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GravitoMagnetic Field in Tensor-Vector-Scalar Theory

We study the gravitomagnetism in the TeVeS theory. We compute the gravitomagnetic field that a slow moving mass distribution produces in its Newtonian regime. We report that the consistency between the TeVeS gravitomagnetic field and that predicted by the Einstein-Hilbert theory leads to a relation between the vector and scalar coupling constants of the theory. We translate the Lunar Laser Ranging measurement's data into a constraint on the deviation from this relation.

The missing mass problem in the galaxies can be solved either by assuming the presence of dark matter, a form of matter yet not included and seen in the standard model of the elementary particles , or altering the newtonian dynamics and gravity. The ΛCDM model buys the first approach. The success of the ΛCDM theory has stemmed various searches to directly/indirectly observe dark matter particles and their signatures which can not be produced by altering newtonian dynamics/gravity. However no conclusive signals for the dark matter particles yet have been reported.
The second approach started by the Milgrom theory of the modified newtonian dynamics (MOND) [1], then changed to the aquadratic Lagrangian model for gravity [2] (AQUAL). This approach also has signatures that can not be produced by the dark matter paradigm, including the detection of the gravitomagnetic force in the MOND regime [4]. Now a generally covariant realization of the AQUAL model, the TeVeS theory [3], is known to represents the modified gravity's approach.
TeVeS equations of motion are more evolved compared to those of the Einstein-Hilbert theory. Though its spherically static solution [5] -analog to the Schwarzschild solution-is known, the exact Kerr-like solution of the theory so far has not been reported. In one hand no solution of TeVeS possessing non-trivial gravitomagnetic field is known. In the other hand, we have acquired very precise empirical knowledge on the gravitomagnetic field around the spacetime geometry of the earth. These include the Lunar Laser measurements [6] and the gravity probe B [7]. In this paper, we investigate how measuring the gravitomagnetic force near the earth constraints the TeVeS parameters. In so doing, we first review the field equations of the TeVeS theory. We next solve these equations at the linear order around the space-time geometry of the earth. We show that the TeVeS gravitomagnetic force is less than the Einstein-Hilbert gravitomagnetic field by the factor of 3k 4π . Noting that the LLR measurements require the consistency with the Einstein gravitomagnetic field with the accuracy of 0.1% we bound the scalar coupling k 4π to be weaker than 2.3 × 10 −4 at the standard confidence level. This value of k constraints the dimension-full parameter of the TeVeS to l > (1.65 ± 0.35) × 10 25 m. These are the first constraints on l and k parameters of the theory.

I. EQUATIONS OF MOTION OF TEVES
The dynamical fields in the Tensor Vector Scalar (TeVeS) theory [3] are the Einstein metric g ab , a time-like vector A a , and a scalar φ. There is also a non-dynamical scalar field σ. The Einstein metric couples to the physical energy momentum tensor represented by T ab . The physical metric is defined bỹ The vector field is forced to be the unite timelike vector with respect to the Einstein metric We assume that matter is an ideal fluid. So the physical energy momentum tensor takes the form of where ρ is the proper energy, p is the pressure and u a is the 4-velocity, all three expressed in the physical metric.
The metric equation of motion reads where G ab is the Einstein tensor constructed out from the metric and its derivatives, round bracket represent symmetrization with respect to indices, e.g.
while τ ab and Θ ab stand for the contribution of the scalar and vector fields: where F ab represent the field strength of the vector field and λ is a Lagrange multiplier, K and k are respectively the vector and scalar couplings, l is a parameter and F stands for the TeVeS function. It is useful to define a function µ(y) by using which expresses the equations of the scalar fields to where The equation of motion for the vector field reads Recalling (2), one can derive the equation of motion for the Lagrange multiplier by contracting the vector equation with A a . The λ's equation is a linear algebraic equation in term of λ.

II. TEVES THEORY IN THE SPACE-TIME GEOMETRY AROUND THE EARTH
The TeVeS theory coincides to the general relativity in two limits The Bekenstein limit can be understood as the perturbative regime of the Milgrom limit. Here we consider the Bekenstein limit. We shall consider the Newtonian regime of TeVeS theory wherein µ = 1 and the contribution of F to the equation of motion of the metric can be neglected [3]. Note that the term neglected does not contribute to 1 r 2 behavior of the fields around spherical mass distribution we are interested in. So we have We are interested in the solution representing the space-time geometry around the earth. The linear equations of motion suffice to describe the dynamics of gravity near the earth with the precision required. So we ignore all the non-linear terms in the field equations. To this aim we consider that and we keep the linear terms in h ab . We also utilize the fact that h ab ∝ T ab , φ ≪ 1 and keep the linear leading terms: Contracting (17) with A a and recalling A a A a = −1, results inserting which back into (17) gives a second order non-linear equation for A: We require spatial isotropy far away from the matter distribution. The isotropy demands The presence of the matter adds a correction to this value of A a (0) : where δ n A a represents the term at the order of (h ab ) n . We are interested only in the linear correction to A : δ 1 A.
Since the field strength of theĀ vanishes, δ 1 A a does not couple to h ab . Therefore δ 1 A a = 0 which in turn implies that λ is vanishing at the linear order approximation to the equations of motion: Due to the value of λ (25), the equation of motion for the metric (16) simplifies to the ordinary Einstein equation: We are interested in the gravitomagnetism approximation to general relativity. So we consider the physical energy momentum of a pressure-less slow moving mass distribution where the non-zero components of the energy momentum tensor read where ρ is physical density and v i represents the physical velocity. Defining the Einstein gravitoelectric and gravitomagnetic potential as usual These quantities are solved to where Φ N is the Newtonian field potential and A EH i is the gravitomagnetic potential of the Einstein-Hilbert theory. In the absence of the mass distribution, we require the physical metric to coincide to the Einstein metric. This sets the cosmological value of the scalar field to zero. This choice corresponds to the cosmological definition of time where in the physical time and the Einstein time coincides to each other. In the presence of mattter φ also gets a series expansion in term of the deviation from the Minkowski metric where δ n φ represents the term at the order of (h ab ) n . We assume that v i ≪ 1 so the equation of motion of δ 1 φ can be approximated to This implies that where Φ N as in the previous case represent the Newtonian gravitational potential. Due to the way that the scalar appears in the physical metric, beside the fact thatĀ has only a temporal component, the contribution of J to δ 1 φwhich we have neglected -does not contribute to theg 0i components of the physical metric. The physical gravitomagnetic and gravitoelectric fields are defined by the metric that a test particle probes, or the metric that appears in the worldline action of a particle where τ is an affain parameter. It is the physical metric that leads to the physical gravitomagnetic and gravitoelectric force. Inserting (33), (29) and (30) into the definition of the physical metric (1), the non-vanishing components of the physical metric at the gravitomagnetism approximation follows Therefore the gravitomagnetic and gravitoelectric field of the TeVeS theory read The value of the Newtonian constant is identified by fitting 1/r 2 function on the gravitoelectric acceleration around stationary mass distribution. So the reported value of G is in fact G obs = G(1 + 3k 4π ). Let the gravitomagnetic/electric field be expressed in term of the observed value of the gravitational Newton constant: This means that the gravitomagnetic field of the TeVeS solution representing the space-time geometry around the earth is weaker than the Einstein-Hilbert theory's prediction by the factor of 3k 4π . Ref. [6] considers the lunar laser ranging measurements and reports that the Einstein-Hilbert gravitomagnetic force is consistent with the observation with 0.1% accuracy. This bound requires at the standard confidence level. This bound is two orders of magnitude weaker than what is commonly expected for k in order to keep the TeVeS theory in its MOND regime consistent with the cosmological/astronomical observation. However, a priori contradicting this expectation does not diminish the possibility to retain the theory consistent with the nature. The Gravity Probe B [7] through the Pugh-Schiff effect [8,9] provides independent measurements of the geodetic and frame-dragging effects at an accuracy of 0.28% and 19%, respectively, neither of which improves the LLR measurements' constraint on k. Now that we have a direct constraint on k, we can constraint the parameter of l in TeVeS theory. Noting that K < 10 −2 , the MOND limit of the TeVeS theory for the choice of the F function of [3], gives to the following critical acceleration The MOND terminology requires [10] a 0 = (1.0 ± 0.2) × 10 −10 m s 2 .
Our constraint on k then demands that l > (1.65 ± 0.35) × 10 25 m (44) at the standard confidence level. This is the first constraint on the dimension full parameter of the TeVeS theory.