In Special Relativity (SR) the Lorentz Transformation (LT) equations have been interpreted as a quasi-coordinate conversion, relating the locations and times of occurrence of events observed in one frame of reference to those that could be observed by another observer in uniform relative motion.  An intuitive understanding of this interpretation of the equations is obtained by considering coordinate conversions generally, including both the translation and rotation of coordinates in relatively stationary coordinate frames.  The literature of SR has tended toward graphic analogies with the rotation of coordinates using diagrams of the temporal and a single spatial coordinate axis appropriate to the nontrivial equations.  Because the nature of the phenomena portrayed in such a diagram is not visually observable in actuality, rotation analogies (although soundly based in the formalism) fail to elicit the intuitive understanding that goes with a depiction of what could (at least in principle) be set up for direct viewing.  For this and other reasons we will break with that graphic tradition, preferring the analogy with the translation of coordinates. The analogy of the LT with coordinate rotation breaks down with regard to the illustrated axes being "skewed" towards each other rather than "rotated" in the same direction.  In the translation analogy, relativistic aberration and parallax will be demonstrated to exhibit comparable roles that will add to our visual understanding. But here again, the analogy breaks down with respect to an assumed scale factor along the direction of motion.

Figure 1 illustrates the relationship of measured coordinate values of an event seen by two displaced but relatively stationary observers situated at respective origins of their frames of reference.  The appropriate coordinate "transformation" equations are provided in the box at the left of the figure. The parameter x₀ is the permanent separation of the two observers, |ct| is the distance to the observed phenomena, |t| is the time interval between occurrence of the observed event and its observation by the unprimed observer, O₁ and c is the speed of light.  Primed parameters indicate a corresponding distance or coordinate values relative to the "primed" observer, O₂.  All values of t are negative since observed events occurred prior to the observation that is assumed to take place at t = 0.

In the general translation situation where an event occurs at clock time t_e, the observations will occur at unique clock times t_o1 and t_o2.  (Actually either their observations take place at different times, t_o1 and t_o2, or they pertain to different events occurring at the unique times, t_e1 and t_e2, because of the unique distances from the event and the assumed universality of the speed of light for relatively stationary but displaced observers.)  So that using times, t_e, t_o1 and t_o2, obtained from synchronized clocks and observations of a single event, the coordinate time interval parameters are defined by, t = t_e - t_o1 and t’ = t_e - t_o2.  Using coordinate translation equations, two observers can bring their observations into complete agreement by recreating a single point and time in a common space-time at which each observed event took place.  Thus, the two observed events can be reconciled with a single exterior event observed uniquely by each observer.  This is in spite of the fact that the event will have occurred in the more distant past for one of the observers than for the other with respect to the time of their respective observations.
 
It was presumed for centuries that the appropriate transformation of coordinates for two frames in uniform relative motion would be very like that realized for stationary displaced observers.  The apparent displacement would be a function of the distance to the observed event, the relative velocity of the observers,  and the speed of light.  The implications of these assumptions are shown in figure 2 where the relationship to figure 1 is readily apparent.  Again the appropriate equations for this case are included in the box at the left of the figure.  The distances vt and ct in the equations were retained without canceling factors of t in order to illustrate the similarities to the final two equations in figure 1.  The time difference equation was no longer considered to represent the relationship of the time intervals between occurrence of the event and its respective observations, however, since it was naturally assumed that observation would take place at t = t’ =0 when the two observers are in coincidence.  That the speed of light might be a universal constant with respect to each observer (rather than with respect to absolute space) for which case the direct analogy to that equation would seem to apply, was not considered a legitimate possibility.  In that case both observers would not be observing the same events when they were in coincidence, an unthinkable consequence at the turn of the twentieth century and to virtually everyone who matters even now.
 

Coordinate Translation Transformation Equations
x  = c t cos q
y  = c t sin q
x'  = c t' cos q ' = x - x_o
y'  = c t' sin q ' = y
where
    c t' = c t ( 1 - 2 b x / c t  + b 2 )^1/2
   b = (x_o / c t)
 
 

Figure 1: Coordinate Translation between Relatively Stationary Observers
 
 

 

Galilean Transformation Equations
x  = c t cos q
y  = c t sin q
x'  = c t' cos q ' = x - v t
y'  = c t' sin q ' = y
where
    c t' = c t ( 1 - 2 b x / c t  + b 2 )^1/2
   b = (v t / c t)

Figure 2: Coordination of Observations by a Stationary and Moving Observer According to the Galilean Transformation
 
 

Lorentz Transformation Equations
x  = c t cos q
y  = c t sin q
x'  = c t' cos q ' = g (x - v t)
y'  = c t' sin q ' = y
where   c t' =g  c t ( 1 - 2 b x / c t  + (b x / c t) 2 )^1/2
and for similarity we show b = (v t / c t)
and  g = 1/ ( 1 - b ² )^1/2

Figure 3: Demonstration of g Stretching Factor of the Lorentz Transformation Relative to the Galilean Transformation
 

This Galilean transformation (GT), as it was called did not account for an increasing number of optical experiments performed before the twentieth century.   A transformation developed by Lorentz (the LT) did.  A direct comparison of the geometric relations implied, respectively, by the LT and GT equations is illustrated in figure 3.  In figure 4, a mapping of events in the x,y plane is demonstrated; the same sort of mapping applies to the x,z plane.  Clearly, the LT equations imply a stretching along the x direction of the spatial displacement with respect to what had been anticipated for the GT.  In figure 3 the apparent differences in the times of occurrence of the observed events (if one embraces the universality of the speed of light as Einstein did) is easily demonstrated by a straight-forward application of the Pythagorean theorem to the distances traversed by light in the two reference frames.

In this way the postulated universal speed of light relative to every observer is shown to be associated with the temporal Lorentz equation for which there is no direct analog in the GT equations.  However, it is easily seen that the effectual difference between the Lorentz temporal equation shown in the box in figure 3 and the Galilean speed of light equation shown in figure 2 involves only second and higher order terms in b.  The effect of relative motion on observation is apparently much more similar to the situation of two relatively stationary observers separated by the distance vt than anticipated by the GT equations.  The primary difference between the sets of equations pertains to whether the differences of |c’ t| or |c t’| from |c t| of two co-located observers are to be presumed to reflect a difference in when separate events occurred or in how fast the light has propagated.  The basic interpretation of the Lorentz equations is the former.  As we will see, Einstein’s interpretation accepted this basic clock time value difference preserving the value of the speed of light.  He also accepted a common sense ether theory constraint of a single event observed by both; but, common sense does not apply in quantum domains.

The two events E₁ and E₂ in figure 4 are representative of an entire class of simultaneous events in the unprimed reference frame.  They are equidistant from an observer in reference frame K and map into another class of events (E’₁ and E’₂) that could be observed by another relatively moving coincident observer in frame K’ in accordance with the LT equations.  (This simultaneity condition in K is realized for reflection from the various interior points on a concave spherical mirror of radius equal to cT if an instantaneous flash of light had occurred at the center of the apparatus.)  Although the spatial mapping of this class of simultaneous events on the surface of a sphere is easily verified to generate an elongated ellipsoid of associated events as shown, the corresponding times of occurrence for the two sets of events are rather more difficult to visualize.  It should be noted that events for which x is less than zero (E₂ in the figure) are deferred in K’ relative to their times of occurrence in K, i. e., they occur at later times.  For events for which x is greater than zero (E₁ in the figure), the opposite is the case.  Circular cross sections of spheres of radius g | cT | whose centers are, respectively, at the origin of K’ at times t’= g | T | have been drawn as a standard temporal reference to illustrate this point in figure 5.  When an arriving (departing) photon is at this distance, i. e., leaving (arriving at) the surface of such a sphere, its observation (emission) will (have) occur(red) +/- t’ =  g | T | seconds later (earlier).  In this figure the time differences given by +/- t_i’ =  g | T | - t_i’ for the events E’₁ and E’₂ (which both occur at t = 0 in K) are illustrated using line segments of length, d_i’, such that +/-t_i’ = d_i’/c, as graphic temporal references.  With less emphasis the time differences are shown for the entire class of such events occurring at t = 0 in K by the lengths of corresponding line segments.  The physical significance of these segments can be seen by noticing the equivalence of related segments inside and outside an ellipsoidal (as if only half-silvered) reflection surface.
 

 

 

In figure 6, the generalized mapping of coordinates from one frame of reference to another in uniform relative motion in accordance with the Lorentz equations is illustrated.  The indicated circles correspond to cross-sections of spherical surfaces of simultaneously occurring events.  The ellipses correspond to the cross sections of ellipsoidal surfaces of events corresponding to the simultaneous events in the other frame of reference.  Notice that the symbols, L_v , L_v ^-1, L_-v , and L_-v ^-1  all refer to the LT equations or the inverse (which is also an LT associated with an oppositely directed relative velocity) associated with a relative velocity of v or -v.

(x’, y’, z’, t’) = L_v (x, y, z, t) ,

where the values of x’, y’, z’ and t’ are given respectively by the LT equations provided in the box in figure 3.  In these equations it is easily shown that, L_v ^-1= L_-v and L_-v ^-1 = L_v .

Figure 7 includes nine panels that depict the progress of a wave surface initiated in panel a. at time t' = - g T on the X’ axis at x' =  g b c T.  In figure 7.a and in subsequent panels the progress is depicted at intervals of Dt'= ¼ g  T.  Black circles indicate reflectors on a cross section of a rigid spherical mirror where reflection events occur in K and the smaller open circles in panels 7.c through 7.g indicate where the corresponding events will have occurred in K’.  The points on the ellipsoid do not correspond to a rigid body in the same sense as the spherical apparatus of K.  The set of points indicated by the smaller open circles corresponds instead to a mere ephemeral mapping of events, i. e., the spatial locations of the reflection events in K’ independent of when they may have occurred.  The wave front is not simultaneously reflected in K’ as it is in K, some are reflected before, and others after the instant of coincidence of the observers as was shown above.

The reflection events from a sphere could be directly correlated with events situated at the surface of a stationary ellipsoid in the other frame.  But, paradoxically, reflection phenomena on the surface of the rigid sphere would not be coincident with the events on the associated ellipsoid at the time the events occurred except for a ring of events for which x = x' = 0 at t = t'= 0. Refer, for examples, to the current reflection events depicted in figure 7.e.  Divergence and convergence of the light shown in figures 7.a and 7.i, respectively, coincide at the respective times, i. e.,  t' = - g  T and t' = + g  T  in K', and are coincident with the origin of K at these times, i. e.,  x' = + g  v T and x' = - g  v T.  So that for these two significant events, according to the relatively moving observer, it is possible for the material object to be co-located with the respective events at the respective times of observation and emission.  But the time of occurrence of emission being different for the two observers (i. e., t' = - g  T rather than t = - T ), therefore, still imposes the condition that the position of the center of the apparatus at the time of the emission must have been different (i. e., x_d'= + g v T rather than x_d = + v T ).  This still produces the seemingly paradoxical situation where the divergence event (for example) would seem to have occurred twice although each of the two similar events would only be observable by the respective observer!
 
 

 

Notice that in the panels of figure 8 reflection occurs when and where the contracted rigid sphere intersects the elongated ellipsoidal surface.   Thus, we arrive at the Lorentz contraction and time dilation hypotheses that have been accepted as integral parts of SR dogma.

Any explanation of why such a contraction and time dilation should occur has been considered "off limits" by the "Received View" of physical philosophy that continues to be in vogue.  According to this philosophy scientific theories merely describe phenomena.  Explanations are held to be superfluous.  But the only role of this hypothesis is as a metaphysical explanation to accompany the strange phenomena described by the LT.  In GoF #95, the author showed that supposed confirmations of time dilation (by ignoring quantum energy level effects that alter transition rates in based on altered energies in accordance with E=m c² instead of the "dilation" formula) may actually refute the hypothesis instead. Clock time dilation is not required to effect measured decay time differences of high and low energy states.  If physical explanations were allowed, the alternative interpretations might well account more consistently for the same phenomena; such explanations would relate to electromagnetic absorption and quantum theories with the Transaction Interpretation rather than being mere inexplicable counterparts of the mathematics.

NOTE: This article first appeared in issue Number 97 of Gift of Fire, (Journal of the Prometheus Society), July 1998.