# The Mechanical Behavior of a Multispring System Revealing Absurdity in the Relativistic Force Transformation

Authors: Mohammad Javanshiry
Journal: International Journal of Mathematics and Mathematical Sciences
 (2021)

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## Abstract

The mechanical motion of a system consisting of simple springs is investigated from the viewpoint of two inertial observers with a relativistic relative velocity. It is shown that the final displacement of the springs is not measured the same by the observers. Indeed, it is demonstrated that there is an incompatibility between kinematics and dynamics in Einstein’s relativity regarding the force transformation.


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## Body

## 1. Introduction

This article represents an advanced version of the author’s spring paradox [1] in which it was shown that the final displacement of two relatively moving springs is measured differently from the standpoint of different observers as soon as the springs meet each other. Here, we try to make the possible effect of the signal delay due to the constancy of light speed of little or no consequence as a cornerstone in resolving the paradox.Similar to our previous works on the subject [1, 2], we insist here, too, that the relativistic dynamics are not easily reconcilable to the relativistic kinematics since there are fundamental deficiencies with the Lorentz transformation for force. Moreover, it is worthwhile to note that some other works show paradoxes of special relativity regarding rotating reference systems for only kinematic effects [3], which is related to the subject of this article.Although the analysis demonstrated in the article is based on the well-known dynamics of special relativity, other dynamics have been introduced in some references of the literature. For instance, it has been shown that different dynamics can be derived for the kinematics of special relativity [4], and thus, our multispring system paradox analysis can be performed under other dynamics too. It is interesting whether the shown paradox holds for all possible dynamics.In addition, the studies in [5, 6] develop new mathematical formalisms on special relativity, and hence, some theoretical research may investigate the application of these formalisms to the analysis of paradoxes such as the paradox discussed in this article.On the other hand, there are alternative theories for special relativity [7, 8], and the continuation of the research presented in this article may concern checking whether the multispring system paradox also applies to these theories.


## 2. The Multispring System Paradox

Too many very thin identical springs, each with a similar constant ofkP″, are attached at one end to the circumference of a thin solid cylindrical plate, all being perpendicular to the plane that passes through the plate, and in the other end, the springs touch the floor. Since these springs are fused to the thin plate, we denote them by P. Another spring S with a greater spring constant of KS′ attaches the center of the plate to the ceiling of the compartment in which the experiment is carried out (see Figure 1).Figure 1
A multispring system as viewed in the lab frame of reference. TheP springs are set in parallel to each other and in series with S. The rotating plate makes the P springs rotate along with it at u′.The distance between the floor and ceiling, as well as the free lengths of the springs, isd0′. On the other hand, it is assumed that the constants of the springs have the following relation:(1)kP″=KS′n′,wheren′ is the number of thin P springs. In that the thin springs are set in parallel to each other, their net constant is simply calculated to be(2)kP″¯=n′kP″=n′KS′n′=KS′.In other words, the net constant of theP springs is equal to that of S, and thus, it is anticipated that the upward forces of the P springs and the downward force of S are balanced, so that, from the viewpoint of the lab observer M, the thin plate remains motionless at a distance d0′/2 from the ceiling as well as the floor level (see Figure 2(a)). Now, suppose that the plate starts to rotate about its axis of symmetry (z′) along with the thin springs P fused to its perimeter. The surface of the floor is considered to be frictionless so that the other ends of the thin springs can easily slide over it, and the springs are not bent or deformed (see Figure 2(b)). If the tangential velocity u′ of the cylinder’s perimeter—to which the thin springs are attached—is a significant portion of light speed, the constant of each thin spring is reduced by the reciprocal of the Lorentz factor αu′ [1, 9]. Therefore, we can write (see Appendix A)(3)kP′=αu′kP″=αu′KS′n′,where kP′ is the reduced constant for each of the rotating thin springs measured by the lab observer M and αu′=1−u′2/c2. Moreover, it is worthwhile to mention that kP″ is the constant for each of the P springs measured either in its rest frame before the rotation, or in the frame momentarily at rest relative to the spring in the process of rotation. The net constant for the rotating springs is thus calculated as follows:(4)kP′¯=n′kP′=n′αu′KS′n′=αu′KS′.Figure 2
The multispring system as viewed in planex′z′. (a) Before the rotation of the plate, the plate is balanced at the midpoint between the ceiling and the floor since kP″¯=KS′. (b) In the process of rotation of the plate, the P springs are weakened (kP′¯<KS′), and thus, the plate finds its equilibrium state somewhere below the previous location. In this case, the final displacement of the P springs is supposed to be d0′−ΔzS′ where ΔzS′ is the displacement of the spring S. Recall that the middle spring of the rotating plate is shown thinner in size due to maximum speed and maximum Lorentz contraction along the x′-axis.
(a)(b)Indeed, the rotating springs are weakened due to the relativistic effects, and, as long as the springsP are assembled in series with S, the plate finds its equilibrium state at a distance smaller than d0′/2 from the floor. If we denote by ΔzS′ the final displacement of S, the final displacement of the thin springs would then be d0′−ΔzS′. When the upward force FP′ of the plate springs (P) equals the downward force FS′ of S, the forces are in equilibrium and we have(5)FS′=FP′⟶KS′ΔzS′=kP′¯d0′−ΔzS′.Substituting Equation (4) implies(6)ΔzS′=αu′1+αu′d0′.Now, we are interested in seeing if an observerN, who approaches the lab observer M at v along x′, would measure ΔzS the same as obtained in Equation (6) (ΔzS=ΔzS′); otherwise, relativity encounters a fatal paradox. Indeed, since the lengths in the transverse directions to the velocity v are left unchanged according to the relativistic kinematics, it is expected that the measurements made by MandN be the same regarding the final displacement of the springs.However, observerN asserts that each thin spring travels in a trochoid curve though the resultant velocity w of each P spring is always perpendicular to the spring’s alignment. Indeed, N observes that S approaches him at v, while each of the P springs, according to their angular position in the plate, approaches or recedes from him at w so that we can write(7)w2=wx2+wy2,where wxandwy are the components of the resultant velocity w both complying with the relativistic velocity addition formula. Therefore, if M measures the velocity u′ of a specific P spring at an angular position of θ′ to have the components of ux′=u′sinθ′anduy′=u′cosθ′ (see Figure 3(a)), the relativistic velocity addition suggests that N measures the corresponding velocities as follows [10] (see Figure 3(b)):(8)wx=ux′+v1+ux′v/c2,(9)wy=αvuy′1+ux′v/c2.Figure 3
(a) The angular position of theithP spring is shown as Pi on the plate being observed by the lab observer M in plane x′y′. Because the tangential velocity of the plate’s perimeter is u′, the velocity of Pi is decomposed to ux′anduy′. (b) The angular position of the spring as viewed by the moving observer N. The plate is Lorentz contracted due to its relative velocity of v. Indeed, N asserts that the velocity w of the ith spring has two components of wxandwy complying with the relativistic velocity addition formula.
(a)(b)Inserting Equations (8) and (9) into Equation (7), we obtain(10)w2=u′2+v2+2ux′v−v2uy′2/c21+ux′v/c22.Since all springs’ alignments are perpendicular to the velocities ofvandw, their constants would be reduced by their corresponding reciprocal Lorentz factor. In other words, the constant of S is reduced by αv and the constants of the P springs are decreased by αw as seen by N. Now, if, for simplicity, the number of the P springs n′ tends to infinity, observer N can easily use integration to calculate the resultant upward force of the P springs as follows:(11)dFP=dkPd0′−ΔzS=αwdkP″d0′−ΔzS,where dkP″ is the infinitesimally small constant of each of the infinite number of the P springs measured either in the spring’s rest frame before the rotation of the plate, or in the frame momentarily at rest with respect to the spring during the rotation. Moreover, dkP is the infinitesimally small constant of that specific P spring as measured by N, and, as stated earlier, ΔzS is the final displacement of S measured by N. Remember that the displacement of the P springs would then be d0′−ΔzS.On the other hand, the number of theP springs can be calculated by dividing the length of the plate’s perimeter by the infinitesimal width of each spring:(12)n′=2παu′r′r′dθ′=2παu′dθ′,where r′ is the radius of the plate measured by M. Recall that the plate’s perimeter is Lorentz contracted by αu′ during the rotation. Inserting Equation (12) into Equation (1), the differential form of each of the P springs constant is obtained:(13)dkP″=KS′2παu′dθ′.Substituting Equation (13) into Equation (11), we get(14)dFP=d0′−ΔzSαwKS′2παu′dθ′,for which the integration implies(15)FP=d0′−ΔzSKS′2παu′∫02παu′αwdθ′.Remember that it is rational to use2παu′ instead of 2π for the upper bound of the integrations over θ′ (see Appendix B). In that spring S has a velocity v from the viewpoint of N, its constant would reduce to KS=αvKS′. The corresponding spring force would thus be(16)FS=KSΔzS=αvKS′ΔzS.ObserverN claims that the upward force of FP should balance the downward force of FS in order for the plate to remain in a static situation. Using Equations (15) and (16), we have(17)FP=FS⟶d0′−ΔzSKS′2παu′∫02παu′αwdθ′=αvKS′ΔzS⟶ΔzS=1/2παu′∫02παu′αwdθ′αv+1/2παu′∫02παu′αwdθ′d0′,where αw=1−w2/c2. On the other hand, using Equation (10), αw can be simplified to(18)αw=αvαu′1+ux′v/c2.See Appendix C for the proof. Substitutingux′=u′sinθ′ together with Equation (18) into Equation (17) yields(19)ΔzS=1/2π∫02παu′1+u′vsinθ′/c2−1dθ′1+1/2π∫02παu′1+u′vsinθ′/c2−1dθ′d0′.As stated earlier,ΔzS which is measured by N must equal ΔzS′ measured by M; otherwise, relativity results in a paradox. Comparing Equation (19) with Equation (6), if ΔzS=ΔzS′, we indeed get(20)12π∫02παu′dθ′1+u′vsinθ′/c2=αu′.Unfortunately, the above formula is not always valid for all arbitrary values ofu′andv, and thus, it seems that relativity includes a null result, at least, in this example. To prove, it suffices to substitute v=0.6candu′=0.8c and do the calculations numerically. In this case, the left-hand side of Equation (20) equals 0.506, whereas the right-hand side equals 0.600, which are not equal to each other. This counterexample shows a deficit in special relativity. However, one also can take the integral analytically to show that Equation (20) is not valid for all arbitrary values of u′andv.An important point with this problem is that if the forces are transmitted via some sort of signaling from theP springs towards the center of rotation of the plate to which one end of S is attached, the arrival of the signals to the center is simultaneous from the viewpoint of both MandN. This simultaneity makes the spring S react to all of the signals sent by the P springs instantly as viewed by both of the observers; otherwise, it is expected that the plate is deformed in shape due to the signal delays.


## 3. Important Notes regarding This Paradox

To reduce the reader’s confusion, we gather some important remarks concerning this problem and the possible resolutions:(1)
Remember that this problem is not connected to some aspects of the Ehrenfest paradox [11] according to which a fast-rotating disc cannot approach the light speed since the centrifugal pressure exceeds the shear modulus of the material of which the plate is made. In our problem, indeed, it is not necessary for the tangential velocity u′ of the plate to have a value close to c in order to encounter a paradox. That is, if u′ is much smaller than the speed of light, the paradox is still valid though the difference in ΔzandΔz′ is very small.(2)
The centrifugal force exerted on theP springs due to the rotation of the plate may bend the springs slightly outward the center of rotation; however, this phenomenon can be neglected by assuming that the said force is not great, or the springs’ stiffness is great so that they are not easily bent out of shape.(3)
The uniform distribution of theP springs as seen by M is no longer uniform from the viewpoint of N (see Figure 4). Remember that this phenomenon has already been discussed in the literature [12]; however, it is unlikely that this nonuniformity can cause the plate’s normal to incline relative to the z direction as viewed by N. In fact, one can claim that the resultant force of the upper springs would certainly balance that of the lower ones measured by M (see Figure 4(a)) such that the plate remains parallel to the floor level, whereas N may claim that, due to the nonuniform distribution of the P springs, the mentioned forces are not balanced, which makes the plate become oblique (see Figure 4(b)). Recall that if the plate is inclined in the rest frame of N, in that it is not measured inclined in that of M, this can bring about another paradox besides the main paradox discussed earlier. However, the author guesses that the increase in the density of the springs takes place for those having greater tangential speeds from N’s point of view, and thus, the spring constants would have smaller values. On the other hand, the lesser the springs' density, the slower they move and the greater their constants. Therefore, it is possible that the increase in springs' density compensates for the decrease in their constants, and vice versa, so that the upper and lower resultant forces would finally balance each other, which prohibits the plate from additional rotation. It is also possible that this inclination is somehow related to the disputatious arguments about Mansuripur’s article where a similar nonuniformity in the distribution of some point-like electrical charges causes the moving observer to detect a possible torque on a current-carrying loop of wire, whereas the lab observer does not, due to the uniform distribution of the charges [13–16]. A comprehensive discussion is beyond the scope of this article.(4)
It is not mandatory to consider an infinite number of theP springs. One can repeat the calculations using any finite number of springs.(5)
Instead of involving the viewpoint of the observer in the rest frame of theP springs, one can directly apply the Lorentz transformation for force between MandN to finally reach Equation (20) (see Appendix D).(6)
Remember that this article does not question the relativistic version of Hook’s law, but rather the relativistic transformation for force in its general form. Hence, one can replace the springs with electromagnetic fields and electrical charges in a way similar to [2] in order to rewrite the paradox (see Appendix D).Figure 4
(a) The distribution of theP springs looks uniform from the viewpoint of the lab observer M. (b) The distribution is no longer uniform as viewed by the moving observer N. The density of the P springs' distribution, as well as the mass density of the plate, is higher where the tangential velocity of the plate (w) is greater, and vice versa.
(a)(b)


## 4. Conclusion

As a complementary to the author’s previous works regarding the relativistic force transformation, this article shows, too, an inconsistency between the kinematics and dynamics in relativity.



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