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Rafael Bautista-MenaCEIBA Center, Universidad de los Andes, Bogot´a, ColombiaNovember 16, 2018
Abstract
This paper offers an alternative approach to dis-cussing both the principle of relativity and the deriva-tion of the Lorentz transformations. This approachuses the idea that there may not be a preferred in-ertial frame through a privileged access to informa-tion about events. In classroom discussions, it hasbeen my experience that this approach produces somelively arguments.
Suppose that two inertial “observers”, from now onnamed Alice and Bob, each attached to a referenceframe in one spatial dimension and time, need to ex-change information about an event E that is a partof an information set to which both have access to.In discussions about kinematics, the information setabout E reduces to three items: 1.- The fact that ittook place (for instance, as registered by a “click” instandard detectors that both Alice and Bob are en-dowed with.) 2.- The spatial coordinate for E , and3.- the associated time. In such discussions, the firstitem is almost always taken for granted. For the pur-poses of this work, the only proviso I will make is tocall “events” only those described in item 1, and avoidthe more widespread usage, where “event” tends tobe more or less automatically identified with a space-time point. Items 2 and 3 are the usual concern ofkinematics, and as is well known, the communicationof these data between Alice and Bob goes via a set ofsimple, linear transformation equations, namely theLorentz transformations. Another element necessary for the discussion thatfollows is to assume the existence of some standard“messenger”, which may be produced by any event,and is by definition the fastest entity known to theobservers to be apt to carry encoded messages acrossempty space. As an integral part of the definitionof their frames, Alice and Bob are capable of eitherencoding or decoding messages, using some universalcode. Such messages always carry the informationabout those events registered in their frames. Byassumption, any event that Alice is capable of reg-istering with her detector will also be detectable byBob.Bob’s frame of reference will be drawn as a Cartesianframe, with the vertical time axis perpendicular tothe horizontal space axis. By definition, the origincorresponds to the event of coincidence with that ofAlice’s frame. This event is labelled O in Figure 1.Alice’s choice of axes for encoding her informationabout events is completely defined by the angles (seeFigure 1) α and γ that her time and space axes re-spectively make with those of Bob’s frame. I adoptthe convention that, as drawn, these angles are posi-tive, and satisfy the constraint α + γ ≤ π β .All lines parallel to Alice’s x axis will be called “eq-uitemps”, and all lines parallel to her time axis will As measured in each of their frames. gb P E H
Alice's equitempMessenger line t Bob t Alice x Alice x Bob O Figure 1: Angle conventions between Alice’s andBob’s framesbe called “equilocs”, after the denomination used inMermin [1].It is important to emphasize that the particularchoice of perpendicular axes for Bob’s frame is madeonly for ease of exposition. In fact, all that mattersis the relative angle between Bob’s and Alice’s corre-sponding axes. In what follows, methods of Euclideangeometry will be used within a context that is usu-ally associated with Minkowski’s space-time. Furtherdetails as to why and how this may be done can befound in Brill and Jacobson [2].
The act of interpreting, or decoding, a message is,by assumption, local in character. I will model thisassumption by locating Alice at some definite, fixedpoint. By convention, Alice will be located at thespatial origin for all time. As a consequence, Alicewill learn about the occurrence of event E , that tookplace say at frame coordinates ( x E , t E ) only at a later time t E + x E /c , where c is the speed of the standardmessenger in her frame. In a symmetric fashion, ifAlice wanted to be causally connected with event E ,then the latest moment at which she could send a“triggering” message would be t E − x E /c . This par-ticular event, Alice’s delivery of the latest signal thatcould connect her to E , is shown at the vertex P in Figure 1. As an obvious extension, Alice could becausally connected to any event that may be triggeredalong the world-line connecting Alice with event E .Another event of interest is the “horizon” event H .This is the event simultaneous with E , triggered by amessenger sent out from O along the positive x axis.Notice that this is the farthest point to which Alicemay expect to be causally connected with before orsimultaneously with event E . An accessible event, conditional on events E and O , isone for which Alice may be able to have a causal con-nection with before or at most simultaneously withthe occurrence of event E . The accessible set A , con-ditional on events O and E , is the set of all space-time points with which Alice could establish a causalconnection, right after her time t = 0 and up untilthe frame time at which E takes place. From this de-scription, it is clear that E must lie within the “causalcone” defined by the messenger. In the space-time di-agram shown in Figure 1, the polygon OP EHO cor-responds to the accessible set conditional on events O and E .The intuition for A is to think of it as composed bymember “sites”, all identical in their properties, anddistinguished only by their space-time coordinates.Each member of A is equally capable of hosting asingle event. Therefore, A describes Alice’s capacityto influence events along the positive x axis between t = 0 and the time corresponding to event E . Alter- Here the “site” of an event is assumed to correspond toa single member of the set. In principle, one could considerthe possibility of a larger subset of A as the site of a singleevent, in which case the individual space-time coordinates ofthe elements in the subset would fail to provide any meaningfulinformation about the event. But this case looks more likequantum physics. The present discussion is fully containedwithin a classical context. A may also be seen as the set containing themaximum amount of information (potential events)generated between those times, that Alice may ex-pect to collect. Given the same constraints: a common event O ,and an external, independent event E , no inertialobserver may expect to be causally connected tomore sites, or to be able to have access to moreinformation than any other. In more mundaneterms, given the same prior information, Alice maynot know anything that Bob wouldn’t know too, norvice versa.One way to make operative this form of the relativityprinciple is to assign a measure I ( A ) to set A . I willmake the following assumption: The Euclidean area of the accessible set boundedby the polygon
OP EHO is a direct measure of themaximum number of events about which Alice mayhave knowledge, conditional on events O and E . Without providing a proof, it seems reasonableto suppose that this statement is fully consistentwith the properties of homogeneity and isotropy offlat space-time.
Let’s begin by computing the coordinates for all fourevents defining Alice’s accessible set, as determinedby Bob. Figure 1 shows the polygon
OP EHO andthe associated angles.In self-evident notation, the coordinates for each ver-tex as functions of Bob’s coordinates (
X, T ) for event I am aware that this implies attaching to the set A , andto space-time in general, a topology different from the usuallyassumed for continuous space-time. In fact, it would have tobe based on finite, or at most, countable sets. But the size ofthe corresponding “space-time cells” could be made as smallas desired, as long as they were finite. E are given by: x P = sin α sin β h T sin( α + β ) − X cos( α + β ) i , (2) t P = x P cot α, (3) x H = T − X tan γ cot( α + β ) − tan γ , (4) t H = x H cot( α + β ) . (5)The measure of the accessible set, corresponding tothe area bounded by OP EHO is I ( A ) = 12 h Xt P − T x P + x H T − t H X i . (6)Equation (6) may be rewritten as follows:2 I ( A ) = h T + h X + h XT. (7)The principle of relativity, as stated here, now re-quires that this measure be frame invariant. In otherwords, the h i ’s in (7) ought to be universal constants.It is not meant here that those coefficients are newphysical constants, in the sense that Planck’s con-stant or the charge of the electron are. But rather,that the corresponding algebraic expressions for the h i ’s must reduce, in a trivial way, to simple numeri-cal values. Therefore, as an immediate consequenceof the relativity principle, there follows: h i = constant. (8)Their explicit forms are the following: h = − sin α sin β sin( α + β )+ 1cot( α + β ) − tan γ , (9) h = − cos α sin β cos( α + β )+ cot( α + β ) tan γ cot( α + β ) − tan γ , (10)3 = cos α sin( α + β ) + sin α cos( α + β )sin β − cot( α + β ) + tan γ cot( α + β ) − tan γ . (11)These rather lengthy expressions may be more easilyhandled using the following shorthand notation: v ≡ tan α ; w ≡ tan( α + β ); z ≡ tan γ . Now (9), (10)and (11) look as follows: h = − wvw − v + w − zw , (12) h = − w − v + z − zw , (13) h = w + vw − v − zw − zw . (14)Inspection of (12), (13) and (14) leads to the followingidentity: h w + h w + h = 0 . (15)From the statement of the relativity principle in (8),equation (15) implies that w is equal to some constantvalue. Therefore: α + β = constant. (16)Then, irrespective of the choice of frame, the relativeslope associated with the speed of the messenger isfixed.Equation (12) can be rearranged as follows: vz ( w − h w ) + w h z + ( h − w ) v + w − h w = 0 . (17)Since both v and z represent trigonometric functions,and since (17) must be an identity, quadratic termsshould be linearly independent from linear terms,therefore the coefficient in the quadratic term mustvanish, leaving as its only feasible solution: w = h . (18)Notice that w = 0 is not a feasible solution, for itdoesn’t solve (15). With this result, equation (17)reduces to: h z − v = 0 . (19) From this relation follows that, if the principle asstated by (8) is to be upheld, then the choice of axesby Alice is constrained by (19). This relationship is,by the way, the best justification of why h must bedifferent from zero, for otherwise, Alice wouldn’t havea choice at all, or put another way, it would deny theexistence of any reference frame.Using (18), equation (15) becomes:1 + h h + h = 0 . (20)Substitution of (18) and (19) into (13) yields: h h = − . (21)This last result, combined with (20) produce: h = 0 . (22)These findings for the h i ’s lead back to (7), whichnow reduces to:2 I ( A ) = h X − T h . (23)In this expression, it is always possible to set h = 1,because this is just a rescaling of the ruler and the“tick” of the clock used by Alice. Then (23) is easilyrecognizable as the Minkowski square of the space-time interval. This same choice makes w = 1, which,going back to (16), produces the neat result: α + β = π . (24)Then, the messenger’s slope must cut in halves thequadrant of Bob’s frame. Finally, (19) simplifies to: v = z. (25)Using the convention established earlier for α and γ ,the last equation is equivalent to say that they areequal. Therefore, Alice’s axes are also placed sym-metrically around the line of the messenger. Thisgeometrical arrangement is well known: Bob’s andAlice’s frames are connected by the Lorentz transfor-mation.The second consequence that follows from (25) is thatit makes obvious that in Alice’s frame the speed ofthe messenger is the same as in Bob’s frame. There-fore, there exists one messenger whose speed is thesame in all frames of reference.4 Discussion
In the present work I have derived both the necessityof the existence of a messenger with an invariantspeed in all frames of reference and the Lorentztransformations, starting from the principle ofrelativity, stated as a symmetry in the access tocausal connections. In more relaxed terms, thisapproach establishes the impossibility to tell thestate of inertial motion via the access to different“amounts of information” between reference frames.This approach relies on two assumptions:1. The local character of any encoding/decodingcapable “observer”.2. The measure I ( A ) as the correct invariantquantity.Traditionally, the question ‘Why the Lorentztransformation?’ has been answered with ‘Becauseit is the only solution consistent with the relativityprinciple.’ The present work instead addresses thequestion ‘Which way to the relativity principle?’Within the context of this paper, the principlehas been spelled out through the invariance of themeasure of the conditional accessible set I ( A ).A question raised by this approach may be whyit works. It is not new to obtain the Lorentztransformations from the relativity principle, plusadditional assumptions about the properties of flatspace-time, as it has been shown in several excellentarticles (see, for instance, L´evy-Leblond [4], Mermin[3], Lee and Kalotas [5].) The only difference inmy approach is the expression of the principle interms of a kind of information democracy, which iscloser in spirit to the intend in Field [7], who arrivesat the Lorentz transformations from a postulatedspace-time exchange invariance. To see the connec-tion with other treatments, recall that a universalmessenger generates a causal ordering on the futurecone. Therefore, the relativity principle imposes acausal structure on set A . Turning this argumentaround, suppose now that we would want to have aset A with a postulated causal structure. Suppose See an interesting approach to the Lorentz transformationsfrom this angle in [6]. also that Alice triggered an event timed betweenevents O and E . Since Alice and Bob are equivalentin their capacity to register events, Bob would learnabout such event. But he could not register thisevent as having occurred either before O or after E , because that would violate the assumption of acausal structure for A . Therefore, all the events thatAlice would trigger between O and E , are the sameones that Bob could detect too, no more and no less.This argument sheds light on why the invarianceof the measure I ( A ) acts as a substitute for theconventional statement of the relativity principle.Then, the main contribution of this particularformulation is its approach to relativity from anevent-counting concept, represented (as proxy) bya Euclidean measure. On the other hand, onelimitation is that it starts from the assumption thatthe correct transformation relation between inertialframes is linear.This approach is open to criticism, among otherreasons, on the basis that it looks like a step backtoward anthropocentrism, through my recourse toterms such as “information”, “encoding”, “decod-ing”, and others of a similar nature. I always bear inmind the now famous retort ‘Whose information?’Nevertheless, I believe that my use of such termsonly highlights the limitations of language. After all,in the case of, say, an elastic collision between twoelectrons, we use terms such as “interaction” to referto the exchange of momentum between the particles,only out of well established tradition. I wish to thank Ana Rey, Juan Restrepo, AlonsoBotero and Jorge Villalobos for their useful commentsand their kind help.