Newton's trajectories versus Mond's trajectories

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Physics

Letters

B

www.elsevier.com/locate/physletb

Newton’s

trajectories

versus

MOND’s

trajectories

E. Gozzi

a,b,

∗

a_{DepartmentofPhysics(MiramareCampus),UniversityofTrieste,StradaCostiera11,34151Trieste,Italy} b_{IstitutoNazionalediFisicaNucleare,SezionediTrieste,Italy}

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory: Received25May2016

Receivedinrevisedform25December2016 Accepted4January2017

Availableonline9January2017 Editor:S.Dodelson

MOND dynamicsconsists ofa modification ofthe acceleration with respect to the one provided by Newtonianmechanics.Inthispaper,weinvestigatewhetheritcanbederivedfromavelocity-dependent deformation ofthe coordinatesof the systems. The conclusion isthat it cannot bederived this way becauseoftheintrinsicnon-localcharacterintimeoftheMONDprocedure.Thisisafeaturepointedout sometimeagoalreadybyMilgromhimself.

©2017TheAuthor(s).PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense (http://creativecommons.org/licenses/by/4.0/).FundedbySCOAP3_.

1. Introduction

Astrophysics andcosmology haveundergone great experimen-taladvancements inthe last fortyyears.Anice,recent review of all this can be found in ref. [1]. The theoretical explanations for several ofthese phenomena are still open like, for example,the anomalous rotation velocity of stars in galaxies, and of galaxies themselves in their cluster, the acceleration of the expansion of the universe, the inflationary scenario and even, perhaps, some possibleanomalies in thesolar system(for thislast see thenice review[2]).

In this paper we will address the first of the open problems listedaboveforwhichessentiallytwoapproacheshavebeenused byphysicistssofar.ThefirstapproachistheDarkMatter hypoth-esiswhichpostulates theexistenceofnon-baryonicmatterwhich doesnotinteractwithelectromagneticradiationwhilethesecond approachisbasedontheMONDhypothesis;inthispaperwewill concentrateonthislastone.

The MOND hypothesis [3–5] is a modification of Newtonian

mechanicswhich shouldapply to bodiesmoving withaslow ac-celeration(lessthanthethresholdvalue a0

≈

1

.

2

×

10−10m s−2).

Thismodificationshoulddescribe theanomalousrotationofstars around the center oftheir galaxies which seems to occur at the samespeedwhateveristhedistanceofthestarsfromthecenter andthesamefortherotationofgalaxiesaroundthecenteroftheir cluster[6].Actuallytherearecounter-examplesandlimitationsto thislast statement andalso towhat MOND achieves,see for ex-ample[7,8].Ingeneral MONDworksifwelimititsapplicabilityto

*

Correspondenceto:Department ofPhysics(MiramareCampus),Universityof Trieste,StradaCostiera11,34151Trieste,Italy.

E-mailaddress:[email protected].

the interior ofgalaxies and not to galaxies in their cluster or to largerscalephenomena.

On a larger scale, for example the rotation of galaxies inside theirclusterorlensingoraccelerationofthegrowthofprimordial perturbationsetc.,abetterexplanationisgivenbytheintroduction ofdarkmatterwhichaccountsformanyotheranomalous phenom-ena.

It isactuallyemergingthat acompletedescriptionofallthese

phenomena seems to require a mixture of MOND dynamics and

Dark Matter (seeforexamplethe recentpaper[9]).Up tonow a huge effort has been put in exploring the Dark Matter scenario whilemuchlesshasbeendoneforMOND.Ifattheendamixture ofthemwillemerge,wethinkitisworthtoputnowsome more effortinstudyingtheMONDhypothesis.

Weknowthatmanydonot liketochangetheNewtonianlaws ofmotion,likeMONDactuallydoes,butthisissomethingthathas beendonebeforeinphysicsmanytimes.Lookattheatoms:in or-der todescribe themotion of theelectron around the proton in anhydrogenatomwehadtoabandontheNewtonianlawsof mo-tionandbuildanewtheorycalledQuantumMechanics(QM).Not only the laws of motion changed in QM with respect to Classi-cal Mechanics (CM) buteven the basic variables to describe the systemhadtobe modified.Imagineif, inatomicphysics, wehad resistedfromchangingthelawsofCManddecidedinsteadto in-troduce someDark Matterbetweentheelectron andthe nucleus.

Maybe we could cook up a distribution of Dark Matter which

would keep the electronin orbit. Imagegenerations and genera-tionsofchemistsforcedtodescribealltheperiodictableelements notviaQMbutusingCMplussomeAtomicDarkMatter.

There havebeen other changes inNewtonian mechanics (like, for example, special and general relativity whose centenary oc-curredthisyear[10]) butletusstickheretothechangesbrought

http://dx.doi.org/10.1016/j.physletb.2017.01.002

infrom QM. Passingfrom the earth–moon systemwhich is cor-rectlydescribed byCM down to an atom (correctly described by

QM) the jump is roughly of 20 orders of magnitude. In going

upwardsfrom the earth–moon system to the stars rotating in a galaxythejumpisalsoof20ormoreordersofmagnitude,sowe feelthatalsointhissecondjumpwemayhavetochangethe New-tonianlawsofmotionlikewedidinthefirstjumpfromCMtoQM. ThisiswhatMOND’s theorydoesandwearegoing todescribeit insection 2.MONDhasbeenthefirstattemptinthisdirection.It mayneedseveralimprovementsbutitisworthtobestudied fur-therandwedothatinsection3.Wemayevenhavetochangethe basicvariables withwhom we describe the systems like it hap-pensinQMandsomeideasonthisaspectwillbepresentedinthe conclusions.

2. Review of the MOND dynamics

ThebasiclawofMONDdynamics,likeinNewton’s mechanics, istheone which givesthe accelerationa thatabody ofmass m feelsunderaforce F (wewillsticktoonedimensionalmotionfor simplicity).WhileinNewtonianmechanicsthislawis:

F

=

ma

,

(1)

inMOND dynamicstheequation above is deformedintothe fol-lowingone: F

=

m

μ

(

a a0

)

a

,

(2) where

μ

(

a a0

)

≡ (

1

+

a0 a

)

−1

_.

₍₃₎

a0 is a fixed and very small acceleration of the order given

be-fore.The function

μ

()

iscalledinterpolatingfunctionbecausefor a0

→

0 itgoesto1andreproducestheusualNewton’slaw.There

areother interpolatingfunctionswiththisfeature andthey have beenusedintheliterature[4,5].Wewillsticktotheoneabove.

Eq.(2)cannotbeconsidered asa deformationofthemass be-causewe will seelater on that we usethe same massm in the centrifugalforce.So the

μ

hastobeconsidered asa deformation oftheacceleration:

a

−→

μ

(

a

a0

).

Theuseofthisdeformedaccelerationturnsouttobevery use-fulwhenwe studytherotationofthe starsaround thecenterof theirgalaxy.Thismotionhasaverysmallacceleration:a



a0.In

thislimiteq.(3)gives:

μ

(

a

a0

)

=

a

a0

,

andMOND’slaw(2)becomes F

=

ma

2

a0

.

(4)

Applyingthisformulawhen F isthegravitationalforce gener-atedby a mass M and indicating withG the Newton’sconstant, weget: ma 2 a0

=

GmM r2 orequivalently: a

=

√

G Ma0 r

.

(5)

The expressionofthe centripetalaccelerationofabody likea starrotating withvelocity v around the centerofits galaxy ata distancer is:

a

=

v

2

r (6)

ifweput(5)equalto(6),weget v

= (

G Ma0

)

1 4

_.

From this expression we seethat the velocity is the same at everydistancefromthecenter.Thisiswhatreallyhappensin na-ture[6]andforwhichpeoplehavecomeupwiththeideaofdark matter.

The MOND approach has unfortunately several problems like the non-conservation ofmomentum [11]. Thisproblem could be overcomeifone restrictsthemodified accelerationtobe onlythe gravitational one and if the associated Poisson equation is also modified [12]. One could then say that at those scales there is a modification of Newtonian gravity and nothing else. This sec-ondapproachhasbeenchristenedbyMilgromasModified-Gravity MOND (MG-MOND)while theoriginal one, inwhichthe acceler-ationsof all forces were modified, it was called ModifiedInertia MOND (MI-MOND) [13]. In this paper we will stick to the MI-MOND.The factthat theclassical momentumisnot conservedis something that hashappenedbefore inpassing froma theory to

a newone for examplewhen we passed from Classical

Mechan-ics to Quantum Mechanics. In this last theory the analog ofthe “momentum”[14]associatedtoawave-function

ψ(

q

)

isgivenby: P

≡ −

i

_¯

h

∂ψ

∂

q

=

Pcl

+

O

(

h

¯

).

(7)

Thisquantum P is conservedbutitisdifferentfromtheclassical Pcl becausethereare O

(

h

¯

)

corrections. Sothesamemayhappen

inMONDwheremaybetheconservedmomentumcouldbe

some-thing different than theold classical one. In ref. [15] we proved thatquantummechanicscouldbeobtainedfromclassical mechan-icsviaadeformation ofitsvariablesandthismostprobablyisat the origin ofthe fact that themomenta in the two theories are different. Maybewecouldtry thesamefortheMI-MONDtheory, thatmeanscheckifwecanobtainitviaadeformationofthe ba-sicvariablesofclassicalmechanics.Thatiswhatwewilltryinthe nextsection.

3. Configuration-space analysis

Wehaveseenso farthatMONDdynamicsisadeformation of the acceleration.Itseems naturalto expect that thiseffectis in-ducedbya deformationoftheconfiguration-space ofthesystem. Thisisnotonlynaturalbut,aswesaidbefore,italreadyhappened when we had to do the firstmodification of classical mechanics to pass toquantum mechanics. In fact it was proved inref. [15]

Analogously, mayit bethat ingoing nowfromCM totheMOND theorywehavetodeformQ

(

t

)

toanewvariable



Q

(

t

)

?.Ifsothen

wesummarizethewholechainfromQMtotheMONDtheory as

follows:startingwithQManditsvariablesq

(

t

)

we madea defor-mationandpassedtoCManditsvariables Q

(

t

)

[15]andnextwe gettotheMONDtheoryviaafurtherdeformationwhichbringsus tothe



Q

(

t

)

likeinthepicturebelow:

q

(

t

)

−→

Q

(

t

)

−→ 

Q

(

t

).

(8) Somehowitislikeif,goingtolargerandlargerscales,wehad toprogressivelydeformtheconfigurationalvariablesweuse.

Forthe momentlet uslimit ourselfto theq

(

t

)

component of the Q

(

t

)

variables ofCM (fordetails seeref.[15])andlet ussee iffortheMOND theory wecan finda



q

(

t

)

whoseaccelerationis equaltotheMONDaccelerationam which,accordingtoeq.(4),is: am

≡ (

d2_q

₍

_t

₎

dt2

)

2

₍

1 ao

).

(9)

Sowewouldliketofinda newconfiguration-spacevariable,



q

(

t

)

, suchthat: d



q

(

t

)

dt2

= (

d2q

(

t

)

dt2

)

2

₍

1 ao

).

(10)

Letusindicatetherelationbetweenq

(

t

)

and



q

(

t

)

asfollows:



q

=

F

(

q

,

q

˙

)

(11)

where

F

isafunctionwhichshouldbedeterminedusingeq.(10). Wehavechosen thefunction

F

todependnotonlyonq butalso onq otherwise,

˙

asitwillbeclearfromthecalculationswhich fol-low,therewillbenochanceofsatisfyingeq.(10).Letusnowuse

(11)toderivetheL.H.S.of(10): d2



q

(

t

)

dt2

=

∂

2

F

∂

q2q

˙

2

₊

₂

∂

2

F

∂

q

˙

∂

qq

˙

q

¨

+

+

∂

2

F

∂

q

˙

∂

q

˙

(

q

¨

)

2

₊

∂

2

F

∂

q

˙

∂

q

¨

q

¨

... q

₊

+

∂

F

∂

qq

¨

+

∂

F

∂

q

˙

... q

.

(12)

ComparingtheR.H.S ofeq.(12)withthe R.H.Sof eq.(10)we seethatonlythethirdtermintheR.H.Sof(12)hasaformsimilar totheR.H.S.ofeq.(10)andfromthisweget:

∂

2

F

∂

q

˙

∂

q

˙

=

1

a0

.

(13)

Looking at thisequation it is clearwhywe hadto choose an

F

dependingalsoonq.

˙

Ifwestart“integrating”eq.(13)weget:

∂

F

∂

q

˙

=

1 a0

˙

q

+

G

(

q

)

(14)

where

G

isa functiontobe determined.The remainingtermson theR.H.S ofeq.(12),besidesthethirdone, mustsumupto zero andusinginthemtherelation(14),weget:

∂

2

G

∂

q2q

˙

2

₊

∂

G

∂

qq

¨

+ [(

1 a0

)

q

˙

+

G

]

...q

₌

0

.

(15)

Ifweconsiderthisasa differentialequationfor

G

,thefactthat as“coefficients”inthisequationwehavetermsdependingonq,

˙

q,

¨

...

q ,impliesthat asa generalsolution for

G

we willgetafunction thatwilldepend,besidesq,alsoonq,

˙

q,

¨

...q .Butthisiscontradictory becausein(14)the

G

was dependent onlyon q.Sowe conclude

thatthereisnosolutiontoourequation(11),i.e.thereisno man-nertobuilda



q

(

t

)

fromaq

(

t

)

whichisthethingwewantedtodo in(11).

The reader may think that by choosing a more general

F

in(11),dependingalsoonq and

¨

...q ,thingscouldbe fixedup,but this is not true. In fact it turns out that the analog of eq. (15)

wouldthendependalsoon....q implyingthat

F

wouldhaveto de-pendalsoonthisvariableleadinginthiswaytoacontradiction.

The reader maythink that another wayout could be to have no

G

at allinequation(14).Inthiswaymanytermsin(12)would disappearbutthen,besidesthethirdterm, wewouldbeleftwith thelast onethatis ∂F

˙ q

...

q whichisnotzeroandsotheproblemis notsolved.

Anotherattemptcouldbebasedongivingmorefreedomtoour equationsbylettingeventhetimet changeandnotjustq

(

t

)

.We wouldthenhavetrajectoriesindicatedby



q

(

t

)

wheretis:

t

=

E

(

t

)

(16)

with

E

afree function to bedetermined together withthe func-tion

F

fromtheanalogofeq.(10).Itisnotdifficulttoprovethat eveninthiscasewewouldendupinsomecontradictionlikewith eq. (15).Letusstartby simplifyingthecalculationsviaan

F

de-pendingonlyonq.

˙

Theequationwegetasanalogof(12)is:

d2



q

(

t

)

dt2

=

∂

2

F

∂

q

˙

∂

q

˙



1

E



−

E



(

E



)

2 dq dt

+

+

1

(

E



)

2 d2_q dt2



2

+

∂

F

∂

q

˙



d3_q

₍

_t

₎

dt3



.

(17)

Remember:wewantedthattheexpressionabovebeequalto: 1 a0



d2q dt2



2 (18) andthisimpliesfrom(17)that:

∂

2

F

∂

q

˙

∂

q

˙

1

[

E



_]

4

=

1 a0 (19) while all the other terms on the R.H.D. of (17) must sum-up to zero,i.e.:

∂

2

F

∂

q

˙

∂

q

˙

1

E

6

(

E



)

2

₍

dq dt

)

2

₋

₂

∂

2

F

∂

q

˙

∂

q

˙

E



E

5q

˙

q

¨

+

∂

F

∂

q

˙

d3q

(

t

)

dt3

=

0

.

(20) Thecoefficientsofthisequationdependnotonlyonq but

˙

alsoon

¨

q and...q andasaconsequencealsoits solution

F

willdependon thesehigherderivativesofq whilewehadmadethechoiceatthe beginning ofhavingan

F

dependingonlyonq.

˙

Wecouldthinkof playing withthe

E

in order tocancel the first two termsof the equation (20),butthereisnowaytoeliminatethethirdterm. So thisisthe contradictionwhich cannotbe removed evenwiththe presenceofthere-parametrizationfunction

E

.

We feel that what we have derived hasa lotto do with the “non-locality”inthetime“t”whichMilgrom [18]discoveredasa peculiarfeatureoftheMI-MOND.Thisnon-localityimpliesthe de-pendenceofthetransformationsonallthehigherorderderivatives ofq.Thisisthesamewewouldbeforcedtodoinourcasein or-dertohaveaconsistentsolutiontoourequations.Todiscoverthat

non-locality Milgrom instead had to impose that the MI-MOND

4. Conclusions and outlook

Theconclusionswecandrawfromthecalculationsofthe pre-vioussectionsontheMI-MONDtheoryseemratherdepressingbut theremaybeawayout.AswesaidintheIntroduction,inref.[15]

weshowedthatthetrueimportantcoordinatesofCMseemtobea setofvariableswhoseconfigurationalpartweindicatedwithQ

(

t

)

in orderto distinguish it from q

(

t

)

. The Q

(

t

)

is a full multiplet whichcontainsnot onlytheq

(

t

)

butalsotheJacobifields,which we indicated withthe symbol c

(

t

)

,its symplectic dual,c

¯

(

t

)

, and whatisknowninstatisticalmechanicsastheresponsefield

λ(

t

)

. Formoredetailsonthesevariablesseeref.[17].

Weindicated informula (8)that itis this Q

(

t

)

which should bedeformedintoa



Q

(

t

)

inordertogointotheMI-MONDtheory. Thiswouldimply that thetransformation ofthe firstcomponent of



Q

(

t

)

whichis



q

(

t

)

wouldnotbe anymoreoftheform(11)but wouldhavein

F

alsoadependence ontheother componentsof Q thatwe indicatedabove withc,

¯

c and

λ

.Thismoregeneral

F

mayleadtoasolutionofourproblem.

Thereadermay wonder whywe insist somuch in deforming the Q . These variables were studied in ref. [15] froma mathe-maticalpointofview butwefeeltheyhavealsoaveryimportant physicalmeaning whichis now underinvestigation. What seems toemergeisthat Q ,togetherwithitsmomenta P ,donot repre-sentpoints inphase-space likeq, p but blocksofphase-spaceof dimension

¯

h. Theseblocks are thetrue degreesoffreedom of CMbecauseCMcannothandleobjectsinphase-spacewhichhave a volume less than h.

_¯

Applications of this idea have been done inref. [19]. Beingthesethe true degreesoffreedom of CM they are the correct objects which had to be deformed to get to the MI-MONDtheory.Mostprobablywe were notgettingthecorrect MI-MONDvariablesbecausewedidnotstartfromthetrue-degrees offreedomofCM.

Inref.[15]wedescribed amathematicalproceduretogofrom QM(whosepath-integral variableswereindicatedwithq

(

t

)

, p

(

t

)

) toCMwhosepath-integralvariableswere Q

(

t

)

,P

(

t

)

.Astheselast seem tobe blocksofphase space[19],theoriginal mathematical steps[15]togofromQMtoCMcanmostprobablyberewrittenin physicaltermsasa“renormalization-group-like”procedurewhere theQ(t)aretheanalogoftheblock-spinvariablesinthe renormal-izationprocess. Thisprocedure requiresthe construction ofwhat areknownasthe

β

and

γ

functions(forareview seefor exam-ple[20]).Inref.[15] webasicallyproved(in adifferentlanguage) that the

β

-function is identicallyzero. Thisis relatedto the fact thatthemassesandthecouplingsofthetheoryremaininvariant, i.e.theydonotchangeingoingfromQMtoCM.Infactinref.[15]

we provedthat the path-integralsin CM andQMhave thesame weightbutwithdifferentfields.Thisactuallyimpliesthatwhatis differentfromzeroisthe

γ

-function.Infact the

γ

-functiontells ushow the field changes under thisrenormalization group. Our field changes fromq

(

t

)

inQM to Q

(

t

)

inCM so our

γ

-function mustbedifferentfromzero.

We feel that QM must be an ultraviolet fixed point of this

γ

-function because up to now no violations to the laws of QM have beenfoundsoweshouldconsideritthecorrecttheoryatthe smallestpossible scale (in phase-space) andthis is equivalent to sayingthatQMisanultravioletstablefixedpoint.ForCMinstead therearemany indications(likeforexampletherotationcurvesof stars) that it maynot be the correcttheory at thelargest possi-blescale,somostprobablyitisnottheinfraredstablefixedpoint ofthe

γ

function that wementioned before. Thisinfraredstable fixedpointwillbe givenbytheMONDtheory(orsome modifica-tions ofit). We say“some modifications ofit” becausewe know thattheMONDtheory doesnot workwellatscales largerthana galaxy.

Letus summarize:we start fromthe variables q

(

t

)

ofQM on the ultraviolet fixed point andthen proceed withthe block-spin procedure to the variables Q

(

t

)

of CM. By continuing with the block-spinprocedurewe shouldarriveattheinfraredstablefixed pointwhichisMI-MOND(orsomemodificationofit)withits de-greesoffreedom



Q

(

t

)

.Fromthisrenormalization-groupprocedure itisclearwhywesaythat,togettotheMI-MONDtheory,wehave todeform thewhole Q

(

t

)

andnotjustitscomponentq

(

t

)

aswe didinthispaper.Theyareinfact the Q

(

t

)

thevariables thatwe getinCM viatherenormalization-groupandfromthesewe have tostarttocontinuewiththesameprocedureinordertogettothe infrared stablefixedpoint, that isthe MI-MOND(or some modi-fication ofit). Asthe Q ofCM areblocksofphase-space

¯

h we envisionthattheMOND-Q willbehugeblocksofphasespacelike those ofa starora galaxy.To understandhow todeform the Q weneedtoknowthe

γ

functionandatthemomentwe haveno ideaonhowtogetit,butitmustbeaderivationassimpleasthe derivationofthe

β

functionthatweimplicitlyperformed(without realizing)in[15]andprovedtobeidenticallyzero.

InthispictureCM issomehowinbetweenQMandMI-MOND

theoryanditisnottheinfraredstablefixedpoint ofthe

γ

func-tion.IfsothenCM shouldbe unstableundertherenormalization group flow. Actually It seems to be unstable under the change of some extra parameters [15] entering Q

(

t

)

, parameters which are formallytwo partnersoftime butwhoseproduct hasthe di-mensionoftheinverseofanaction.Theseparameterswere never before introduced inCM andthat is the reasonnobody realized thatCMwasunstable.Thiswholeanalysisisalsobringingtolight the special role that the action plays in physics. We know that when the action gets very small, of the order of h,

_¯

we have to change the lawsof motionfrom those ofCM to the one of QM. Thestudyweareperformingonthe

γ

functionindicatesthatalso forverylargevalueoftheactionwemayhavetochangethelaws

ofmotionandpassfromCMtotheMI-MOND(orsome

modifica-tionofit).Theverylargeactionwetalkaboutarethoseofagalaxy oraclusterofgalaxieswhichhavesomeofthelargestvalueofthe actionamongtheobjectspresentintheuniverse.

Anotherprojectworthpursuingisthefollowingone. The MG-MOND[12,13] theorydoesnot presentthenon-localityproblems of the MI-MOND besides being a theory where all conservation lawsarerespected.Soitwouldbenicetoseeifitcanbeobtained via a deformation ofthestandard Poissonequation. Basicallythe MG-MOND [12] postulates a different Poisson equation with re-spect to the Newtonian one relating the gravitational potential U

(

x

)

tothematterdensity

ρ

(

x

)

.Ithastheform:

∇ ×



μ

(

∇

U



a0

)

∇

U

(

x

)



=

4

π

G

ρ

(

x

).

(21)

G is Newton gravitational constant,

μ

(

·)

is an interpolating functionlikeintheMI-MONDtheoryandwhen

μ

=

1 wegetthe Poisson equation of the Newton Theory. Eq. (21)is a non-linear equation whichgivesrisetovery peculiarphenomenainphysics. Themostinterestingoneisthesocalled“ExternalFieldEffect”or (EFE) [12].It basically saysthat,differentlythan inNewtonian or Einsteingravity,anobjectatadistancer fromacenterOdoesnot only feelthe gravitationalforce createdby the matter insidethe sphereofradius r withcenterinObutalsotheexternal gravita-tionalfield createdby other objectslying outsidethe sphere.For exampleforaplanetofthesolarsystemitwouldfeelalsothe ex-ternalfieldcreatedbythegalaxy.Notethatthiseffectwouldtake placeonobjects,likethesolarplanets,whichhaveanacceleration which is not small withrespect to a0 of the MOND laws.There

featuresofthisformulation,itismoredifficulttocalculateitand insomemodelitcouldevenbebroughttozero[13].

UsingMG-MONDaneffectofEFEontheinternalplanetsofthe solarsystemwouldmanifest itselfforexample witharetrograde motionoftheperihelionofSaturnandextensivenumerical calcu-lationsofthemodifiedPoissonequationhavebeenperformed[22]. FromthelatestdataonSaturntheeffectisunfortunatelyveryvery small.WhereinsteadtheeffectshouldnotbesmallisontheOort cloudandsome veryinteresting workhasbeendone[23].In par-ticular it turns out that the orbits of objects in the Oort cloud wouldbequitedeformed.

Weshouldremindthereaderthattherearealsoother MOND-likeeffects[24].IngeneralmanyoftheseMOND-likeeffectscould besimilartothosegeneratedbyadistantplanet[25]andalotof interestingworkisbeingperformedatthemomentonthisissue.

AsmostoftheworkontheseMG-MONDeffectsisnumerical,it wouldbenicetofindtheanalyticdeformationtransformationthat bringthemodified Poissonequationintothe standardNewtonian one.Allthephysicaleffectsthat wementionedhavetobeburied intothistransformation.

This projectand the previous one presented in these conclu-sions are under investigations and we hope to come back soon withmoredetails.

Acknowledgements

I wish to thank all the participants to the many seminars I gaveonthesetopicsoverthelast threeyearsfortheir non-trivial questions.Ialsowishtothanktherefereeforprovidingmany ref-erences of which I did not know the existence before and from whichIlearneda lot. Thiswork hasbeensupported by the Uni-versityofTrieste(FRA2014)andbyINFN (geosymqftandgruppo IV).

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