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International Workshop on Paolo FarinellaPisa, June 14-16, 2010
Constraining spacetime torsion with the Moon and MercuryGiovanni Bellettini
Math. Department, Univ. Roma Tor Vergata & INFN Frascati, Italy
International Workshop on Paolo Farinella Pisa, June 14-16, 2010
Collaborators
- S. Dell’Agnello (INFN Frascati),
- R. March (IAC CNR, Roma),
- R. Tauraso (Univ. Roma Tor Vergata)
Some mathematical references on torsion:
E. Cartan, Ann. Ec. Norm. Sup. 41 (1922, 1923, 1924).
I. Agricola , Archivum Matematicum Brno (2006), p. 30.
W. Kuhnel, Differentialgeometrie, p. 156.
M. Nakahara, Geometry, Topology and Physics, p. 258.
M.M. Postnikov, Geometry VI. Riemannian Geometry, p. 19.
Motivation: [MTGC]: Y. Mao, M. Tegmark, A.H. Guth, S. Cabi,Phys. Rev. D 76, 104029 (2007).
International Workshop on Paolo Farinella Pisa, June 14-16, 2010
Plan of the talk
short introduction to torsion
derivations of equations of motion
constraints on torsion parameters
International Workshop on Paolo Farinella Pisa, June 14-16, 2010
Torsion of ∇
(M, g) semi-riemannian manifold∇ connection on the tangent bundle of M compatible with g , i.e.∇Xg(Y ,Z ) = g(∇XY ,Z ) + g(Y ,∇XZ ).Therefore, ∇ and g are not completely independent:
∇g = 0
(i.e., ∇-parallel transport conserves the g -scalar product).[X ,Y ] commutator between X and Y∇XY −∇Y X − [X ,Y ] =: T (X ,Y ) torsion tensor of type (1, 2)This is a concept independent of the metric g .Write in coordinates ∇eµeν = Γλ
µνeλ
International Workshop on Paolo Farinella Pisa, June 14-16, 2010
Lemma
T ≡ 0 if and only if ∇ is symmetric, i.e. Γλµν = Γλ
νµ
In coordinates the torsion tensor T becomes:
T λµν :=
1
2
(Γλ
µν − Γλνµ
)In case ∇ = ∇g is the Levi-Civita connection of the metric g , thenthe corresponding torsion is zero.
International Workshop on Paolo Farinella Pisa, June 14-16, 2010
Geometric meaning: an attempt to construct an infinitely smallparallelogram with sides dλ spanned by two vectors X ,Y , leads toa pentagon whose closing side is an infinitesimal equal toT (X ,Y )(dλ)2 + O((dλ)3)
)
p0
Y X
pqλ λY(λ )
X(λ
International Workshop on Paolo Farinella Pisa, June 14-16, 2010
Example
(E. Cartan, 1922) M = R3, g = 〈·, ·〉,
∇XY := ∇gXY +
1
2X × Y , X ,Y ∈ TR3
- ∇ is metric
- ∇ has nonvanishing torsion T (X ,Y ) = X × Y
- ∇ has nonvanishing Riemann tensorRiem∇(ei , ej)ek = 1
4(ej〈ei , ek〉 − ei 〈ej , ek〉)- ∇ has the same geodesics (that will be called autoparallel)
than ∇g , since in this case the contortion tensorK (X ,Y ) := 1
2 X × Y is antisymmetric. These are curvesalong which the velocity vector is transported parallel to itselfby the connection.
- but ∇ induces a different parallel transport.
International Workshop on Paolo Farinella Pisa, June 14-16, 2010
Indeed: consider the z-axis γ(t) = (0, 0, t), a geodesic, and thevector field X which, at every point γ(t), consists of the vector(cost, sint, 0). Then ∇g
γX − γ × X = 0 = ∇γX . Hence X isparallel transported for ∇ according to a helicoidal movement.
X(t) is parallel transported according to a helicoidal movement of constant width
π2t =
t = π3
X
International Workshop on Paolo Farinella Pisa, June 14-16, 2010
Example
E. Cartan (1923) M = S2 ⊂ R3, g induced metric. We call twovectors parallel if their angles wih the meridian through thatpoint coincide. Then
- ∇ is metric- ∇ has nonvanishing torsion- ∇ has vanishing Riemann tensor- the autoparallels of ∇ are different from geodesics. The
autoparallels are the loxodromes, which intersect themeridians at a constant angle.
International Workshop on Paolo Farinella Pisa, June 14-16, 2010
- Einstein’s general relativity (1915): ∇ = ∇g is the Levi-Civitaconnection, Riem∇ 6= 0, T ≡ 0. There is an action functional.
- Hayashi-Shirafuji’s new general relativity (1979): Riem∇ ≡ 0,T 6= 0. There is an action functional. A static massive bodygenerates a torsion field.
The ingredients that we have are a metric g and a connection ∇with
∇λgµν = 0
but Riem∇ 6= 0, and we do not have an action, so that we do nothave the field equations. We will use the system of ODE’sexpressing autoparallel trajectories:
d2xλ
dτ2+ Γλ
µν
dxµ
dτ
dxν
dτ= 0,
where τ can be taken to be the proper time. This system ofequations expresses the fact that the velocity vector is paralleltransported to itself.
International Workshop on Paolo Farinella Pisa, June 14-16, 2010
Derivation of the equations of motion
In coordinates, the connection Γλµν is determined uniquely by gµν
and by the torsion tensor as follows:
Γλµν =
λ
µν
+ K λ
µν ,
where · is the Levi-Civita connection, and
K λµν := T λ
µν + Tλνµ + Tλ
µν
is the contortion tensor (and recall that T λµν := 1
2
(Γλ
µν − Γλνµ
)).
International Workshop on Paolo Farinella Pisa, June 14-16, 2010
Assumptions:
M = R× R3
weak field approximation
slow motion
spherical symmetry: metric and torsion around a sphericallysymmetric body (Sun/Earth) in spherical coordinates(t, r , θ, φ)
bodies move along autoparallel trajectories, and not alonggeodesics
in the computation of the geodetic precession (a three-bodyproblem), we assume that the we can superimpose linearlytwo spherically symmetric fields to obtain the global field
International Workshop on Paolo Farinella Pisa, June 14-16, 2010
ds2 = −h(r)dt2 + f (r)dr2 + r2[dθ2 + sin2 θdφ2].
to second order in m/r we have, as in the PPN framework,
h(r) = 1− 2m
r+ 2(β − γ)
m2
r2, f (r) = 1 + 2γ
m
r,
where all other PPN parameters are supposed to be negligible.Symmetry arguments imply:
T ttr =t1
m
2r2+ t3
m2
r3,
T θrθ =S φ
rφ = t2m
2r2+ t4
m2
r3,
and t4 will not enter in our computations.t1 = 0 will be fixed by the imposing the Newtonian limit.Therefore Γλ
µν becomes an explicit function of four independentparameters,
Γλµν = Γλ
µν (γ, β, t2, t3, r , θ, φ) .
International Workshop on Paolo Farinella Pisa, June 14-16, 2010
Correction to precession of pericenter (Mercury)
The equations of autoparallel trajectories
d2u
dφ2+
(1− A
h2m2
)u =
m
h2+
3
2Fmu2,
where u = 1/r , h = r2 dφdτ exp(t2
mr ), and
A = 4− 2γ − 2β + 4t2 + 2t3.
Notice that A = 0 in the case of general relativity (γ = β = 1 andt2 = t3 = 0).If ω is the longitude of the pericenter, the secular contribution(δω)sec reads as
(δω)sec = (2 + 2γ − β + 2t2 + t3)m
a(1− e2)v ,
where a is the semimajor axis of the satellite orbit, e is theeccentricity, and v is the true anomaly.
International Workshop on Paolo Farinella Pisa, June 14-16, 2010
Remarks:
If t2 = t3 = 0 then
(δω)sec = (2 + 2γ − β)m
a(1− e2)v ,
which is the usual precession of pericenter. Hence whenγ = β = 1 we find the usual expression of given by generalrelativity.
Our result is not in agreement with the result on theprecession of pericenter obtained by [MTGC].
International Workshop on Paolo Farinella Pisa, June 14-16, 2010
Correction orbital elements (Sun, Earth, satell.)
We assume
gµν = (gµν)0 + (gµν)01, T λ
µν = (T λµν )0 + (T λ
µν )01,
where
- (gµν)0 and (T λµν )0 are the metric and the torsion tensors,
taking into account the Sun only, supposed at rest;- (gµν)0
1 and (T λµν )0
1 are the metric and the torsion tensorstaking into account the Earth only; these tensors are given ateach time by the previous expressions, computed as if theEarth were at rest (at that time).
We also assume, beside the spherical symmetry of the Sun and theEarth,
motion of the Earth is circular.
International Workshop on Paolo Farinella Pisa, June 14-16, 2010
Now we follow the de Sitter’s computations (1916). Secularcontributions to the variation of the longitude Ω of the node of thesatellite orbiting around Earth:
(δΩ)sec =1
4
mν0
ρ(2 + 4γ + 3t2) t, (1)
where m is the mass of the Sun, ν0 is the angular velocity of theEarth, and ρ is the distance between the Sun and the Earth. Ift2 = t3 = 0 then
(δΩ)sec =1
2
mν0
ρ(1 + 2γ) t,
which is the usual geodetic precession. Hence, when γ = β = 1 wefind the usual formula of geodetic precession found by de Sitter
(δΩ)GRsec =
3mν0
2ρt.
International Workshop on Paolo Farinella Pisa, June 14-16, 2010
Concerning the longitude ω of the pericenter of the satellite: wefurther assume
small eccentricity of the satellite around Earth.
Again we find the secular contribution
(δω)sec =1
4
mν0
ρ(2 + 4γ + 3t2) t.
Note that
- (δΩ)sec and (δω)sec are independent of the details of thesatellite motion.
- The analogous result obtained by [MTGC] in the case ofgyroscopes is slightly different: the parameter t2 has adifferent multiplicative constant in front.
International Workshop on Paolo Farinella Pisa, June 14-16, 2010
Constraints on torsion parameters
Consequences of the computations on the geodetic precession.
Using the PPN formalism, and
the Lunar Laser Ranging data
the above mentioned formula (1) for (δΩ)sec
the Cassini spacecraft result γ = 1 + (2.1± 2.3)× 10−5,
we find|t2| < 0.0128.
International Workshop on Paolo Farinella Pisa, June 14-16, 2010
Consequence of the computations on the perihelion. Using
the planetary radar ranging data
γ = 1 + (2.1± 2.3)× 10−5
we get|(1− β) + 2t2 + t3| < 0.003.
Our conclusions are:
|t2| < 0.0128, |(1− β) + 2t2 + t3| < 0.003.
International Workshop on Paolo Farinella Pisa, June 14-16, 2010
If in addition we use the estimate β = 1 + (1.2± 1.1)× 10−4, wededuce
|2t2 + t3| < 0.003,
and|t3| < 0.0286.
International Workshop on Paolo Farinella Pisa, June 14-16, 2010