On Newton's Third Law

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by  Mario J. Pinheiro

New Journal of Phys. - IST/IPFN 2009-Pinheiro
On Newton’s Third Law
Mario J. Pinheiro∗
Department of Physics and Institute for Plasmas and Nuclear Fusion,
Instituto Superior T´ecnico, Av. Rovisco Pais, 1049-001 Lisboa, Portugal†
(Dated: January 26, 2009)
The law of action-reaction is thoroughly used in textbooks to derive the conservation laws of
linearandangularmomentum,anditwasconsideredbyErnstMachthethecornerstoneofphysics.
We give here a background survey of several questions raised by the action-reaction law, and in
particular, therole ofthephysicalvacuumisshown toprovidean appropriateframework toclarify
9 theoccurrenceofpossibleviolationsoftheaction-reactionlaw. Itisalsoobtainedanexpressionfor
0 thegeneral linear momentumof abody-particleinthecontextof statistical mechanics. It isshown
0 that Newton’s third law is not verified in systems out of equilibrium due to an additional entropic
2 gradient term present in theparticle’s momentum.

n
a PACSnumbers: 04.20.Cv;03.50.De;05.20.-y;05.70.Ln
J Keywords: Fundamental problems and general formalism; Classical electromagnetism, Maxwell equa-
tions;Classicalstatisticalmechanics;Nonequilibriumandirreversiblethermodynamics
3
2

I. INTRODUCTION We can find in Cornille [6] a review of applications of
]
h action-reactionlawinseveralbranchesofphysics. Inad-
p dition, Cornille introduced the concepts of spontaneous
The law of action-reaction, or Newton’s third law [1],
- force (obeying to Newton’s third law) and stimulated
s is thoroughly used in textbooks to derive the conserva-
s force (which violates it).
tion laws of linear and angular momentum. Ernst Mach
a
l consideredthe thirdlawas“his mostimportantachieve- In this paper we intend to show that generally in
c ment with respect to the principles” [2, 3]. However, any system out of equilibrium, when entropy is velocity-
.
s the reasoning used primarily by Newton applies to point dependent, Newton’s third law is violated. The need for
c
particleswithoutstructureandisnotconcernedwiththe re-examination of this problems is pressing since long-
i
s motionofmaterialbodiescomposedwithalargenumber term exploitation of the cosmos face serious difficulties
y
of particles, in or out of thermal equilibrium. duetotheoutdatedspacecrafttechnologiesmankindpos-
h
sess.
p Ernst Mach sustained that the concept of mass and
[ Newton’sthirdlawwereredundant;thatinfactitshould Sec. II discusses the general issues in mechanics and

be enough to define operationally the mass of a given electromagnetism related to the action-to-reaction law.
1
v body as the unit of mass to be sure that “If two masses Sec. III discussesthe possible role ofphysicalvacuum as
6 1 and 2 act on each other, our very definition of mass a third agent that might explain action-to-reactions law
2 asserts that they impart to each other contrary acceler- violations. Secs. IV and V discusses the intrinsic viola-
7 ations which are to each other respectively as 2:1” [2]. tionofNewton’sthirdlawinsystemsout-of-equilibrium.
3 Yet philosophy has delivered us extraordinary new in- Sec. VI presents the conclusions that follow logically
.
1 sightstoabasicunderstandingofthe underlyingphysics from the previous discussion.
0 of force. For example, F´elix Ravaisson [4] in the XIX
9 century sustained that within the realm of the inorganic
0
world action-equals-reaction; they are the same act per-
:
v ceived by two different viewpoints. But in the organic II. BACKGROUND SURVEY
i world, whenever more complex systems are at working,
X
“Ce n’est pas assez d’un moyenterme indiff´erentcomme
r The usual derivation of the laws governing the linear
a le centre des forces oppos´ees du levier; de plus en plus, and angular momentum presented in textbooks is as fol-
il faut un centre qui, par sa propre vertu, mesure et dis-
lows. The equation of motion of the ith particle is given
pense la force” [5]. So, there is in Nature the need of
by:
an “agent” that control and deliver the action from one
body to another and this is, as we will see, the role of
dp
the physicalvacuum, orbarelyjust the environmentofa F + F = , (1)
i ij
body. Xj6=i dt
whereF isanexternalforceactingontheiparticleand
i
F representsthe internalforceexertedonthe particlei
∗Electronicaddress: [email protected] ij
bytheparticlej. Inthecaseofcentralforcestherelation
†WewouldliketothankpartialsupportfromtheFundac¸a˜oparaa
F =−F isverified,amanifestationofNewton’sthird
Ciˆencia e a Tecnologia (FCT) and the Rectorate of the Technical ij ji
UniversityofLisbon. law. Summing up over all the particles belonging to the
2
system we have from Eq. 1: An ongoing debate on the validity of electrodynamic
force law is still raging,with experimental evidence that
dp
F = i. (2) Biot-Savart law does not obey action-reaction law (see
i
dt
Xi Xi Ref. [16, 17] and references therein). The essence of the
problem stands on two different laws that exist in mag-
Podolsky [7] called attention to the discrepancies ob-
netostatics giving the force between two infinitely thin
tained using directly Newton’s second law, or using in-
line-current elements ds and ds through which pass
stead the invariance of the lagrangian under rotations. 1 2
currents i and i . The Amp`ere’s law states that this
In the case of non-central forces, like a system subject 1 2
force is given by:
to a potential function of the form V = r−1cosϑ, we
might expect a deviation from Newton’s third law. In- µ i i r 3
deed, angle-dependent potentials, long-range (van der d2F2,A =− 04π1 2r132[2(ds1·ds2)−r2 (ds1·r12)(ds2·r12)].
Waals) forces describe rigorously the physical properties 12 12
(3)
of molecular gases. We can wonder from which mecha-
This means that the force between two current elements
nism it comes the unbalance of forces.
depended not only on their distance, as in the inverse
We might expect that thermodynamics and statistical
squarelaw,butalsoontheirangularposition(inparticu-
mechanics provide a better description of macroscopic
lar,implicatingtheexistenceofalongitudinalforce,con-
matter. The internal energy and in particular the aver-
firmedexperimentallybySaumont[19]andGraneau[20],
age total energy of a system E = U , which includes
i i and discussed by Costa de Beauregard [21]). The other
summingupalltheparticlesconstiPtutingthesystemand forcegenerallyconsideredisgivenbytheBiot-Savartlaw,
allstoragemodes,playsafundamentalroletogetherwith
also known as the Grassmann’s equation in its integral
an equally fundamental, although less understood func-
form:
tion, the system entropy. Interesting enough a micro-
scopic model of friction have shown that the irreversible µ i i 1
d2F =− 0 1 2 [(ds ×(ds )×r )]. (4)
entropy production is drawn from the increase of Shan- 2,BS 4π r3 2 1 12
12
non information [8].
This question is related to the fundamental one still Here, r is the position vector of element 2 relative to
12
not answered in physics and biophysics: how chaos in 1. While Amp`ere’s law obeys Newton’s third law, Biot-
various natural systems can spontaneously transform to Savart law does not obey it (e.g., Ref. [22, 23, 24, 25]).
order? The observation of various physical and biologi- The theory developed by Lorentz was criticized by H.
cal systems shows that a feedback is onset according to: Poincar´e[26],becauseitsacrificedaction-to-reactionlaw.
“The medium controls the object-the object shapes the Theproblemoflinearmomentumofstationarysystem
medium”[9]. Atthemicroscopiclevelalargeclassofsys- of charges and currents is faraway of a consensus too.
temsgeneratedirectedmotionthroughtheinteractionof Costa de Beauregard [27] pointed out a violation of the
a moving object with an inhomogeneous substrate pe- action-reaction law in the interaction between a current
riodically structured [10]. This is the ratchet-and-pawl loop I flowing on the boundary of area A with moment
principle. M=IAandanelectricchargeconcludingthatwhenthe
The apparent violation of Newton’s third law that we moment of the loop changes in the presence of an elec-
can find in some systems, e.g., when two equal charged tric field a force must act on the current loop, given by
bodies having equal velocities in magnitude and oppos- F=[E×M˙ ]/c2. ShockleyandJames[28]attributeFto
ingdirections,iswell-known. TheLorentz’sforceapplied achangein“hiddenmomentum”G =−[E×M]/c2 car-
l
to both charges do not cancel each other since the mag- ried within the current loop by the steady state power
netic forcesdo notactalong a commonline (see alsothe flow necessary to balance the divergence of Poynting’s
Onoochin’s paradox [11]). The paradox is solved intro- vector. The total momentum is p = G +G , where
l b
ducingtheelectromagneticmomentum[E×H]/c2(values G = m < r˙ > is the body momentum associated
b CM
in SI units will be used throughout the text) [12]. with the center of mass m [29, 30]. In particular, it
In the domain of astrophysics, the same problem ap- was shown [29] that the “hidden linear momentum” has
pears again. For instance, based on unexplained astro- as quantum mechanical analogue the term α·E, where
physicalobservations,suchasthehighrotationofmatter α are Dirac matrices appearing in the hamiltonian form
around the centers of galaxies, it was proposed a mod- Hψ =i~∂ψ/∂t,whereH =−ic~α·∇·isthehamiltonian
ification of Newton’s equations of dynamics [13], while
operator (e.g., Ref. [31]). Although certainly an impor-
more recently a new effect was reported, about the pos- b b
tant issue,the conceptof “hiddem momentum” needs to
sibility of a violation of Newton’s second law with static
be further clarified [32].
bodies experimenting spontaneous acceleration [14]. In
Calkin [33] has shown that the net linear momentum
the frame of statistical mechanics, studying the effective
of any closed stationary system of charges and currents
forces betweentwo fixedbig colloidalparticles immersed
is zero and can be written:
in a bath of small particles, it has been shown that the
nonequilibriumforcefieldisnonconservativeandviolates u˙
P= d3rr( )=Mr , (5)
action-reaction law [15]. Z c2 CM
3
where u is the energy density, M is the total mass, III. INTERACTION WITH THE VACUUM
M = d3r(u/c2) and r is the radius vector of the
CM
centerRof mass. He has shown, however, that the linear Although Newton’s third law of motion apparently
mechanical momentum P in a static electromagnetic
ME does not hold in some situations, it is likely action and
field is nonzero and it is given by: reactionalwaysoccursby pairs anda kind ofaccounting
balance such as F=−F′ holds.
P =− d3rρAT. (6) According to the Maxwell’s theorem, the resultant of
ME
Z K forces applied to bodies situated within a closed sur-
face S is given by the integral over the surface S of the
Here, AT denotes the transverse vector potentia−→l given Maxwell stresses:
by AT = (µ /4π) d3rJ/r. Eq. 6 shows that ρA is a
0
measure of momentRum per unit volume. T(n)dS = fdΩ=K. (8)
Similar conclusion were obtained by Aharonov et Z Z
al. [34] showing, in particular, that the neutron’s elec-
Here, f is the ponderomotive forces density and dΩ is
tric dipole moment in a external static electric field E
0 the volume element. The vector T(n) under the integral
experiencesaforcegivenbyma=−(v·∇)(v×E ). The
0 in the left-hand side (lhs) of the equation is the tension
experimental verification of the Aharonov-Casher effect
force acting on a surface element dS, with a normal n
would confirm total momentum conservation in the in-
directed toward the exterior. In cartesian coordinates,
teractions of magnets and charges [35].
each component of T(n) is defined by
Breitenberger [36] discusses thoroughly this question
showingthedelicateintricaciesbehindthesubject,point-
T (n)=t cos(n,x)+t cos(n,y)+t cos(n,z), (9)
x xx xy xz
ingouttheconservationofcanonicalmomentumandthe
“extremelysmall”effectofmagneticinteractionsthrough
with similar expressions for T and T . The 4-
y z
the use of the Darwin’s lagrangian derived in 1920 [37].
dimensionalmomentum-energytensorisageneralization
Boyer [38] applying the Darwin’s lagrangian to the sys-
ofthe3-dimensionalstresstensorT . Ifelectriccharges
lm
tem of a point charge and a magnet have shown that
are inside a conducting body in vacuum, in presence of
the center-of-energy has uniform motion. Darwin’s la-
electric E and magnetic H fields, then Eq. 9 must be
grangianiscorrecttothe order1/c2 (remainingLorentz-
modified to the form:
invariant) and the procedure to obtain it eliminates the
radiation modes, and thus describing the interaction of 1 ∂[E×H]
T(n)dS−K= dΩ. (10)
chargedparticlesin the frame onanaction-at-a-distance Z Z 4πc(cid:18) ∂t (cid:19)
electrodynamics. It can lead, however, to unphysical so-
lutions [39]. In the right-hand side of the above equation it now ap-
Hnizdo [40] have shown that at nonrelativistic veloci- pears the temporal derivative of G = gdΩ, the elec-
ties Newton’s third law is verified in the interactions be- tromagnetic momentum of the field in thRe entire volume
tween current-carrying bodies and charged particles be- contained by the surface S (with g its momentum den-
cause the electromagnetic field momentum is equal and sity).
opposite to the hidden momentum hold by the current- In the case the surface S is filled with a homoge-
carrying bodies; the mechanical momentum of the en- neous medium without true charges, Abraham proposed
tire closed system is conserved. Hnizdo have also shown to write instead the following equation:
that,however,thefieldangularmomentuminasystemis
∂ εµ
notcompensatedbyhiddenmomentumandthustheme- T(n)dS = [E×H]dΩ, (11)
chanical angular momentum is not conserved alone, but Z ∂tZ 4πc
had to be summed up with the field angular momentum
with ε and µ the dielectric constant of the medium and
in order to be a conserved quantity.
its magnetic permeability.
In fact, the “magnetic current force”, produced by
Eq.11 can be written onthe form of a generalconser-
magnetic charges that “flow” when magnetism changes,
given by f = ε E ×(B˙ − µ H˙)[41] is the “Abraham vation law:
m 0 0
tdeirffmer”faropmpeathrienMg iinnktohwesAkybrdaehnasmitydfeonrscietyfMfortcherfoAugwhhtihche ∂∂σxαβ − ∂∂gtα =fα (12)
β
expression:
where α = 1,2,3, σ is the stress tensor, g is the mo-
αβ α
∂
fA = [gM −gA]. (7) mentum density of the field, and fα is the total force
∂t density. After some algebra this equation can take the
final form:
Here,gM =[D×B]is the Minkowsky’smomentumden-
sity of the field and gA = [E×H]/c2 is the Abraham’s ∂σαβ =fL+ 1 ∂ [D×B] +f′ . (13)
momentum density. ∂x α 4πc∂t α m,α
β
4
Here, f′ is the total force acting in the medium (see
m
Ref.[42]),fL =ρ E+1[j×B]istheLorentzforcedensity
e c
with ρ denoting the charge density and j the current
e
density.
Of course, field, matter and physical vacuum form to- MATTER FIELD
gether a closed system and it is usual to catch the mo- TbaMbbaaba TbaFbbaaba
Abraham’s Minkowsky’s
mentumconservationlawinthegeneralform[43,44,45]:
term term

∂(TField+TMatter+TVacuum)
αβ αβ αβ
=0. (14)
∂x
β
Table I shows the different expressions for the energy-
momentum tensors of Minkowksy, TM and Abraham,
α,β
TA .
α,β
ThegeneralrelationbetweenMinkowskiandAbraham
momentum, free of any particular assumption, holding
VACUUM
particularly for a moving medium, is given by TbaVbbaaba
PM =PA+ fAdtdV. (15)
Z
For clearness, we shall distinguish the following differ-

ent parts of a system: i) the body carrying currents and
thecurrentsthemselves(thestructure,forshort,denoted
herebyK),ii)thefields,andiii)thephysicalvacuum(or
the medium).
On the theoretical ground exposed above,the impulse
transmitted to the structure should be given by the fol-
lowing equation:
PK = fAdtdV =PM −PA. (16)
Z
Here, fA denotes the Abraham’s force density [46, 47]: FIG. 1: Conservation law for the closed system: Matter +
Field + Physical Vacuum.
ε µ −1∂[E×H]
fA = r r . (17)
4πc ∂t
ε µ ∂[E×H]dV in order Newton’s third law be pre-
This is in agreement with experimental data [48] and ∞ 0 0 ∂t
sRerved. It is important to remark that the field momen-
was proposed by others [49, 50]. As this force is act- tum[D×B]is equivalentto ρA,the firsttermis related
ingoverthemedium,itisexpectednonlinearitiesrelated
to the stress-tensor representation, while the second one
to the behavior of the dielectric to different applied fre-
is related to the “fluid-flow” representation [54].
quencies, temperature,pressure, and large amplitudes of
Asiswellknown,Maxwell’sclassicaltheoryintroduces
the electric field when a pure dielectric response of the
the idea of a real vacuum medium. After being consid-
matter is no longer proportionalto the electric field (see
ereduselessby Einstein’s specialtheoryof relativity,the
Ref. [51] on this topic).
“ether” (actually replaced by the term vacuum or physi-
The momentumconservationlawcanbe rewrittenun-
cal vacuum) was rehabilitated by Einstein in 1920 [55].
der the general form (e.g., Ref. [42]):
In fact, general theory of relativity describes space with
physical properties by means of ten functions g (see
∂σαβ =fL+ 1 ∂ [D×B] +f′ , (18) also [56]). According to Einstein, µν
∂x α 4πc∂t α m,α
β
The “ether”ofgeneralrelativityis amedium
with f denoting the force acting on the medium. The
m that by itself is devoid of all mechanical and
second term in the r.h.s. of above equation could pos-
kinematic properties but at the same time
sible be called vacuum-interactance term [52] - in fact,
determinesmechanical(andelectromagnetic)
Minkowski term. Already according to an interpreta-
processes.
tion of Einstein and Laub [53], the integration of above
equation over all space, the derivative over stress ten- Dirac felt the need to introduce the idea of “ether” in
sor gives a null integral and the Lorentz forces summed quantum mechanics [57]. In fact, according to quantum
over all the universe must be balanced by the quantity fieldtheory,particlescancondenseinvacuumgivingrise
5
TABLE I:Expressions for the energy-momentumtensors of Minkowksy TαM,β and AbrahamTαA,β, using i,k=1,2,3,4; x1=x,
x2=y,x3=z, x4 =ict. ThePoynting’s vector is S=[E×H] and the energy for a system at rest is w= 81π(ǫE2+µH2).
Minkowsky Abraham
σ −icgM σ −icgA
TM = α,β TA = α,β
α,β „ −iS w « α,β „−iS w «
c c
gM = ǫµ[E×H] gA = 1[E×H]
c2 c2
to space-time dependent macroscopic objects, for exam- physical vacuum fluctuations to the motion of dielectric
ple, offerromagnetictype. Besides,stochasticelectrody- fluids in crossed electric and magnetic fluids communi-
namicshaveshownthatthevacuumcontainsmeasurable cating to matter velocities of the order of 50 nm/s [66],
energycalledzero-pointenergy(ZPE)describedasatur- although this result was contested by van Tiggelen et
bulent sea of randomly fluctuating electromagnetic field. al. [67].
Quite interestingly, it was recently shown that the inter- The exploration of these ideas to propel a spacecraft
action of atoms with the zero-point field (ZPF) guaran- has been advanced in the literature, e.g., see Refs. [68,
tees thestability ofmatter and,inparticular,the energy 69, 70, 71].
radiated by an accelerated electron in circular motion is
balancedbytheenergyabsorbedfromthe ZPF [58]. An
attempttoreplaceafieldbyafinitenumberofdegreesof
freedom was done by Pearle [59]. In this theory N par- IV. DEDUCING THE LINEAR MOMENTUM
ticles were supposed do not interact directly with each FOR A MATERIAL BODY ON THE BASIS OF
other, but interact directly with a number of dynamical STATISTICAL PHYSICS
variables (called the “medium”) carrying the ”informa-
tion” from one particle to another.
When two bodies of matter collide, the repulsive force
Graham and Lahoz made three important experi-
on them is equal whenever no dissipative process is at
ments [60, 61, 62]. While the first experiment provided stake. When a ball rebound on the floor it has the same
anexperimentalobservationofAbrahamforceinadielec-
total mechanical energy before and after the collision,
tric,thesecondoneprovidedameasurementofareaction except for a loss term which is due to the fact that the
forcewhichappearinmagnetite. Thethirdoneprovided
bodies have internal structure. At a microscopical level,
thefirstevidenceoffreeelectromagneticangularmomen- bodies are aggregates of molecules. When the body col-
tum created by quasistatic and independent electromag-
lides,moleculesgainaninternal(random)kineticenergy.
netic fields E and B in the vacuum [63]. Whereas the Macroscopicallythis generates heat, and therefore raises
referred paper by Lahoz provided experimental evidence
the system entropy. In global terms, some fraction of
forAbrahamforceatlowfrequencyfields,itstillremains
heat does not return to the particle’s collection consti-
to gather evidence of its validity at higher frequency do-
tutingtheballandtheentropyoftheuniverseultimately
main,althoughsomemethodsarepresentlyoutlined[64].
increases.
All this is known since a long time and we only try
Let’s consider an isolated material body composed by
to put more clear the theoretical framework, that only
a great number of macroscopic particles (let’s say N)
needs to be experimentally tested for proofof principles.
possessing an internal structure with a great number of
In view of the above, we will write the ponderomotive
degreesoffreedom(tovalidatetheentropyconcept)with
force density actingonthe composite body ofarbitrarily
momentump , energyE andwith intrinsic angularmo-
large mass (formed by the current configuration and its i i
mentum J , all constituted of classical charged particles
supporting structure) in the form i
withchargeq andinertialmassm . Usingtheprocedure
i i
dV ∂ outlined in Refs. [72, 73] we can show that the entropy
ρ =ρ E+[J×B]+∇·T+ (ε µ [E×H]). (19)
dt c ∂t 0 0 gradient in momentum space is given by
Hence, the composite body is acted on by Minkowski
force in such a way that ∂S
p =m v +q A+m [ω×r ]−m T . (21)
i i e i i i i i∂p
MV=−GM +GA. (20) i
The Minkowski momentum is transferred only to the It was assumed that all particles have the same drift ve-
field in the structure and not to the structure and the locity and they turn all at the same angular velocity ω.
field in the medium [42, 62, 65]. In summary, to move The center of mass of the body moves with the same
a spacecraft forward, the spacecraft must push “some- macroscopicvelocityandthe bodyturnsatthesamean-
thing” backwards; and this “something” might be the gular velocity [44]. The last term of Eq. 21 represents
physicalvacuum. Thiseffectwasshowntobemadefeasi- the gradientofthe entropyinanonequilibriumsituation
ble, the Abraham’sforcerepresentingthe reactionofthe and S is the transformed function defined by:
6
N p2 J2
S = S E − i − i −q V +q (A ·v)]+(a·p)+b·([r ×p ]+J ) , (22)
i i i i i i i i i i i
(cid:26) (cid:20) 2m 2I (cid:21)(cid:27)
Xi=1 i i
where a and b are Lagrange multipliers. on the other. These are examples of irreversible (out of
Wheneverthesystemisinthermodynamicequilibrium equilibrium)phenomenathatdonotcomplywithaction-
the canonicalmomentumis obtainedforeachcomposing reaction law.
particle: At this stage, we can argue that the momentum is al-
waysaconservedquantityprovidedthatweaddtheright
pi =prel+mi[ω×ri]+qiAi. (23) term,makingNewton’sthirdlawverified. Thisapparent
“missing symmetry” might result because matter alone
Otherwise, when the system is subjected to forced con- does not form a closed system, and we need to include
straintsinsuchawaythatentropicgradientsinmomen- the physical vacuum in order symmetry be restored. So,
tumspacedoexist,thenanewexpressionfortheparticle when we have two systems 1 and 2 interacting via some
momentum must be taken into account, that is, Eq. 21. kind of force field F, the reactionfrom the vacuum must
Summing up over all the constituents particles of a be included as a sort of bookkeeping device:
given thermodynamical system pertaining to the same
aggregate (e.g., body or Brownian particle), we obtain: Fmatter =−Fmatter +Fvacuum. (26)
12 21
∂S Wemayassumetheexistenceofaphysicalvacuumprob-
P=Mv + m [ω×r ]+QA− m T . (24)
e i i i i∂p ably well described by a spin-0 field φ(x) whose vacuum
Xi Xi i expectation value is not zero:
To simplify we can assume that all the particles inside
vacuum∼φ(x), (27)
the systemhavethe samerandomkineticenergy,T =ζ:
i
andatits lowest-energystateto havezero4-momentum,
∂S
P=Mv + m [ω×r ]+QA−ζ ne, (25) k =0 (e.g., Ref. [45]).
e i i µ
∂r˙
Xi Xi i This new state out of equilibrium can be constrained
by applying an external force on the system (e.g., set all
where by Sne we denote the entropy when the system system into rotation about its central axis at the same
is in a state out of equilibrium. The first term on the angular velocity ω).
right-handside is the bodily momentum associatedwith It was shown that the entropy must increase with
motion of the center of mass M; the second term rep- a small displacement from a previous referred state
resents the rotational momentum; the third is the mo- [44, 80]. Considering that the entropy is proportional
mentum of the joint electromagnetic field of the moving tothe logarithmofthe statisticalweightΩ∝exp(S/k )
B
charges [74, 75]; finally, the last term is a new momen- and considering that S = S +S we can expect an
eq ne
tumterm,thatcanbephysicallyunderstoodasakindof increase of the nonequilibrium entropy S with a small
ne
“entropicmomentum”sinceitisultimatelyassociatedto increase of the ith particle’s velocity v = r˙ , since with
i i
the information exchanged with the medium on the the an increase of particle’s speed (although in random mo-
physical system viewpoint (e.g., momentum that even- tion) the entropy must increases altogether. Therefore,
tually is radiated by the charged particle). Lorentz’s we must always have:
equations don’t change when time is reversed, but when
retardedpotentials are applied the time delay of electro- ∂S
T ne ≥0,∀i=1,...N. (28)
magnetic signals on different parts of the system do not ∂r˙
i
allow perfect compensation of internal forces, introduc-
ing irreversibility into the system [76]. This is always Inconditionsofmechanicalequilibriumtheequalitymust
true whenever there is time-dependent electric or/and hold,otherwisecondition 28canbeconsideredauniver-
magnetic fields [77]. Cornish [78] obtained a solution sal criterium of evolution. Considering that the entropy
of the equation of motion of a simple dumbbell system is an invariant [81] there is no extra similar term when
held at fixed distance and have shown that the effect of themomentumistransferredtoanotherinertialframeof
radiation reaction on an accelerating system induces a reference.
self-accelerated transverse motion. Obara and Baba [79] Quitewithstanding,thereisanimportanttheoremde-
have discussed the electromagnetic propulsion of a elec- rived by Baierlin [82] showing that the Gibbs entropy
tric dipole system and they have shownthat the propul- for a system of free particles with kinetic energy K,
sion effect results from the delay action of the static density ρ and absolute temperature T, S(K,ρ,T), is
and inductive near-field created by one electric dipole greater than the entropy associated to the same system
7
subjecttoarbitraryvelocity-independentinteractionsV,
S(K+V,ρ,T), such as S(K+V,ρ,T)≤S(K,ρ,T).

At the electromagnetic level, Maxwell conceived a dy-
Thermal Thermal
namical model of a vacuum with hidden matter in mo- reservoir reservoir

tion. As it is well-known, Einstein’s theory of relativ-

ity eradicated the notion of “ether” but later revived T2 V1 T3
its interest in order to give some physical mean to g .
ij
Minkowski obtained as a mathematical consequence of
the Maxwell’s mechanical medium that the Lorentz’s
force should be exactly balanced by the divergence of
the Maxwell’s tensor in vacuum T minus the rate of
vac
change of the Poynting’s vector:
∂
ρE+µ [J×H]=∇·T − ε µ [E×H]. (29)
0 vac 0 0
∂t
EinsteinandLaubremarked[53]thatwhenEq.9isinte-
gratedallovertheentireUniversetheterm∇·T van-
vac
ishwhichmeansthatthesumofallLorentzforcesinthe
Universemustbeequaltothequantity ε µ ∂/∂t[E×
∞ 0 0
H]dv in order to comply with NewtonR’s third law (see
Ref.[83]). But,thislongrangeforcedependsonthecon-
stant of gravitation G. Einstein accepted the Faraday’
FIG. 2: Schematic of the self-accelerated device.
point of view of the reality of fields, and this gravita-
tionalfieldaccordingtohimwouldpropagatealloverthe
entire space without loss, locally obeying to the action- V. IS IT VERIFIED THE
reactionlaw. Butnothing canreassureus thatthe prop- ACTION-EQUALS-REACTION LAW IN A
agating wave through the vacuum will be lost at infinite THERMODYNAMICAL SYSTEM
distances [84]. Poincar´e[85] also arguesabout the possi- OUT-OF-EQUILIBRIUM ?
bledissipationoftheactiononmatterduetotheabsorp-
tion of the propagating wave in the context of Lorentz’s The maximizing entropy procedure proposed in
theory. Ref. [72, 73] suggest the following “gedankenexperi-
ment”. This problembearssome resemblanceto the Leo
By Noether’s theorem, energy conservation is related Szilard’s thermodynamic engine with one-molecule fluid
to translational invariance in time (t → t+a) and mo- (e.g.,Ref.[87]),althoughwearenotconcernedherewith
mentum conservation is related to translational invari- neguentropy issues.
ance in space (ri → ri +bi). This important theorem Letus supposeasystemconsistingofasphericalbody
thus implies that the law of conservation of momentum made ofN number of particles closedin a box and mov-
(notequivalenttotheaction-equals-reactionprinciple)is ing along one direction (see Fig. 2). The left side is at
always valid, while the law of action and reaction does temperatureT ,therightsideisattemperatureT ,while
2 3
not always holds, as shown in the previous examples. thebodyparticleitselfisattemperatureT (andinequi-
1
librium with their photonic environment). Furthermore,
Some kind of relationship must therefore exists be- we assume that both surfaces and the body particle are
tweenentropyandNewton’sthirdlaw,asitwasthrough allthermalreservoirs,andhencetheirrespectivetemper-
the combined equation with the first and second law atures do not change. Let us suppose that the onset of
of thermodynamics that our main result were obtained. nonequilibrium dynamics can be forced by some means
ThisideawasverifiedrecentlythroughastandardSmolu- in the previously described device. When the particle
chowski approach and on Brownian dynamic computer collides with the top its momentum varies according to:
simulation of two fixed big colloidal particles in a bath
of small Brownian particles drifting with uniform ve- δp =−mv ”+mv +(T −T )∂ S. (30)
↿ 1 1 3 1 v
locity along a given direction. It was shown that, in
striking contrastto the equilibrium case, the nonequilib- Here,∂ S denotesthe(nonequilibrium)entropygradient
v
riumeffectiveforceviolatesNewton’sthirdlaw,implying in velocity space. After the collision the particle goes
the presence of nonconservative force showing a strong back to hit the right surface at temperature T . The
3
anisotropy [86]. This result reminds our Eq. 26. momentum variation after the second collision is given
8
by: self unidirectional propulsion. It seems now certain that
depletion forces between two fixed big colloidal particles
δp⇃ =mv1−mv1”+(T2−T1)∂vS. (31) in a bath of small particle exhibits nonconservative and
strongly anisotropic forces that violate action-reaction
We assume that the body attain thermal equilibrium
law[86](seealsoRef.[89]). Also,internalCasimirforces
with the environment (which must remain at constant
between a circle and a plate in nonequilibrium situation
temperature T ) fast enough before the next hit against
1 violates Newton’s law [90].
the wallofthe thermalreservoir. Thetotalbalanceafter
a complete loop, back and forth, is given by
δp⇃ =−δp↿−∂vS(T2+T3−2T1)=−δp↿−∆ζ∇vS. (32) VI. CONCLUSION
To make it more clear, we might write Eq. 32 under the
Thepurposeofthisstudyistoexaminehowtheaction-
form
reactionlawisfacedintheliteratureinthedomainofme-
δp =−δp −δpis, (33) chanics, electrodynamics and statistical mechanics, and
⇃ ↾ ↾
to offer a methodological approach in order to tackle
where we denote by δpis ≡ ∆ζ∇ S, the change in mo- the fundamental aspect of the problem suggesting that
↾ v
mentum by the physical vacuum (others, would call “in- a third part should be included in the analysis of forces,
ertialspace”). Therefore,itisclearfromtheaboveanaly- whatwecalledhereforthesakeofconciseness,thephysi-
sis that in systems out of equilibrium Newton’s third law calvacuum. Furthermore,ageneralprocedureleadusto
is not verified. The conservation of canonical momen- an expression for the general linear (canonical) momen-
tum, however, is well verified, as it must be according tum of a body-particle in the framework of statistical
to Noether’s theorem. Otherwise, when the tempera- mechanics. Theoretical arguments and numerical com-
tures are equal for all thermal bath in contact, such as putations suggestthat Newton’s third law is not verified
T =T =T , Newton’s third law is verified: in out-of-equilibrium systems due to an additional en-
1 2 3
tropic gradient term present in the particle’s canonical
δp =−δp . (34) momentum. Although Noether’s theorem guaranty the
⇃ ↿
conservation of canonical momentum, the actions-equal-
In the frame of nonlinear dynamics and statistical ap- reactionprinciplecanberestoredinnonequilibriumcon-
proach Denisov has shown [88] that a rigid shell and a ditions only if a new force term representing the action
nucleus with internal dynamic asymmetric can perform of the medium on the particles is taken into account.
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