Analyses of Scissors Cutting Paper at Superluminal Speeds
AAnalyses of Scissors Cutting Paper at Superluminal Speeds
Neerav Kaushal a) and Robert J. Nemiroff b) Department of PhysicsMichigan Technological University1400 Townsend Drive, Houghton, MI 49931 (Dated: 24 September 2019)
A popular physics legend holds that scissors can cut paper with a speed faster than light. Herethis counter-intuitive myth is investigated theoretically using four simple examples of scissors.For simplicity, all cases will involve a static lower scissors blade that remains horizontal justunder the paper. In the first case, the upper blade will be considered perfectly rigid as itrotates around and through the paper, while in the second case, a rigid upper blade willdrop down to cut the paper like a guillotine. In the third case, the paper is cut with a laserrotating with a constant angular speed that is pointed initially perpendicular to the paperat the closest point, while in the fourth case, the uniformly rotating laser is pointed initiallyparallel to the paper. Although details can be surprising and occasionally complex, all casesallow sections of the paper to be cut faster than light without violating special relativity.Therefore, the popular legend is confirmed, in theory, to be true.
I. INTRODUCTION
A common expression is that nothing can travel fasterthan light – but this is not strictly true. Shadows, for ex-ample, are a well known counter example . Conversely,it is common lore in physics that the vertex of a pairof scissors can exceed c but no comprehensive pedagog-ical account of this exists. In this work we will explorethe truth of this physics legend by analyzing four specificexamples.Special relativity states that any object possessing afinite mass cannot be accelerated from below the speedof light to above, because then its momentum and energywould diverge . It is also well documented that locallycreated information cannot be transmitted faster thanlight. Light, neutrinos, relativistic particles, and gravi-tational waves; are all limited to travel in vacuum at, orvery near, c .In this paper, it is shown that the cutting vertex ofscissors can, in theory, move faster than light, relativeto the inertial frame of the paper. In all of the exam-ple cases considered here, the upper edge of the lowerscissors blade will be considered to lie just under the pa-per and directly along the x -axis, where y is always zero.This axis extends along the paper and also along the lineon which the paper will be cut. The thickness of bothscissors blades will not be considered important. There-fore, the cases considered are actually variations of thegeometry and movement of the upper blade.Sections II and III will define and theoretically analyzefour example cases. In the first two cases, the upper bladewill be considered a rigid material. In Section II.A theupper blade will be considered to rotate about a pivot,while in Section II.B it will be considered to drop likea guillotine. In the second two cases, the upper scissorsblade will be considered to be a uniformly rotating laser.In the third case analyzed in Section III.A, this laser will a) Electronic mail: [email protected] b) Electronic mail: nemiroff@mtu.edu point initially at a direction perpendicular to the paperit cuts, while in the fourth case, in Section III.B, thelaser will point initially parallel to the paper. Section IVsummarizes the findings and gives some discussion.
II. RIGID BLADE SCISSORSA. Rotating Upper Blade
The case of a rigidly rotating upper scissors blade willbe considered first. The upper blade starts by spanningthe y -axis while being able to pivot at the position of itshinge, below the x -axis, at ( x, y ) = (0 , − L ). Here L iscalled the “hinge distance”. The blade rotates clockwiseabout the hinge starting at t = 0. The upper scissorsblade will be considered to close with a constant angu-lar velocity ω on the fixed lower blade. Therefore, thepoint of intersection of the two blades, the vertex, be-ing confined to the x -axis, moves from the origin towardsincreasingly positive x at speed v . FIG. 1. (Case II A) A scissors is depicted with hinge distance L and rigid upper blade rotating about the hinge with con-stant angular velocity ω . The angle the upper blade makeswith the x -axis is θ , while the distance to the vertex – theplace where the paper is being cut – is designated x . a r X i v : . [ phy s i c s . pop - ph ] S e p The upper edge of the lower blade and the lower edgeof the upper blade are the only edges considered hereto be important. Each edge is described by a line, andthe vertex is the point where these lines intersect. Thescissors is assumed to be entirely rigid so that when theupper blade closes with constant angular velocity ω onthe lower blade, all of it translates at once, without flex-ing. The geometry is depicted in Figure 1.Simple Euclidean geometry shows that x = L cot θ. (1)The velocity v with which the vertex advances in thepositive x direction is given by v = ωL sin − θ. (2)The vertex velocity varies with the angle between theblades as shown in Figure 2 . FIG. 2. (Case II A) The velocity at which the vertex advancesincreases with decreasing angle between the blades. The angleat which this velocity exceeds the speed of light c depends onthe hinge distance of the scissors and the speed of rotation ω of the upper blade. The vertex moves towards increasingly higher values of x with increasing speed. After a finite time, the scissorswill close completely, meaning that the upper scissorsblade will become parallel to the lower blade. This willoccur at time t close = ( π/ /ω . Considering both scissorsblades to be infinitely long, the vertex will cross the entire infinite length of the positive x -axis in this finite time.Clearly, to accomplish this, it must approach an infinitespeed. Therefore, the vertex speed will eventually exceedthe speed of light.The value of θ where the vertex speed v exceeds c willbe called θ c and can be found by setting Eq. (2) equalto c . Straightforward algebra shows that θ c = arcsin (cid:114) ωLc . (3)Similarly, the value of x where the vertex speed v exceeds c will be called x c and can be found by integrating Eq. (2). The result is x c = (cid:114) Lω ( c − ωL ) . (4)Last, the time t c when the vertex exceeds c can be simplyfound from Eq. (3) as t c = ( π − θ c ) ω . (5)Clearly, higher angular speeds ω create proportionallylower c -crossing times. This type of scissors can cut apiece of paper superluminally even with finite values ofangular velocity.The strict rigidity of the rotating scissors, however,makes this case unphysical in theory but there are possi-ble ways to get around this which are discussed in detailin the last section. B. Falling Guillotine
Another form of scissors is a guillotine – a falling rigidupper scissors blade that makes a constant angle θ withthe x -axis. The guillotine does not rotate – it moves downat a constant “blade speed” u in the negative y direction.For didactic purposes, the guillotine is considered to beinfinitely long, and only the lower edge of the guillotineblade is important. This edge will be considered perfectlystraight . As before, the paper to be cut is laid across the x -axis. The vertex in this case moves along the positive x -axis with velocity v as shown in Figure 3. FIG. 3. (Case II B) A guillotine-type scissors blade movesdownwards so that its vertex moves along the x axis. The velocity v of vertex of the inclined surface of guil-lotine and the paper is given by v = u cot θ. (6)For a given downward speed u , the critical guillotine an-gle below which the blade cuts the paper faster than lightis θ c = cot − ( c/u ). Slower guillotine speeds need a shal-lower angle to cut the paper faster than light.If the guillotine speed u is constant, then the speed ofthe vertex v is also constant. Therefore, this guillotinescissors either always cuts the paper superluminally, orit never does.The variation of the vertex velocity with the guillotineangle for three different blade velocities is shown in Fig-ure 4. Note that the speed of the vertex varies linearlywith the guillotine angle. For example, for a guillotinedropping with a speed of u = 1 m s − , the vertex reaches c at an angle of 1 . × − degrees with respect to thehorizontal. It is not feasible in practice to create a guillo-tine with such a small angle because atomic scales – toofine to control – become involved. FIG. 4. (Case II B) A plot of vertex velocity versus the guil-lotine angle for different blade speeds.
III. ROTATING LASER SCISSORSA. Starting Orientation: Beam perpendicular to the paper
A different type of scissors is now considered: a laserscissors. Cutting paper with a laser is now relativelycommon in industry as lasers are inexpensive and can bepointed with high accuracy . Furthermore, in terms ofthe physics, laser beams do not have rigidity concernsthat regular material scissors have shown in the previousanalyses, yet the beam has well defined characteristicsthat can be analyzed. For didactic purposes, this laserbeam will be considered to be perfectly collimated – whenstationary, the laser emits a thin stream of photons lin-early out from the its base. The scissors vertex will beconsidered as the intersection point of the laser beamwith the paper. It will be assumed that whenever laserlight touches the paper, it cuts the paper, even thoughthis may be impractical when far from the laser source.As before, the paper will be considered laying flat in the x − z plane with the laser constrained to cut the paperalong the positive x axis.Consider now a specific laser emanating light from( x, y ) = (0 , − L ) towards the positive y -axis. For simplic-ity, the laser is considered hinged at the point of lightemanation, and rotated about this hinge clockwise atconstant angular velocity ω . Therefore, the laser cuts the paper such that the vertex moves away from the ori-gin toward positive x values, as shown in Figure 5. FIG. 5. (Case III A) A laser source at (0 , − L ). At t = 0, thelaser is switched on, and given an angular velocity ω so thatthe vertex moves along + x -axis. At t = 0, the laser is switched on so that the first laserphoton travels towards the origin. Also at t = 0, thelaser begins pivoting about the hinge at angular velocity ω . There will be a single spot of light cutting the paperalong the x -axis while the laser is being rotated from 0 ◦ through 90 ◦ with respect to its initial orientation. Thetime it takes for the laser to rotate through an angle θ isgiven by t rotate = θω . (7)Once the laser has turned to angle θ , the photons ema-nating from it will take a finite time to traverse the spacebetween the laser and the paper, ultimately reaching thepaper at angle θ on the x -axis. This time is given by t traverse = Lc cos θ . (8)The total time from t = 0 to when a laser photon touchesand cuts the paper at angle θ is given by the sum of thesetwo times such that t total = t rotate + t traverse = θω + Lc cos θ . (9)When the laser finally points at 90 ◦ , the beam takes in-finite time to reach infinitely far down paper.Following Nemiroff , the speed of this real beam spotor vertex along the surface of paper, v is given by v = L cos θ dθdt total = ω c Lc cos θ + ω L sin θ . (10)A plot of v versus θ is shown in Figure 6. At the origin,the initial vertex speed is v ( initial ) = ω L . Note thatthis speed is not constrained to be slower than light: ifeither ω or L is fixed, the other variable can be increasedso that v ( initial ) > c .The behavior of v is a surprisingly complicated func-tion, dropping initially as t traverse increases relatively FIG. 6. (Case III A) Velocity of vertex v vs the angle θ withrespect to the vertical rapidly for small θ . The fastest v will occur at the originunless v ( initial ) < c . If v ( initial ) is sufficiently large,then v will drop below c before recovering at large x .Regardless of v ( initial ) and its low θ behavior, the scis-sors vertex speed v will always asymptote to c as x goesto infinity, far down the paper because there the laserphotons travel nearly parallel to the paper. B. Starting Orientation: Beam parallel to the paper
Now consider a laser emanating light from ( x, y ) =(0 , L ) in the positive x direction, parallel to the x axis. Asbefore the laser is hinged at the point of light emanationand the paper is laid across the x axis in the x − z plane.Now, however, the laser will not start cutting the paperat the origin, but at a point down the positive x axis.The initial geometry is shown in Figure 7. FIG. 7. (Case III B) A laser source residing at (0 , + L ), at t = 0 is given an angular velocity ω . If the previously analyzed cases of scissors cutting pa-per were more surprising and complex than one might naively expect, then this case is even more surprising,and perhaps even bizarre! Here, the total time for thebeam to illuminate and hence cut through a point at an-gle θ is given by the sum of the time it takes for the laserto rotate to angle θ and the time it takes for light to gofrom the laser to the point on x -axis at angle θ , suchthat t total = t rotate + t traverse = θ + π/ ω + Lc cos θ . (11)The beam starts at t = 0 by pointing along θ = − π/ x -axis, and then rotates at con-stant angular speed ω so that θ increases through zeroand ends when θ = + π/
2, where it points along the neg-ative x -axis. When the laser is pointing towards the neg-ative θ , then sin θ will be negative. The speed of thevertex along the surface of paper or the speed at whichpaper is cut, v is given by v = L cos θ dθdt total = ω c Lc cos θ + ω L sin θ . (12)When v is positive, the location where the laser beamhits the x -axis – the vertex – moves in the direction ofpositive x , while a negative v indicates vertex motion inthe negative x direction. A plot of | vc | vs t total is shownin Figure 8 FIG. 8. (Case III B) Variation of | vc | vs t total of the laserscissors vertex. From the complexity of the graph, it is clear that thiscase is conceptually distinct. Perhaps the first surpriseis the first place the paper is cut. Although the laserpoints first at a location infinitely far down the x axis,the first point on the paper which is cut is not ∞ becauseit will take an infinite amount of time for laser photonsto reach that far. The angle where the paper is first cutis determined by the θ where dt total /dθ = 0. This createsa hole in the paper with a location found fromsin θ cos θ = − cL ω . (13)The solution is θ fp = arcsin (cid:32) Lw c − (cid:114) L w c + 1 (cid:33) . (14)The plot of | vc | vs angle θ of the laser beam with respectto the vertical is shown in Figure 9 . FIG. 9. (Case III B) Variation of | vc | vs angle θ of the laserscissors vertex. After the first time the paper is cut, there are two locations on the paper that are associated with eachone t total time in Eq. (11). These two locations depictthe paper being cut in two directions at once – bothtoward the origin at x = 0, and also toward x = ∞ .Two different vertices move at once! All of the photonscome from the laser, but starting from the initial hole inthe paper, those photons that cut the paper toward theorigin are emitted after ( t rotate is greater) the photonsthat cut the paper toward infinity. Furthermore, the setof photons cutting the paper towards the origin have ashorter travel time ( t traverse is lesser) – from laser topaper – than the other set of photons cutting towardinfinity. This type of superluminal pair splitting has beenverified experimentally in another context .The vertex cut that moves toward the origin startswith formally infinite speed but then slows. Whether itsspeed drops below c before reaching the origin dependson ω and L . The vertex cut that moves away from theorigin also starts cutting the paper with formally infinitespeed, and then it also slows down . The speed of thisvertex, though, always approaches c as it moves towardinfinite x , and is always greater than c . IV. SUMMARY AND DISCUSSION
It has been shown that the popular physics legend ofscissors being able to cut paper faster than light is ac-tually true, in theory, for several simple cases. A scis-sors vertex moving superluminally does not violate spe-cial relativity because it carries no energy or momentum.Therefore, the rip in the paper could start or stop at any time with no change in the energy or momentum to thescissors blades or the paper, and neither of these motionalparameters would diverge as the rip approached c .In the first case, it was shown that when the rotationspeed ω of the upper scissors blade is constant along itsinfinite length, then for it to close in a finite time, thespeed of the vertex between the two scissors blades mustapproach infinity – which exceeds c . This case is unphys-ical for rigid scissors, however, if one assumes that theinformation that the upper blade is rotating moves outfrom the hinge. This is because this information – andthe force it carries – can only move along the upper bladeat speed c at the most. It will therefore not be possiblefor the upper blade to remain rigid – it must either flexor break .However, it is possible that the upper blade’s mo-tion does not only originate from a single torque appliedaround the hinge. It could be that the upper blade’sangular motion derives from forces distributed along itsentire length. If so, then the previous objection will notapply. However, even if the upper blade’s rotation arisesfrom a distributed force, it will still not be possible forany part of it to move faster than c relative to the paperand the hinge. For a rigidly rotating upper blade, at somedistance d > c/ω from the hinge, the blade past d mustmove faster than light to remain rigid, which is unphysi-cal. However, if d < x c , as defined in Eq. (4), then it ispossible for a rigid scissors to cut paper superluminally.In the second case of guillotine scissors, when the anglebetween the blades was small enough, and the downwardspeed was high enough, the vertex can move – and thepaper can be cut – superluminally. However, in this case,the guillotine angle at which paper is cut superluminallyis actually so small that atomic vibrations and materialimperfections in the guillotine blade do not allow thiscase to be realized practically.The third and fourth cases involve cutting the paperwith a laser. In the third case, the initial vertex speedscales linearly with the angular speed with which thelaser turns, as well as the distance between the swivelpoint and the paper. Therefore, since these parametersare not limited, the vertex speed can be arbitrarily fast –even exceeding c . Regardless of the initial speed, it wasshown that the vertex speed will approach c infinitely fardown the paper.The last case gives the most interesting results. Here,a hole first forms in the paper at the point of the ini-tial laser contact. From this initial hole, the paper ripsoutward in two directions simultaneously. Each vertexmoves superluminally to start, although the speed of cut-ting may drop to subluminal. A realistic caveat is thatfar down the paper the laser photons will become toosparse to cut the paper continuously.In sum, it is of educational interest that such a com-mon object as a scissors can, in theory, display suchan uncommon attribute as superluminal motion. Thiscounter-intuitive behavior does not violate special rel-ativity and is derivable from straightforward kinemat-ics and Euclidean geometry prevalent in undergraduatephysics curricula. ACKNOWLEDGMENTS
The authors thank Lucas Simonson for initial com-ments, and Jacek Borysow and an anonymous referee fora critical reading of the manuscript.
REFERENCES D. J. Griffiths,
Introduction to quantum mechanics / DavidJ. Griffiths. Englewood Cliffs, N.J.: Prentice Hall, c1995. (1995). R. J. Nemiroff, Q. Zhong, and E. Lilleskov, Physics Education , 043005 (2016), arXiv:1506.02643 [physics.pop-ph]. S. I. Chase, “The Superluminal Scissors (Updated),” http://math.ucr.edu/home/baez/physics/Relativity/SR/scissors.html (1993). “The scissor paradox: can we pass the information fasterthan light?” https://physics.stackexchange.com/questions/106718 (2014). R. J. Nemiroff, “Extraordinary Concepts in Physics,” http://asterisk.apod.com/viewtopic.php?f=39&t=21005 (2010). “Dr. SkySkull and the mystery of the subluminal superluminallight!” https://skullsinthestars.com/2015/02/17 (2017). A. Einstein, Annalen der Physik , 639 (1905). “In the NEC ’faster than light’ experiment, did they re-ally make something go faster than light?” (2010). “OR Laser: Coherent Technologies: Laser Cutting Pa-per,” (2019). R. J. Nemiroff, Publications of the Astronomical Society of Aus-tralia , e001 (2015). M. Clerici, G. C. Spalding, R. Warburton, A. Lyons, C. An-iculaesei, J. M. Richards, J. Leach, R. Henderson, andD. Faccio, Science Advances (2016), 10.1126/sciadv.1501691,https://advances.sciencemag.org/content/2/4/e1501691.full.pdf. D. Faccio and A. Velten, Reports on Progress in Physics81