 2R2k ∆t. u= 2MG/c2, − 2R − 2 − − r (28) p=1, Hence for one complete orbit about the gravi- v =0=k=r′ =s, (38) tating mass the precessionof the gyroscopewith while the Kerr-de Sitter solution is obtained respect to the rotating frame is when k = Λ/3 and p = 1 kj2; Λ being the α′ =Ω∆τ cosmologicalconstantandwi−ththeotherparam- eters taking the same values as in (38). For the 3β(2 3βγ) γR =2π 1 − 3βγ+ 2R2k. conformal case the additional parameters must − 2R − 2 − r be constrained through observation. (29) Similar to the general relativistic case we take This results in a precession angle per orbit of the transformation α=2π α′ y cosθ, (39) − → 3β(2 3βγ) 3βγ γR 2π − + +R2k , in order to obtain Boyer-Lindquist-like coordi- ≈ 4R 2 − 4 (cid:18) (cid:19) nates. Noting that dy = sinθdθ the metric (30) − turns out to be 5 dr2 sin2θdθ2 ds2 = r2+j2cos2θ + j2+ur+pr2+vr3 kr4 1+r′cosθ pcos2θ+s cos3θ j2kcos4θ (cid:18) − − − (cid:19) + (cid:0) 1 (cid:1) 1+r′cosθ pcos2θ+scos3θ j2kcos4θ j2+r2 dφ jdt 2 r2+j2cos2θ − − − j2+ur+pr2h+(cid:0) vr3 kr4 jsin2θdφ dt 2 . (cid:1)(cid:0)(cid:0) (cid:1) (cid:1) (40) − − − (cid:0) (cid:1)(cid:0) (cid:1) i The Sagnac effect is now investigated by con- set r =R so that sidering corotating and counterrotating circular ltihgehtcebnetarmalsoinbjtehcet.eqAufatteorriaalcpirlcaunlearθo=rbπ2itatbhoeuset ds2 = dRt22 j2+R2 ω0−j 2 light beams reach a detector (that can also dou- j2+h(cid:0)u(cid:0)R+pR2(cid:1)+vR3(cid:1) kR4 (jω 1)2 . ble as a source) which is assumed to be rotating 0 − − − with uniform angular velocity ω0, such that its (cid:0) (cid:1) (42i) rotation angle is φ =ω t. (41) Nowgiventhatwearedealingwithraysoflight 0 0 we take ds=0. Eq.(42) then gives two solutions Since we are considering circular orbits, we also for ω , where 0 1 Ω = 2j uR+(p 1)R2+vR3 kR4 ± 2(R4+j2( uR+(2 p)R2 vR3+kR4)) − − − − − − h (cid:0) (cid:1) 4j2(uR+(p 1)R2+vR3 kR4)2+4(R4+j2( uR+(2 p)R2 vR3+kR4))(uR+pR2+vR3 kR4) , ± − − − − − − q (43) i these two solutions refer to the rotating and which when substituted back into Eq.(45) gives counterrotating orbits. For light the rotation angles are given by φ =Ω t. (44) ± ± Ω ± φ =φ 2π. (48) 0 0 ω ± In conjunction with Eq.(41) the t coordinate 0 − can be eliminated Ω ± φ = φ . (45) ± 0 ω 0 Solving for φ 0 The first intersection of the rays of light with the positionofthe observerafter emissionatt= 0, occurs when 2πω φ+ =φ0+2π, (46) φ0± =±Ω 0ω =±2πω0ξ±, (49) ± 0 φ =φ 2π, (47) − − 0 where − 6 ξ = 2 R4+j2 uR+(2 p)R2 vR3+kR4 2j uR+(p 1)R2+vR3 kR4 ± − − − − − − 4(cid:2)j2(cid:0)(uR+(p(cid:0) 1)R2+vR3 kR4)2+4(R4+(cid:1)(cid:1)(cid:3)j2h( uR(cid:0)+(2 p)R2 vR3+kR4))(u(cid:1)R+pR2+vR3 kR4) ± − − − − − − q −1 2ω R4+j2 uR+(2 p)R2 vR3+kR4 . (50) 0 − − − − (cid:0) (cid:0) (cid:1)(cid:1)i CombiningthiswiththemetricinEq.(42),and canthusbedeterminedbyusingthedetectorco- reverting to physicalunits the proper time delay ordinate relation in Eq.(41) dφ dτ = (j2+uR+pR2+vR3 kR4)(jω 1)2 ((j2+R2)ω j)2 (51) 0 0 cRω − − − − 0 q Integrating between φ and φ gives the Sagnac time delay between the two light beams 0− 0+ φ φ δτ = 0+ − 0− (j2+uR+pR2+vR3 kR4)(jω 1)2 ((j2+R2)ω j)2 0 0 cRω − − − − 0 q 2π(ξ +ξ ) = + − (j2+uR+pR2+vR3 kR4)(jω 1)2 ((j2+R2)ω j)2. (52) 0 0 cR − − − − q In order to calculate the correctiondue solely to obtaining a generalization of Kepler’s third law. the rotationofthe sourcethe detector rotational The main results of this paper are obtained parameter, ω , is set to zero, such that in Eq.(30) and Eq.(55), which represent the 0 geodetic and Lense-Thirring effects in conformal 4πjuR+(p 1)R2+vR3 kR4 δτ = − − , (53) gravity respectively. 0 cR uR+pR2+vR3 kR4 Apart from the Einstein term 3πβ/R, Eq.(30) − which can bepexpressed in terms of a Lense- also includes terms due to the linear term γr Thirring precession velocity ω , by in the Mannheim-Kazanas metric as well as LTCG the cosmological term kr2. We note that as in ω πR2 δτ =8 LTCG , (54) the case of the other already studied classical 0 c2 p+ u +vR kR2 tests, namely the bending of light [26], time R − delay [27] and perihelion precession [28], the where p contribution to the geodetic precession from the u+(p 1)R+vR2 kR3 linear term in the metric has an opposite sign to ω =jc − − . LTCG 2R3 that of the Einstein term, so that it reduces the (cid:18) (cid:19) (55) amount of precession per orbit. Moreover this Using the values for the constants given in contribution increases linearly with the radius R Eq.(38), this gives the general relativistic result of the circular orbit, such that at larger radii its obtained in Ref. [36]. effect can cancel that of the Einstein term. The Lense-Thirring precession velocity was obtained in terms of the Sagnac time delay for a V. DISCUSSION AND CONCLUSION stationary observer which occurs solely due to the drag of spacetime by the rotating central In this paper, we first considered how circular object. Assuming that the constants u, p and k orbits behave in conformal gravity, thereby in (55) take the same values as in the Kerr-de 7 Sittersolution,weseethattheconformalgravity of γ, which in our case is substantiated by the correction through the constant v [which corre- fact that its contribution to the geodetic preces- sponds to γ when j = 0, θ = π/2 in (40)] has sion in (30) is independent from the mass of the againadiminishingeffectonthe totalprecession central object, just like the cosmological term. velocity. However in this case this correction Howeverhaving saidthis, there is nothing in the is inversely proportional to the radius of the theory that forbids γ from being also system de- circular orbit. pendent, in which case it may provide the neces- sary changes in the spacetime geometry to allow For a gyroscope in the Earth’s orbit, such as theembeddingofasphericallysymmetricmatter GP-B, the extra contribution to the geodetic distribution in a cosmologicalbackground. precession from the linear and cosmological To conclude, this paper gives a derivation of terms inthe metric areinsignificantforpractical Kepler’s third law and the gravitomagnetic ef- purposes, when the value of γ is taken as the fects in the alternative gravitational theory of reciprocal of the Hubble radius, as required for conformal gravity. On the solar system scale, the fitting of galactic rotational curves shown where conformal gravity (like other alternative in Ref. [1]. In earlier studies such as those in gravitational theories) agrees with general rela- Refs.[11, 28] constraints were obtained for the tivity, the corrections to the Einstein precession conformalgravityparameterγ. Howeverinboth areverysmalltobemeasuredbyanyspace-based cases of geodetic and Lense-Thirring effects the experiment. However these may become signifi- uncertainties in available GP-B data are too cant on much larger scales, especially when the large to obtain reasonable constraints. So, for geodetic effect is considered. example, using the uncertainty in the observed value for the GP-B geodetic drift mentioned in the introduction, and the Weyl’s gravity correction to geodetic effect in (30), one gets ACKNOWLEDGMENTS γ 1.5 10−20 cm−1, which is eight orders of ≤ × magnitude larger than the value obtained from J. L. S. wishes to thank the Physics Depart- galactic rotational curves. ment at the University of Malta for hospitality during the completion of this work. 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