Gravitomagnetism in MOdified theory of Gravity Qasem Exirifard The Abdus Salam International Centre for Theoretical Physics (ICTP), Strada Costiera, Trieste, Italy ∗ We study the gravitomagnetism in the Scalar-Vector-Tensor theory or Moffat’s Modified theory of Gravity(MOG). We compute the gravitomagnetic field that a slow-moving mass distribution pro- duces in its Newtonian regime. We report that the consistency between the MOG gravitomagnetic field and that predicted by the Einstein's gravitional theory and measured by Gravity Probe B, LAGEOS and LAGEOS 2, and with a number of GRACE and Laser Lunar ranging measurements requires jαj < 0:0013. We provide a discussion. The human civilisation still lacks a conclusive knowl- where LM is the Lagrangian density of the matter and edge about the physics at galaxies and beyond, in the sense that either dark matter exists or the dynamics p 1 1 L = −g [ gµν r Gr G − V (G)] of gravity is more involved than that predicted by the S G3 2 µ ν Einstein's gravitional theory. In the first approach, we 2 1 µν have not yet found any dark matter. The later approach +µ G[ g rµµrν µ − V (µ)] ; (2) 2 started by the Milgrom's theory of the Modified Newto- nian Dynamics (MOND)[1], changed to the a-quadratic 1 p 1 µν 1 2 µ LΦ = −g B Bµν − µ Φ Φµ + V (Φ) ;(3) Lagrangian model for gravity (AQUAL)[2]. A generally 4π 4 2 covariant realisation of the AQUAL model is the TeVeS 1 p L = −g(R + 2Λ) ; (4) theory [3]. TeVeS is in agreement with the Gravitomag- G 16πG netism and the near horizon geometry of super massive black hole [4]. It has been shown that changing EH grav- are the Lagrangian densities of the scalar, the vector and ity to TeVeS can not reproduce the CMB [5]. Though the tensor, respectively. G(x) is related to the Newton's adding a 4th sterile neutrino for 11 − 15 eV to matter constant. µ(x) represents the mass of the vector field, content of TeVeS theory fits the spectrum in a LCDM Bµν = @µΦν −@ν Φµ, and V (G);V (µ);V (Φ) are potential. cosmology [6], it wouldn't form the large structure of the Following [13], \we neglect the mass of the Φµ field, universe [7]. TeVeS also appears not to be consistent with for in the determination of galaxy rotation curves and GW170817 [8]. There is also Moffat’s theory of gravity galactic cluster dynamics µ = 0:042 (kpc)−1, which cor- −28 [9] that passes many tests and observations when TeVeS responds to the vector field Φµ mass mΦ = 2:6 × 10 fails; particularly it produces the acoustic peaks in the eV [17{19]. The smallness of the Φµ field mass in the cosmic microwave background radiation and the matter present universe justifies our ignoring it when solving the power spectrum [10], and it is consistent with gravita- field equations for compact objects such as the vicinity tional wave's data [11, 12]. We would like to study the of Earth, neutron stars and black holes. However, the gravitomagnetism in Moffat’s theory. In the first sec- scale µ = 0:042 kpc−1 does play an important role in fit- tion we review Moffat’s theory of gravity around the ting galaxy rotation curves and cluster dynamics". Ref. Earth space-time geometry[13]. In the second section [20] provides the latest most detailed fitting of MOG to we present the gravitomagnetism approximation to Mof- galaxy dynamics. Around the Earth space-time geom- fat's theory around the space-time of the Earth. We re- etry GN is constant. So we set @µG = 0. Since µ is port how Gravity Probe B [14], LAGEOS and LAGEOS very small we ignore @µµ at the vicinity of the Earth. 2[15], and with a number of GRACE and Laser Lunar We ignore the cosmological constant as well. Since the ranging measurements [16] constrain the α parameter to vector field around the space-time geometry of the Earth arXiv:1906.02013v2 [gr-qc] 15 Jun 2019 jαj < 0:0013. In the last section we provide a discussion. is small, and assuming that V (Φ) has a Taylor expan- sion around Φµ = 0, we can ignore V (Φ) as well. These simplify Moffat's action at the vicinity of the Earth to [13]: I. THE ACTION OF MOG THEORY AROUND THE EARTH 1 Z p S = d4x −g(R + GBµν B ) : (5) 16πG µν The action of MOG theory [9] is given by: The matter current density is defined in terms of the Z 4 matter action SM : S = d x(LG + LΦ + LS + LM ) ; (1) 1 δS p M = −J µ ; (6) −g δΦµ ∗Electronic address: [email protected] where J µ is proportional to the mass density. A test 2 particle action is given by where ρ is the local density of the fluid and uµ is the local Z Z four velocity of the fluid. Notice that the fluid interaction p µ ν µ STP = −m dτ gµν x_ x_ − km dτΦµx_ ; (7) with itself is not a functional of φµ. In so doing the source current for the vector/gauge field reads: where dot is variation with respect to τ and m denotes µ µ the test particle's mass. This means that geodesics in J = kρu : (18) the Randers's geometry describes orbits of particles in The equation of motion for the gauge field, in the gauge the MOG [21]. The coupling constant is assumed to be of r Φµ = 0, reads proportional to the mass of the test particle and the pro- µ µ µ portionality factor reads: Φ = −4πkJ : (19) p k = ± αGN ; (8a) G − G II. GRAVITOMAGNETISM IN MOG AROUND α = N ; (8b) GN THE EARTH where GN is the gravitational Newton's constant. Let an auxiliary field e(τ) be defined on the world line In order to address the gravitomagnetism, we look at of the test particle. Then consider the following action: a small deviation from the flat space-time geometry: Z Z g = η + h ; (20) 1 µ ν e µ µν µν µν STP [x; e] = −m dτ( gµν x_ x_ + ) − km dτΦµx_ ; 2e 2 Φµ = 0 + φµ ; (21) (9) the variation of which with respect to e(τ) yields: where in the natural unites it holds jh j 1 ; (22) δSTP [x; e] p µ ν µν = 0 ! e(τ) = gµν x_ x_ (10) δe jφµj 1 : (23) inserting which in (9) reproduces (7). So (9) is equiv- We then consider a slow moving test particle: alent to (7). Notice that (9) is invariant under the reparametrization of the world-line: jt_j ≈ 1 ; (24) i τ ! τ~ =τ ~(τ) ; (11) jx_ j c : (25) dτ e(τ) ! e~(~τ) = e(τ) : (12) These simplify the action of the test particle (15) to dτ~ Z 1 1 The reparametrization invariance allows to set S = m dt( jx_ ij2 + h + kφ + (h + kφi)_xi) ; TP 2 2 00 0 0i e(τ) = 1 ; (13) (26) µ ν in analogy with electromagnetism, the gravitoelectric and gµν x_ x_ = 1 (14) gravitomagnetic potentials are then identified to: as the standard parametrization of the world-line. Doing 1 so yields: φ = h + kφ ; (27a) Phys 2 00 0 Z 1 S [x] = −m dτ( g x_ µx_ ν + kmΦ x_ µ ); (15) APhys = h + kφ : (27b) TP 2 µν µ i 0i i We would like to study the theory around slow moving The equation of motion derived from (26) can be rewrit- mass distribution. Consider that the we have a mass ten as follows v distribution composed of N particles. The action of the x¨ = −∇Φ + × (r × A ) : (28) these particles follow from (15): Phys c Phys N This allows interpreting r×A as a gravitomagnetic field. X X SM = STP [mi; xi] + V (i; j) ; (16) r × A causes precessions of the orbits of a test particle. i=1 i6=j This precession is referred to as the Lense-Thirring pre- cession [22]. Ref. [23] provides a decent recent review where STP [mi; xi] for each particle is given in (15) and on Lense-Thirring precession for planets and satellites in V (i; j) defines the interaction between particles. In the the Solar system. The similarity between the gravito- continuum limit, where we have a fluid at equilibrium, magnetic field and magnetic field beside the spin preces- this suggests that _ e sion formula in electrodynamics (S = µ × B; µ = 2m S) Z dictates that the spin of a gyroscope precesses by [24] 4 p 1 µ ν ν SM = − d x −g( ρgµν u u + kρΦν u ) (17) 2 1 Ω = − r × A : (29) +Interaction of fluid with itself ; LT 2 Phys 3 This precession is called the Pugh-Schiff frame-dragging III. CONCLUSION AND DISCUSSION precession [25, 26]. The Pugh-Schiff frame-dragging pre- cession due to the rotation of the earth recently has been We have reported that the consistency between the measured by the gravity probe B with the precision of MOG gravitomagnetic field and that predicted by the 19% [14]. The gravitomagnetic field due to geodesic ef- Einstein's gravitional theory and measured by Gravity fects are measured at an accuracy of 0.28%[14]. GIN- Probe B, LAGEOS and LAGEOS 2, and with a num- GER, aiming to improve the sensitivity of the ring res- ber of GRACE and Laser Lunar ranging measurements onators, plans to measure the gravitomagnetic effect with requires jαj < 0:0013.