CONSTRAINING SPACETIME TORSION WITH THE MOON , MERCURY AND LAGEOS

We consider an extension of Einstein General Relativity where, beside the Riemann curvature tensor, we suppose the presence of a torsion tensor. Using a parametrized theory based on symmetry arguments, we report on some results concerning the constraints that can be put on torsion parameters by studying the orbits of a test body in the solar system.


Introduction
One of the various generalizations of the Einstein theory of General Relativity is the so-called Einstein-Cartan theory [11]: it consists of a spacetime endowed with a locally Lorentzian metric g λµ and with a compatible connection Γ ν λµ , not assumed to be symmetric, so that Γ ν λµ is not equal, in general, to Γ ν µλ .The connection Γ ν λµ has a curvature tensor, and its lack of symmetry amounts to the additional presence of a torsion tensor [4] 1 S ν λµ := 1 2 Γ ν λµ − Γ ν µλ .Γ ν µν is determined uniquely by g µν and by the torsion tensor, as follows: Γ ν λµ = ν λµ + S ν λµ + S ν µλ + S ν λµ , where {•} is the Levi-Civita connection 2 .
Usually, the torsion tensor is related to the intrinsic spin of matter [10,11] 3 .Since the spins of elementary particles in macroscopic matter are usually oriented in a random way, such theories predict a negligible amount of torsion generated by massive bodies.As a consequence, spacetime torsion would be observationally negligible in the solar system.On the other hand, as pointed out in [15], the existence or nonexistence of a torsion tensor in the solar system should be tested experimentally4 : for this reason the authors of [15] developed a parametrized theory based on symmetry arguments and, computing the precessions of gyro-scopes, put constraints on torsion with the Gravity Probe B experiment.
The aim of this short paper is to report on the results of [16,17] concerning the constraints on torsion that can be put by studying the orbits of a test body in the solar system, following the nonstandard parametrized approach of [15].The computations concern torsion corrections to: (i) the orbital geodetic (or de Sitter) precession, (ii) the precession of the pericenter of a body (a planet) orbiting around a central mass, (iii) the orbital frame-dragging (or Lense-Thirring) effect.We then use the measured Moon geodetic precession, Mercury's perihelion advance and the data from the measurements of LAGEOS satellites5 , to put contraints on torsion, looking at the secular perturbations of the orbits.

Working assumptions and autoparallel trajectories
Our main working hypotheses are the following: (i) Weak field approximation and slow motion of the test bodies, assumptions probably sufficiently accurate for solar system experiments.(ii) Spherical or axial symmetry, depending on the situation at hand (in the case of the Lense-Thirring effect, we suppose also Earth uniformly rotating, while in the computation of the de Sitter effect Earth and Sun are supposed to be nonrotating).For example, if m is the mass of the body, the torsion tensor around a spherically symmetric body (Sun/Earth) in spherical coordinates (t, r, θ, φ) can be parametrized to second order in m/r 1 as where t 1 , t 2 , t 3 , t 4 are dimensionless torsion parameters (all other components of the torsion tensor vanish.Moreover, it turns out that t 4 will not enter into our computations6 ).In case of a uniformly rotating spherical body, the expression of the torsion tensor reads (to second order in m/r) as where w 1 , . . ., w 5 are other torsion parameters, m is the mass of the rotating body and J its angular momentum.(iii) Bodies move along (causal) autoparallel trajectories, namely they satisfy and not along geodesics.This latter assumption is more questionable; on the other hand, assuming that the test bodies move along geodesics does not give any constraint on the torsion parameters.For example, indicating with m the mass of the Sun, the system of autoparallel trajectories (of the Moon) in the case of computation of the geodetic precession reads as where ξ α , ρ are the heliocentric rectangular coordinates of the Earth, X α , ∆ are the heliocentric rectangular coordinates of the Moon, and x α , r are the geocentric rectangular coordinates of the Moon.In the case of the Lense-Thirring effect, the system of autoparallel (of a satellite) reads as where m ⊕ is the mass of the Earth, J its angular momentum, It should be noted that from assumption (iii) it follows that the antisymmetric part of the torsion tensor cannot be measured (the torsion tensors that we will consider are not totally antisymmetric).(iv) In the computation of the geodetic precession we assume that we can superimpose linearly the metric and torsion fields of Sun and Earth to obtain the global fields.(v) All computations have been performed by Taylor expanding in m/r at the required order7 .

Description of the results
Using perturbative methods in Celestial Mechanics, in the case of the three-body problem we can constrain the torsion parameters with the Moon as follows.The secular precession of node Ω of the satellite orbiting around Earth turns out to be: where ν 0 is the angular velocity of the Earth and t is time.We observe that (δΩ) sec is independent of the details of the satellite8 .The same right hand side of Eq. 1 is obtained for the secular precession of the lunar perigee (δ ω) sec .From these computations, we find b ≡ geodetic precession with torsion geodetic precession in GR Using the Lunar Laser Ranging data giving the relative deviation from GR [23] we find |b − 1| < 0.0064.

Using the Cassini measurement
vol.
supplement/ Constraining Spacetime Torsion with the Moon, Mercury and LAGEOS Using a two-body computation, we can now constrain the torsion parameters with Mercury as follows.If ω is the longitude of the pericenter, the secular contribution (δ ω) sec reads as where a is the semimajor axis of the orbit, e is the eccentricity and v the true anomaly.Then B ≡ perihelion precession with torsion perihelion precession in GR Using the planetary radar ranging data giving the relative deviation from GR of 10 −3 [22], we find |B − 1| < 0.001.Using the Cassini measurement one gets |1 − β + t 3 | < 0.0286.If in addition we assume In the case of the Lense-Thirring effect, the precession of the node Ω of LAGEOS reads as where a the semimajor axis of the satellite, e the eccentricity of the orbit and t is time.We then have b Ω ≡ precession of the node with torsion precession of the node in GR Using |α 1 | < 10 −4 [24] and the measurements of [5], we get |b Ω − 0.99| < 0.10 and Reasoning similarly for the perigee, we eventually have It is worthnothing that Eqs. 4 and 5 are obtained making use also of the previously obtained estimates Eqs. 2 and 3.

Discussion
Beside the assumptions listed in Section 2, the bounds Eqs. 2, 3, 4 and 5 rely on the combination of a certain number of experimental estimates on the PPN parameters and on the various precessions.They can be improved as far as the experimental estimates improve.Estimates Eqs.2-5 should be coupled with the estimates obtained in [15] obtained analyzing the precession of gyroscopes of GPB.These latter estimates turn out to be different from ours, see [16,17] for the details.

Future prospects
Before the end of the decade, robotic missions on the lunar surface could deploy new scientific payloads which include laser retroreflectors and thus extend the Lunar Laser Ranging reach for new physics (and possibly for torsion).In particular, the single, large, fused-silica retroreflector design developed by the University of Maryland and INFN-LNF [6] could improve on the performance of current Apollo arrays by a factor 100 or more.
After the end of this decade, results from the Bepi-Colombo Mercury orbiter are expected to improve the classical test of the perihelion advance [19].The latter measurement can be cross-checked by new planetary radar ranging data taken simultaneously with Bepi-Colombo's ranging data.Mercury's special role in the search for new physics effects, and for spacetime torsion in particular, is due to the relatively large value of its eccentricity and to its short distance to the Sun.
Eventually we observe that the recently approved JUNO mission to Jupiter [18] will make it possible, in principle, to attempt a measurement of the Lense-Thirring effect through JUNO's node.Hence such a mission may yield an opportunity to improve of the constraints on torsion parameters.

Conclusions
Estimates Eqs.2-5 give an order of magnitude of the torsion tensor; they neither prove nor disprove the existence of a non-vanishing torsion tensor in the solar system.A more definite answer could be given by refining these estimates, taking advantage of future missions.