Generalized uncertainty principle, classical mechanics, and general relativity

Contents lists available atScienceDirect

Physics

Letters

B

www.elsevier.com/locate/physletb

Generalized

Uncertainty

Principle,

Classical

Mechanics,

and

General

Relativity

Roberto Casadio

a

,

b

,

∗

,

Fabio Scardigli

c

,

d

a_{DipartimentodiFisicaeAstronomia,AlmaMaterUniversitàdiBologna,viaIrnerio46,40126Bologna,Italy} b_{I.N.F.N.,SezionediBologna,ISFLAGvialeB.Pichat6/2,I-40127Bologna,Italy}

c_{DipartimentodiMatematica,PolitecnicodiMilano,PiazzaLeonardodaVinci32,20133Milano,Italy} d_{Institute-LorentzforTheoreticalPhysics,LeidenUniversity,P.O.Box9506,Leiden,theNetherlands}

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory: Received22April2020

Receivedinrevisedform7June2020 Accepted10June2020

Availableonline16June2020 Editor:B.Grinstein

The GeneralizedUncertaintyPrinciple (GUP)hasbeendirectlyappliedto themotionof(macroscopic) test bodies ona given space-time in order to compute corrections to the classical orbits predicted in Newtonian Mechanics or General Relativity. These corrections generically violate the Equivalence Principle. The GUP has also been indirectly applied to the gravitational source by relating the GUP modifiedHawkingtemperaturetoadeformationofthebackgroundmetric.Suchadeformedbackground metricdeterminesnewgeodesicmotionswithoutviolatingtheEquivalencePrinciple.Wepointouthere that thetwo effectsare mutuallyexclusivewhen comparedwithexperimentalbounds. Moreover,the former stems frommodified Poissonbrackets obtainedfrom awrongclassical limitof thedeformed canonicalcommutators.

©2020TheAuthor(s).PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense (http://creativecommons.org/licenses/by/4.0/).FundedbySCOAP3_.

1. Equivalence Principle and diffeomorphism invariance

Itiswellknown [1] thatthe(weak)EquivalencePrinciple(EP; namelytheequalitybetweengravitationalandinertialmass) dic-tatesthattheequationofmotionoftestparticlesinagravitational fieldbeoftheform

d2_xλ dτ2

+ 

λ μν dxμ dτ dxν dτ

=

0

.

(1)

Ontheotherhand,Eq. (1) turnsalso outtodescribegeodesicsina manifold with metric gμν and the Levi-Civita connection



λ

μν

=

1 2gλσ



gμσ,ν

+

gνσ,μ

−

gμν,σ



.1 In his foundational paper [2] of GeneralRelativity(GR),AlbertEinsteinproposedthatthegeodesic equation (1) playedtheroleoftheequationofmotionforapoint particleinthe gravitationalfield gμν ,whichin turnshould obey thecelebratedfieldequations

*

Correspondingauthor.

E-mailaddresses:casadio@bo.infn.it(R. Casadio),fabio@phys.ntu.edu.tw (F. Scardigli).

1 _{Asusual,commasdenotepartialderivativesw.r.t. thecoordinatesx}μ_and

semi-colonsthe covariantderivativesinthe metric gμν; Rμν isthe Riccitensorand

R theRicciscalar;weshallalsouseunitswithc=1 butdisplaytheBoltzmann constantkB,thePlanckconstanth,¯ theNewtonconstantGN andthePlanckmass mp=  ¯ h/GNexplicitly. Gμν

≡

Rμν

−

1 2R gμν

=

8

π

GNTμν

,

(2)

where Tμν istheenergy-momentumtensorofthematter source. In that original formulation, theidentification of theequation of motion with the geodesic equation was seen as an independent axiomofthetheory,inparticularindependent fromthefield equa-tions (2).Fromthispointofview,onecansaythatthecontentof theEPispreciselythattheequationofmotionisthegeodesicequation.

Insuccessivestudies [3,4],Einsteinandcollaboratorsobtaineda resultofconsiderableimportance:theequationofmotionofpoint particles,that isthegeodesic equation (1), caninfact bederived fromthegravitationalfieldequations (2).2_{Inotherwords,thefield} equationsdetermineuniquelytheequationofmotionforbodiesin a gravitationalfield which are not subjected toother forces,and theensuingtrajectoriesaregeodesicsofthecorrespondingmetric. This finding is in full agreement with the postulate of geodesic motion, which therefore appears as a consequence of the field equations,andnotasanindependentaxiomofthetheory.

AnexplicitderivationcanbefoundforinstanceinRefs. [6,7].It isimportantheretoremarkthatthestarting pointisthe conser-vationoftheenergy-momentumtensor,towit

2 _{Strictlyspeaking,theargumentappliestodust(asmoothfluidwithzero} pres-sure),sincepoint-likesourcesareknowntobemathematicallyincompatiblewith Eq. (2) [5].

https://doi.org/10.1016/j.physletb.2020.135558

Tμν_;ν

=

0

.

(3)

This continuity condition can be obtained directly from Eq. (2), using the Bianchi identity for the Einstein tensor, 0

=

Gμν_;ν

=

8

π

GNTμν_{;ν .}Inthisway,itappears asaconsistencycondition for

thefield equations.More generally,Eq. (3) can be derived by re-quiringthe diffeomorphisminvariance ofthe matter action [1,8]. Infact,underageneric(infinitesimal)changeofcoordinates,xμ

=

xμ

+ ξ

μ

(

x

)

, themetric tensorchanges by

δ

gμν

= −(ξμ

;ν

+ ξν

;μ

)

,

andthematteractionvariesas

δ

SM

=

1 2



d4x

√

−

g Tμν

δ

g_μν

= −



d4x

√

−g T

μν

ξ

μ;ν

=



d4x

√

−

g Tμν_;ν

ξ

μ

.

(4)

Sincethevariation

ξ

μ isarbitrary,requiringthat

δ

SM

=

0 is

equiv-alenttorequireEq. (3).Inconclusion,geodesicmotionandtheEP are deeplyrooted into the field equationsof GRand, evenmore fundamentally,they stemfrom the diffeomorphism invarianceof thematteraction(whichisdemandedbythePrincipleofGR).One thereforecannotmodifyorrenouncetoeitherofthemeasily. 2. Generalized Uncertainty Principle

Muchefforthasbeenputintotryingtoincorporatetheeffects ofgravityinquantumphysicsbymeansofaGUPoftheform [9–

17]

x

p

≥ ¯

h 2



1

+ β

0

p2



,

(5)

where x and p are the position andconjugate momentum of a particle,withthecorresponding quantumobservablesdenotedby

ˆ

x and p,

ˆ

O2

≡ 

O

ˆ

2



− 

O

ˆ



2 foranyoperator O ,

ˆ

and

β

0

= β/

m2p

isadeformingparameterexpectedtoemergefromcandidate the-oriesof quantum gravity. Uncertaintyrelations can be associated with(fundamental)commutatorsbymeansofthegeneral inequal-ity

A

B

≥

1

2



 [ ˆ

A

, ˆ

B

] 



_{ .}

(6)

Forinstance,onecanderiveEq. (5) fromthecommutator

ˆ

x

,

p

ˆ

=

ih

_¯



1

+ β

0p

ˆ

2



,

(7)

forwhichEq. (6) yields

x

p

≥ ¯

h 2



1

+ β

0

 ˆ

p2





= ¯

h 2



1

+ β

0



p2

+  ˆ

p



2



.

(8)

Thisimmediatelyimpliesthat theGUP (5) holdsforanyquantum state, since



p

ˆ



2

_≥

_{0 always. In particular for mirror-symmetric}

states

ψ

mssatisfying

 ψ

ms

| ˆ

p

| ψ

ms

 =

0

,

(9)

one has

p2

= 

ψ

ms

| ˆ

p2

| ψ

ms



and theinequality (8) coincides

withtheGUP (5).WealsorecallthatEq. (5) impliestheexistence ofaminimumlength

= ¯

h

√

β

0 whichoneexpectsoftheorderof

thePlancklength.

TheoreticalconsequencesoftheGUPonquantum(microscopic) systemshavebeenextensivelyinvestigatedbyvariousauthors(see e.g. [18–20]).Inaddition,severalexperimentshavebeenproposed

to testdifferentGUP’sinthe laboratory [21–23],aswell assome groundandspacebasedexperimentscould alsobeableto reveal GUPeffects(seee.g. [24]).Itisveryimportantthatthesizeofsuch modifications canbe constrainedalsowithmacroscopictest bod-ies byexisting astronomicaldataemployed forthe standardtests ofGR.Constrainingthedeformingparameter

β

usingastronomical data requires toestimate the effect ofthe GUP (5) in the classi-cal limit.Intheexistingliterature,thishasbeendoneindifferent ways.Inthefollowing,wecriticallyreviewandcomparetwo com-plementaryapproachesofparticularrelevance.

3. GUP and classical mechanics

WorksdevotedtoevaluatetheimpactoftheGUPonthemotion ofclassical(macroscopic)bodiesusuallyemployamodificationof theclassicalPoissonbracketswhichresemblesthedeformed quan-tumcommutator (7) (see,e.g. [25–31]).Theyessentiallyimplement theclassicallimitastheformalmappingintoPoissonbrackets

1 ih

_¯

ˆ

x

,

p

ˆ

=



1

+ β

0p

ˆ

2



→ {

x

,

p

} =



1

+ β

0p2



.

(10)

Such deformed Poisson brackets are then used to determine or-bitsintheSolarsystemandderiveperturbativecorrectionstothe Newtoniantrajectories.

ThetypicalformforthecorrectioncomingfromEq. (10) canbe foundinAppendix AofRef. [32].Tokeepthecalculation transpar-entandfocusontheconcepts,we justconsiderapoint-likemass

m falling radially towards a mass M

m. From the Newtonian Hamiltonian H

=

p 2 2 m

−

GNM m r

≡

p2 2 m

+

m VN (11)

andthePoissonbrackets (10) withx

=

r,the canonicalequations read

˙

r

= {

r

,

H

} =



1

+ β

0p2



_p m (12)

˙

p

= {

p

,

H

} = −



1

+ β

0p2



_{GNM m} r2

,

(13)

whereadotstandsforthetimederivative.Tofirstorderin

β

,one thenobtainstheequationofmotion

¨

r

 −

GNM r2

1

+

4

β

m 2 m2 p

˙

r2



.

(14)

Equivalently, one can proceed like in Ref. [25], starting from Eq. (12). The conservation of the total energy E

=

m

E

then im-plies p2

=

2m2

(

E

−

VN

)

,usingwhichonecanfinally write(for a

particlewithzeroangularmomentum)

˙

r2



2

(

E −

VN

)



1

+

4

β

m 2 m2 p

(

E −

VN

)



,

(15) againtofirstorderin

β

.

Thetermsoforder

β

inbothEqs. (14) and(15) dependonthe massm ofthetestbody andonitsvelocity

˙

r

∼ (E

−

VN

)

1/2.Itis

Difficultiesastheabovearefullyconfirmedalsowhenthe mod-ifiedclassical Poissonbracketsare formulatedina covariantway, on a fixed background metric [29,30]. A slightly different path is followed in Ref. [28], where the EP is recovered even for the GUPmodifiedclassical mechanics,by consideringcomposite bod-ies andpostulating that thekinetic energyis additive. The price to pay in this case is a different deformation parameter

β

0i for

eachspecie i of(elementary)particlesofmassmi composingthe macroscopicbody. Correspondingly, there wouldexist a different minimal length

i

= ¯

h

√

β

0i for each elementary particle. For

in-stance,theminimallengththatcanbeprobedbyaprotonshould besmallerthanthatprobedbyanelectron.Thisfeatureisclearly atoddwiththeuniversalityofgravitation,andwiththefact that thePlancklengthcanbecomputedinawaythatdoesnotdepend atallontheparticleconsidered(seee.g. [33]).

Whatistheoriginofsuchblatantlyunphysicalpredictionsand potentialviolationoftheEP?Theerrorcanbetracedbacktothe implementationof theclassical limit in Eq. (10) for objects with strictly non-vanishing momentum.In fact, fora generic (normal-ized) state

ψ

with



p

ˆ



=

0, the classical limit of the commuta-tor (7) isformallygivenby

{

x

,

p

} =

lim ¯ h→0

 ψ | [ˆ

x

,

p

ˆ

] | ψ 

ih

_¯

=

lim ¯ h→0



1

+ β

GN

¯

h



 ˆ

p



2

+

p2



.

(16)

However, classical (macroscopic) bodies with non-vanishing mo-mentum should be more precisely represented by semiclassical states

ψ

cl,for which we expect the classical limit can be

gener-icallydefinedbythetwoproperties3

lim

¯

h→0

 ψ

cl

| ˆ

p

| ψ

cl

 =

p

,

(17)

wherep istheclassicalmomentum,and

lim ¯ h→0

p 2

_≡

_lim ¯ h→0



 ψ

cl

| ˆ

p2

| ψ

cl

 −  ψ

cl

| ˆ

p

| ψ

cl



2



=

0

.

(18)

Therefore, even under the stronger condition

p2

_/

_h

_¯

_→

_{0, the}

limit (16) becomes

{

x

,

p

} =

lim ¯ h→0



1

+ β

GNp 2

¯

h



,

(19)

whichdivergesbadlylikeh

_¯

−1.4_{Ofcourse,thisdivergencedoesnot} occur formirror symmetricstates,forwhich Eq. (9) impliesthat theclassicalmomentump

=

0.InfactEq. (19) yieldsthestandard Poissonbracketswithoutcorrectionsifweset p

=

0 beforetaking thelimit. Inother words,since mirrorsymmetricstates canonly representobjectswith zeromomentum, thecommutator (7) and thecorrespondingPoissonbrackets (10) shouldbeappliedonlyto classicalbodiesstrictlyatrest.ItisthenobviouswhyEq. (10) can-notdescribethedynamicsofplanetsorbitingtheSun!

ApossiblewayoutofthisconundrumistoderivetheGUP (5) fromthe(explicitlystatedependent)deformedcommutator

3 _{Ofcourse, the wholetopic ofhow theclassical behavior emergesin} quan-tumphysicsisfarricherthanwhatweneedtodiscusshere(forarecentreview, seeRef. [34]).For instance, the condition (18) forthe states ψcl could be im-plementedbyrequiringp∼ ¯hα,withα>0.Sinceforsuchsemiclassicalstates wecanalso assumex∼ ¯hγ, withγ>0,thenHeisenberguncertaintyrelation

xp∼ ¯hα+γ≥ ¯h/2 wouldcontinuetoholdthroughoutthelimitingprocessfor

¯

h→0 ifα+γ≤1.However,thisisonlyanaivewaytoenforceEqs. (17) and(18) andnotnecessarilyausefulone.

4 _{Thedivergenceobviouslydisappearswhengravityisswitchedoff(G} N=0) be-foretakingthelimit.

ˆ

x

,

p

ˆ

=

ih

_¯



1

+ β

0



ˆ

p2

−  ˆ

p



2



,

(20)

whichindeedleads totheGUP (5) for anyquantum statevia the inequality (6), and it further reduces to the commutator (7) for mirror symmetric states. The commutator (20), for semiclassical statessatisfyingtheconditions (17) and(18),implies

{

x

,

p

} =

lim ¯ h→0

 ψ

cl

| [ˆ

x

,

p]

ˆ

| ψ

cl



ih

_¯

=

1

+ β

GN

0

,

(21) where

0

≡

lim ¯ h→0

(

p

2

_/

_h

_¯

₎

_{dependson the state}

_ψ

cl andcan take

thefollowingvalues:

i)

0

=

0 andtheclassicallimit (21) yieldsthestandardPoisson

bracketswith

{

x

,

p

}

=

1;

ii)

0

>

0 and finite.The limit in Eq. (21) then yields the

con-stantC2₀

=

1

+ β

GN

0,whichcanbesimplyusedtorescalex

and p sothat thestandard Poissonbracketsare again recov-ered;

iii)

0

= ∞

andthecommutator (20) doesnot yieldaconsistent

classical limit.Hence, thecorresponding states

ψ

cl shouldbe

avoided.

Summarizing: the classical limit is either badly defined [be-cause Eqs. (19) or (21) diverge], or is just given by the classical Poisson brackets with

{

x

,

p

}

=

1 without corrections. Therefore, along thisway, it is clearly impossible to estimate any effect of the GUPon macroscopicbodies.To thisaim,we should followa completelydifferentpath.

4. GUP and General Relativity

In order to compute GUP effects on macroscopic bodies, we mayrely on theindirect argumentillustrated inRef. [32]. Letus consider a Schwarzschildblack holeof mass M, whose metric is givenby

ds2

= −

f

(

r

)

dt2

+

dr 2

f

(

r

)

+

r

2_d

_

2

_,

₍₂₂₎

with f

(

r

)

=

1

−

2GNM

/

r. From the inequality (5), one can

de-rive a modified Hawking temperaturewhich, to first order in

β

, reads [33,35–37] T



h

¯

8

π

GNkBM

1

+

β

m 2 p 4

π

2_M2



.

(23)

Wethenintroduceamodifiedmetricfunction f

(

r

)

+ δ

f

(

r

)

=

1

−

2 GNM

r

+

ε

G2_NM2

r2

,

(24)

and compute the correction

δ

f

(

r

)

which can reproduce the re-sult (23) bymeansofastandardQuantumFieldTheorycalculation. We thusfind arelationbetweenthe deformationparameter

ε

of themetricandthedeformationparameter

β

oftheGUPas

β

 −

M

2

m2 p

ε

2

.

(25)

backgroundmetric,5 _{weexpect noviolationoftheEP by} construc-tion,andobtainatypicalcorrectiontotheNewtonianpotential of theform [32]6

VGUP

=

ε

G2_NM2 2 r2





|β|

mp M V 2 N

.

(28)

UnlikeEqs. (14) and(15), thiscorrection doesnotdependonthe massorspeedoftheorbitingobjectatall,infullagreementwith the EP. Moreover, it becomes vanishingly small for macroscopic sources of mass M

mp (as one should reasonablyexpect). For

the sake of completeness, we remark that there are other ap-proaches which avoid anyEP violation by construction, like that of Ref. [43], where gravitational waves are used for constrain-ing a GUP-modified dispersion relationfor gravitons,and that of Ref. [44], where a GUP-deformed background metric is used to compute corrections to the black hole shadow. Furthermore, ex-tensive discussions of precision tests of the EP, and its possible violations,indifferentcontexts(e.g.inscalar-tensorgravityandat finitetemperature)canbefoundinRefs. [45,46].

5. Experimental bounds and conclusions

AsidefromthepreviousconsiderationsontheEPandthe clas-sical limit, the correction termproportional to

β

inEq. (15) can alsobequantitativelyconfrontedwiththecorrection (28), assum-ing of course that the deforming parameter

β

is universal and appliesto bothtest bodiesandgravitationalsourcesofanyscale. For macroscopic objects and, in particular, for consistence with SolarSystem tests, the correction in Eq. (15) requires an incred-ibly smallGUPparameter

β



10−66 _{[25,29].Consequently, using}

thisboundinthecorrection (28) fortheextremecaseofaPlanck sizesourceofmass M



mp,onefinds

VGUP



10−33VN2,which

is essentially zero.This appears ratherodd, since one introduces the GUP (5) precisely for describing quantum gravity effects at the Planck scale. For instance, one expects a minimum measur-able length

∼

p

√

β

comparable to the Planck length, rather

than many orders of magnitude shorter. On the other hand, if one accepts the Solar Systembounds on

β

coming from

VGUP

inEq. (28),thatis

β



1069_{[32,42],thecorrectionfora}

hypothet-icalPlancksizesourcecanstillbeveryrelevant(asexpected). Since thecorrections ofthe formin Eq. (15) are irrelevant at thePlanckscale,violatetheEP,growlargerandlargerforplanets intheSolarSystem,moreovertheystemfromacommutatorwhich isincompatible with theproper classicallimit for anystate with non-vanishingclassicalmomentum,weconcludethatthe dynami-calequations (14) and(15),andthemodifiedPoissonbrackets (10) shouldbeviewedasbothconceptuallywrongand phenomenolog-icallyunviable.

Declaration of competing interest

Theauthorsdeclarethattheyhavenoknowncompeting finan-cialinterestsorpersonalrelationshipsthatcouldhaveappearedto influencetheworkreportedinthispaper.

5 _{FordetailsaboutorbitsinGR,seee.g. Ref. [41].} 6 _{Adeformationofthemetricfunctionoftheform}

δf(r)=εf(r)  2 GNM r 2 (26)

wasusedinRef. [42],wheretheauthorsobtainaGUPparameter

α0 −

M mp

ε, (27)

whichis relatedto β byβα2

0. The experimentalboundson α0 obtainedin Ref. [42] arethereforeequivalenttothoseonβderivedinRef. [32].

Acknowledgements

R.C. ispartiallysupportedbytheINFNgrantFLAGandhiswork hasalsobeencarriedoutintheframeworkofactivitiesofthe Na-tionalGroupofMathematicalPhysics(GNFM,INdAM).

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