Relativity of Motion in Vacuum

6 Pages·0.11 MB·English

by  Marc-Thierry Jaekel

Relativity of motion in vacuum
a b b
Marc-Thierry Jaekel , Astrid Lambrecht et Serge Reynaud
a Laboratoire de Physique Th´eorique de l’ENS ∗, 24 rue Lhomond, F75231 Paris Cedex 05
b Laboratoire Kastler Brossel †, UPMC case 74, Jussieu, F75252 Paris Cedex 05
(Contribution for Vacuum, eds E. Gunzig and S.Diner)
The principle of relativity is one of the main general observable effect when it is composed with a second mo-
laws of physics. Since it applies in particular to motion tion. This is the deep meaning of the famous discussion
in empty space it is related to the symmetries of this of motion in a moving boat used by Galileo in the Dial-
space. Thoughtsaboutthissubjectwithintheframework ogotoemphasizethe propertyofrelativity[4]. Ofcourse
ofclassicalphysicshaveledtothetheoryofrelativity[1]. this property is rigorously true only when the resistance
8 The emergence of quantum theory has then profoundly of air to motion can be ignored. Newton stressed this
9
altered our conception of empty space by forcing us to factinthe Principia inordertoidentifythe emptyspace
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consider vacuum as the realm of quantum field fluctua- in which motion takes place with the vacuum which had
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tions. The notion of quantum vacuum and the existence justbeenthesubjectofthefirstexperimentsofTorricelli,

n ofvacuumfluctuationsunavoidablyleadustoreconsider Pascal, von Guericke and Boyle. As we know today, the
a
the question of relativity of motion. The present arti- Newtonianlawsdescribinginertiaorgravityalreadycon-
J
cle is devoted to this aim with a main line which can tain the Galilean principle of relativity, as it was named
0
be formulated as follows: “The principle of relativity of byEinstein,althoughNewtonhimselfhadarguedforthe
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motion is directly related to symmetries of quantum vac- absolute character of his space reference. The theory of
1 uum”. Keeping close to this statement, we discuss the relativity has freed space from this absolute character
v controversial relation between vacuum and motion. We and one of the reasons for that was precisely that it was
1 introduce the question of relativity of motion in its his- contradictory with Galilean relativity. After the advent
7 toricaldevelopmentbeforecomingtotheresultsobtained ofthistheory,therelationbetweenmotionandspacecan
0
more recently. be made explicitas follows: the expressionofthe lawsof
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Movement in space cannot be defined otherwise than motionaresymmetrypropertiesofspaceor,equivalently,
0
8 movement in vacuum. This assertion was formulated by thesymmetriesofemptyspaceimplythelawsofmotion.
9 LeucippusandDemocritusmorethan2000yearsago. In The formulation of the theory of relativity has largely
/ the context of that time, it has to be understood as a been built upon the symmetries of Maxwell’s equations.
h
logical necessity rather than a physical statement in the Indeedthe empty space ofclassicalphysics is a reference
p
- modern sense. The existence of movement is a matter for writing not only the laws of mechanics but also the
t of evidence which forces us to conceive a space in which propagation of the electromagnetic field. However, the
n
a movement takes place [2]. The absence of resistance to development of classical physics was based on the ide-
u motionis a keycondition inthis respectand in factcon- alization that space can be thought as being absolutely
q stitutes the main argument for the definition of the con- empty. This classical idealization could not be main-
:
v cept of vacuum. At the same time vacuum is a positive tained, not even as a limiting case, when it was realized
i physicalrealitywhichclearlydiffersfromnothingness. It that space is always filled with freely propagating radia-
X
isinparticularthereferencewithrespecttowhichmove- tionfieldsafterthebirthofstatisticalmechanicsandthen
r ment has to be identified. These properties are certainly of quantum mechanics. Black body radiation is present
a
paradoxicalatfirstsight. Thisparadoxhasraisednumer- at every non-zero temperature and it undoubtedly pro-
ousdiscussionsandithasconserveditswholepertinence duces real mechanical effects. It exerts a pressure onto
ever since the birth of modern physics until today. thereflectingboundariesofasurroundingcavityandpro-
After a long eclipse dominated by Aristotelian con- ducesafrictionforcewhenreflectingsurfacesaremoving.
cepts, the question of relativity of motion was brought Those two effects are analogous to the pressure exerted
anew to the fore by Galileo [3]. His central argument, by air molecules on the walls of a container as was made
“Movement is like nothing”,notonlysignifiesthatamo- clear by Einstein’s theory of Brownian motion [5]. It is
tion with uniform velocity is indistinguishable from rest. precisely for explaining the properties of black body ra-
Italsoimpliesthatamotionwithuniformvelocityhasno diation that Planck introduced the first quantum law in
1900. At zero temperature this law describes a region
of space limited by a cavity and entirely emptied out of
radiation. In order to approach a practical realization
∗LaboratoireduCNRSassoci´e`al’EcoleNormaleSup´erieure of vacuum, it is not sufficient to remove all matter from
et `a l’Universit´e Paris-Sud thisenclosuresinceonealsohastolowerthetemperature
†Laboratoire de l’Ecole Normale Sup´erieure et de l’Univer- down to zero to eliminate thermal radiation.
sit´e Pierre et Marie Curie associ´e au CNRS Later on in 1912, Planck was led to modify his law
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in order to improve its agreement with known results in longer emit photons, the atom is still coupled to vac-
the classical regime reached at high temperatures. The uumfluctuationsandthis couplingresultsinmeasurable
modified law predicts that fluctuations remain present effects like the Lamb shift of the atomic absorption fre-
atzerotemperature,fromwhichNernstdeducedin1916 quencies. Two atoms in vacuum are coupled to the field
that space is permanently filled with an electromagnetic fluctuations which produce an attractive force between
field propagating with the speed of light [6]. The ex- them,theso-calledVanderWaals force. Thisforceplays
istence of these zero-point fluctuations or vacuum fluc- a very important role in physicaland chemical processes
tuations, which correspond to half of the energy of one anditsquantumtheoreticalinterpretationhasbeenstud-
photonperelectromagneticfieldmode,wasconfirmedby iedsincethe firstyearsofquantumtheory. While study-
the full quantum theory developed after 1925 [7]. Fluc- ing this problem, Casimir discovered in 1948 that there
tuations thus appear as an inescapable consequence of exists also a force between two mirrors placed in quan-
Heisenberg’sinequalities. Tosumupthis lessonofquan- tum vacuum [11]. The vacuum fluctuations are modified
tum theory, one may say that it establishes a necessary by the cavityformedby the mirrorsandtheir energyde-
identity between the potentiality of movement and the pends on their relative distance. Hence vacuum exerts a
presence of movement. As soon as space allows move- force which mutually attracts the mirrors to each other.
ment, which is of course an essential feature of space, This Casimir force, as it was called later on, depends
then movement exists and in particular subsists at zero only on the distance and on two fundamental constants,
temperature. In the same way, as soon as space allows the speed of light c and the Planck constant h¯. This
field propagation, which is equally a fundamental prop- is a remarkably universal feature in particular because
erty of space, then fields are present. Quantum physics the Casimir force is independent of the electronic charge
forbids the absence of movement as well as the absence in contrast to the Van der Waals forces. We have al-
of propagating fields. The ground state of a mechanical ready noticed that any pragmatic definition of vacuum
oscillatoris definedasthe statewhere its mechanicalen- necessarilyinvolvesaregionofspacelimitedbysomeen-
ergy is minimal. In the same way, quantum vacuum is closure. The Casimir effect signifies in fact that the en-
definedasthe fieldstatewherethe energyoffieldfluctu- ergyofvacuumdepends onthe configurationofthis cav-
ations is minimal. ity from which it follows that its boundaries experience
Then naturally the question arises whether the pres- forces arising from radiation pressure of vacuum fluctu-
ence of fluctuations in quantum vacuum is compatible ations. Although the Casimir force is relatively small it
with the principle of relativity of motion. It is this dis- has been observed in several experiments [12].
cussion which we will sum up in a non-technical manner Vacuum fluctuations thus play a central role in the
inthis article. More technicalargumentsas wellasa list modern description of the structure of matter. They ex-
of references can be found elsewhere [8]. plain numerous novel effects which have been observed
Some preliminary remarks should still be mentioned. without any ambiguity. Their existence is directly asso-
Mostofthe time the discussionsdevotedtoHeisenberg’s ciated with a fundamental property of quantum physics,
inequalitiesstressthelimitstheyimposeontheprecision namelytherepresentationofanyphysicalquantitybyan
ofameasurement. Insistingonthenotionsofuncertainty observabledefinedwiththehelpofnon-commutingmath-
or even of indeterminacy however presents the serious ematical objects. Their status nevertheless continues to
disadvantagethatitfixesthe ambitionofphysicistswith raise intricate questions. One reason for this situation
respect to a classical description of physical phenomena, is the obvious fact that their representations are often
although quantum theory has been developed precisely incompatible with the intuitions inherited from classical
in order to remove the deficiencies of this description. physics. Morefundamentalreasonsarerelatedto the se-
The successes of quantum theoretical description of nat- riousdifficultieswhichhaveremainedunsolvedforalong
uralphenomena plead for a quite renewedpoint of view. time and, for some of them, still remain unsolved at the
Quantum fluctuationsare intrinsic properties ofphysical interfacebetweenthephysicsofvacuumfluctuationsand
quantities which display their inherent quantum nature. the laws of mechanics or gravitation.
Today’s knowledge allows for a theoretical understand- The archetype of these difficulties is the relation of
ing and also for an experimental study of these proper- vacuumenergyto gravitation. The totalvacuumenergy,
ties. Incertainexperimentsonecanevenmanipulatethe that is to say the energy summed over all field modes in
quantum fluctuations in order to reduce the noise they their vacuum state, takes an infinite value. This implies
produce on a particular signal [9]. that vacuum energy does not contribute in a standard
The fluctuations of the electromagnetic field which re- way to gravitation since the universe would have a very
main in the vacuum state have well known observable different appearance otherwise. One simple way to deal
effects [10]. An isolated atom in vacuum interacts only with this problem is to set the vacuum energy equal to
withvacuumfluctuationsandthisinteractionisresponsi- zero and, therefore, to use it as a reference for all other
ble for the spontaneous emission processes during which energies [13]. In contrastto a widely spreadopinion this
the atom changes its internal state by falling from a prescription does not allow to ignore the gravitational
higher energy level to a lower one. When fallen in the effects of vacuum. Indeed vacuum energy is modified in
ground state, the state of lowest energy where it can no a space curved by the gravitational field. Furthermore,
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even if the mean value of the vacuum energy does not to renounce to general principles of mechanics at the el-
contribute to gravitation its variations necessarily con- ementary level of microscopic physics whereas the me-
tribute to it. Yet, the spatial distribution of vacuum en- chanicaldescriptionofnaturehas a validity restrictedto
ergy changes continuously as it is composedof field fluc- themacroscopicdomain[20]. Thisaprioriseparationbe-
tuations which propagate with the speed of light. The tweenmicroscopicandmacroscopicdomainshasnowlost
energydensityofvacuumshowsfluctuationswhichman- thepragmaticalpertinenceithadwhentheargumentwas
ifest themselves as fluctuations of space-time curvature. formulated, due to the progress towards highly sensitive
It is possible to discuss some of these questions in anal- measurements approaching the quantum level of sensi-
ogywiththestandardformalismofquantumfieldtheory tivity for macroscopic objects [21]. From a more funda-
(a discussion and references can be found in [14]). How- mentalpointofview,thissolutioncannotbesatisfactory
everafinalanswertothese questionswillnotbe reached sinceitrejectsthemechanicalquestionswhichconcernin
until asatisfactoryquantumdescriptionofgravitationis particular inertia and gravitation outside the framework
available. of quantum theory. Although the problems of inertia
As already stated in the introduction, from the mere and gravitation differ, they are directly related to each
point of view of logics movement in space must be un- other. In the domain of gravitation also, vacuum field
derstoodasmovementinvacuum,sothatthepresenceof fluctuations modify the laws of motion by the introduc-
field fluctuations in quantum vacuum forces us to recon- tion of dissipative reaction terms which lead to stability
sider the notion of motion itself. This question has been problems [22]. Both cases point at difficulties generated
discussedatlengthinconnectionwithattemptstoobtain by the mixture of classical and quantum theoretical de-
a consistent description of motion for elementary quan- scriptions, namely Einstein or Newton equations on one
tum objects like the electron [15]. It has been learned hand and quantum field theory on the other hand. It is
fromclassicalelectrodynamicsthat the expressionofthe knownmoregenerallythatanydescriptionbuiltonatoo
forceactingonamovingchargedparticlecontainsacon- crude mixing between classical and quantum formalisms
tribution, knownasthe Abraham-Lorentzforce,describ- is necessarily plagued with inconsistencies [23].
ing the reaction to motion entailed by the electromag- The general problem of quantum field theory with
netic field emitted by the particle. The modern quan- moving boundaries has emerged as an outcome of the
tum theory tells us that the radiation reaction force is thoughtsaboutpossible approachesto this problem[24].
directly related to the vacuum field fluctuations through This subject may be considered as an extension of the
thefluctuation-dissipationrelations,whicharethequan- study of Casimir force, precisely of vacuum radiation
tum generalizations of the classical results of Brownian pressure on reflecting boundaries, to the case of moving
motion theory [16]. The occurence of a dissipative force boundaries. At the same time, it is directly related to
induced by motion of the electron and its direct associa- the questions raised by quantum descriptions of motion,
tion with vacuum fluctuations forbid to obtain any con- inertia or gravitation. As it will become apparent in the
sistent description of atoms within the classical frame- following, this problem has been given formulations well
work. In the quantum theory, this crisis was only solved adaptedtothepresenttheoreticalframeworkofquantum
at a very high price since any mechanical description of physics and has led to satisfactory solutions of some of
movement was abandoned for elementary quantum ob- the general difficulties mentioned above.
jects [17]. Before discussing this point in more detail Let us first consider the simple case of a mirror mov-
in the next paragraphs, let us notice that this was not ing with a uniform velocity. Clearly there exist two very
the end of the story. Vacuum field fluctuations were different situations depending on whether motion takes
also shown to modify the inertial mass of a point-like place in vacuum or in a thermal field. The principle of
scatterer and, furthermore, to lead to an infinite mass relativity of motion applies to the first situation but not
correction. Arenormalizationprescriptionwasdesigned, to the second one. Should vacuum exert a force on a
which states that the real mass of the particle is finite, mirror with a uniform velocity, the reaction of vacuum
as the result of adding the infinite positive correction to would distinguish between inertial motion and rest. As
a bare mass which is itself infinitely negative. However, expected, quantum theory predicts the friction force to
other difficulties emerge as outcomes of this approach. vanish in this case so that the principle of relativity of
In particular, the mechanical response of the particle to uniform motion is valid in quantum vacuum. Besides,
an applied force shows instability and violates causality quantumtheorygivesaninterestinginterpretationofthis
[18]. In fact, the renormalizationprocedure usedto keep propertyaccordingtowhichvacuumfluctuations appear
theparticlemassfiniteinspiteofaninfinitecorrectionis exactly identical to an inertial observer and to an ob-
incompatible with a causal mechanical description [19]. server at rest. The invariance of vacuum under Lorentz
Thesedifficultieshaveledmostofthetheoreticalphysi- transformationsisanessentialconditionfortheprinciple
ciststoadoptapragmaticpointofviewwhichhasproved ofrelativityofmotionanditestablishesapreciserelation
itself useful in the microscopic domain. Any mechan- between this principle and a symmetry of vacuum.
ical description is then given up in this domain where We now come to the general case of arbitrary motion
physical descriptions are instead built on quantum field in vacuum. In 1976, Fulling and Davies showed that a
theory. Taken seriously, this approach forces physicists perfectly reflecting mirrormoving in vacuum emits radi-
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ation as soon as the mirror has a non uniform accelera- ancientatomists aswell aswith the Galileanprinciple of
tion [25]. Meanwhile, the motion tends to be reduced to relativity of motion.
a uniform acceleration by the reaction of vacuum. The In this context, it would be extremely interesting to
change of mechanical energy associated with damping is obtainexperimentalevidencefor motionaldissipativeef-
dissipated in form of emitted radiation and the reaction fects. Although these effects are extremely small for any
force follows from the associated momentum exchange. motion which could be achieved in practice for a single
This motional force is directly related to fluctuations of mirror, an experimental observation could turn out to
the vacuum radiationpressure as they may be evaluated be conceivable with a cavity, instead of a single mirror,
for a mirror at rest, in full consistency with the quan- oscillating in vacuum. Due to a resonance enhancement
tum fluctuation-dissipation relations [26,27]. When the of the emission of motional radiation the experimental
equations of motion in vacuum are modified in order to figures become indeed much more promising in this case
take the motional force into account, it turns out that [30].
mirrorspossesscausalmechanicalresponsefunctionsand The radiationreaction force calculated by Fulling and
hence stable motions in vacuum. It has to be noted that Davies for a single mirror in vacuum is proportional to
the expression of the reaction force coincides with the the Abraham-Lorentz derivative associated with its mo-
Abraham-Lorentz derivative in the limit of a perfectly tion. Hence, it vanishes exactly for a uniformly accel-
reflecting mirror which thus raises the same problems erated motion. One question then arises, which natu-
as in the case of the electron. This difficulty is solved rally generalizes the one already met when considering
by a careful description of the reflection properties of uniform motion. How do vacuum fluctuations appear
the mirror,accountinginparticularforthe factthatany for an observer with uniform acceleration? The absence
real mirror is certainly transparent at high frequencies. of reaction of vacuum againstuniformly accelerated mo-
The motion of a real mirror in vacuum is described in tion would suit well the invariance of vacuum fluctua-
a consistent manner once these points are satisfactorily tions under a group of transformations corresponding to
dealtwith[28]. Furthermore,afulltreatmentofthecou- observers with uniform acceleration as well as uniform
plingbetweenthemirror’smotionandthefieldscattering velocity. This question has raisedand still raises numer-
showsthat the vacuum fluctuations ofthe quantumfield ous controversialdiscussions between physicists [31]. To
are eventually transmitted to the position of the mirror, understand this controversy it is necessary to describe
even if the latter was originally considered as classical. whatis often calledthe Unruh effect which predicts that
In any realistic situation, the mechanical effect of vac- vacuum fluctuations should appear as thermal fluctua-
uum upon the mirror may be considered as vanishingly tionsinauniformlyacceleratedreferenceframe[32]. The
small and the mirror is found in this limit to obey the effective temperature, which is proportional to Planck
standard Schr¨odinger equation [29]. These results prove constant and to acceleration, is analogous to the Hawk-
thattheobjectionsagainstthepossibilityofgivingacon- ing temperature of the field radiated by the surface of
sistentquantumrepresentationofmotioninvacuum can a black hole [33]. This analogy plays an important role
be bypassed. They also show that the solution involves in the arguments pleading in favor of the physical real-
subtle correlationsbetweenthe descriptionsofmotionin ity of the Unruh effect but it can be questioned as both
vacuum and of vacuum itself. For instance, the fluctu- situations are clearly different within the framework of
ations associated with motion in vacuum can in no way general relativity. The curvature of space-time is not in-
be treated as uncorrelated with the fields fluctuations of volved in a change of reference frame whereas it plays a
vacuum in which motion takes place. central role in the description of a black hole.
The existence of motional effects also questions the Concerning the problem of relativity of motion, the
principle of relativity of motion applied to arbitrarymo- Unruh effect shows serious drawbacks. If vacuum is re-
tions in vacuum. As the reactionof vacuum vanishes for ally different for accelerated and inertial observers, this
uniform velocities it does not raise any problem to the certainly makes it difficult, if not impossible, to describe
theory of special relativity. In contrast a non-uniformly inertia in a consistent quantum formalism. In particular
accelerated motion produces observable effects, namely difficulties should arise when analysing situations which
theresistanceofvacuumagainstmotionandtheemission involve composed motions. For example a motion with
ofradiationbythemovingmirrorintovacuum. Quantum uniform velocity in an acceleratedframe should give rise
theorythusallowsustosketchtheoutlineofaframework to a vacuum reaction like uniform velocity in an ordi-
where the questions raised in the introduction find con- nary thermal field whereas a uniformly accelerated mo-
sistentanswers. Spaceisnotempty sincevacuumfluctu- tion in a frame obtained from a Lorentz transformation
ations are always present. These fluctuations effectively would be free from dissipation. Contradictions are also
representareferenceforthe definitionofmotionbecause met concerning energy conservation for an accelerated
they give rise to real dissipative effects in the general detector. Furthermore, the interpretation of detection
case of an arbitrary motion. At the same time, they do processes appear to depend on the reference frame. As
notdampuniformmotionssincequantumvacuumobeys far as this point is concerned different results have been
Lorentz invariance. The physical properties of quantum obtained depending on the model used for the detector
vacuum are thus consistent with logical requirements of [31]. For those models involving detectors which click
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when acceleratedinvacuum it canevenhappenthat the ments do not hold for Rindler transformations. The
detection process is found to be related now to absorp- situation is much more favorable for conformal coordi-
tion then to emission of a photon. More generally it nate transformations which form a group including the
is impossible to preserve the usual understanding of the Poincar´e transformations besides the conformal changes
principle of relativity because the notions of vacuum or to accelerated frames. In particular, the composition of
photon number are not the same for different observers a uniformly accelerated motion with an inertial motion
[34]. results in another uniformly accelerated motion. Asso-
These difficulties have their originin a too rapididen- ciated with preservation of Maxwell equations and of
tification of uniformly accelerated reference frames with electromagnetic vacuum, these group properties amply
their Rindler representation. This particular represen- justify the use of conformal transformations to repre-
tation of accelerated frames favors a criterion of rigidity sent uniformly acceleratedreference frames. The confor-
in the transformation of solid bodies. A consequence of malrepresentationisthe naturalchoicewhenrelativistic
this choiceis that Maxwellequations aremodified inthe propertiesareinterpretedasinvariancepropertiesrather
change of reference frame. The Unruh effect essentially than mere form-invariance relations [39].
tells us that the definition of vacuum is also altered in a Furthermore, a clear understanding of the notion of
Rindler transformationand thatthe acceleratedvacuum quantumparticlesrequiresthatthenotionofvacuumhas
obtained after the transformation appears as a thermal been understoodbefore. This pointhas been thoroughly
field. But the Rindler representation of accelerated ref- discussedinthepresentpaperforwhatconcernstheprin-
erence frames is not the only possible one. In fact an cipleofrelativityofmotion. Itmustequallybetakeninto
infinite number of representations exist when one only account when considering the extension of the equiva-
imposes the condition that a given point with uniform lence principle to the quantum domain. The thoughts
acceleration is brought to rest. Furthermore, the the- presented here plead for bringing out the symmetries of
ory of generalrelativity does not provide any manner to vacuum,thequantumversionofemptyspace,inthefore-
privilegiate one particular representation of accelerated front of primary questions. As it has been shown here,
frames since it convincingly argues that physical results thisshouldensurecompatibilityoftheconceptionofmo-
cannotdependonthechoiceofaparticularmapofspace- tion with the principle of relativity. It has also been
time. shownelsewherethatthisapproachfitsperfectlywellthe
In quantum theory in contrast, motion must be un- description of localization in space-time built upon sym-
derstood as taking place in a space filled with vacuum metries of quantum fields. Precisely, conformal symme-
fluctuations. This feature implies to attach a particu- tryallowstodefinequantumobservablesassociatedwith
lar importance to those transformations to accelerated positions in space-time and to obtain their transforma-
frameswhichleavevacuuminvariant. Thisleadstofavor tionsunderLorentztransformationsandtransformations
conformal coordinate transformations which have been to accelerated frames [40].
known for a long time to leave Maxwell equations in- Ithasoftenbeen notedthatthe seriousproblemsaris-
variant [35]. These tranformations represent a natural ing in the mechanical description of electrons is directly
extension of Lorentz transformations which also include associated with the fact that they are treated as point-
uniformly accelerated frames [36]. An essential property like structures although such a treatment certainly con-
ofconformaltransformationsistopreservethedefinition tradicts their quantum nature. More generally, the clas-
of vacuum fluctuations [37] and, more generally,the def- sical conception of space as a set of points upon which
inition of particle number [38]. Hence there is no more classicalrelativityisbuiltwiththehelpofdifferentialge-
difference between the viewpoints of accelerated and of ometry is challenged by the quantum nature of physical
inertial observers as far as their perception of vacuum observables. The design of new conceptions of motion
is concerned,providedthat acceleratedobserversare de- and localization in space could therefore reveal itself a
fined through conformal transformations. The fact that preliminary step to the solution of yet unsolved prob-
vacuum exerts no force on a uniformly accelerated ob- lems lying at the interface between quantum theory and
ject, already discussed for electrons and for mirrors, is gravitation.
in fact a direct consequence of this symmetry property
of quantum vacuum. In this sense, the principle of rela-
tivity which was known for uniform velocities now has a
domainofapplicationextendedtouniformaccelerations.
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