Lorentz Transforms

Today, many only consider the Lorentz transforms in the context of Einstein’s subsequent model of special relativity (SR), published in 1904. However, this perspective does not properly represent the history of developments, which date back to 1892, when Lorentz started to describe the propagation of light in a reference frame linked to a luminiferous aether. At this initial stage, Lorentz was only trying to explain the Michelson–Morley experiment by proposing that moving bodies might be subject to a length contraction along the axis of motion as a function of velocity [β], although the idea was still conceptual in the absence of any causal mechanism. However, Lorentz also realised that coordinate mapping from one reference frame to another could be simplified by using a transform that included a ‘local time’ variable, such that coordinates between frames [S’] and [S] could be more easily explained, if not proved. While, Lorentz did not initially consider ‘local time’ to have any physical significance, it appeared to help explain the aberration of light and the result of the Fizeau experiment. Later, in 1900, Poincaré used Lorentz’s idea of local time to forward the idea that clocks in a moving frame could be synchronized by exchanging light signals that had constant velocity [c] in all frames. Eventually, in 1904, Lorentz added the idea of time dilation to his transformations.

Note: In retrospect, the 1905 edition of special relativity was possibly little more than a consolidation of Lorentz and Poincare’s ideas. However, today, a wave model might question some of the subsequent interpretations of the Lorentz transforms, if a ‘privileged frame’ of reference could be shown to exist, even though the null results of the Michelson experiment appeared to provide initial evidence that this ‘ privileged frame’ did not exist.

So, having briefly provided some historical background, it seems an appropriate point to outline the mathematical logic that supports SR in the form of the Lorentz transforms. Again, one of the most basic points of confusion when discussing SR relates to the notation that identifies the reference frames being discussed, i.e. the stationary [S] and moving [S’] frames. Part of this confusion is that the moving [S’] frame can assume itself stationary and, as such, would have no awareness of any relativistic effects and can therefore assume the position of the stationary frame. As such, the stationary [S] frame is a somewhat arbitrary concept from which local measures of time [t] and length [x] can be mapped to complementary measures of time [t’] and length [x’] in the moving [S’] frame having velocity [v].

Note: The fact that SR allows the ‘moving frame’ to assume itself stationary means that any relativistic effects would take place in the other frame simultaneously. For, in essence, SR allows the definition of the moving and stationary frames to simply be reversed.

As outlined in the introduction entitled Mathematical Transforms, we shall adopt a somewhat different interpretation of the [x,t] and [x’,t’] variables, although the diagram right still applies. Here the main difference is explained in terms of ‘who sees what and when’ when orientated to the perspective of observer-S and observer-S’. In this context, only observer-S perceives the relative velocity [v] and therefore it is the only observer that can be aware of any relativistic effects taking place in the moving [S’] frame. So, while contrary to the inference of the diagram, observer-S’ in the moving [S’] frame is effectively assuming the role of a stationary observer, while only observer-S can measure any relativistic effects from its position in the stationary [S] frame. For this reason, the values of [x’,t’] reflect the measures of observer-S’, while the values of [x,t] reflect the measures of observer-S an any relativistic effects.

Note: So, in terms of the Lorentz transforms, the revised orientation maps the [x’,t’] to the [x,t] coordinates, such that it  reflects the mapping of the stationary geometry to the moving geometry, as per Configuration-1. However, it will also be highlighted that without supporting causal mechanisms, we should not simply accept the assumptions of length contraction and time dilation on the basis of a mathematical formulation, especially as the perspective of observer-S’ in a relativistic moving frame has not been directly verified.

However, the coordinate mapping to be developed might be initially anchored to the Galilean transforms in [1].

[1]      

In order to compare coordinate change in both frames, it is useful to anchor the spatial and time coordinates of both frames [S] and [S’] to a common origin from which changes in distance [x,x’] and time [t,t’] can then be quantified using the Lorentz transforms. However, unlike the Galilean transforms, SR assumes time to be a more subjective measure that depends on the velocity of the inertial frame, which can be reversed. The other important postulate of SR to highlight at this stage is the assumption that the speed of light [c=1] is constant in all reference frames, such that we might use [2] as the starting point for the Lorentz transforms.

[2]      

By equating the expressions in [2] to zero allows these two equations to also be equated to each other. On the assumption that length contraction is restricted to the [x] axis of motion, the form of [3] can be further simplified on the basis that [y=y’] and [z=z’], if the velocity [v] is constrained to the [x] axis.

[3]      

However, the form of [3] appears problematic if we assume [t=t’], as per [1], and try to substitute for [x’=x±vt], as shown in [4], although the discussion of Configuration2 suggested that other transforms may be possible.

[4]        

In order to address the inequality highlighted in [4], the difference in [x] and [x’] caused by velocity [v] might be reconciled by introducing the factor [ϒ] adopted by the Lorentz transforms.

[5]       

It is highlighted that the two forms in [5] are essentially the Lorentz transform for [x] and the inverse transform for [x’]. The symmetry of these transforms is required by the first postulate, such that the laws of physics are compatible in each frame except for the direction of the velocity [±v].

Note: As previously highlighted, a wave model might question the thinking behind the inverse form of the Lorentz transforms by considering the results of [5], when [t’] and [t] are both zero. In this case we see that the symmetry requires length contraction in either frame relative to the other at the same time.

Despite the note above, this discussion will continue with the accepted postulates of SR, where the speed of light [c] is common to both reference frames, such that [x=ct] and [x’=ct’], such that we can insert these assumptions into [6].

[6]      

We can now proceed by dividing both expressions in [6] by [c].

[7]      

Using [7], we can solve for [ϒ] as follows:

[8]     

We now have a solution for the measure of [x] and [x’] based on the form of [5] using the factor [ϒ] shown in [7]. However, we might proceed from this point by substituting for [x’=ct’] and [x=ct], as shown in [6]. However, if we re-arrange [6], we can obtain equations for [t] and [t’], which were previously shown in [7].

[9]     

However, we now seek the Lorentz transform for time [t,t’] by dividing through the first expression in [9] by [c].

[10]     

Finally, we might expand [ϒ] as defined in [7], such that we have the expanded Lorentz transforms between [t’,t]

[11]      

Using the simplified form to the right of [11] by adopting velocity [β], we might now table the Lorentz transforms for [x,t] and the inverse transforms for [x’,t’], which by the argument of the first postulate simply requires the sign of velocity [β] to be reversed.

[12]    

Note: The inclusion of the [±] sign within [12] highlights that either frame can assume the role of the stationary [S] and moving [S’] frame by simply reversing the direction of velocity [±β]. Again, a wave model might contradict the symmetry assumption of SR by its own assumption that velocity [v] is relative to the wave propagation media, such that length contraction and time dilation would only occur in the moving frame [S’], but only observed from the stationary [S] frame.

At this stage, we might table a few additional issues associated with the Lorentz transforms. If we return to the diagram at the start of the discussion, we might assume that the moving [S’] frame has a relative velocity [β=0.5] as it passes a stationary [S] frame in close proximity, such that the origins of both coordinate systems might be synchronised at [x,t=0] and [x’t’=0].

From the initial point of synchronization, what would change after 1 second?

As highlighted, the symmetry of the Lorentz transforms can make the identification of the stationary [S] frame and the moving [S’] frame somewhat arbitrary as a comoving observer-S’ in the moving [S’] frame can assume itself to be stationary. However, we are assuming that the values of [x’,t’] correspond to observer-S’ in the moving [S’] that assumes itself stationary.

β ϒ g x’ t’ x t x’ t’ t=[13]
0.5 1.155 0.866 0.00 0.00 0.000 0.000 0.00 0.00 0.000
0.5 1.155 0.866 1.00 1.00 0.577 0.577 1.00 1.00 0.577
-0.5 1.155 0.866 1.00 1.00 1.732 1.732 1.00 1.00 1.732

In the table above, the first two shaded columns indicates the value of stationary values of [x',t'], which are then transformed to the moving values [x,t] that reflect the relativistic effects assumed to be taking place in the moving [S’] frame, as observed by the stationary observer-S. The second two shaded columns reflect the required inverse transform of the [x,t] values back to [x',t']. The final column right, replicates an equation used in Configuration-1 to calculate the light propagation time between [A] and [B] based on the direction of [β=c±v], as shown in [13].

[13]           

While the units of the equation resolve to time, the propagation distance [s=ct] is determined based on the constancy of the velocity of light [c=1] in all frames. While the values in [13] can be correlated to the table above, it is highlighted that the light propagation times [tAB] and [tBA] are based on the contracted length [gx’=0.866], which can only be observed from a stationary [S] frame that perceives velocity [β]. Again, we might return to the question that has remained open from the start.

Who sees what, when and how?

From the perspective of the Lorentz transforms, it has been argued that only observer-S in the stationary [S] frame is ever aware of the relativistic effects implied by its local ‘measure’ of [x,t], irrespective of what orientation is adopted. Conceptually, the ‘when’ issue is more complicated as the relativistic events are assumed to be taking place in the moving [S’] frame, although the relativistic effects on these events can only be recorded by observer-S in terms of its local coordinate values [x,t]. In this respect, the conceptual mapping of event [x’,t’] to [x,t] would only be physically meaningful, if there was no significant propagation delay between [x’] and [x] plus [t’] and [t].

Note: The reason why any inference to a measurement of [x’,t’] is preceded by the word ‘conceptual’ is simply because the measurement of length and time in a frame moving with significant relativistic velocity has not been directly verified. In this respect, any measurement assigned to observer-S’ or observer-S are questionable. While measuring the one-way speed of light is often considered problematic, the discussion Clock Synchronisation questioned the scope of this problem. For if a one-way measure of the time between [AB] and [BA] could be made in the moving frame by a comoving observer-S’, the SR assumption that any inertial frame can simply assume itself to be stationary might be called into question.

Finally, there is the issue of ‘how’ the relativistic effects of length contraction and time dilation are explained in terms of any physical causal mechanism. As indicated, the Lorentz transforms are essentially a mathematical formulation that simply assumed both length contraction and time dilation, but does not forward any causal explanation of either. However, the next discussion of the Ivanov Transforms will consider length contraction in terms of a standing wave compression.