These sorts of questions concerning the notion of 'time dilation' eventually demand an explanation in physics or mechanics for the phenomenon originally proposed by Einstein in his seminal paper ''On the Electro-Dynamics of Moving Bodies'' -- in this instance the speedometer is just another variety of clock. The mathematical argument devised by AE is not difficult to follow after all, provided that, having conceded the postulate of invariant c, the premise is accepted; namely that the light paths ostensible to observers in the moving and stationary frames k and K in AE's scenario actually differ.
Considering then that in all the myriad ways this problem has been posed and addressed since that scenario of a moving light source was first imagined, and its description elaborated mathematically using Lorentz Transformation based on such a postulate of invariant maximal c, no adequate explanation in physics or mechanics has to date been able to account for the effect that, while observers of the moving frame k and its light source situated in the stationary frame K perceive the entire effect of the transit of that light source over its actual distance in K, those moving with that light source in the frame k perceive only that component within that frame k itself, the following seeks briefly to understand this anomaly that these light paths ostensible in those respective frames actually differ.
Clearly then, in order to account for the effect observed in the frame K, that component of light transit constituted by the velocity v of the light source is somehow incorporated into the velocity c of light itself (which one is entitled to suppose is intrinsic to the essence of space), from which moreover it might be concluded that such material motion is in fact an aspect of light, merely a variation on the mechanics of space from which light itself arises as an innate effect; and that this incorporation manifests itself differently for observers in the frames k and K.
The reason that this component v is not apparent to those observers in the frame k is of course that they too are moving with that same velocity, and this is no doubt the original intuition of AE. What this inevitably suggests however is that these observers, clocks, the light source and the frame k itself constitute a dynamic entity whose capacity to sustain its form and integrity in that motion at v at once impedes the possibility for any objective perception of it with respect to the transit of light itself at c.
What then is the essential sense in which this dynamic entity of an inertial frame in motion, indeed of a singular universal frame of which it is a mere aspect, is to be understood? Here one simply postulates as the context for this imagination the existence of a Unitary Universal Substance [UUS] of which all reality is comprised, including observers in any frame, within which the motion of material bodies, as in this case, is effectively a progressive wave effect -- viz, the progressive aspect of an oscillatory wave dynamic -- deriving in the capacity for that substance continuously to reconstitute itself in a given spatial direction. In the case of a light source for example, this implies a corresponding dynamic in the motion of electrons.
The postulate of such a UUS need not of itself be troublesome to the receptive intellect. What is required is only a consistent model of its singular dynamic based on certain corollaries to that proposition itself; namely that in the first place such a substance holds together and coheres with what amounts to a universally 'cohesive force' (moreover with an evident and self-evident proclivity to stability, thus of form, further implying the tendency for cyclical recurrence of phenomena) which, when construed in concert with a singular impetus implicit in such an inviolate universal entity or unity, may be conflated with a property of inertia; a priori this derives in the corollary axiom of a singular universal process or intention implicit in a singular cause.
This 'cohesive force' then may be understood to operate from within the interior spatial depth of such a UUS at every point and importantly in perpetually disparate components whose interaction inclines naturally and inexorably towards a condition of parity or 'cohesive equilibrium': components of cohesive force may be imagined and modeled to operate in axes between such loci of action, therefore distributed in planes* at right angles to such axes at the point of such cohesive interaction, generally corresponding to the geometric relation between a magnetic force and associated E-field.
All discernible physical forces are only aspects of this fundamental and quite exclusive 'cohesive force', and even 'electric charge' itself is eventually a component of its distribution in planes of cohesive interaction. *As a guide, imagine the intersection of two expanding spheres at a planar interface.
The property of inertia implicit in the process towards cohesive equilibrium or symmetry, and in the constant irreversible maintenance and enhancement of a condition of universal 'cohesive resonance' or 'cohesive equilibration' between disparate components of cohesive force equivalent to this very tendency, corresponds to 'mass'; so that, for example, the essential relation between that property and the cohesive force itself whose distribution in a given region implies such mass/inertia, and broadly 'energy', is only the function of the rate at which motion within the UUS changes: viz, acceleration.
It is this cohesive disparity between any and all components of cohesive force which imparts to the UUS its oscillatory or vibrational character, eventually a progressive wave principle as more-or-less regularly incremental resultants in such an effect tend in given spatial directions. This can be modeled with respect to a condition of the perfect oscillatory regularity of cohesive equilibrium represented by a cubic lattice structure which is therefore appropriate in particular to the analysis of states tending to approach that ideal condition of cohesive symmetry such as H1 gas.
This is the essence of the process of 'reconstitution' of the UUS: that a given configuration of the oscillatory dynamic of cohesive relations will tend naturally to recur with an incrementally progressive aspect which depends on resultants in cohesive disparity -- viz, 'cohesive polarities' which are components of cohesive force --, the limits of the regularity in which define the wavelengths of visible light (essentially an 'interference effect' between fundamental vectors of cohesive polarity in such a model).
Obviously then, nothing may move relative to a unitary universal substance; all motion and relative motion is of and within that substance itself. This motion, indeed all motion, is inherently wave motion -- which is to say that all motion within the UUS arises as an oscillatory effect between points of disparate cohesive potential in transient existence as disparate resultants of prior cohesive interaction within the UUS; such that the more-or-less regularly incremental progression in such an effect constitutes wave motion and corresponding velocity in a given direction, maximally c in the limiting case in which this incremental progression in a fundamental interference relation between two disparate archetypal components of cohesive force corresponds to a specific wavelength of visible light in vacuo.
It is for this reason then, in the cohesive mechanics of such a UUS and its unitary wave (interference) dynamic or principle that the resultant motion of electrons constituting a light source moving in a given direction cannot be apprehended by observers moving in the same way with the same relative resultant; while to those observers stationary in the frame K relative to that source, the entire effect is perceptible as an integrated resultant within the limiting context of a maximal resultant c. The velocity v of the frame k and the light source becomes an aspect of c, the wave velocity intrinsic to the UUS, as if overlaid upon it; while the velocity c of light is itself eventually only the manifestation of the optimising tendency of the cohesive dynamic to attain to an ideal condition of cohesive equilibrium or symmetry between any pair of two mutually cohesive loci within the UUS, a metaphysical impossibility
It is important to recognise here that since we too are composed entirely of such a substance, its ultimate nature can never be objectively discerned; in a general sense its optimal condition of homogeneity is appreciable as 'empty space' (with its VEV etc.). At the same time however, since the essential principle inhering in such a substance is one of 'cohesive resonance' between its constituent components, effectively components of 'cohesive force', its dynamics are readily perceptible at every turn, and similarly amenable to understanding. Indeed, since the perception of the operation of the fundamental cohesive force through mechanisms of 'perceptual resonance' -- viz. perceptual consciousness -- represents the sole means of apprehension of this UUS, then that substance or fabric may be regarded as effectively equivalent to that ubiquitous force. This intuition was also expressed by Faraday.
Clearly though, nothing in the preceding argument permits that the regularly oscillatory mechanism of clocks of any kind actually proceeds more slowly when in motion; and nothing of the sort is to be inferred from AE's original scenario. This idea has somehow gathered conventional and popular credence for want of the capacity for the orthodox perspective of physics as it stands -- that is, a body of convention which refuses to countenance the notion of a UUS requisite to any comprehensive theory of reality -- to explain the effect that observers of a light source from diverse inertial frames in relative motion differ (in the manner described by Lorentz transformation of coordinates between them).
The reasoning here constrains itself to explaining the mechanics of that phenomenon according to a consistent model of the cohesive dynamics of a UUS incorporating all such frames. That model is a dynamic purely geometric model based on the ideal of 'cohesive equilibrium' towards which the unitary 'cohesive dynamic' within that substance inherently inclines -- according to a principle of increasing 'cohesive resonance' between constituent components --, to which an appropriate time scale, essentially a fundamental oscillatory period which must be argued to inhere within such a universal unity, is then ascribed.