A look at Einstein s clocks synchronization

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1 A look at Einstein s clocks synchronization ilton Penha Departamento e Física, Universiae Feeral e Minas Gerais, Brasil. nilton.penha@gmail.com Bernhar Rothenstein Politehnica University of Timisoara, Physics Department, Timisoara, Romania. brothenstein@gmail.com Abstract While Einstein s clocks synchronization process is performe, one has a well efine region in which the clocks are synchronize an another one in which the clocks are not yet synchronize. The frontier between them evolves ifferently from the perspective of observers in relative motion. A iscussion is conucte upon irect observation of the phenomenon an Minkowski iagrams. Introuction Special theory of relativity relies upon two postulates, state by Einstein in his famous 1905 paper 1. The first is the so calle principle of relativity which asserts that the laws of physics hol the same in every inertial reference frame. This leas to the important outcome that no experiment in one inertial frame can istinguish it, intrinsically, from any other. The secon postulate asserts that light is always propagate in empty space with a efinite velocity c which is inepenent of the state of motion of the emitting boy. Einstein s strongest justification for this postulate came from Maxwell s electroynamics. That theory ha ientifie light with waves propagating in an electromagnetic fiel an conclue that just one spee was possible for them in empty space, c = x 10 8 m/s, the upate value, no matter the motion of the emitter. An event is any physical occurrence taking place at a given point in space at a given instant of time. To establish the coorinates of an event one shoul choose a reference frame. Let K be a reference frame accoring to which the spatial position of an arbitrary event is given by cartesian coorinates (x,y), in the case of a twoimensional space, for example. The instant of time ct at which the event happens is the reaing of a clock place exactly at its spatial position (x,y). So (x,y,ct) are the coorinates of the referre event accoring to the reference frame K. Similarly, if the reference frame is K, the coorinates of the same arbitrary event are (x,y,ct ) where ct is the reaing of a clock place at spatial position (x,y ) in K. Notice that we treat time as ct instea of t. This has the avantage of improving the transparence of the symmetry that exists between space an time in special relativity. We can see this clearly in the Lorentz transformations equations. It worth mention the famous assertive 2 from Herman Minkowski: Henceforth space by itself, an time by itself, are oome to fae away into mere shaows, an only a kin of union of the two will preserve an inepenent reality. Formally, one can represent a given event as E(x,y,ct) or E (x,y,ct ) epening on it is escribe in a K or K frame. The totality of possible events constitutes what is calle a spacetime an x,y,ct an x,y,ct are spacetime coorinates accoring to two ifferent reference frames. One set of spacetime coorinates can be mappe into the other through the so calle Lorentz transformations. Suppose that you, the reaer, are in the inertial frame K, at its spatial origin (x,y,ct) = (0,0,ct) an some frien of yours is in K, also at its origin (x,y,ct ) = (0,0,ct ) which moves away with constant spee V, along x- axis. The x -axis an y -axis are assume to have the same irection as the x-axis an y-axis, respectively. You may consier yourself that the inertial frame K where you are in is stationary while your frien is in a nonstationary frame K. Since K an K are both inertial frames, your frien may also, by himself (herself), consier that he (she) is in a stationary frame K an, juge that you are in a non-stationary inertial frame K. 1

2 In every inertial frame it is highly convenient to imagine a stanar lattice of stationary observers as small as they can be, each one with a stanar clock, being all the clocks alike an stationary. You are one of those stationary observers, the one who sits at origin of K. It is common, among physicists, to refer figuratively to such clocks as wristwatches. We will follow suit here. It is also convenient to have all the wristwatches synchronize. To achieve such synchronization, one can use a proceure propose by Einstein which goes as follows. Consier first the K frame. Initially all the wristwatches are stoppe. An then one shoul choose a master wristwatch, usually the one at the spatial origin, the place where you are. Your wristwatch shoul be set to start running at an arbitrary time ct o an all the others set to start running at time ct = ct 0 + r where r is the spatial istance from you. Then a light source previously place at your place emits a pulse when it starts running at time ct 0. The pulse propagates through the lattice an triggers off each one of the wristwatches which start reaing ct = ct o + r as previously settle. Once all the wristwatches are running, they are all synchronize. But, while this process is not finishe, all the lattice wristwatches, incluing yours, insie a circle of raius 2 2 r = x + y = ct (1) are synchronize an those outsie the circle are not synchronize yet; actually they are all stoppe, by construction, waiting for the light pulse to reach them. So one has a circular frontier between the synchronize an unsynchronize wristwatches regions; such frontier propagates outwars, with constant spee c, from the spatial origin. Consier now the K frame. All the wristwatches in K shoul be initially stoppe. A master wristwatch shoul usually be chosen as the one at the spatial origin of K, which means your frien s wristwach. It shoul be set to start running at an arbitrary time ct o an all the others set to start running at time ct = ct 0 + r where r is the spatial istance from your frien s wristwatch. Then a light source previously place at the master wristwatch lattice site emits a pulse at time ct 0. The pulse propagates through the lattice an triggers off each one of the wristwatch which start reaing ct = ct o + r as previously settle. Once all the wristwatches are running they are all synchronize. But, while this process is not finishe, all the lattice wristwatches, incluing that of your frien, insie a circle of raius 2 2 r ' = x' + y' = ct', (2) are synchronize an those outsie the circle are not synchronize yet; actually they are all stoppe, by construction, waiting for the light pulse to reach them. So one has a circular frontier between the synchronize an unsynchronize wristwatches regions; such frontier propagates outwars, with constant spee c, from the spatial origin. Important: with no loss of generality one can assume that ct 0 = ct 0 = 0 an that both origins O an O are coincient, i.e., (x,y,ct) = (0,0,0) = (x,y,ct ) when a unique light source emits a pulse which propagates through K an K synchronizing all the wristwatches. Accoring to the secon postulate it oes not matter whether the light source is at rest in K or at rest in K. Such event may be represente by E O (0,0,0) an E O (0,0,0) accoring to K an K frames. Light clocks A light clock is a evice mae of two parallel mirrors, say, M O an M, just in front of each other separate by a fixe istance. A light source place at one of the mirrors emits a pulse which shoul be bouncing between them. It is common to refer to a clock by mentioning its tick-tacks. Let the tick be moment at which the light pulse is at the position of M O mirror an let the tack be the moment at which the light pulse is at 2

3 the position of the mirror M. The elapse time between two consecutive ticks in the same given light clock shoul be a characteristic of such light clock. Such perio is what one usually calls a proper time. Let us represent such proper time as 2cτ = 2. Since the spee of light oes not epen on the irection, cτ is the elapse proper time for both up an own propagation of the light ray insie the light clock. Consier a light clock at rest in the inertial reference frame K. Let us call it K light clock. Let M O be place at the origin O such that its plane is perpenicular to y-axis an M place at (x,y,ct) = (0,,ct) parallel to the first. You, along with your wristwatch, are at the position of the M O mirror. Consier also another of the same light clock, at rest in the inertial reference frame K. Let us call it K light clock. Let the mirror M O be place at the origin O such that its plane is perpenicular to y -axis an M place at (x,y,ct ) = (0,,ct ) parallel to the first. Your frien, along with his (her) wristwatch, are at the position of the M O mirror. The proper perio in this case is 2cτ = 2. Assuming that both light clocks are alike,, the mirrors spatial separation while the K clock is at rest, shoul be taken equal to, the same separation, the mirrors separation while the K clock is at rest, an consequently, they both have a same proper time (cτ = cτ ). In particular we shoul be intereste in comparing the elapse time between two given events as seen by you at rest in K an by your frien at rest in K. Again notice that the clocks are mae alike. They are mae to measure the time in the same way. In fact, if they were sie by sie at rest with respect to each other one woul not etect ifference in their measurements. The viewpoints of K an K Let us return to above synchronization proceure an let O an O be the master wristwatches of K an K locate at their respective spatial origins O an O. They are respectively yours an your frien s wristwatches. Let us consier the moment at which the synchronization proceure starts for both inertial frames an consier your point of view. For our purposes here let the mirrors M an M to be half-silvere. See Figure 1. Light is emitte at the common spatial origins O an O, where there are the mirrors M O an M O when wristwatches at such spatial position rea ct 0 = 0 = ct 0 by efinition. This correspons to a K an K light clocks tick. Figure 1 Your point of view in K. While K is assume stationary, K is moving right with constant spee V = β c at time ct 0 = 0 = ct 0 when a light pulse is emitte at the common origins O an O. All wrist watches in both frames are stoppe. See Figure 2. Light reaches position A where there are a half-silvere mirror M an a wristwatch A, which just reas ct A = cτ. Let such event be represente by E A (x A,y A,ct A ) = E A (0, cτ ). This correspons to a K 3

4 light clock tack. All wristwatches at rest in K which are insie a circle of raius r A = cτ show the same time reaing cτ an those outsie the circle are still stoppe (iling). Figure 2 Your point of view wavefront reaches the halfsilvere mirror M an wrist watch A, both at rest in K. All wristwatches at rest in K which are insie a circle of raius r A = cτ show the same time reaing cτ. All wristwatches outsie the circle in K are still stoppe. Time ilation See Figure 3. Light emitte at O (an O ) keeps expaning its wavefront an reaches the half-silvere mirror M at which position there is a wristwatch A, both at rest in K, an the wristwatch B, at rest in K. Such event may be represente by E A (x A,y A,ct A ) = E A (0,,cτ ), in K, an by E B (x B,y B,ct B ) =E B (βct B,,ct B ), in K. The wristwatches A an B, they momentarily face each other; while B reas ct B > cτ, A just reas ct A = cτ = cτ. This correspons to a K light clock tack. Also, light previously reflecte at the mirror M is going on its way back the origin O, the place where you are still. All wristwatches at rest in K which are insie a circle of raius r B = ct B show the same time reaing ct B an those outsie the circle are still stoppe. Figure 3 Your point of view wavefront reaches the halfsilvere mirror M an wristwatch A, both at rest in K, an wristwatch B,which is at rest in K; B an A are just facing each other at the moment the light pulse strikes them. All wristwatches at rest in K which are insie a circle of raius r B = ct B show the same time reaing ct B. All wrist watches outsie the circle in K are still stoppe. A simple calculation, base on Pithagoras Theorem, leas to 4

5 ct B = γ cτ (3) where 1 γ =. (4) 2 1 β Since cτ = cτ (by construction), we have ct B = γ cτ (5) ' ' This means that the elapse time between same two events which is measure by your frien in K seems larger to you in K. To you the elapse time is larger by the γ factor. Although the light clocks are mae alike they show ifferent reaings when they are moving with respect to the observer. This is what one calls time ilation: ct= γ ct' (6) Length contraction At this point we assume the existence of a rigi ro, extene along the x -axis, at rest in K, such that the left en is at O (0,0,ct ) an the right one is at C (x C,0,ct ). Let its length be (x C - 0) = as measure by your frien. He (she) actually can measure it by using a stanar meter. So the still ro has a proper length equal to L' = cτ ' (7) ' Figure 4 Your point of view, propagating irectly from O (O ) reaches the right en of the ro. Accoring to you light spens a time equal to (1+β)γcτ to perform the way to the right en of the ro. All wristwatches at rest in K which are insie a circle of raius r C = (1+β)γcτ show the same time reaing (1+β)γcτ. All wristwatches outsie the circle in K are still stoppe. See Figure 4. Part of light reflecte at mirror M follows its way to the mirror M O at the spatial origin O of K while this one moves along x-axis with imensionless spee β. Also the light ray propagating along x- axis, from O(0,0,0) (an O (0,0,0)) reaches the right en of ro (that we assume to exist at rest in K ) which has a proper length L = cτ. At the moment light reaches the ro right en, a wristwatch C, at rest in K, show the reaing ct C, ct C = ct + β ct = γ cτ + β γ cτ (8) B B 5

6 while C, at rest in K, which is momentarily just face to face with C, shows ct C =(1+β)cτ. To you, the observer at rest in K, light runs through a istance, along x-axis, just equal to x C x = ( 1+ β )γ cτ (9) O before reaching the right en of the ro at (x C,y C,ct)=((1+β)γ cτ, 0,(1+β)γ cτ ). Let us refer to such event as E C (x C,y C,ct C )= E C ((1+β)γcτ, 0,(1+β)γcτ ) when escribe in K an as E C (x C,y C,ct C )= E C ((1+β) cτ, 0,(1+β) cτ ) when escribe in K. While light propagates from O irectly to C, the left en of the ro moves to (β(1+β)γ cτ, 0, (1+β) γ cτ ) along the x-axis. The spatial istance between these two points is the length that the ro appears to have to you: L= ct β ct = ( 1+ β ) γ cτ β (1+ β ) γ cτ (10) C C 2 2 ( 1 ) c = (1 ) c ' ' L== β γ τ β τ (11) 1 L = L'. (12) γ The ro looks shorter if it belongs to moving inertial frames. This is the so calle relativistic length contraction. Figure 5 Your point of view, reflecte by mirror M reaches back your position. This correspons to a tack in your reference frame K an closes a cycle. All wristwatches outsie the circle in K are still stoppe. To unerstan the viewpoint of your frien you just have to use -β instea of β, an change prime for unprime variable. You will see that he (she) also thinks that your clock works slower (you are moving an he (she) is at rest); also if you have a ro at rest in K it will seem shorter to him (her). The situation is symmetric. Lorentz Transformations From what is iscusse above it is possible to infer the so-calle Lorentz Transformations. From (12) one can write 6

7 1 ( x ) = ( ' C β ctc x C ' 0) (13) γ Although the proper length of above rigi ro was assume to be cτ it actually coul have any length an the conclusion about the length contraction woul have been the same. So the position C mentione above is arbitrary. Then expression (13) can be put in the following way: x' = γ ( x β ct). (14) This gives the spacetime coorinate x in terms of spacetime coorinates x an ct. If one change β for -β an prime for unprime coorinates one gets x= γ ( x' + β ct' ). (15) Now if we insert (14) into (15) one has x= γ ( γ ( x β ct) + β ct' ), (16) 2 β γ ct' = (1 γ 2 ) x+ βγ ct, (17) ct' = γ ( ct β x). (18) This expression gives spacetime coorinate ct in terms of spacetime coorinates x an ct. Again by changing β for -β an prime for unprime one has ct= γ ( ct' + β x' ). (19) Expressions (14) an (18) together with y = y are the Lorentz Transformations x' = γ ( x β ct) (20) y ' = y (21) ct' = γ ( ct β x), (22) an expressions (15) an (19) are the inverse Lorentz Transformations x= γ ( x' + β ct' ) (23) y= y' (24) ct= γ ( ct' + β x' ). (25) Important: By exchanging x for ct an x for ct in the Lorentz Transformation (inverse Lorentz Transformation) it becomes clear the symmetry between space an time spacetime coorinates (refer to Minkowski). 7

8 Again the viewpoint of K an K In the process of synchronization, while the pulse propagates raially triggering off all the wristwatches on its way one has an inner region where the wristwatches are all synchronize an an outer region where the wristwatches are not yet synchronize. To you, at rest in the inertial frame K, the frontier between the mentione regions propagates with no istortion at constant spee c accoring to secon postulate of special relativity an has the shape of a circle. The raius r of such circular frontier is given by expression (1).To your frien at rest in K, the circular frontier has a raius r that satisfies expression (2). However the frontier propagation that happens in K appears to you istorte; your frien also sees the frontier propagation that happens in your frame as istorte. Let the set of simultaneous (fixe ct ) events happening on the circular frontier accoring to observers at rest in K be represente by E (x, y, ct ) in Cartesian coorinates, an E ( r cosθ, r sinθ, ct ) in polar coorinates as well. By applying appropriately Lorentz transformations to the simultaneous events spacetime coorinates in K, an setting r = ct one shoul get how these same events are seen by stationary observers in K: where E( x, y, ct) = E( r cosθ, r sinθ, ct) (25) x = r cosθ = γ r (cosθ + β ), (26) y = r sinθ = r sinθ, (27) ct = γ r ( 1+ β cosθ ). (28) The polar coorinates r, θ in K are explicitly expresse as r = γ r ( 1+ β cosθ ), (29) sinθ ( ) θ = arctan. (30) γ cosθ + β The circular wavefronts in K are seen as ellipses in K. In the sequence of figures 6-10 we plot the K an K frontiers between the synchronize an the not yet synchronize regions as seen by you, at rest observer in K, for the same ifferent stages shown by figures 1-5. All the wristwatches insie the circles are alreay synchronize in K frame. All the wristwatches insie the ellipses are alreay synchronize in K. In the overlappe regions all the wristwatches in K show the same reaing ct an those in K show the reaing ct ; the relation between the reaings is ct = γ ct. On the left upper corner of each figure we schematically show the wristwatches reaings for both frames. 8

9 Figure 6 Just for an easy comparison with the figures ahea we show here the scenario at ct = 0 = ct. Figure 7 Your point of view wavefront reaches the halfsilvere mirror M an wristwatch A, both at rest in K. All wristwatches at rest in K which are insie a circle of raius r A = cτ show the same time reaing cτ an those outsie the circle in K are still stoppe. The circular wavefront in K is seen as an ellipse in K. Those wristwatches in K which are alreay synchronize shoul be showing cτ /γ. Figure 8 Your point of view wavefront reaches the halfsilvere mirror M at rest in K an wristwatch A, at rest in K. All wrist watches at rest in K which are insie a circle of raius r A = γcτ show the same time reaing γcτ an those outsie the circle in K are still stoppe. The circular wavefront in K is seen as an ellipse in K. Those wristwatches in K which are alreay synchronize shoul be showing cτ. 9

10 Figure 9 Your point of view, propagating irectly from O (O ) reaches the right en of the ro. Accoring to you light spens a time equal to (1+β)γcτ to perform the way to the right en of the ro. All wristwatches at rest in K which are insie a circle of raius r C = (1+β)γcτ show the same time reaing (1+β)γcτ. While this happens, the origin O is isplace by the istance β(1+β)γcτ. Figure 10 Your point of view, in K, at the moment the light propagating along the y- axis reaches back the origin O closing a cycle (tick-tack-tick). The circular wavefront in K is seen as an ellipse in K. Minkowski iagrams In the sequence of figures we show the Minkowski iagrams for every stage shown in figures There is a region in which the two lightcones overlap. All the wristwatches in K insie that region exhibit the reaing ct = γ ct, where ct is the reaing of those alreay synchronize wristwatches in K. The first ticks of the K an K light clocks are simultaneous events, just by construction. The subsequent ticks an tacks are not simultaneous. Simultaneity is a relative concept. 10

11 Figure 11 Your point of view, in K, at the moment that the synchronization proceure starts in both reference frames. (y = 0 = y ) Figure 12 Your point of view, propagating irectly from O (O ) reaches mirror M, which is at rest in K. All wristwatches at rest in K which are insie a circle of raius r = cτ show the same time reaing cτ. Those wristwatches in K which are synchronize show cτ /γ. The K an K wrist watches can only be compare in the overlappe region which goes from x = (-1+β) cτ to x = cτ. (y = 0 = y ) Figure 13 Your point of view, propagating irectly from O (O ) reaches M at rest in K. One can see that to your frien, light has alreay reache the right en of the ro. To you light has not reache that point yet. This is an event which is not simultaneous to both K an K. All wrist watches at rest in K which are insie a circle of raius r = γcτ show the same reaing γcτ. Those wristwatches in K which are synchronize show cτ. The K an K wristwatches can only be compare in the overlappe region (from x=(-1+β)γcτ to x=γcτ ). (y = 0 = y ) 11

12 Figure 14 Your point of view, propagating irectly from O (O ) reaches the right en of the ro (C ). Accoring to you light spens a time equal to (1+β)γcτ to perform the way to the right en of the ro. All wristwatches at rest in K which are insie a circle of raius r C = (1+β)γcτ show the same time reaing (1+β)γcτ. Those wristwatches in K which are synchronize show (1+β)cτ. The K an K wrist watches can only be compare in the overlappe region (from x = -cτ /γ to x=(1+β)γ cτ ). The right en of the ro is reache at time (1+β)γcτ accoring to you; all the wristwatches in K which are insie a circle of raius (1+β)γcτ show the same time reaing (1+β)γcτ. Those wristwatches in K which are synchronize show (1+β)cτ. Figure 15 Your point of view, in K, at the moment the light which was reflecte by mirror M reaches back the origin O, closing a cycle (tick-tack-tick). The K an K wrist watches can only be compare in the overlappe region (from x = 2(-1+β) cτ to x=2cτ ). Discussion We have iscusse Einstein s clock synchronization as applie to two inertial reference frames with relative movement. Observer at rest in each of the frames realize they are in a stationary system an see the light propagating from the master wristwatch as wavefronts which are circular (2+1 spacetime) an so the frontier between the alreay synchronize wristwatches region an that of the not yet synchronize watches is also circular. When he (she) attempt to follow the frontier in the non stationary frame he (she) realizes that the referre frontier is elliptical. Pictures are plotte to explicit the evolution of such frontiers an the concept of relative simultaneity. Also with the help of two ientical light clocks, one in each frame, we infer time ilation, length contraction an Lorentz Transformations. 1 A. Einstein, Ann. Phys. (Leipzig) 17, 891 (1905). 2 H. Minkowski, Space an Time, Cologne Conference, September 21, 1908 (reprint in English, "The Principle of Relativity", (Dover Publications, New York, 1923), p