Action–reaction symmetry breaking by induced internal forces

Ioannis Xydous1∗118532 Piraeus, Greece, ORCiD: 0000-0001-5460-2531

Abstract. Newton’s third law of motion would be the cornerstone of physics were it not forcertain experimental results that demonstrate a conflict with conservation of momentum undercertain special circumstances. Such conceptual conflicts appear in systems in which the interactingparts of the system are mediated by a nonequilibrium environment that gives rise to nonreciprocalforces. The present theoretical study of a new mechanism of motion that utilizes induced internalforces in an isolated system (internally powered), addresses the probable cause behind the breakingof action–reaction symmetry and explores the potential implications of these findings for Einstein’sspecial relativity and Lorentz transformations.

PACS numbers: 45.20.df, 45.20.D-, 03.30.+p

Introduction. In classical mechanics, any realizationof the action–reaction principle presupposes two bodiesexerting equal and opposite forces on each other. The no-tion of an internally powered isolated system moving inthe absence of external forces, touches upon the extraor-dinary idea that object A is attracted to object B whileobject B is simultaneously repelled by object A. Nonre-ciprocal forces associated with the breaking of action–reaction symmetry is a subject being addressed by vari-ous disciplines in physics as in statistical mechanics [1–3], condensed matter [4–9], quantum mechanics [10–12],high energy physics [13], relativistic mechanics [14–16]and optics [17–21]. The present paper proposes and re-veals the potentially missing part of Newton’s [22] thirdlaw of motion that enables the realization of the aboveidea by simultaneously establishing the breaking of New-ton’s action–reaction symmetry as a fundamental law ofnature. The acceleration of an isolated system that uti-lizes induced internal forces leads inevitably to a newphenomenon, namely the reduction of the effective iner-tia, with profound implications for Einstein’s [23] specialrelativity.

Methods. As shown in FIG. 1, an applied internalforce FA exerted on part mA of an ideal isolated sys-tem causes a reaction internal force on part mR alongthe line connecting them (the mutual actions are linearlyentangled). Thus,

∑Fext = 0 and

−→︷︸︸︷FA +

←−︷︸︸︷FR = 0, (1)

Us =

∫ d

0

FAds and∑

Fint + (FA + FR) = 0, (2)∫ d

0

∑Fintds+

(Us +

∫ d

0

FRds

)= 0, (3)

where Us is the energy stored within the system.

∗Electronic address: ioannisxydous@gmail.com

mR mA mR′

mR mA mR′~FA

~FR

~FAI~FRI

~FR~FA

~FRI~FAI

~rA × ~FA~rR × ~FR

~rR

~rA

~FA

~FR

FIG. 1: Action–reaction forces in isolated systems. Upper:Induced internal forces (FAI and FRI). The mutual actionsof masses mR and mA are helically entangled with each other,allowing the development of one-way (reaction and action mayhave the same magnitude and direction) induced forces (right-wards) that, by nature, depend on the direction of helix ro-tation. Lower: Collinear internal forces (FA and FR). Themutual actions of masses mR and mA are linearly entangledwith each other with no direction limitations (reaction andaction have same magnitude but opposite direction).

When the mutual actions are not linearly but helicallyentangled, the net internal force (induced) is expected tobe not null. Starting from the conservation of angularmomentum, the net external and internal torques are

∑τext = 0 and

−→︷︸︸︷FA +

←−︷︸︸︷FR = 0, (4)

rA 6= rR ⇒ τA 6= 0 and τR 6= 0, (5)∑τint + τA + τR = 0, (6)∑

τint + (rA × FA) + (rR × FR) = 0, (7)∑τint + (rA × FA) + (rR × (−FA)) = 0, (8)

x︷ ︸︸ ︷∑τint +

y︷ ︸︸ ︷((rA − rR)× FA) = 0. (9)

A translation mechanism (translation screw in FIG. 2)can maintain, amplify or reduce the magnitude of the in-put force by delivering the same amount of energy (ide-ally) entering the system (energy conservation).

2

mA

~pmA~p

~FA

~FA′

~FAI~FRI

mA

~pmA~p

~FA~FR

FIG. 2: Proof of concept. Action–reaction forces in isolatedsystems. Upper: Induced internal forces (FAI and FRI). Theapplication of a clockwise torque through a couple (FA =−FA′) on the translation screw about the axis of rotation, in-duces a counter-clockwise net torque (conservation of angularmomentum) that is converted (translation screw mechanism)to unidirectional induced forces FAI and FRI . Consequently,any change in momentum of mass mA results in the non-zerochange in momentum of system as a whole. Lower: Collinearinternal forces (FA and FR). Newton’s laws of motion for-bid a system to acquire momentum through collinear internalforces.

At this point, developing a general expression for thenet induced force requires the introduction of the dimen-sionless factor nr (mechanical advantage) along with adefinition of the net induced torque. Hence,

nr =|ω × (rA − rR) |

|uI|=

2π|rA − rR||dI|

, (10)

x︷ ︸︸ ︷nr∑

τint +

y︷ ︸︸ ︷nr ((rA − rR)× FA) = 0, (11)∑

τind = nr∑

τint, (12)

x︷ ︸︸ ︷∑τind +

y︷ ︸︸ ︷nr ((rA − rR)× FA) = 0. (13)

Dividing Eq.(13) by the position-vector magnitude|rA − rR| yields

←−︷ ︸︸ ︷∑τind

| (rA − rR) |+

−→︷ ︸︸ ︷nr ((rA − rR)× FA)

| (rA − rR) |= 0, (14)

Us =

∫ 2π

0

((rA − rR)× FA) dφ, (15)

Us is the energy stored (1 cyc) within the system.

The net induced force (internal) is not null, which im-plies that the isolated system starts to accelerate fromrest, something that Newton’s laws of motion do not an-ticipate. Thus,∑

Find =dp

dt= −nr ((rA − rR)× FA)

| (rA − rR) |6= 0, (16)

∑Find = −nr ((rA × FA) + (rR × FR))

| (rA − rR) |6= 0, (17)∑

Find =dp

dt= − (FAI

+ FRI) 6= 0, (18)∫ dI

0

∑Findds+ Us = 0, (19)

(rA − rR)

{∦ FA : dp

dt = − (FAI+ FRI

) 6= 0,

‖ FA : dpdt = − (FA + FR) = 0.

(20)

An ideal isolated system (see FIG. 1 and FIG. 2) couldacquire non-zero momentum if and only if, any change inmomentum of mass mA results in the non-zero change inmomentum of system as a whole. In the case of collinearinternal action–reaction forces, the conservation of mo-mentum is

dp

dt= − (FA + FR) = 0 (collinear), (21)

p = (mA +mR +mR′)ucm, (22)

p = − (p′mA− pmA

) , (23)

p = − (mAucm′ −mAucm) , (24)

p = −mA (ucm′ − ucm) , (25)

dp

dt= − (ucm′ − ucm)

dmA

dt= 0⇒ ucm′ = ucm. (26)

Then, for the induced internal action–reaction forces,the conservation of momentum becomes

dp

dt= − (FAI

+ FRI) 6= 0 (induced), (27)

uI = ucm′ − ucm, (28)

p = −nrmA (ucm′ − ucm) = −nrmAuI, (29)

dp

dt= −nruI

dmA

dt6= 0⇒ ucm′ 6= ucm. (30)

When the change in velocity uI has constant mag-nitude, the effective inertia of the system becomes

|uI| = const.⇒ ucm′ − ucm 6= 0, (31)

m = mA +mR +mR′ (32)

dm/dt = dmA/dt+ dmR/dt+ dmR′/dt (33)

but dmR/dt = dmR′/dt = 0⇒ dm = dmA, (34)

dm = dmA ⇒dp

dt= −nr (ucm′ − ucm)

dm

dt, (35)

dp = −nr (ucm′ − ucm) dm (36)∫ mi

m

dm = − 1

nr (ucm′ − ucm)

∫ p

0

dp, (37)

mi = m

(1− p

nr ·m (ucm′ − ucm)

)(38)

mi = m

(1− p

nr ·m · uI

). (39)

3

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

mi/m

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

β = v/c

SRnr = 1nr = 2nr = 3nr = 4

FIG. 3: Relativistic inertia.

Let us suppose that there is a theoretical quasiparti-cle (system) that exhibits the property whereby its effec-tive inertia decreases as its velocity usw or momentump increases. Setting uI to be equal to the speed of light,Eq.(39) becomes the nonrelativistic inertia of the system.Thus,

uI = c⇒ mi = m

(1− p

nr ·m · c

). (40)

Alternatively, because of energy conservation, Eq.(40)becomes

usw = u⇒ mic2 −mc2 = −

(p

nr

)2

· 1

2m, (41)

mic2 −mc2 = − 1

n2r

mu2

2= −Uk

n2r⇒ Us =

Uk

n2r, (42)

mi = m

(1− u2

n2r · 2c2

). (43)

The classical limit of the relativistic inertia for chargedparticles is obtained using the Taylor-series expansion ofthe Lorentz factor:

γ =

(1− u2

c2

)−1/2=

∞∑n=0

(uc

)2n n∏k=1

(2k − 1

2k

), (44)

γ = 1 +1

2

(uc

)2+

3

8

(uc

)4. . .⇒ u << c, (45)

γ ≈ 1 +1

2

(uc

)2, (46)

mi = mγ ≈ m(1 +u2

2c2). (47)

Similarly, Eq.(43) is the classical limit of the relativisticinertia for the theoretical quasiparticle.

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

β=v/c

0 2 4 6 8 10 12 14 16

Uk/mc2

SRnr = 1nr = 2nr = 3nr = 4

FIG. 4: Group velocity.

The binomial series expansion of the inverse Lorentzfactor yields

un = u/nr, (48)

1

γr=

(1− un

2

c2

)1/2

= 1− 1

2

(unc

)2− 1

8

(unc

)4.. (49)

un << c⇒ 1

γr≈ 1− 1

2

(unc

)2, (50)

mi =m

γr≈ m(1− u2

n2r · 2c2). (51)

Consequently, the relativistic inertia of the theoreticalquasiparticle is

mi =m

γr= m

(1− u2

n2r c2

)1/2

= ξc ·m(

1− u2

n2r c2

)−1/2.

(52)A general expression that incorporates Einstein’s spe-

cial relativity (see FIG. 3) is derived as follows:

ξc =ucc

=

(1− u2sw

n2r · c2

)⇒ mi = ξc ·mγr,

where usw is the travelling speed of the trans-lation mechanism in the quasiparticle structure.

For a particle in Einstein’s theory of special rel-ativity (see FIG. 3 and FIG. 4), we have∑

Find = 0 and∑

Fext ≥ 0, (53)

usw = 0⇒ ξc = 1⇒ nr = 1, (54)

uc = c⇒ 0 ≤ u < c,mi = m

(1− u2

c2

)−1/2, (55)

mic2 −mc2 = Uk, (56)

4

mc2(

1− u2

c2

)−1/2−mc2 = Uk, (57)

u = c

√1− 1(

1 + Uk

mc2

)2 . (58)

For the theoretical quasiparticle (see FIG. 3 andFIG. 4), we have∑

Find ≥ 0 and∑

Fext = 0, (59)

usw = u⇒ ξc ≤ 1⇒ nr ≥ 1, (60)

0 ≤ u ≤ nr · c and 0 ≤ uc ≤ c, (61)

mi = m

(1− u2

n2r · c2

)1/2

, (62)

mic2 −mc2 = −Uk/n

2r , (63)

mc2(

1− u2

n2r · c2

)1/2

−mc2 = −Uk/n2r , (64)

u = nr · c

√1−

(1− Uk

n2r ·mc2

)2

. (65)

The Lorentz transformations for particle motion in thex direction are

x′ = a1x+ a2t where a1 = γr and a2 = −γru, (66)

y′ = y and z′ = z, (67)

t′ = a3x+ a4t where a3 = − uc2γr and a4 = γr. (68)

For the Lorentz transformations to support the exis-tence of such a theoretical quasiparticle, the parametersa1 and a2 of Eq.(66) must be multiplied by ξc and those(a3 and a4) of Eq.(68) must be divided by ξc, leading to

ξc =ucc

=

(1− u2

n2r · c2

)= 1/γ2r , (69)

b1 = ξca1 = ξcγr and b2 = ξca2 = −ξcγru/nr, (70)

x′ = b1x+ b2t, (71)

y′ = y and z′ = z, (72)

t′ = b3x+ b4t, (73)

b3 =a3ξc

= − u

nrc2γrξc

and b4 =a4ξc

=γrξc. (74)

Consequently, the general Lorentz transformations formotion in the x direction can be written in a more com-pact form as

x′ = ξcγr(x− (u/nr)t), (75)

y′ = y and z′ = z, (76)

t′ =γrξc

(t− u

nrc2x

). (77)

The proper time is defined as the time measured in therest frame of the theoretical quasiparticle, thus

dx′ = 0⇒ dx = (u/nr)dt, (78)

dtτ = dt′ =γrξc

(1− u2

n2r c2

)dt =

dt

ξcγr= γrdt. (79)

Similarly, the proper length is

dt = 0⇒ dxτ = dx′ = ξcγrdx =dx

γr. (80)

The general Lorentz transformations [Eqs. (75) and(77)] can be verified by deriving the momentum and en-ergy using the proper time, hence

usw = u, (81)

p =dr

dt· dt

dtτ=mu

nr· ξcγr, (82)

p =1

nr

mu

γr=mu

nr

√1− u2

n2r · c2, (83)

mic2 = mc2 · dt

dtτ, (84)

mic2 = mc2 · ξcγr =

mc2

γr= mc2

√1− u2

n2r · c2. (85)

Note that the original expressions of Einstein andLorentz are recovered when there are no induced internalforces (no translation mechanism) in the system, thus∑

Find = 0 and∑

Fext ≥ 0, (86)

usw = 0⇒ ξc = 1⇒ nr = 1⇒ γr = γ. (87)

Conclusions. Newton’s laws of motion and Einstein’stheory of special relativity overlook the nature of the in-duced mechanical force and apparently fail to anticipatethe motion of an isolated system due to induced inter-nal forces. Besides the present results imply a paradigmshift in the way that motion is conducted, the new phe-nomenon whereby the effective inertia decreases whilethe speed of a system increases led to the discovery of awider framework—which predicts the existence of quasi-particles with group velocities that may reach and evensurpass the speed of light.

Acknowledgments

I thank my wife, who encouraged me to initiate andcomplete this work.

5

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