Test particles behavior in the framework of a lagrangian geometric theory with propagating torsion

0.17 MB·English

by  G. Aprea

Test particles behavior in the framework of a
lagrangian geometric theory with propagating
torsion
G. Aprea, G. Montani, R. Ruffini
ICRA-InternationalCenter for Relativistic Astrophysics,
4
DipartimentodiFisica,Universita`degliStudi diRoma“LaSapienza”,
0
0 PiazzaleAldoMoro5,00185Roma,Italy.
2
April 10, 2003
n
a
J
3 Abstract
1
Working in the lagrangian framework, we develop a geometric theory

in vacuum with propagating torsion; the antisymmetric and trace parts
1
of the torsion tensor, considered as derived from local potential fields,
v
4 are taken and, using the minimal action principle, their field equations
5 are calculated. Actually these will show themselves to be just equations
0 for propagating waves giving torsion a behavior similar to that of metric
1 which,asknown,propagatesthroughgravitationalwaves. Thenweestab-
0 lishaprincipleofminimalsubstitutiontoderivetestparticlesequationof
4 motion,obtaining,asresult,thattheymovealongautoparallels. Wethen
0 calculate the analogous of the geodesic deviation for these trajectories
/
c and analyze their behavior in the nonrelativistic limit, showing that the
q torsion trace potential φ has a phenomenology which is indistinguishable
- from that of the gravitational newtonian field; in this way we also give a
r
g reason for why therehaveneverbeen evidencefor it.
:
v PACS number: 04.50.+h
i
X Keywords: Torsion, Alternative theories of gravity.
r
a
1 Introduction
As well known, in non flat spaces the concept of parallel transport of vector fields
needs the introduction of a connection which is also needed to define the covariant
derivative. In fact, by means of a connection, we can define the equation of parallel
transport in the following way: on a manifold M, given a curve γ(t) passing for a
pointP M,theparalleltransportedvectorfieldofthevectorfieldVα(P)alongγ(t)
∈
is the solution of:
dVα
= C αVνγ˙µ. (1)
dt − µν
In regard to the covariant derivative of a vector field Vα(x), instead, it is defined in
this way:
D Vα=∂ Vα+C αVγ (2)
β β βγ
1
The quantities C α are just the coefficients of the affine connection and a space,
µν
endowed with them,is called an affinespace and usually indicated bythesymbolL ;
4
in general these coefficients are non tensorial quantities but that is not true for their
antisymmetric part called torsion:
S =C α, (3)
αβ [αβ]
which indeed transforms like a tensor. Just because of this property, the presence of
torsioninspace-timedeniestheprincipleofequivalenceofitsimportance;hereweare
notreferring totheequivalencebetweeninertial andgravitational mass, whichispre-
served since the theory remains geometric, but to the formulation of the equivalence
principle[19]accordingtowhich,oncedefinedaninertial1 frameinapoint,therethe
laws of physics are the same as those of special relativity. In the case of presence of
torsion, the latter, being a tensor, can’t be set to zero by a convenient choice of the
coordinatesso,sinceweexpecttorsion tobesourceofsomeforce, itisnotpossibleto
defineaninertialframeinanypointwhichisanecessaryconditionfortheapplicability
of the principle.
In order to calculate lengths and angles, we also need to introduce in our affine
space L a metric and that can be done defining the square modulus of a vector Vα
4
as:
V 2 =g VαVβ; (4)
αβ
k k
here g is just the symmetric metric tensor. With the help of the tensor of non-
αβ
metricity:
Qαβγ Dαgβγ, (5)
≡−
it is possible to establish a relation between the connection coefficients and torsion,
metric, and nonmetricity:
1
Cαβγ = [∂αgβγ ∂γgαβ+∂βgαγ]+[Sαβγ Sβγα+Sγαβ]+
2 − −
(6)
1 1
+ [Q Q +Q ] Γ +K + [Q Q +Q ].
2 αβγ− γβα βγα ≡ αβγ αβγ 2 αβγ− γβα βγα
The new quantities we have introduced here are the Christoffel symbols 2 Γ and
βαγ
thecontortiontensorK ;theformeraresymmetricinthefirsttwoindiceswhilethe
αβγ
latter is antisymmetric in the last two. In the rest of this paper, in order to preserve
lengths and angles under parallel displacement, we will assume the metric postulate
according towhich nonmetricityisvanishing; suchaspace isindicatedbythesymbol
U :
4
GenerallinearspaceL4 Qαβγ Einstein-CartanspaceU4 Sαβγ RiemannianspaceV4
−−−−−→ −−−−−→
CompletelyneglectedinthefirstformulationofthetheoryofGeneralRelativity(GR)
byEinstein,theintroductionoftorsion waslatertakenintoconsideration byEinstein
himself [7], Eddington [6], Schr¨odinger [24] and principally Cartan [3][4] [5], who had
theideaofatheoryinwhichtorsionwasconnectedwithintrinsicangularmomentum.
Later this idea was shelved and only since the fifties the possibility of introducing
torsion into GR had been revalued. Utiyama [30], Kibble [14] and Sciama [25][26],
inspired bythework of YangandMills [31]on gauge theories, formulated a theoryof
gravitation as a gauge theory in which thepresenceof torsion was necessary; thenby
1Inaninertialframe,abodyatrestremaininsuchastate.
2Althoughthetensorialformalismused,theyarenontensorialquantities
2
themidseventiesHehletal. [8]managedtosetupaPoincar´egaugetheoryofgravita-
tion, theU theory,with torsion corresponding to therotation gauge potential. They
4
assumed the geometric lagrangian density was the curvature scalar and that matter
could be taken into account simply through the minimal coupling rules: ηµν gµν,
→
∂µ Dµ;inthistheoryintrinsicangularmomentumcreatestorsionwhichcan’tprop-
→
agate through empty space; in brief, this is due to the fact that the field equations
that relate torsion and spin are algebraic and don’t involve derivatives, allowing the
substitution of torsion in the matter Lagrangian leading to an effective contact in-
teraction. Since, in the first instance, it is reasonable to expect torsion to behave as
any other interaction field, i. e. propagating into vacuum, this aspect of U theory is
4
unsatisfactoryandinthispaperatheorytoovercomethisproblemisdiscussed. Inthe
next section some assumptions are made about the form of the torsion tensor which
arenecessarytoobtainapropagatingtorsionwithoutchangingtheformoftheaction,
namely thecompletely antisymmetric and trace part of thetorsion tensorare consid-
ered derived from two local torsion potential. Then, in section 3, by the principle of
least action we determine the field equations for these potentials which indeed reveal
themselves to be wave equations. In the second part (sections 4 and 5) we discuss
the problem of determining the equation of motion of test particles and establish a
principle of minimal substitution which leads usto say that test particles movealong
autoparallels; finally we calculate the nonrelativistic limit of these trajectories and of
thetidaleffectsandshowthatthetorsiontracepotentialφentersinalltheequations
inthesamewayas,inthislimit,thegravitationalpotentialdoes. Concludingremarks
follow.
2 The form of the torsion tensor
As we have seen, torsion is a three indices tensor which is antisymmetric in the first
two; according to the rules of group theory it can be decomposed in a completely
antisymmetricpart,atracepartandathirdpartwithnospecialsymmetryproperties
[9]. In our analysis we consider only the first two terms and, in addition, we assume
that they can be derived from theexterior derivative of two potentials, in thisway:
{}
S B ∂ A =D A (7)
[µνλ]→ µνλ ≡ [µ νλ] [µ νλ]
1
Tr(Sµνλ) ((∂µφ)gνλ (∂νφ)gµλ) (8)
→ 3 −
Aµν(x) is an antisymmetric tensor while φ(x)is a scalar.
These potentials play a role analogous to that of metric in the symmetric part of the
connection:
1
Γαβγ = [∂αgβγ ∂γgαβ+∂βgαγ]. (9)
2 −
By virtue of (6), we can also write the expression for the connection coefficients and
contortion:
K S S +S =
µνλ µνλ νλµ λµν
≡ −
2 (10)
=∂[µAνλ]+ 3((∂λφ)gµν−(∂νφ)gµλ)
C =Γ +K =
µνλ µνλ µνλ
2 (11)
=Γµνλ+∂[µAνλ]+ 3((∂λφ)gµν −(∂νφ)gµλ)
Theintroductionoftorsion potentialsfortheantisymmetricpartoftorsion isalready
present in the literature 3 and has its main motivation just in obtaining a torsion
3See,forexample,[11]
3
propagatingintovacuum. Asfarastheexpression(8)forthetracepartisconcerned,
itisworthnotingthatthesameexpressionispresentin[12]butinadifferentcontest;
in fact, in Hojman et al’s article it was introduced to get a coupling of torsion to
electromagnetic field which didn’t break gauge symmetry. Really there is another
way available to obtain propagation; it consists, in analogy to Yang Mills theory, in
introducingsquaretermsincurvatureandtorsionintheEinstein-Hilbertaction. This
approach is discussed, among others, in [27],[20],[28], [29],[18],[2]; here we make the
different choice of using torsion potentials which, we believe, has these advantages:
1) we can preserve the simplicity of the Einstein-Hilbert action with the minimal
substitution Γ G +K ; 2) we have put both riemannian connection and
αβγ αβγ αβγ
→
torsion on the same level since as the former is derived from metric, the latter is
derived from potentials; 3) in the limit of small and slow varying φ the action (13) is
equivalenttothelowenergylimitofstringtheorylagrangian,asalreadymentionedin
[11](and reference therein), suggesting torsion potentials to be a necessary ingredient
in more general theory.
3 Field equations
According to what we have said in the previous sections our Lagrangian density is of
theHilbert-Einstein form 4:
1
= dx √ g R(x)
A −2k Z − ≡
(12)
1
≡−2k Z dx √−g gβγδδα(cid:16)∂αCβγδ−∂βCαγδ−CαγηCβηδ+CβγηCαηδ.(cid:17)
Wewill obtain the field equations with theleast action principle calculating the vari-
ationswithrespecttothemetricandboththetorsionpotentials. Inordertosimplify
the variational calculation we proceed now in splitting the action in its riemannian
part plustorsion-depending terms; with thehelp of (11), we get 5:
(13)
1 {} 2
= dx √ g R(x) BαβγBαβγ (∂αφ)2 ; (14)
A −2k Z − (cid:18) − − 3 (cid:19)
Variations respect tog , A and φ yield:
αβ αβ
{} 1
−Gαβ− 2gαβBµνσBµνσ+3BανσBβνσ+ (15)
8 1
gαβ(∂µφ)2 gαµgβν(∂µφ)(∂νφ) =0
− 3(cid:18)2 − (cid:19)
{}
DµBµαβ =0 (16)
{}
Dµgµν∂νφ=0. (17)
Inthefirstequation (15) thereistheRiemannian Einstein tensorasinGR;moreover
inthiscasewehavefourtermsallquadraticinthetorsionpotentials. Byvirtueofthis
4Here k is a constant related to Newton’s gravitational constant G = 6.670 ×
10−8cm3g−1s−2 via:
1 c3
=
k 8πG
5Thesymbol“{}”stands forriemannian
4
if weareinterested in solving it at firstorderfor little valuesof thetorsion potentials
we can neglect those quadratic terms and fall back in theGR field equations; we can
solvethemandgetthemetrictobeputinequations(16)and(17)tofind,atthefirst
order, the torsion potentials. As for equations (16) and (17), it can be seen that the
goalofapropagatingtorsionhasbeenreached,infactwehavetwosecondorderPDE
forboththepotentials. Equation(16)canbesimplifiedusingitsinvarianceunderthe
gauge transformation:
{} {}
Aαβ A′ αβ =Aαβ+DαYβ DβYα; (18)
→ −
in fact, if we choose Y such that
{}
DαA′ αβ =0. (19)
aftersomecalculation involvingcommutationrulesforcovariantderivatives,itcanbe
put in the form:
{} {} {} {} {} {}
DµDµA′ αβ−RµαβρA′ ρµ+RαρA′ βρ+RβρA′ ρα−RµβαρA′ µρ ≡∆DR(A′)=0.
(20)
Here∆DRistheDeRhamoperatorwhichgeneralizetheLaplacianoperatorinnonflat
spaces. It is easy to show that a field A′ obeying (20) and (19) has only one freedom
degree,or,inotherwords,itcanhaveonlyonedirectionofpolarizationintheiperplane
normal to its propagation direction. As far as equation (17) is concerned, it is worth
noting that it is a massless Klein-Gordon field equation so that we can consider the
potential φ as a geometrical manifestation of thisfield.
4 Test particles paths
The problem of the determination of the equation of motion of a test particle can
be approached in a number of ways; one is that proposed by Papapetrou [21] which
consists in obtaining theequation of motion from theconservation law of theenergy-
momentum tensor. According to us this approach has some unsatisfactory aspects:
thefirst is that we can havesome ambiguity on theright way to get theconservation
law of the energy-momentum tensor since we can start both using N¨oether theorem
and Ricci identities, but, in spaces with torsion, the results can be different [10];
in addition, once we have the conservation law, we must have the expression of the
energy-momentum tensor which is rather difficult especially in the case of presence
of nonriemannian quantities as torsion; in this case, in fact, a anti-symmetric part of
the energy-momentum tensor, probably depending on spin, is involved and it is not
clear either whether it is correct to give a semiclassical expression of it, being spin a
purelyquantumquantity,oritsexplicitform. Than wehavetheapproachof Hojman
[13]whichconsistsin definingallthepossible scalar quantitiesthatcanbeinthetest
particle action; then the action is build up and the equation of motion are obtained
by variations with respect to the particle coordinates. According to us this approach
has one of the same unsatisfactory aspects of the previous, since, taking into account
the spin of the test particle, again we need to have a semiclassical expression for the
spin depending part of the test particle action although spin should be treated only
in quantum mechanics. Another way to get the test particle equation of motion is
to make use of the principle of the shortest path which assumes that a test particle
moves from a point A to another B in space in a way such that its trajectory has
the least length among all the curves joining A with B. Although this method seems
simpleandnice,itiscompletelyregardlessofthepresenceoftorsion becausethelast,
5
not contributing to the length of a curve, neither appears in the equation of motion
of any test particle.
According to us, instead, the presence of a tensorial quantity as torsion, which has
a role in the parallel transport of vector fields in space, should have some effects on
the motion and so we assume as the correct method to have the equation of mo-
tion in curved space from the knowledge of that in flat space the following minimal
substitution rule:
d D
Ordinary derivative Covariant derivative . (21)
(cid:18)dτ(cid:19)→ (cid:18)dτ(cid:19)
According to this rule the equation of motion in curved space is obtained from the
analogous oneof special relativity
duα
=0, (22)
dτ
where uα is the 4-velocity via the (21):
Duα
=0 (23)
dτ
which can berewritten as:
duα 2
dτ =−Cµναuµuν =−Γµναuµuν−gαλ3((∂λφ)gµν−(∂νφ)gµλ)uµuν. (24)
Thisistheautoparallel equationwhich,togetherwith thegeodesic equation,isaspe-
cial curvethat can bedefinedin non flatspaces; while thelatter istheshortest curve
joiningtwopoints,theformeristhecurvewhosetangentvectorisparalleltransported
alongit. Theautoparallelcurveisthesimplestgeneralizationoftheflatspaceequation
ofmotion(22)whichissuitabletotakeintoaccountoftorsionorothernonriemannian
quantities 6.
It is worth noting that it is also possible to introduce a new action principle such
that, starting from theaction
M τ2
M = dτ x˙2, (25)
A − 2 Z
τ1
where τ is the proper time, we can have autoparallels as the right trajectories. This
approach is proposed in [15],[16], [17] and here we summarize it briefly. The key
point is that a spacetime with torsion, which can be obtained from a flat spacetime
by a nonholonomic mapping, is affected by a closure failure of parallelograms; as a
consequence,variationsoftestparticletrajectoriescannotbeperformedasintheusual
wayinflatspacetime,i.e. keepingδxa(τ)vanishingatendpoints. Infact,thevariations
δSqµ(τ), images of δxa(τ) under a nonholonomic mapping, are generally not closed;
sotheycanbechosentobezeroattheinitialpointbutthentheyarenonvanishingat
thefinal point, thisfailure being proportional to torsion. The variational calculation,
then, can be done as follows; first we rewrite explicitly the relation between the old
(xa(τ)) and thenew (qµ(τ))paths in integral form:
τ
qµ(τ)=qµ(τ )+ dτ′e µ(q(τ′))x˙a(τ′), (26)
1 a
Z
τ1
where e µ(q(τ′)) is the nonholonomic mapping; then we calculate the variation asso-
a
ciated to qµ(τ):
τ
δSqµ(τ)= dτ′ δSeaµ(q(τ′)) x˙a(τ′)+eaµ(q(τ′))δx˙a(τ′) (27)
Zτ1 h(cid:16) (cid:17) i
6Inthiscontestitisobvioushowtoincludenonmetricity.
6
After havingintroduced a further quantitycalled auxiliary nonholonomic variation:
¯δqµ(τ) e µ(q(τ))δxa(τ), (28)
≡ a
which,incontrasttoδSqµ(τ),vanishesatendpointsandformsclosedpathsinq-space,
we derivethe relation
d
dτδSqµ(τ)=(cid:16)δSeaµ(q(τ))(cid:17)x˙a(τ)+eaµ(q(τ))δx˙i(τ)= (29)
d
=[δSeaµ(q(τ))]x˙i(τ)+eaµ(q(τ)) [eaν(q(τ))¯δqν(τ)],
dτ
which, after thesubstitutions:
d
δSeaµ = CλνµδSqλeaν , eiν =Cλνµq˙λeiµ, (30)
− dτ
can berewritten:
d d
δSqµ = C µδSqλq˙ν +C µq˙λ¯δqν + ¯δqµ, (31)
λν λν
dτ − dτ
or, better,
d
δSbµ= C µδSbλq˙ν+2S µq˙λ¯δqν, (32)
λν λν
dτ −
where we have introduced δSbµ, the difference between δSqµ and ¯δqµ. Now we can
calculatethevariationoftheactionanalogousof(25)underanonholonomicvariation
δSqµ = ¯δqµ+δSbµ:
M τ2 τ2 1
δS M δS dτ gµνq˙µq˙ν = M dτ gµνq˙νδSq˙µ+ ∂µgλκδSqµq˙λq˙κ ;
A ≡ (cid:18)− 2 Z (cid:19) − Z (cid:18) 2 (cid:19)
τ1 τ1
(33)
using [δS,d/dτ]=0 which follow from (26), we can partially integrate the δSq term
−
and, bythe identity∂µgνλ Cµνλ+Cµλν, get:
≡
τ2 d
δSAM =−MZτ1 dτ(cid:20)−gµν(cid:16)q¨ν+Γλκνq˙λq˙κ(cid:17)¯δqµ+(cid:18)gµνq˙νdτδSbµ+CµλκδSbµq˙λq˙κ(cid:19)(cid:21).
(34)
Now it is straightforward to obtain first the equation of motion in absence of torsion
(δSbµ(τ) 0):
≡
q¨ν +Γ νq˙λq˙κ=0, (35)
λκ
that is the geodesic equation; then we can consider all the affine connection and get,
with thehelp of (32), theautoparallel equation:
q¨ν+C νq˙λq˙κ =0. (36)
λκ
Asforthepossibilitytogettheautoparallelmotionfromtheenergy-momentumtensor
conservation, wenowgiveapossible modification of thetestparticleaction suchthat
this result could bepartially obtained. Weassume thetest particle action to be:
M = dτ uµuν e−41φgµν; (37)
A Z
7
then we can calculate its variations with respect to gµν and φ:
gTµν = δδAgµMν =Z dτ u√µugν e−41φ δ(x−x0),
− (38)
φT = δAδφM =−14Z dτ u√µugν e−14φ gµν δ(x−x0),
−
from comparison with
δ = d4x√ g (gTµν δgµν +φT δφ). (39)
A Z −
Now,followingthesameprocedureofHammond[11],weconsiderthemotionofatest
particle in a background geometry which is perturbed in a negligible way by it and
start from theidentity:
(√−g gTµν),ν =√−g gTµν;ν−√−gΓαβµ gTαβ; (40)
now we integrate it over a small volume dV where the only appreciable energy mo-
mentum tensor is that of the test particle and, discarding all surface terms, with the
help of theconservation law in ourcase:
8
gTµν = ∂µφ φT, (41)
;ν 3
we get:
1 d 8
dV(√ g gT(µ0))= ∂µφ dV(√ g φT) Γ µ dV(√ g gT(αβ)).
u0dτ Z − 3 Z − − αβ Z −
(42)
This, with thehelp of (38), can be rewritten in the form:
duα 2
dτ =−Γµναuµuν − 3gαλ(∂λφ)gµνuµuν. (43)
If we multiply both members of this equation by uα we get:
0=uα∂αφ. (44)
Nowifweconsiderourautoparallel equation(24)weseethatit isinaccordance with
these last two equations (43),(44), meaning that the Papapetrou motion is included
as a special case.
5 Autoparallels, Autoparallel deviation and their
nonrelativistic limit
Attheendofthelastsectionwehaveconcluded,onthebasisofourminimalsubstitu-
tion rule, that test particles follow autoparallel trajectories whose equation we recall
now:
d2xα dxβ dxγ dxβ dxγ
= Γ α K α . (45)
dτ2 − βγ dτ dτ − βγ dτ dτ
Itiseasy toseethatin thisexpression theantisymmetricpartoftorsion contribution
vanishes;itonlycontributesasasourceforthemetricthrough(15). Inthissectionwe
will study the nonrelativistic limit of autoparallels and in addition we will calculate
theanalogous of thegeodesic deviation and wewill seetherole oftorsion in thetidal
forces.
8
5.1 Nonrelativistic limit of autoparallels
In order to calculate the nonrelativistic limit of (45), we make the following assump-
tions:
a) 3-velocity much smaller than c so that we can put ui vi;
≃
b) the gravitational field is static and weak, i. e. the quantities h =g η
αβ αβ αβ
−
(hereη is theMinkovsky metric) are verysmall;
αβ
c) thetorsion potential φ is static and weak.
By virtue of these assumptions, since we want to keep only first order terms, we
will eliminate all those terms containing φ, h e vi more than once. After some
αβ
calculations we obtain:
dd2xti =−κ2∂ih00− 23∂iφ. (46)
Now we recall that in GR we had:
dd2xti =−κ2∂ih00, (47)
that allowed us to identify h with thegravitational potential Φ:
00
κ
h =Φ. (48)
2 00
As we can see from (49) the “force” due to the torsion potential is present in the
sameform ofthegravitationalfieldh ;inaddition,asfortheorderweareinterested
00
in, and reminding of the supposed field’s static nature, equation (17) for the field φ
reduces to:
∆φ(~x)=0, (49)
which is thesame of thegravitational field:
∆h (~x)=4πρ. (50)
00
5.2 Deviation of autoparallels
Followingstandardscalculations[22]andremindingthatnowtestparticlesmovealong
autoparallels,ifwetaketwooftheminitiallyverycloseeachother,wefindtheirrelative
acceleration tobe:
{}
D2sα {}
= R αsδuβuγ+ (51)
dτ2 − δγβ
dsγ dsβ {}
K α uβ+ uγ D K α sδuβuγ. (52)
− βγ (cid:18) dτ dτ (cid:19)−(cid:18) δ βγ (cid:19)
Here sα is an infinitesimal vector representing the relative displacement between the
twoparticles. Nowwecansubstituteourexpressionforthecontortiontensor(10)and
get:
{}
D2sα {}
= R αsδuβuγ+
dτ2 − δγβ
−23hδβα(∂γφ)+gαηgβγ(∂ηφ)i(cid:18)ddsτγuβ+ ddsτβuγ(cid:19)+ (53)
2 {} {}
−3(cid:20)δβαDδ(∂γφ)+gαηgβγDδ(∂ηφ)(cid:21)sδuβuγ.
9
This equation represents the generalization to a theory with torsion of the geodesic
deviation of GR:
{}
D2sα {}
= R αsδuβuγ. (54)
dτ2 − δγβ
It is easy to see that the difference between the two expressions consists in two lin-
ear terms in torsion and torsion derivative, the first of which is multiplied by a term
of relative velocity. Once again we see completely antisymmetric part of torsion not
contributing, if not as a source in thefield equation (15), to thisexpression.
To calculate the nonrelativistic limit of (53), we will make the same assumptions
b),c) made in the last paragraph plus:
a`) involvedvelocities are muchsmaller than c and:
dxα
(1,0,0,0);
dτ ≃
d) s0= ds0 =0,whichsimplymeansthattheparticlesaccelerationsarecompared
dτ
at the same time.
Inthiscase,byvirtueofourassumptions,wewillkeeponlytermscontainingatmost,
as factors multiplied by si, one between h and φ. Then, (53) reduces to:
αβ
d2si {} 2
R isj ηij(∂ φ)sk; (55)
dt2 ≃− j00 − 3 kj
so the tidal field is:
{} 2
Gi=−Rj00isj− 3ηij(∂k∂jφ)sk. (56)
Now, from GR,it is known that in thenonrelativistic limit:
{}
Rj00i =∂j∂iΦ, (57)
where Φ is the gravitational potential.
Taking that into account we can rewrite the finalexpression for thetidal field:
2
i= (∂i∂jΦ)sj (∂i∂jφ)sj. (58)
G − − 3
So, in the nonrelativistic limit, torsion produces a tidal forces effect analogous to the
one produced by thegravitational field.
Itmustbenoted,now,thatsincethefieldsh andφ,inthenonrelativistic limit,
00
both obeya Poisson PDE((49) and(50)) andenterin equations(46) and(58) in the
sameway,itisimpossibletodistinguishtheeffectofthetorsionfieldfrom thatofthe
gravitational one unless we know exactly the source and the initial condition for the
latter; this, together with the small intensity of the torsion forces, makes them even
more difficult to bedetected.
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