The Symmetry of the Lorentz Distortions in Pairs of Inertial Frames. The four Lorentz distortions make it impossible to detect absolute rest (or absolute motion) by any local experiment, that is, by ordinary interactions among material objects on moving inertial frames, such as interferometers and comparing light clocks with dynamic clocks. But as Einstein's argument emphasized, the empirical equivalence of inertial frames implies that they are equivalent globally as well as locally. It is also impossible to detect absolute motion by experiments involving the relationships between inertial frames with high relative velocity, for example by comparing how fast their clocks are ticking or how long their measuring rods are. And as the symmetry of the two sets of Lorentz transformation equations implies, what makes it impossible to detect absolute motion by such global experiments is that the Lorentz distortions always appear to be occurring in the other inertial frame as a function of the velocity of the two references relative to one another. Thus, in order to explain the empirical equivalence of inertial frames ontologically, we must explain this symmetry in the members of any pair of inertial frames as an appearance.

The first step in that explanation is to take note of how clocks on inertial frames are mis-synchronized by using Einstein’s definition of simultaneity at a distance, if the velocity of light is actually due to an inherent motion in space itself.

The second step is to show how that mis-synchronization of clocks on inertial frames moving rapidly across space combines with the Lorentz distortions that they are actually suffering as a result of their absolute motion to make it appear that Lorentz distortions are always in the other inertial frame (and that the rate seems to be a function of their relative velocity).

The mis-synchronization of moving clocks. The strategy of spatiomaterialism is to explain the truth of the principle of relativity on the assumption that all forms of matter, including light and material objects, coincide with parts of space. The assumption that both matter and space are substances enduring through time makes it possible to explain presentist change, but it also entails that space and time are absolute. Thus, it must reject Einstein’s definition of simultaneity at a distance.

Einstein stipulates that a local event is simultaneous with the moment of reflection of a light signal from a distant mirror when that local event occurs halfway through the total period required for the signal to travel there and back. That is to assume that the velocity of light is the same in both directions. This assumption is true on inertial frames at absolute rest, but it is not true on objects moving through absolute space. If light everywhere has a fixed velocity relative to absolute space, the velocity of light relative to a moving frame is slower traveling outward in the direction of forward motion and faster in the opposite direction. Thus, clocks on moving frames that are synchronized according to Einstein’s definition of simultaneity at a distance will be actually mis-synchronized. It is important to be clear about the nature and amount of the error introduced, because mis-synchronization plays a crucial role in causing the appearances that make absolute motion undetectable by comparing inertial frames with one another, or the symmetry of Lorentz distortions in pairs of inertial frames.

The most revealing way to show the mis-synchronization is to use a diagram to represent the spatial and temporal relations among the relevant events. This is to use the Newtonian diagram of space and time, which is the spatiomaterialist counterpart to Minkowski’s “graphical method” of using spacetime diagrams for “visualizing” what is going on, and it is both simpler and easier to understand. Since spatiomaterialism assumes that space is a substance and, thus, absolute, the argument may begin with the coordinate frame at rest in absolute space. Nothing precludes representing time as an axis perpendicular to spatial dimensions, as long as we do not assume that anything exists but what is located on lines parallel to our absolute space-axis (horizontal lines in the diagram) for each moment. We can refer to events in the past and future, even though they do not exist, because they can be interpreted as references to space and matter which have, as substances, an existential aspect that entails that they did exist and will exist. We can also represent the motion of the other inertial frame as a timeline whose slope depends on its velocity (t = x/v), as Minkowski did. Furthermore, we can take this timeline to be the time-axis of the moving inertial frame, because that involves only a simple Galilean coordinate transformation of the kind used in Newtonian physics. So far, this is equivalent to Minkowski’s spacetime diagram.

Spatiomaterialism cannot, however, go on to assume that the moving frame has a space-axis that is inclined relative to our absolute space-axis, as Minkowski's spacetime diagram does. We must assume that moving measuring rods always lie parallel to the absolute space-axis, since all parts of moving rods are particular substances and must exist at the same time. But spatiomaterialism does hold, following Lorentz, that moving measuring rods lying in the direction of motion are contracted, and so we must recognize that the moving measuring rod is shorter than it would be if it were at absolute rest. Now, to see the significance of Einstein’s definition of simultaneity at a distance, we need only consider the geometry of synchronizing clocks in absolute space and time, that is, from the point of view of the absolute frame depicted below. (See the diagram below comparing the synchronization of both forward and afterward clocks on the absolute and moving inertial reference frames.)

The foregoing diagram depicts the general nature of the mis-synchronization, but we will need to know just how much clocks are mis-synchronized. Thus, consider the following diagram in which the moving measuring rod is depicted as L'.

The length of the contracted moving measuring rod in absolute space is L'. It is depicted at four locations that it occupies at crucial moments during the process of synchronization. The thinner inclined lines trace the path of each end of the rod where clocks are located. The thin dotted-line represents the path of the light used to synchronize the clocks at each end. Following Einstein’s definition of simultaneity, (1) moving observers send a light signal forward from the origin of their frame, (2) the light is reflected from a mirror at the forward end of their measuring rod (and the clock there is set at 0), and (3) they record when it returns. Einstein’s definition requires moving observers to set their clocks on the assumption that the light was reflected halfway through the total period required for its round trip. Since the light signal reaches the mirror in the period T₁ and returns to the observers in the period T₂, they assume it was reflected at (T₁ + T₂)/2 after the light was sent. Thus, they set their nearest clock so that it would have read 0 at that moment. But since the measuring rod is actually in absolute motion, the light does not reach the mirror at the far end until it has passed both the length of the measuring rod and whatever distance the rod travels during the first leg (T₁). And on the return leg (T₂), light does not have to travel the whole distance of the measuring rod, since the other end is also moving toward the light. But since moving observers assume that the reflection occurs halfway through the period required for the round trip, they are, in effect, assuming that the set of simultaneous events lies on the line that runs through the halfway point on the timeline for the clock at the observers’ end of the measuring rod and the point of reflection at the mirror on the timeline for the clock at the forward end of the measuring rod. That is what moving observers take to be their space-line as seen by us from the frame at absolute rest.

The result of mis-synchronizing clocks is precisely the same diagram for the moving frame that Minkowski constructed from his hyperboloid curve, representing the conclusion of Einstein’s special theory. (The same results would also follow from the Lorentz transformation equations.) However, we have derived the moving observers’ apparent space-axis (or space-line), not from a mysterious equation, but in a perfectly intelligible way. The moving space-line is rotated upward in the diagram, because the moving clocks have been mis-synchronized. And they have been mis-synchronized because the moving observers have followed Einstein’s definition of simultaneity at a distance, which assumes that the velocity of light is the same both ways in every direction relative to any inertial frame.

The amount of the error introduced by mis-synchronization will be as important as its cause in the next step of this argument, so bear with me for one final point. The home clock reading 0 is one event in absolute space and time, and the forward clock reading 0 is another event. The separation between them in the absolute frame has a curious value, both in space and in time. The moving measuring rod has a length of L', but the distance in absolute space between these two events turns out to be L'/(1 - v²/c²), which means that the mis-synchronization makes it seem that the moving measuring rod is expanded at the square of the usual rate (see above diagram). The length of time between the two events can be derived from the slope of the moving space-line in the diagram for absolute space and time (that is, v/c²).^[1] This is the slope of the tangent to Minkowski’s mysterious curve at the point of intersection with the timeline for the observers’ nearest clock,^[2] and it occurs in the second expression in the numerator for the Lorentz transformation for time.^[3] But in this context, the slope means that the difference in time between the events is v/c²[L'/(1 - v²/c²)] (or the product of the slope of the moving space-line and the distance between the points on it in absolute space). We will use these values shortly.

The Cause of the Apparent symmetry of Lorentz distortions. Attempt to detect absolute motion by measuring the rate of clocks and the length of measuring rods on the other inertial frame are "global experiments," and the reason that absolute motion cannot be detected is that the Lorentz distortions appear to be symmetrical. Since transformation equations must work both ways between any two inertial reference frames, this symmetry is entailed by Einstein's argument for the Lorentz transformation equations in his special theory of relativity. And this symmetry is an essential part of the empirical equivalence of inertial frames that Poincaré called the "principle of relativity."

If the clocks and measuring rods were material objects in absolute space, this symmetry would imply that clocks on two inertial frames passing one another in space are both going slower than the other and that their longitudinally-oriented measuring rods are both shorter than one another. It is one of the reasons that Einsteinians must give up the belief in absolute space and time. By the same token, spatiomaterialism must explain this symmetry about pairs of inertial frames as a mere appearance of space and matter as substances enduring through time, just as the local equivalence was.

This is the part of the explanation of the empirical equivalence of inertial frames that Lorentz left out of his Newtonian theory. But it is readily supplied by the geometry of events in absolute space and time. The apparent symmetry of the distortions is a result of the actual Lorentz distortions suffered by the moving frame, together with the mis-synchronization of moving clocks, as we can see by considering how the measurements of the others’ clocks and rods are made.

Length contraction. Consider first the apparent symmetry of length contraction. The most direct way to measure the others’ standard of length is to make simultaneous marks from both ends of one’s own measuring rod onto the other inertial frame as it passes by and compare that distance with the others’ measuring rod. This works fine for absolute observers; they mark off a distance longer than moving measuring rods lying in the direction of motion, indicating that the moving measuring rods are contracted. But it also seems to moving observers that absolute rods are contracted in the direction of motion, and we can see why by considering what takes place in making the measurement.

It is because (1) clocks on the moving frame have been mis-synchronized and (2) moving measuring rods are contracted. We have just seen that moving observers mis-synchronize their clocks when they accept Einstein’s definition of simultaneity: the distance in absolute space between the events at which moving clocks at both end of a moving measuring rod read the same time is equal to an expansion of the actually contracted measuring rod at the square of the usual rate, that is, (1 - v²/c²). Thus, when moving observers make what they think are simultaneous marks on the absolute measuring rod that is passing by, they mark off a distance on the absolute frame that is longer than their actually contracted measuring rod by the square of the usual rate, and since that distance is longer than the absolute measuring rod by the usual rate, the absolute measuring rod seems to be contracted at the usual rate.^[4]

In other words, as the absolute inertial frame comes toward them, the mis-synchronization of their clocks leads moving observers to make a mark from the afterward end of their own measuring rod first and then, after the moving frame has traveled some distance, they make a second mark from the forward end, so that distance marked off on the absolute frame includes both the length of the contracted moving measuring rod and all the distance that the absolute frame travels between making the two marks. That virtual expansion of the moving measuring rod makes it appear that the absolute measuring rod is contracted.^[5]

The error introduced by mis-synchronization is, in short, a virtual distortion at the square of the usual rate, but in the opposite direction, so that when the method of measuring combines it with the actual shrinkage of the moving measuring rod, the effect is to make absolute measuring rods seem distorted at the usual rate relative to the moving rod. This same “geometrical mechanism” is at work in the measurement of how fast the other’s clocks are ticking.

Time dilation. The most direct way for us to measure the speed of clocks on the other inertial frame is for us to move in our inertial frame along with one of the others’ clocks that is passing by and to compare it with the series of clocks on our own frame by which we will be passing. (Observers cannot take a clock with them as they move through their own frame, because that would make it a clock on the other frame. But nothing precludes observers from keeping up with the other inertial frame and using clocks already located at various points on their frame for the comparison.) When observers on the frame at absolute rest keep up with the moving clock and compare it with a series of their absolute clocks, they observe the real slowing down of the others’ clock caused by its absolute motion. The symmetry of the distortions means, however, that when observers on a frame in absolute motion keep up with an absolute clock and compare it with the series of their own moving clocks by which they pass, the absolute clock seems to be slowed down.

But in the latter case, it is because (1) clocks on the moving frame have been mis-synchronized, (2) the moving observers are moving backwards on their own moving frame (-v) to keep up with the absolute clock, and (3) clocks on the moving frame are slowed down. The amount of deviation of a distant moving clock from absolute simultaneity with a local moving clock is, as we saw, a function of the distance in absolute space between the events at which two moving clocks have the same readings, namely, v/c² times the absolute distance (the slope of the rotated space line). In this measurement, that distance depends on how long the moving observer has been traveling at -v, that is, the distance -vt'. Thus, the deviation of the next clock from absolute simultaneity will be VT times v/c², or -t'(v²/c²). That amount of time plus the time that elapses during the moving observers trip from one clock to the next (that is, t') yields a total apparent time period of t' - t'(v²/c²), or t'(1 - v²/c²), which is a virtual speeding up of moving clocks at the square of the usual rate of distortions. Thus, since (1), the mis-synchronization of moving clocks, combines with (2), the moving observers’ motion on the moving frame, to produce, in effect, a virtual speeding up of moving clocks at the square of the usual rate, the result, when combined with (3), the actual slowing down of moving clocks at the usual rate, is that the absolute clock being compared with them appears slowed down at the usual rate.^{[6], [7]}

In sum, given how the measurements are made, the mis-synchronization of moving clocks introduces a virtual distortion through which the moving observers’ own distortions are projected onto the absolute inertial frame. This can be seen in our diagram of events happening to particular substances in absolute space and time, for as we found, the mis-synchronization shows up as a rotation of the moving space-line that involves both a virtual speeding up of moving clocks and a virtual lengthening of moving measuring rods. Thus, to see how it gives rise to the apparent symmetry of the distortions, consider how the measurement of the others’ clock is represented below.

When absolute observers keep up with the moving clock and compare it with a series of their own clocks, they follow the moving timeline. When the moving clock says t'=1, they compare it with an absolute clock (located on that absolute space-line) which reads t=1/(represented by the horizontal line labeled I in the diagram). And when moving observers travel backwards on their own frame to keep up with the absolute clock, they follow the absolute timeline (x=0). When they pass by their own moving clock reading t'=1, they compare it with the absolute clock which reads t= (represented by the rotated moving space-line labeled II in the diagram). The difference between these two measurements is obviously due to the rotation of the moving space-line, which, as we have seen, comes from mis-synchronizing moving clocks. Notice that the absolute clock’s reading of t=1 lies between these two comparisons. Therein lies the power of mis-synchronization to cause the appearance. Combining the slope induced in the moving space-line by mis-synchronization (v/c²) with the movement of the moving observers in making the measurement (x' = VT, that is, keeping up with the absolute clock) is equivalent to a temporal distortion on the moving frame at the square of the rate of the actual distortion (1-v²/c²), but in the opposite direction. So, it combines with the actual slowing down of moving clocks to make the absolute clock seem slowed down relative to moving clocks.

The diagram also shows how the mis-synchronization is responsible for the apparent symmetry of the contraction of measuring rods. But in this case, it is the virtual expansion of the moving measuring rods induced at the square of the usual rate by the mis-synchronization that is relevant. When absolute observers make simultaneous marks on the moving frame, they find that the moving measuring rod is contracted at the usual rate (labeled III in the diagram). But when moving observers make what they think are simultaneous marks on the absolute frame, they actually mark off a distance that is expanded at the square of the usual rate (labeled IV in the diagram). Once again, the power of mis-synchronization can be seen in how the actual moving measuring rod is contracted relative to the absolute measuring rod and the virtual moving measuring rod is expanded relative to the absolute rod, both at the usual rate.

The symmetry of Lorentz distortions is, therefore, a symmetry betwen real distortions in reference frames in absolute motion and apparent distortions in the reference frame at absolute rest, and it is a thoroughgoing symmetry, which holds for all the basic ways of measuring the other frame's clocks and measuring rods. Indeed, any of the standard measurements can made from either member of the pair of inertial frames, though when they are considered from the point of view of the other inertial observer, they reveal that the other's clocks are speeded up and the other's measuring rods are expanded in the direction of motion. This can be seen in the table of measurements.

This explanation of the apparent symmetry of the kinematic distortions also accounts for the apparent symmetry of the dynamic distortions (though the longitudinal distortion in the force field is not always recognized as such by Einsteinians), for the apparent increase in absolute masses is implied by the false belief that absolute clocks are slowed down and the assumption that Newton’s laws apply the same way on all inertial frames (Einstein’s principle of relativity). Likewise, the apparent decrease in longitudinal forces is implied by Einstein’s principle of relativity and the false belief that absolute measuring rods are contracted in the direction of motion.

The apparent symmetry of the four distortions has been explained for the special case in which one of the inertial frames is at absolute rest, but it can be generalized to explain the apparent symmetry between any two objects moving in absolute space. In the general case, the rate of the apparent distortions is a function of their (apparent) relative velocity, and what is detected on both sides is partly a result of real distortions and partly illusions caused in the way described above.^[8]

Though observers on any pair of inertial frames agree about their relative velocity, it is worth noting that, on the spatiomaterialist explanation of the empirical equivalence, their measurements of relative velocity do not coincide with their real velocity relative to one another in absolute space: the apparent relative velocity is never more than the velocity of light, but the real velocity of inertial frames relative to one another can approach twice the velocity of light, because light moves at that velocity in opposite directions from any given point in absolute space.

Conclusions. One part of the promise made in Spatiomaterialism in order to use this ontology as a foundation for demonstrating necessary truths has been kept. We have seen that spatiomaterialism can explain the truth of Einstein’s special theory of relativity, and means that nothing established empirically by Einstein’s theory forces us to give up spatiomaterialism. Thus, if spatiomaterialism can also explain the truth of Einstein’s general theory of relativity (and quantum mechanics), physics will provide no grounds for doubting that spatiomaterialism is the best ontological explanation of the world. But there are a few implications of this ontological explanation of special relativity that should be noted in conclusion.

First, though we have discovered the power of absolute velocity to cause changes in material objects by following in the footsteps of Lorentz, that does not mean that we must postulate an ether in addition to absolute space.

Lorentz and Poincaré both expected to explain time dilation and length contraction as the result of an interaction between material objects and an ether at rest in absolute space (as if material objects were made of nothing but electrons that interact with the electromagnetic ether as they move through it). Though material objects must also have something to interact with on our explanation of the Lorentz distortions, we can take it to be space itself. We have postulated space as a substance that contains matter, and having already used that relationship to explain the truth of the laws of classical physics, we now use it to explain the Lorentz distortions. Indeed, I have suggested reasons for expecting Lorentz distortions to occur apart from what is necessary to make absolute motion undetectable.

Though there is no luminiferous ether, there is still a medium of light propagation, and it still makes sense to hold that there is an inertial frame in which light has the same one-way velocities in every pair of opposite directions. That will be important in our explanation of the truth of Einstein's general theory of relativity, because we will not always assume that the light medium is at absolute rest in space. The aspect of space by which it serves as the medium of light propagation is more complex than it appears now, because we shall have to assume that the velocity of light varies with location in space in a way that can be seen as depending on the velocity of the light medium relative to space. It is as if the ether were being accelerated in space, but even though that may suggest that the light medium is an ether after all, we will still not postulate an ethereal substance coinciding with space to explain this phenomenon.

Second, the difference between the actual Lorentz distortions in material objects with absolute velocity and the apparent symmetry of Lorentz distortions in pairs of inertial frames revealed by this ontological explanation shows that the mathematical representation of special relativity is hiding an aspect of reality.

The mathematical way of saying that inertial frames are all equivalent is to say that the laws of physics are covariant, or Lorentz covariant. That means that laws of physics that apply in one frame take the same form in any other inertial frame, that is, when they are subjected to the Lorentz transformation. (This equivalence is what is represented by Minkowski’s equation for the absolute separation between any two events and is the foundation for the equations of four-vector physics, which do not mention any specific inertial frame.) Einstein’s original article showed that covariance holds in the case of electromagnetism, and imposing covariance as a requirement on other physical theories has generated predictions that turn out to be true.

Despite the obvious simplicity, comprehensiveness, elegance, and fruitfulness of this mathematical representation of special relativity, however, it is a mistake to take covariance to be the deepest and most complete truth about the real nature of the world. Our ontological explanation of the truth of special relativity reveals that covariance actually represents two different phenomena, with two different ontological causes. There is the local equivalence of inertial frames, which is caused by the actual Lorentz distortions, and there is the global equivalence, which is caused by the mis-synchronization of clocks and how that makes one’s own Lorentz distortions appear to be in the other inertial frame.

Third, this ontological interpretation of the mathematical representations used in special relativity confirms that the method of physics is implicitly skeptical about ontological causes that are not entailed by realism about its efficient cause explanations.

When physics infers to the best efficient-cause explanation, it looks for laws of nature that represent the quantitative aspects of the regularities involved, because such mathematical representations can often be used to predict surprising, precise measurements that confirm their truth. The empirical method of science is so dependent on mathematical representations that, once experiments have confirmed their predictions, physicists are realists about their efficient-cause explanations. They let scientific realism determine their ontology.

Accordingly, the belief in spacetime is simply realism about special relativity. That is, substantivalism about spacetime is the ontology that results from taking the simplest mathematical theory that can predict all the relevant phenomena to correspond to what exists. Since the special relativity holds that all inertial frames are empirically equivalent, scientific realism takes the empirical equivalence among inertial frames to be an ontological equivalence. That is to replace absolute space and time with spacetime. But it is also the leave out an aspect of reality, for it is to ignore the observable fact that only the present exists.

Finally, the principle of relativity itself turns out to be merely a practical principle, without ontological significance. Though as a practical matter, the assignment of coordinates to events can be made only relative to an inertial frame whose absolute motion cannot be known, that does not mean that they do not have actual locations in absolute space as time passes. There is an absolute truth about the dates and places of events. Even though we can never know what they are, we can know that there is a fact of the matter about when and where they occur. That is what is implied by this ontological reduction of special relativity. I have called it an explanation of empirical equivalence, because by explaining the apparent truth of the principle of relativity, it denies that this relativity is a basic principle of physics.^[9]