Einstein’s Errors


Albert Einstein introduced the special theory of relativity (STR) in his paper, On The Electrodynamics of Moving Bodies” (1905). He later explained STR in a somewhat less technical book, Relativity: The Special and General Theory (English translation, 1920). From my reading of the book and other sources, some of which are listed in the bibliography below, I have found what I believe to be errors in STR — or at least in the explanations of it offered by Einstein and accepted by “consensus” among physicists. (The “sneer quotes” are meant to deride the anti-scientific notion of truth by consensus. In any event, there were and are qualified dissidents, including the late Dr. Thomas E. Phipps Jr., who has a place in my bibliography.)

A lot of what I say here will be jarring to readers who accept Einstein’s STR. I welcome their comments, and the comments of others. But readers should know that I will automatically delete any comment unless it is directed specifically at the facts, physical explanations, mathematics, or logic of this page — or which point to useful source materials. General comments (pro or con) are useless, as are comments that simply regurgitate or defend the standard explanation of special relativity without addressing what I say here. I understand the standard explanation, but I believe it to be questionable and even flawed, for the reasons given here.

If comments have been closed, you may send an e-mail message to me at this address: the Germanic nickname for Friedrich followed by the last name of the great Austrian economist and Nobel laureate whose first name is Friedrich followed by the 3rd and 4th digits of his birth year followed by the usual typographic symbol followed by the domain and extension for Google’s e-mail service — all run together.


The Impetus for This Page

Special Relativity: The Standard Explanation

A Fatal Flaw?

A Further Look at Simultaneity

The Velocity Conundrum

Getting Light Right

Further Thoughts on the Meaning of Spacetime and the Validity of STR



Why am I, a mere amateur, writing about Einstein’s special theory of relativity? First, it has long intrigued me because it seems to me counterintuitive, as it does to most people who know at least a little bit about it. If it’s true that the speed of an object affects its length and the rate at which it ages, what is the physical explanation for such phenomena? If there is one, is is buried deeply under reams of mathematical derivations that assume the rightness of STR. Similarly, there are tons of books and articles that take STR as gospel and merely present illustrations of its mysterious effects, without ever really explaining the underlying physical phenomena.

Second, deeper reading into STR has led me to dissidents — not just crackpots, but real scientists who have uncovered flaws in STR. If there are flaws, why is STR and its successor, Einstein’s theory of general relativity (GR), still almost universally accepted and taught as “the truth”? In other words, how dare those skeptics question a “scientific consensus”?

Real science isn’t about consensus, of course, it’s about finding the best possible explanations for what happens in the observable world — without ever knowing with certainty that the most recent explanations represent the “truth.” In fact, the explanations have changed drastically over time because old explanations have been challenged by the facts and logic of newer explanations, often despite what some would have called a “consensus” in favor of the old explanations. (See “The Science Is Settled”.)

In the case of STR, three real scientists (among many) who have uncovered flaws are the late Thomas E. Phipps Jr. and Christoph von Mettenheim. I begin with Mettenheim (2015), who says that

opinions on the meaning of a word can vary even if the word remains the same. The approach thus opens the possibility of ostensibly solving a problem by using some concept in one meaning at the beginning of an investigation and using it in a different meaning at its end…. Einstein repeatedly reached his results by that method…. At this point an example from his mathematics may serve to show its consequences…. The only mathematical knowledge needed for understanding the following is that if we do something on one side of an equation, then we must do the same also on the other side.

In his famous paper “On the Electrodynamics of Moving Bodies,” containing the first presentation of his Special Theory of Relativity, Einstein violated that principle. After introducing the premise that the speed of light is always constant, independent of the motion of its source, he considered a system with the ends A and B and defined the synchronism of two ‘clocks’ located at those points…

Mettenheim then walks through Einstein’s derivation. I will quote here only the key points made by Mettenheim:

The purport of equation (1) is that assuming the speed of light to be constant, a light signal travelling in a system at rest from A to B and back to A will take equal time on both ways….

Two pages further down, in § 2 of his paper, he introduced the following equations [(3) and (4)] for defining the time taken by light in a moving system on its single ways from A to B and back from B to A after reflection in B….

The expressions on the left sides of equations (3) and (4) are the same as those on either sides of (1). Einstein did not introduce new definitions of his symbols. Equation (1) therefore implies that the left sides of (3) and (4) are equal. Their right sides must then also be equal which gives us [equation (5)]….

One glance will show that equation (5) cannot possibly be correct. The numerators on both sides are identical whereas the denominators differ only in the symbols ‘+’ and ‘-’….

This shows that Einstein’s equations either violate the definition of ‘=’ or, alternatively, one of those of ‘+’ or ‘-’. The origin of that self-contradiction lay in equation (2), where he shifted the meaning of one of the symbols he used.

Mettenheim continues:

The fact that a mistake of that importance could have remained undiscovered so long is more interesting than the mistake itself. It casts a strange light on the state of theoretical physics in the past century. Even if Einstein’s self-contradiction were explicable somehow, one still would expect some other physicist to have discussed or even published that explanation somewhere. But that never happened….

… The fact that Einstein had based his special theory of relativity on the mathematical self-contradiction shown above gave me a new argument supporting my own explanation of gravity. And the fact that his mistake had remained undiscovered in a whole century gave me a new argument for demonstrating the weakness of the tradition of theoretical physics in the 20th Century. If a mathematical mistake as obvious as the one just shown could remain undiscovered for a whole century, then what are we to think of the far more complicated calculations of quantum theory or of the general theory of relativity?…

Though criticizing Einstein severely in this essay, I am doing my utmost to be fair to his memory. By neither styling him a genius nor accusing him of dishonesty I think I am not only doing him more justice but also being fairer to him than his uncritical admirers are. All those following him blindly put on him alone the responsibility for leading theoretical physics astray in more than a century. They fail to see that the far more important reason for that crisis is not in Einstein but in his followers. It is in their lack of critical faculties and of independence of thinking, in their exaggerated desire for geniality and in the barren intellectual soil which that attitude left over for permitting creativity to survive also in the field of science.

Here is what Phipps (2012) says about Mettenheim’s Popper versus Einstein:

This seems to me a deeply thought-out critique of Einstein’s theories from an unexpected quarter — that is, from a purely philosophical viewpoint. Von Mettenheim’s approach to time bears essential similarities to the one advanced here, but is supported by entirely different and independent arguments. I find it remarkable that by the application of purely philosophical principles a non-physicist can arrive at a deeper and more practically useful conception of “time” than the physicists have succeeded in doing….

The fallacy … evidently lies in analogical thinking, of the sort Einstein himself employed on many occasions…. [But] the only test of [theory] is empirical…

Philosophers as a profession seem more or less supinely to have joined the chorus of Einstein adulation — he being a thinker [i.e., analogous] after their own heart…. It is good to find an exception…. Apparently not all philosophers stand clueless under the aspect of modern physics.

I must underline Phipps’s point about analogical thinking:

An analogy is a comparison between two objects, or systems of objects, that highlights respects in which they are thought to be similar. Analogical reasoning is any type of thinking that relies upon an analogy. An analogical argument is an explicit representation of a form of analogical reasoning that cites accepted similarities between two systems to support the conclusion that some further similarity exists. In general (but not always), such arguments belong in the category of inductive reasoning, since their conclusions do not follow with certainty but are only supported with varying degrees of strength….

… Steiner (1989, 1998) suggests that many of the analogies that played a major role in early twentieth-century physics count as “Pythagorean.” The term is meant to connote mathematical mysticism: a “Pythagorean” analogy is one founded on purely mathematical similarities that have no known physical basis at the time it is proposed….

… [N]obody has yet provided a satisfactory scheme that characterizes successful analogical arguments. This strategy would face an additional problem even if we could find such a characterization: the appeal to past triumphs provides no insight or explanation for the fact that a certain class of analogies has often been successful. [Paul Bartha, “Analogy and Analogical Reasoning,” The Stanford Encyclopedia of Philosophy, June 25, 2013]

Einstein’s thought experiments are a kind of analogical thinking. In what follows I will sometimes essay a thought experiment of my own, though mainly to suggest the inadequacy of Einstein’s thought experiments or reasoning. The reader, in evaluating what Einstein and I say, should keep this in mind:

[T]hought experimenting seems to be constrained only by relevant logical impossibilities and what seems intuitively acceptable. This is indeed problematic because intuitions can be highly misleading and relevant logical impossibilities fairly ungrounded if they cannot be supplemented by relevant theoretical impossibilities based on current science in order to avoid the jump into futile fantasy. [James Robert Brown and Yiftach Fehige, “Thought Experiments,” The Stanford Encyclopedia of Philosophy , August 12, 2014]

In other words, it ain’t empirical.

As Phipps (2012) puts it, STR

is the precise modern counterpart of Ptolomy’s [geocentric] theory [of the universe]; it is not physics, but mathematical science based on a philosophy picked up in the street — that of spacetime symmetry, which has exactly as much physical legitimacy as Ptolemy’s divinity of the perfect circle. Both “philosophies” are pre-shaped molds into which the human mind, because it likes symmetries, has decided to pour experience.

On with the show.


The principle of relativity predates Einstein. It says that all bodies are in relative motion, and that none of them holds a privileged place in the reckoning of time and space. The translation of time and space between inertial bodies (frames of reference) is straightforward in GalileiNewton relativity, where time is the same in all frames of reference (i.e., there is absolute time) and spatial differences between frames of reference depend simply on their relative velocity (i.e., there is absolute space). In mathematical terms:

t’ = t ,
x’ = x – vt ,
y’ = y ,
z’ = z ,

where t’, x’, y’, and z’ denote the time and position of a frame of reference S’ that is in motion along the x-axis of frame of reference S, the coordinates of which are denoted by t, x, y, and z, and where v is the velocity of S’ relative to S (in the x-direction); and where x’ = x at x = 0 . (I will later explain why it is x’ = x – vt instead of x’ = x  + vt .)

Einstein introduced special relativity to account for the finite speed of light and its effects on time and space. Because light moves at the same, constant speed for all observers, an observer in one inertial frame of reference (S) will not necessarily see an event occur at the same time as an observer in another inertial frame of reference (S’). This, according to Einstein, destroys the idea of absolute time and, therefore, the idea of absolute simultaneity.

Special Relativity” at Wikipedia gives a good account of the alternative devised by Einstein:

Reference frames play a crucial role in relativity theory. The term reference frame as used here is an observational perspective in space which is not undergoing any change in motion (acceleration), from which a position can be measured along 3 spatial axes. In addition, a reference frame has the ability to determine measurements of the time of events using a ‘clock’ (any reference device with uniform periodicity).

An event is an occurrence that can be assigned a single unique time and location in space relative to a reference frame: it is a “point” in spacetime. Since the speed of light is constant in relativity in each and every reference frame, pulses of light can be used to unambiguously measure distances and refer back the times that events occurred to the clock, even though light takes time to reach the clock after the event has transpired.

For example, the explosion of a firecracker may be considered to be an “event”. We can completely specify an event by its four spacetime coordinates: The time of occurrence and its 3-dimensional spatial location define a reference point. Let’s call this reference frame S.

In relativity theory we often want to calculate the position of a point from a different reference point.

Suppose we have a second reference frame S′, whose spatial axes and clock exactly coincide with that of S at time zero, but it is moving at a constant velocity v with respect to S along the x-axis.

Since there is no absolute reference frame in relativity theory, a concept of ‘moving’ doesn’t strictly exist, as everything is always moving with respect to some other reference frame. Instead, any two frames that move at the same speed in the same direction are said to be comoving. Therefore, S and S′ are not comoving.

Define the event to have spacetime coordinates (t,x,y,z) in system S and (t′,x′,y′,z′) in S′. Then the Lorentz transformation specifies that these coordinates are related in the following way:

  \begin{align} t' &= \gamma \ (t - vx/c^2) \\ x' &= \gamma \ (x - v t) \\ y' &= y \\ z' &= z , \end{align}


\gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}

is the Lorentz factor and c is the speed of light in vacuum, and the velocity v of S′ is parallel to the x-axis. The y and z coordinates are unaffected; only the x and t coordinates are transformed….

These effects are not merely appearances; they are explicitly related to our way of measuring time intervals between events which occur at the same place in a given coordinate system (called “co-local” events). These time intervals will be different in another coordinate system moving with respect to the first, unless the events are also simultaneous. Similarly, these effects also relate to our measured distances between separated but simultaneous events in a given coordinate system of choice. If these events are not co-local, but are separated by distance (space), they will not occur at the same spatial distance from each other when seen from another moving coordinate system. However, the spacetime interval will be the same for all observers….

Two events happening in two different locations that occur simultaneously in the reference frame of one inertial observer, may occur non-simultaneously in the reference frame of another inertial observer (lack of absolute simultaneity).

From the first equation of the Lorentz transformation in terms of coordinate differences

\Delta t' = \gamma \left(\Delta t - \frac{v \,\Delta x}{c^{2}} \right)

it is clear that two events that are simultaneous in frame S (satisfying Δt = 0), are not necessarily simultaneous in another inertial frame S′ (satisfying Δt′ = 0). Only if these events are additionally co-local in frame S (satisfying Δx = 0), will they be simultaneous in another frame S′….

The time lapse between two events is not invariant from one observer to another, but is dependent on the relative speeds of the observers’ reference frames (e.g., the twin paradox which concerns a twin who flies off in a spaceship traveling near the speed of light and returns to discover that his or her twin sibling has aged much more).

Suppose a clock is at rest in the unprimed system S. The location of the clock on two different ticks is then characterized by Δx = 0. To find the relation between the times between these ticks as measured in both systems, the first equation can be used to find:

\Delta t' = \gamma\, \Delta t for events satisfying \Delta x = 0 \ .

This shows that the time (Δt′) between the two ticks as seen in the frame in which the clock is moving (S′), is longer than the time (Δt) between these ticks as measured in the rest frame of the clock (S)….

The dimensions (e.g., length) of an object as measured by one observer may be smaller than the results of measurements of the same object made by another observer (e.g., the ladder paradox involves a long ladder traveling near the speed of light and being contained within a smaller garage).

Similarly, suppose a measuring rod is at rest and aligned along the x-axis in the unprimed system S. In this system, the length of this rod is written as Δx. To measure the length of this rod in the system S′, in which the rod is moving, the distances x′ to the end points of the rod must be measured simultaneously in that system S′. In other words, the measurement is characterized by Δt′ = 0, which can be combined with the fourth equation to find the relation between the lengths Δx and Δx′:

\Delta x' = \frac{\Delta x}{\gamma}   for events satisfying \Delta t' = 0 \ .

This shows that the length (Δx′) of the rod as measured in the frame in which it is moving (S′), is shorter than its length (Δx) in its own rest frame (S)….

The usual example given is that of a train (frame S′ above) traveling due east with a velocity v with respect to the tracks (frame S). A child inside the train throws a baseball due east with a velocity u′ with respect to the train. In nonrelativistic physics, an observer at rest on the tracks will measure the velocity of the baseball (due east) as u = u′ + v, while in special relativity this is no longer true; instead the velocity of the baseball (due east) is given by the … equation: u = (u′ + v)/(1 + uv/c2).

All of these effects, even if “real,” are symmetrical. An observer in the “moving” frame (S’) would see the same things happening in the “rest” frame (S). (Technically, anything moving inside a frame of reference is its own frame of reference, inasmuch as it is moving relative to the spacetime coordinates of the frame in which it moves.)

Now for the crucial question: What does the finite speed of light have to do with all of this? The light-clock thought experiment is a good place to start.

Imagine an improbably tall tube with a mirror at each end. A flash of light bounces back and forth from top to bottom. From the standpoint of an observer (S’) standing next to the tube It takes 1 second for the light to go from end to end. Imagine further that the tube is moving uniformly from left to right, as seen by a second observer (S), who can be thought of as stationary, for the purpose of this thought experiment.

As the tube moves, S sees the flashes of light moving diagonally rather than vertically. That is, the distance between the top and bottom of the tube seems to be longer (from the viewpoint of S). S therefore reckons that the light in the moving tube takes more than 1 second to go from top to bottom. In other words, S perceives that the moving light clock is running slow. That is time dilation.

There is a less lucid explanation here, but it gives the mathematical derivation of the time-dilation effect. The resulting relationship is as stated earlier, in the long quotation from the Wikipedia article about special relativity:

\Delta t' = \gamma\, \Delta t for events satisfying \Delta x = 0 \ .

The clock time at S’, as perceived by S, can be expressed as

T’t = T0 + (Δt/γ) , where

T’0 = T0 when the clocks at S’ and S are adjacent in spacetime and are set to the same time

T’t is the time at S’ (according to S) after the elapse of Δt.

Take the case where S’ and S are adjacent at noon, and both of their clocks read 12:00. When it is 13:00 at S (Δt = 1:00) and γ = 2 (at v ≈ 0.87), the clock at S’ will read 12:30.

Length contraction is derived from time dilation by applying a relativistic version of the classical relationship between distance, velocity, and time, d = vt . Given a velocity, time slows as discussed above. As time slows, distance must necessarily become smaller. Distance, in this case, is the span between the endpoints of a rigid object. The mathematical derivation is more complex, but that’s the gist of it. The result is as given earlier:

\Delta x' = \frac{\Delta x}{\gamma}   for events satisfying \Delta t' = 0 \ .

Finally, the finite speed of light is said to undermine the traditional concept of simultaneity. Einstein (1916), draws this conclusion from his train-and-embankment thought experiment (discussed at length, later):

Events which are simultaneous with reference to the embankment are not simultaneous with respect to the train, and vice versa (relativity of simultaneity). Every reference-body (co-ordinate system) has its own particular time; unless we are told the reference-body to which the statement of time refers, there is no meaning in a statement of the time of an event.

We shall see.

One more thing before I proceed. What I have said thus far leaves a lot of questions unanswered. Some of theme are merely what I would call mechanical ones; for example:

  • Is the speed of light really constant?
  • What is the experimental and observational evidence for the constancy of the speed of light, time dilation, and length contraction?
  • Are the relationships in STR real or merely perceptual?
  • If the Lorentz transformation is valid, doesn’t it implicitly admit the absoluteness of time and space?
  • Even if the train-embankment thought experiment is invalid, does that negate STR?
  • Is the light-clock thought experiment a valid application of the constancy and finitenesss of the speed of light?

I will address such questions, either directly or by implication, in the following essays.


Einstein (1920) begins with a discussion of the coordinates of space in Euclidean geometry, then turns to space and time in classical mechanics. In the following passage he refers to an expository scenario that recurs throughout the book, the passage of a railway carriage (train car) along an embankment:

In order to have a complete description of the motion [of a body], we must specify how the body alters its position with time; i.e. for every point on the trajectory it must be stated at what time the body is situated there. These data must be supplemented by such a definition of time that, in virtue of this definition, these time-values can be regarded essentially as magnitudes (results of measurements) capable of observation. If we take our stand on the ground of classical mechanics, we can satisfy this requirement for our illustration in the following manner. We imagine two clocks of identical construction; the man at the railway-carriage window is holding one of them, and the man on the footpath [of the embankment] the other. Each of the observers determines the position on his own reference-body occupied by the stone at each tick of the clock he is holding in his hand. In this connection we have not taken account of the inaccuracy involved by the finiteness of the velocity of propagation of light.

To get to that inaccuracy, Einstein begins with this:

Let us suppose our old friend the railway carriage to be travelling along the rails with a constant velocity v, and that a man traverses the length of the carriage in the direction of travel with a velocity w. How quickly, or, in other words, with what velocity W does the man advance relative to the embankment [on which the rails rest] during the process? The only possible answer seems to result from the following consideration: If the man were to stand still for a second, he would advance relative to the embankment through a distance v equal numerically to the velocity of the carriage. As a consequence of his walking, however, he traverses an additional distance w relative to the carriage, and hence also relative to the embankment, in this second, the distance w being numerically equal to the velocity with which he is walking. Thus in total he covers the distance W = v + w relative to the embankment in the second considered.

This is the theorem of the addition of velocities from classical physics. Why doesn’t it apply to light? Einstein continues:

If a ray of light be sent along the embankment [in an assumed vacuum], … the tip of the ray will be transmitted with the velocity c relative to the embankment. Now let us suppose that our railway carriage is again travelling along the railway lines with the velocity v, and that its direction is the same as that of the ray of light, but its velocity of course much less. Let us inquire about the velocity of propagation of the ray of light relative to the carriage. It is obvious that we can here apply the consideration of the previous section, since the ray of light plays the part of the man walking along relatively to the carriage. The velocity W of the man relative to the embankment is here replaced by the velocity of light relative to the embankment. w is the required velocity of light with respect to the carriage, and we have

w = c − v.

The velocity of propagation of a ray of light relative to the carriage thus comes out smaller than c.

Let’s take that part a bit more slowly than Einstein does. The question is the velocity with which the ray of light is traveling relative to the carriage. The man in the previous example was walking in the carriage with velocity w relative to the body of the carriage, and therefore with velocity W relative to the embankment (v + w). If the ray of light is traveling at c relative to the embankment, and the carriage is traveling at v relative to the embankment, then by analogy it would seem that the velocity of the ray of light relative to the velocity of the carriage should be the velocity of light minus the velocity of the carriage, that is c – v. (Einstein introduces some confusion by using w to denote this hypothetical velocity, having already used w to denote the velocity of the man walking in the carriage, relative to the embankment.)

It would thus seem that the velocity of a ray of light emitted from the railway carriage should be c + v relative to a person standing still on the embankment. That is, light would travel faster than c when it’s emitted from an object moving in a forward direction relative to an observer. But all objects are in relative motion, even hypothetically stationary ones such as the railway embankment of Einstein’s through experiment. Light would therefore move at many different velocities, all of them varying from c according to the motion of each observer relative to the source of light; that is, some observers would detect velocities greater than c, while others would detect velocities less than c.

But this can’t happen (supposedly). Einstein puts it this way:

In view of this dilemma there appears to be nothing else for it than to abandon either the [old] principle of relativity [the addition of velocities] or the simple law of the propagation of light in vacuo [that c is the same for all observers, regardless of their relative motion]. Those of you who have carefully followed the preceding discussion are almost sure to expect that we should retain the [old] principle of relativity, which appeals so convincingly to the intellect because it is so natural and simple. The law of the propagation of light in vacuo would then have to be replaced by a more complicated law conformable to the [old] principle of relativity. The development of theoretical physics shows, however, that we cannot pursue this course. The epoch-making theoretical investigations of H.A.Lorentz on the electrodynamical and optical phenomena connected with moving bodies show that experience in this domain leads conclusively to a theory of electromagnetic phenomena, of which the law of the constancy of the velocity of light in vacuo is a necessary consequence…..

[I]n reality there is not the least incompatibility between the principle of relativity and the law of propagation of light, and that by systematically holding fast to both these laws a logically rigid theory could be arrived at. This theory has been called the special theory of relativity….

Einstein gets to his STR by next considering the problem of simultaneity:

Lightning has struck the rails on our railway embankment at two places A and B far distant from each other. I make the additional assertion that these two lightning flashes occurred simultaneously. If now I ask you whether there is sense in this statement, you will answer my question with a decided “Yes.” But if I now approach you with the request to explain to me the sense of the statement more precisely, you find after some consideration that the answer to this question is not so easy as it appears at first sight.

After thinking the matter over for some time you then offer the following suggestion with which to test simultaneity. By measuring along the rails, the connecting line AB should be measured up and an observer placed at the mid-point M of the distance AB. This observer should be supplied with an arrangement (e.g. two mirrors inclined at 90 ◦) which allows him visually to observe both places A and B at the same time. If the observer perceives the two flashes of lightning at the same time, then they are simultaneous.

I am very pleased with this suggestion, but for all that I cannot regard the matter as quite settled, because I feel constrained to raise the following objection: “Your definition would certainly be right, if I only knew that the light by means of which the observer at M perceives the lightning flashes travels along the length A → M with the same velocity as along the length B → M. But an examination of this supposition would only be possible if we already had at our disposal the means of measuring time. It would thus appear as though we were moving here in a logical circle.”

After further consideration you cast a somewhat disdainful glance at me— and rightly so— and you declare: “I maintain my previous definition nevertheless, because in reality it assumes absolutely nothing about light. There is only one demand to be made of the definition of simultaneity, namely, that in every real case it must supply us with an empirical decision as to whether or not the conception that has to be defined is fulfilled. That my definition satisfies this demand is indisputable. That light requires the same time to traverse the path A → M as for the path B → M is in reality neither a supposition nor a hypothesis about the physical nature of light, but a stipulation which I can make of my own freewill in order to arrive at a definition of simultaneity.”

It is clear that this definition can be used to give an exact meaning not only to two events, but to as many events as we care to choose, and independently of the positions of the scenes of the events with respect to the body of reference (here the railway embankment). We are thus led also to a definition of “time” in physics. For this purpose we suppose that clocks of identical construction are placed at the points A, B and C of the railway line (co-ordinate system), and that they are set in such a manner that the positions of their pointers are simultaneously (in the above sense) the same. Under these conditions we understand by the “time” of an event the reading (position of the hands) of that one of these clocks which is in the immediate vicinity (in space) of the event. In this manner a time-value is associated with every event which is essentially capable of observation.

This stipulation contains a further physical hypothesis, the validity of which will hardly be doubted without empirical evidence to the contrary. It has been assumed that all these clocks go at the same rate if they are of identical construction. Stated more exactly: When two clocks arranged at rest in different places of a reference-body are set in such a manner that a particular position of the pointers of the one clock is simultaneous (in the above sense) with the same position of the pointers of the other clock, then identical “settings” are always simultaneous (in the sense of the above definition).

In other words, time is the same for every point in a frame of reference, which can be thought of as a group of points that remain in a fixed spatial relationship. Every such point in that frame of reference can have a clock associated with it; every clock can be set to the same time; and every clock (assuming great precision) will run at the same rate. When it is noon at one point in the frame of reference, it will be noon at all points in the frame of reference. And when the clock at one point has advanced from noon to 1 p.m., the clocks at all points in the same frame of reference will have advanced from noon to 1 p.m., and ad infinitum.

As Einstein puts it later,

Events which are simultaneous with reference to the embankment are not simultaneous with respect to the train, and vice versa (relativity of simultaneity). Every reference body (co-ordinate system) has its own particular time; unless we are told the reference-body to which the statement of time refers, there is no meaning in a statement of the time of an event.

Returning to the question of simultaneity, Einstein poses his famous thought experiment:

Up to now our considerations have been referred to a particular body of reference, which we have styled a “railway embankment.” We suppose a very long train travelling along the rails with the constant velocity v and in the direction indicated in Fig. 1. People travelling in this train will with advantage use the train as a rigid reference-body (co-ordinate system); they regard all events in reference to the train. Then every event which takes place along the line also takes place at a particular point of the train. Also the definition of simultaneity can be given relative to the train in exactly the same way as with respect to the embankment. As a natural consequence, however, the following question arises:

Are two events ( e.g. the two strokes of lightning A and B) which are simultaneous with reference to the railway embankment also simultaneous relatively to the train? We shall show directly that the answer must be in the negative.

FIG. 1.

When we say that the lightning strokes A and B are simultaneous with respect to the embankment, we mean: the rays of light emitted at the places A and B, where the lightning occurs, meet each other at the mid-point M of the length A → B of the embankment. But the events A and B also correspond to positions A and B on the train. Let M’ be the mid-point of the distance A → B on the travelling train. Just when the flashes of lightning occur, this point M’ naturally coincides with the point M, but it moves towards the right in the diagram with the velocity v of the train. If an observer sitting in the position M’ in the train did not possess this velocity, then he would remain permanently at M, and the light rays emitted by the flashes of lightning A and B would reach him simultaneously, i.e. they would meet just where he is situated. Now in reality (considered with reference to the railway embankment) he is hastening towards the beam of light coming from B, whilst he is riding on ahead of the beam of light coming from A. Hence the observer will see the beam of light emitted from B earlier than he will see that emitted from A. Observers who take the railway train as their reference-body must therefore come to the conclusion that the lightning flash B took place earlier than the lightning flash A. We thus arrive at the important result:

Events which are simultaneous with reference to the embankment are not simultaneous with respect to the train, and vice versa (relativity of simultaneity). Every reference body (co-ordinate system) has its own particular time; unless we are told the reference-body to which the statement of time refers, there is no meaning in a statement of the time of an event.

It’s important to note that there is a time delay, however minuscule, between the instant that the flashes of light are are emitted at A and B and the instant when they reach the observer at M.

Because of the minuscule time delay, the flashes of light wouldn’t reach the observer at M’ in the carriage at the same time that they reach the observer at M on the embankment. The observer at M’ is directly opposite the observer at M when the flashes of light are emitted, not when they are received simultaneously at M. During the minuscule delay between the emission of the flashes at A and B and their simultaneous receipt by the observer at M, the observer at M’ moves toward B and away from A. The observer at M’ therefore sees the flash emitted from B a tiny fraction of a second before he sees the flash emitted from A. Neither event corresponds in time with the time at which the flashes reach M. (There are variations on Einstein’s thought experiment — here, for example — but they trade on the same subtlety: a time delay between the flashes of light and their reception by observers.)

Returning to Einstein’s diagram, suppose that A and B are copper wires, tautly strung from posts and closely overhanging the track and embankment at 90-degree angles to both. The track is slightly depressed below the level of the embankment, so that observers on the train and embankment are the same distance below the wires. The wires are shielded so that they can be seen only by observers directly below them. The shielding doesn’t deflect lightning, so that when lightning hits the wires they will glow instantaneously. If lightning strikes A and B at the same time, the glow  will be seen simultaneously by observers positioned directly under the wires at the instant of the lightning strikes. Therefore, observers on the embankment at A and B and observers directly opposite them on the train at A’ and B’ will see the wires glow at the same time.

Because of the configuration of the wires in relation to the  track and the embankment, A and B must be the same distance apart as A’ and B’. That is to say, the simultaneity of observation isn’t an artifact of the distortion of horizontal measurements, or length contraction, which is another aspect of STR.

Einstein’s version of the thought experiment creates — unintentionally, I assume — an illusion of non-simultaneity. Of course an observer on the train at M’ would not see the lightning flashes at the same time as an observer on the embankment at M: The observer at M’ would no longer be opposite M when the lightning flashes arrive at M. But, as shown by my variation on Einstein’s though experiment, that doesn’t rule out the simultaneity of observations on the train and on the embankment. It just requires a setup that isn’t designed to exclude simultaneity. My setup involving copper wires is one possible way of ensuring simultaneity. It also seems to rule out the possibility of length contraction.

Einstein’s defense of his thought experiment (in the fifth block quotation above) is also an apt defense of my thought experiment. I have described a situation in which there is indubitable simultaneity. The question is whether it forecloses a subsequent proof of non-simultaneity. Einstein’s thought experiment didn’t, because Einstein left a loophole that discredits his proof of non-simultaneity. (I am not the first person who claims to have discovered the loophole.) My thought experiment leaves no loophole, as far as I can tell.

If my thought experiment has merit, it points to an invariant time, that is, a time which is the same for all frames of reference. A kind of Newtonian absolute time, if you will.


Phipps (2012) writes (pp. 166-167):

By destroying the conceptual basis for distant simultaneity Einstein got rid of the “now” that each of us perceives as dividing past from future. In so doing he discredited perception as a criterion of truth, and removed physics from the realm of personal experience by denying the description of Nature as manifested in experience as the goal and definition of physics…. Einstein proclaimed that the world as a progression of experiences from past to future was an illusion existing in a Minkowskian four-dimensional “world” of spacetime symmetry that was the only reality. in that world “now” was physically meaningless and played no role. Differently-moving observers disagreed on it, so it was illusory for all observers and all time…. Subsequently this new spacetime reality acquired a mathematical curvature that made it really real. The physical thus became a metaphor for the mathematical, the ultimate repository of truth; and personal experience no longer entered scholarly discussion except to exemplify the snares besetting the unanointed.

So, I ask, how was this epochpmaking sales job accomplished? In a word, through the example of “Einstein’s train.” This is of course an oversimplification. No single example could achieve such a bouleversement of human thought. But it was the clincher. Before it, legitimate doubt could exist — after it, doubt became aberration or dementia. Why was the train example so convincing? Simply because its homely materials made it directly accessible to the thought of Everyman. It systematic steps of reasoning so eloquently expressed the triumph of rationality that no sane person could resist. Here was pioneering science brought down to a level any attentive student could follow and appreciate. It was a truly definitive sales approach, the master virtue of which was that it left no reply possible, no objection admissible, no disagreement conceivable. At its conclusion the product had to be bought, regardless of cost to preconceptions….

To reiterate, Einstein’s conclusion from his Gedanken train example … is that spatially separated events judged simultaneous by one inertial observer are judged non-simultaneous by another; hence, distant simultaneity is “relative” — which is understood to mean that it does not exist. It is an un-concept.

Phipps then gives several pages to the mathematics of Einstein’s train. He could have saved himself a lot of trouble if he had simply pointed out Einstein’s trick, as I do in “A Fatal Flaw?”. The trick is to deny simultaneity, by calling it relative, because the “moving” observer (M’) sees the flashes of light at different times than the “stationary” observer (M).

I have concocted a thought experiment similar to Einstein’s that restores simultaneity. Here’s a diagram of the setup:

As before, lightning strikes the embankment at A and B, which are equidistant from M, who therefore sees the flashes at the same time. I have introduced an omniscient observer (O), who is on a platform directly above M. O has a mechanism that allows him to trigger the flashes at A and B at a time of his choosing. O can trigger the flashes so that they are seen simultaneously by M’ and M’, that is, when M’ and M are directly opposite one another as the train moves past M.

O triggers the flashes by sending a light-speed signal to high-intensity lamps at A and B. The lamp at A is aimed to the right; the lamp at B is aimed to the left. When the signal from O reaches the lamps, they turn on instantly, and their beams then travel toward M. O can time the sending of his signal so that the flashes arrive at M just as M’ arrives at M. M’ will therefore see the flashes at the same time as M; that is, M and M’ will see the flashes simultaneously and both will perceive that they emanated from A and B simultaneously.

Here is O‘s timing algorithm:

F = L/v – 2L/c , where

F = time at which O sends a signal to A and B, in seconds before M’ reaches A
L = distance A-M = M-B , in light-seconds
v = velocity of train, as a fraction of c
c = velocity of light

Thus if L = 1 and v = 0.5 , F = 0 ; that is, O sends the signal at the instant that M’ is directly opposite A. (It would be trivial to add a constant for any delay between the arrival of M’ at A, O‘s perception of that arrival, and O‘s sending of the signal to A and B.) In the case of v = 0.1 , F would occur 8 seconds before M’ reaches A; in the case of v = 0.9 , F would occur 0.89 seconds after M’ reaches A. In every case, the signal from A would catch up with M’ just as he is opposite M, and the signal from B would arrive at M’ just as he is opposite M.

What about time dilation? Doesn’t M’ “really” take less time to arrive at M than suggested by the algorithm? If that effect were real, it would be trivial to calculate the time-dilation effect and apply it to the estimate of F. Any apparent slowing of the clock at M’ wouldn’t affect O‘s measurement of time, which is what matters here.

Simultaneity is thus rescued from the jaws of special relativity.


STR depends on two postulates:

  1. The laws of physics are invariant (i.e. identical) in all inertial systems (non-accelerating frames of reference).
  2. The speed of light in a vacuum is the same for all observers, regardless of the motion of the light source.

In sum, the speed of light is the same for all observers, regardless of their relative motion and regardless of the motion of the source of light.

The postulates of STR purportedly negate GalileiNewton relativity, where time is the same in all frames of reference (i.e., there is absolute time) and spatial differences between frames of reference depend simply on their relative velocity (i.e., there is absolute space). In mathematical terms:

t’ = t ,
x’ = x – vt ,
y’ = y ,
z’ = z ,

where t’, x’, y’, and z’ denote the time and position of a frame of reference S’ that is in motion along the x-axis of frame of reference S, the coordinates of which are denoted by t, x, y, and z, and where v is the velocity of S’ relative to S (in the x-direction); and where x’ = x at x = 0 .


Source: Susskind Lectures, Special Relativity, Lecture 1 (Galilean relativity).

(It would seem that x’ = x  + vt , but x’ = x – vt because the x’ axis is represented by the vector τ .)

The “kinematical” part of STR is summarized in the equations of the Lorentz transformation:

[T]he Lorentz transformation … relates the coordinates used by one observer to coordinates used by another in uniform relative motion with respect to the first.

Assume that the first observer uses coordinates labeled t, x, y, and z, while the second observer uses coordinates labeled t’, x’, y’, and z’. Now suppose that the first observer sees the second moving in the x-direction at a velocity v. And suppose that the observers’ coordinate axes are parallel and that they have the same origin. Then the Lorentz transformation expresses how the coordinates are related:

 t'={\frac {t-{v\,x/c^{2}}}{{\sqrt {1-v^{2}/c^{2}}}}}\ ,
x'={\frac {x-v\,t}{{\sqrt {1-v^{2}/c^{2}}}}}\ ,
y'=y\ ,
z'=z\ ,

where c is the speed of light.

Here is a graph of the relationship between the velocity of a body (frame of reference) and the distance it travels in, say, a year of proper time (the time that would elapse in a “stationary” body):

Space-time tradeoff vs. velocity

This graph is based on Epstein (2000), specifically, figures 5-6 through 5-10 of and the accompanying discussion on pages 79-85. The graph implies that an object moving at the speed of light would use no time; it would move only along the spaced-used axis, at a constant time value of zero. Here is Epstein’s explanation:

The object does not age at all. The object has the maximum speed through space, the speed of light. Its speed through time is zero. It is stationary in time. “Right now is forever….”

This seems inconsistent with the fact that light has a finite velocity. Even an object that moves at the speed of light would take some amount of time to go any distance, which is what the graph means.

The “stationary” body (an abstract point) would use no space; it would move only along the time-used axis, at a constant space value of zero. The designation of a stationary body is problematic, as discussed below.

In any event, the curve depicts the intermediate relationships between velocity (and space used) and proper time used. For example, a body that moves at 0.7c would travel 0.7 light-year in a year, and would age only a bit more than 0.7 year.

The relationship between v and t’ can be computed as follows:

t’ = (1 – v2)1/2 ,

where t’ is the elapsed time (proper time), as perceived by S, when S’ is traveling at v (expressed as a fraction of c).

The faster a body moves the slower it ages relative to an observer in a “stationary” reference frame; that is, a clock in the moving body is seen by the “stationary” observer to advance at a slower rate than his own, identically constructed clock. (This is a reciprocal relationship, which I address elsewhere in this page.)

The aging rate is determined by the first of the four Lorentz equations given above, where t’ refers to the length of a time interval in S’ relative to the length of a time interval in S . For t > 0 and x = 0 (i.e.. a “stationary” body), t’ is always greater than t ; that is, a clock that ticks every second in S would tick every 1+ seconds in S’ . The greater the interval, the slower the clock runs, which is why, according to STR, time slows as velocity increases.

This is so — according to STR — because of the inextricable relationship between space and time, which is really a unitary four-dimensional “thing” called spacetime. Reverting to the usual conceptions of space and time as separate entities, we are all moving through time, whether or not we are moving through space. (STR is limited to inertial movement on a hypothetical plane, and does not address the gravitational movement of a body or of Earth, the Sun’s solar system, and the Milky Way galaxy.) Some of the time involved in moving through space would have been used anyway, just by standing still. So the time required to move through space is reduced by some amount. The amount depends on how fast a body is moving; the faster it is moving, the greater the reduction in the amount of time required to travel a given distance.

Here is a rough analogy: I can go downstream without paddling my canoe, but I can only go as fast as the current will take me. If I paddle, It will take me less time to travel a given distance. The faster I paddle, the less time it will take. A plot of the time required to go a given distance at various paddling rates would resemble the graph above, though it is impossible to reduce the travel time to zero, and the mathematical relationship between travel time and paddling rate isn’t the same as that represented by the graph.

Putting aside time, which I will address below, STR hinges on the meaning and measurement of velocity. In fact, v is assumed to have an absolute and determinate value for a “moving” frame of reference. Otherwise, the equations of STR are meaningless; that is, if v simply represented the relative velocity of two bodies (both in motion), the solutions for a given body would vary with the body with which it is being compared.

Accordingly, STR assumes that all comparisons are made with a hypothetical stationary body with no velocity — a “fixed point” in the universe, if you will. But the “fixed point” would have to be an object around which the universe revolves, and there is no such body. It is therefore impossible to compute the absolute velocity of a body, which STR requires.

And if one can’t determine the velocity of a body, the Lorentz transformation is really meaningless. The values of t’ and x’ are indeterminate — unless one reverts to Galilei-Newton relativity, which assumes, in effect, that every part of the universe is comprised in a unitary frame of reference, with absolute space and absolute time (and, therefore, absolute measures of velocity).

I will end here with two thoughts. The first is that Phipps (2012) may well have resolved the internal contradictions of STR. Near the end of the book (page 306) Phipps summarizes with this:

By means of CT [collective time, which is not the same as Einsteinian time], the science of mechanics simplifies formally to its nineteenth-century canonical forms … ; … three-space geometry reverts to Euclidean … and … absoluteness of distant simultaneity … [is] restored.

(Regarding the absoluteness of simultaneity, see “A Fatal Flaw?” above.)

More speculatively, it seems to me that there might be a physical phenomenon which serves as the unitary frame of reference implicit in Galilei-Newton relativity: the Higgs field, the lattice of Higgs bosons which is believed to permeate the universe. But this is only a preliminary thought.


“A Fatal Flaw?” exposes the erroneous thought experiment used by Einstein (1920) to demonstrate the (supposed) relativity of simultaneity. “The Velocity Conundrum” points out inconsistencies in the way that STR treats velocity. Here I consider a misleading interpretation of the speed of light, to which Einstein contributed.

Light purportedly moves at the same (constant) speed (in a vacuum) regardless of the motion of its source or sensor. As I will discuss in this section, the meaning of that (purported) fact has been misinterpreted, and then used to draw incorrect conclusions about relativity. Even Einstein did it.

I begin with my own thought experiment. Consider the case of an observer who is standing by a railroad track. On the track, coming toward him from the left, is an open train car — an extra-long one — on which a pitcher throws a ball to a catcher standing 132 feet away. (The pitcher is throwing in the observer’s direction.) Assuming away the problem of air resistance, the observer sees the ball coming toward himself at the sum of two speeds: the speed of the ball (relative to the catcher) and the speed of the train car (relative to the observer). Thus, if the pitcher throws the ball at 90 miles per hour (mph) in a train car that is moving toward the observer at 60 mph, the ball approaches the observer at 150 mph.

In feet per second (fps), the ball moves toward the catcher at 132 fps, the train car moves toward the observer at 88 fps, and the ball therefore moves toward the observer at 220 fps. If the pitcher releases the ball when he is 220 feet from the observer, the catcher is then 88 feet from the observer (220 feet from pitcher to observer – 132 feet from pitcher to catcher). In the 1 second that it takes the ball to reach the catcher, the train moves 88 feet toward the observer. The ball is therefore caught when the catcher is adjacent to the observer. Thus takes the ball 1 second to travel 220 feet from its release point to the observer’s position. This is another way of saying that the ball approaches the observer at 150 mph, which is 220 fps (pitch moving relative to pitcher at 132 fps + train car moving relative to observer at 88 fps). This is the classical addition of velocities according to Galilei-Newton relativity.

I will now rerun the thought experiment, making a slight modification. Instead of a baseball, the pitcher throws an exceedingly slow “ball” of light at the same 132 fps. If light were an ordinary object like a baseball, it would approach the observer at 220 fps. But light isn’t like a baseball in that its apparent speed is the same for every observer: 132 fps. (Remember, this is a thought experiment, and I’m using a very slow speed of light just to keep it simple. I know that light really travels at 2.99792 x 10^8 meters per second in a vacuum.)

At this point I must introduce the crucial misconception about the speed of light. Here is what Einstein says in the second paragraph of the 1905 paper in which he introduces STR:“

[T]he phenomena of electrodynamics as well as of mechanics possess no properties corresponding to the idea of absolute rest. They suggest rather that, as has already been shown to the first order of small quantities, the same laws of electrodynamics and optics will be valid for all frames of reference for which the equations of mechanics hold good. We will raise this conjecture (the purport of which will hereafter be called the “Principle of Relativity”) to the status of a postulate, and also introduce another postulate, which is only apparently irreconcilable with the former, namely, that light is always propagated in empty space with a definite velocity c which is independent of the state of motion of the emitting body.

Clearly, Einstein does not say that the speed of light is the same relative to all observers. He merely says that it is always the same for all observers. I will illustrate the subtle distinction with another thought experiment, which begins with a “true” fact.

Albert A. Michelson and others built a mile-long vacuum tube in which a beam of light was bounced back and forth by an arrangement of mirrors, resulting in a 10-mile journey that could be timed with precision. The experiment was conducted many times from 1930 to 1935. And in 1935, four years after Dr. Michelson’s death, his collaborators reported an estimate of the speed of light that was more than 99.99 percent of the value now deemed correct. (That’s better than Ivory Soap, which was advertised as 99.44 percent pure, though pure what I don’t know.)

Imagine that there was a construction road parallel to the mile-long tube (which there probably was). Suppose that the road wasn’t used during experiments, with one exception. One fine day, Dr. Michelson hopped in his car and was driving parallel to the tube, reaching a steady speed of 50 mph as a beam of light was sent on its 10-mile journey. Imagine, further, that the top of the tube was made of glass, so that Dr. Michelson could catch a glimpse — a truly fleeting one — of the beam of light as it zoomed by him. Did his presence cause the beam of light (however fleetingly) to speed up so that it was going 2.99792 x 10^8 meters per second (the “correct” value) relative to Dr. Michelson? Or, if Dr. Michelson had been able to measure the speed of light at the moment it passed him, would he have found that it was moving at the same 2.99792 x 10^8 meters per second, even though he was moving at 50 mph?

It seems obvious to me that the second possibility in the correct one. The first possibility not only requires light to behave weirdly, but it also requires light to move faster than its own (supposedly limiting) speed. In sum, the speed of light would remain the same for Dr. Michelson (despite the speed of his car). But relative to Dr. Michelson, it would be moving at a speed of 2.99792 x 10^8 meters per second minus the speed of the car (50 mph = 22.35 meters per second).

The moral of the thought experiment: Light (in a vacuum) moves at the same, constant speed for all observers. But it must move at different speeds relative to various observers, depending on the speeds at which they are moving.

I have gone out of my way to make this point because it evidently eluded Einstein. This is from Einstein (1920):

Let us suppose our old friend the railway carriage to be travelling along the rails with a constant velocity v, and that a man traverses the length of the carriage in the direction of travel with a velocity w. How quickly, or, in other words, with what velocity W does the man advance relative to the embankment [on which the rails rest] during the process?… As a consequence of his walking, … he traverses an additional distance w relative to the carriage, and hence also relative to the embankment, … the distance w being numerically equal to the velocity with which he is walking. Thus in total he covers the distance W = v + w relative to the embankment in the second considered.

That is, by the classical addition of velocities formula, the man in the train car is moving at speed v + w relative to a an observers standing by the railroad track. But …

Einstein continues:

If a ray of light be sent along the embankment [in an assumed vacuum], … the tip of the ray will be transmitted with the velocity c relative to the embankment. Now let us suppose that our railway carriage is again travelling along the railway lines with the velocity v , and that its direction is the same as that of the ray of light, but its velocity of course much less. Let us inquire about the velocity of propagation of the ray of light relative to the carriage. It is obvious that we can here apply the consideration of the previous section, since the ray of light plays the part of the man walking along relatively to the carriage. The velocity W of the man relative to the embankment is here replaced by the velocity of light relative to the embankment. V is the required velocity of light with respect to the carriage, and we have

V = c − v .

The velocity of propagation of a ray of light relative to the carriage thus comes out smaller than c .

But this result comes into conflict with the principle of relativity set forth in Section V. For, like every other general law of nature, the law of the transmission of light in vacuo must, according to the principle of relativity, be the same for the railway carriage as reference body as when the rails are the body of reference.

(I have taken the liberty of making a minor change in Einstein’s notation to avoid some confusion, but I haven’t changed his discussion or the relationship depicted by the resulting equation.)

Einstein contradicts himself. He insists that the speed of light relative to the train car should be c. But it can’t be, as I argue above. The speed of light is unaffected by the speed of the train car. It remains c, even though the train car is moving at v. The movement of the train car at v doesn’t affect the actual speed of light, which remains constant — the same for all observers. Therefore, Einstein wrongly rejects V = v – c as the formula for the speed of light relative to the train car. It is exactly the right formula. Rearranging, it says that c = v + V , which preserves the constancy of c (much like preserving its virtue).

I am taking this long detour from the route of my original thought experiment (which I will rejoin) because clarity demands it. So does the evidence, insofar as I am familiar with it. The speed of light has been measured by two basic methods. One kind of measurement involves a stationary apparatus, where the light source and sensor are fixed relative to each other; there is no motion of emitter or observer to account for. The other kind of measurement takes advantage of natural occurrences (e.g., light reflected from distant, moving moons). But in such cases the location of the distant object is accounted for in the computation of the speed of light; what is measured is the speed of light during its transit from a distant object to an earth-bound sensor. This is no different, in principle, than the measurement of the speed of a baseball thrown between pitcher and catcher in a moving train car; there is no “stationary” observer to add the speed of the baseball and the speed of the train car. In other words, such observations do not confirm that the speed of light is independent of the speed of its emitter. But they are consistent with the view that the speed of light “in flight” is constant.

And if the speed of light is constant, it moves at various speeds relative to objects in its vicinity. (For example, if it is emitted from or observed by an object which is moving at 0.9c, it is simply moving 0.1c faster than that object, and so on.) It doesn’t assume different speeds at a whim, or at the whim of whatever it happens to be moving near it. It is one thing to say that the speed of light is constant, regardless of the motion of its source or observer. It is quite another thing — and a prescription for chaos — to say that the speed of light is always greater by c than that of its source or observer (which implies that light somehow travels faster than light). But that is precisely what Einstein implies in his later explanation of STR, and that misstatement — which has been propagated by many expositors of STR — must not be allowed to stand.

I now return, at last, to the thought experiment involving the pitcher, catcher, train car, and observer standing next to the railroad track. The pitcher is throwing a “ball” of (exceedingly slow) light at 132 fps in the direction of the observer while he (the pitcher) is moving toward the observer on a train car that is moving toward the observer (from the observer’s left) at 88 fps. At the point of release, the “ball” of light is 132 feet from the catcher and 220 feet from the observer. But the catcher is moving away from the release point at 88 fps. The “ball” must therefore reach the catcher before it passes by the observer.

The same order of events would obtain in the case of a real beam of light, though many orders of magnitude faster. The constancy of the speed of light is preserved, even though it appears move at less than c relative to the pitcher and catcher, who are moving in the same direction as the “ball” of light.

Here’s the clincher: Einstein’s thought experiment about the relativity of simultaneity, which I discuss in “A Fatal Flaw?,” is built on the same premise. In that thought experiment, the person (at M’) who is analogous to the catcher of my thought experiment, is moving toward the source of light instead of away from it. As a result, he sees a flash of light before the stationary observer (at M) sees it. I am therefore puzzled as to why, in the thought experiment quoted above, Einstein insists that light must move at c relative to the moving train car.


What sets STR apart from the classical Galelei-Newton view of space and time? It’s the intertwining of space and time into four-dimensional spacetime. The four dimensions are the usual spatial dimensions represented by x, y, and z, and the time dimension, represented by t. One aspect of spacetime is the tradeoff between space (distance) and time discussed in “The Velocity Conundrum.” What is the physical meaning of that tradeoff? Is it real or illusory?

It is useful at this point to introduce another thought experiment. Consider a very fast train moving on a track in a long, straight tunnel. The tunnel (including the track) is a stationary frame of reference in relation to the train. That statement should lead you to wonder if the speed of the train is therefore accurately depicted as a fraction of c, the speed of light. It isn’t if the tunnel isn’t absolutely stationary (as it’s unlikely to be, as discussed in “Getting Light Right”), and the speed of the train is really its speed relative to the tunnel, not its absolute speed as a fraction of c.

For the purpose of the thought experiment, the tunnel must be considered absolutely stationary, so that the speed of the train is its speed as a fraction of c. Given that, the can be taken as a proxy for absolute spacetime, just as it should be in Einstein’s famous train-embankment thought experiment (see “A Fatal Flaw?”). But how can there be absolute spacetime if STR says there isn’t? I’ll get back to that.

In any event, the tunnel of this thought experiment is lined with sensors that detect and record the train’s passage at a velocity of 0.6c. The train’s position throughout its journey can therefore be described in terms of the space-time coordinates of the tunnel. Graphically:

The solid black line represents the progress of the train through the tunnel. From the standpoint of observers in the tunnel, the train takes 6 light-seconds to traverse the tunnel. (Note that a beam of light, represented by the dashed line, would reach from one end of the tunnel to the other end in 6 seconds, 4 seconds ahead of the train.) The gray area represents the space-time coordinates occupied by the tunnel during the 10 seconds of the train’s traversal.  The gray area really consists of an infinite number of horizontal lines, each 0.6 light-seconds long, which is the constant length of the tunnel as it moves through time — but (theoretically) not through space.

In this version of the graph, the red dots mark the passage of each second of the train’s trip through the tunnel, as perceived by observers on the train:

Note that there are only 8 red dots along the train’s path, which means that observers on the train experience an 8-second trip through the tunnel, not a 10-second trip. According to STR, the time-dilation effect slows the clocks on the train, relative to clocks in the tunnel. When the train is moving at 0.6c, its clocks “tick” every 1.25 seconds for every 1 second that passes for observers in the tunnel, which means that the train’s clocks advance at 0.8 times the rate of clocks in the tunnel (1/1.25 = 0.8). Further, by the length-contraction effect, observers in the tunnel would perceive that the train has shrunk to 0.8 times its “real” length, that is, the length measured by observers on the train.

This leads to the question why a clock in a moving body (might) move more slowly the faster the body moves, and why a moving body would seem to have shrunk (according to observers in another frame of reference). Are these mathematical illusions or real phenomena? If the Lorentz relationships are symmetrical, as I understand them to be, they would seem to be mathematical illusions similar to the effect of distance on the perceived height of another person. When A and B are together, they can use a tape measure to agree on their heights. If they walk away from each other, they seem smaller to each other, but they know that this is an illusion that will vanish when they are standing next to each other.

Why is there a mathematical illusion? It arises from Einstein’s use of the Lorentz transformation to explain the non-simultaneity of events. (He calls it the relativity of simultaneity, but he means to say that events in different frames of reference can’t truly be simultaneous.) However — as I explain in “A Fatal Flaw?” — Einstein’s thought experiment doesn’t demonstrate non-simultaneity. In fact, as my thought experiment in that essay demonstrates, there is no such thing as non-simultaneity. Which is to say that simultaneity is alive and well.

After all, if there were no such thing as simultaneity, the Lorentz transformation wouldn’t work; that is, there would be no way to translate between frames of reference. But the Lorentz transformation acts, in effect, like an omniscient observer who is able to calibrate events in frames of reference that are moving with respect to each other. And the ability to calibrate implies that there is an absolute standard of space and time. Different clock rates and different length measurements are incidental features. As Buenker (2016) puts it,

Galileo’s Relativity Principle needs to be amended to read: The laws of physics are the same in all inertial systems but the units in which their results are expressed can and do vary from one rest frame to another [emphasis in original].

The real issue is whether different frames of reference have different spacetime coordinates. Do clocks in frames of reference that are moving relative to each other really move at different rates due to their relative velocities (as opposed to their gravitational or energy states)? Does this mean, for example, that a very precise clock stationed at the center of the universe (if there were such a thing) would move faster than every identical clock elsewhere in the universe — all of which are moving at some speed relative to the (hypothetical) center? Does this mean that a clock placed at the edge of the universe — which is constantly expanding outward — would run far, far, far more slowly than the clock at the center of the universe?

If this is so, what is the physical mechanism that causes it? This, I think, is the question that baffles people who struggle to understand Einsteinian special relativity. If STR is true, there must be a physical explanation that can be described in a fairly simple way. But, as far as I am aware, STR is never explained in physical terms. Readers and viewers are simply told that time slows and length contracts as a body goes faster. They are shown equations, graphs, drawings, cartoons, and animations that are consistent with the assertions that time slows and length contracts. But they aren’t given a physical explanation for such phenomena.

Here’s an explanation that I’ve read: Because spacetime is a compound of space and time, a body that moves only in time “uses” no space, other than the space that it occupies. A body that moves in space because it has a velocity “uses” more time the faster it goes (it takes longer for it to reach a given age), and must therefore sacrifice some space. It’s a zero-sum view of spacetime, which accords with this explanation of spacetime:

that measurements of distance and time between events differ among observers, [but] the spacetime interval is independent of the inertial frame of reference in which they are recorded.

Absolute space and absolute time have been replaced with … absolute spacetime. Conceptually, at least. But there’s still a crying need for a deep explanation of spacetime, something beyond its (supposed) mathematical properties.

What is the physical (material) mechanism at work when the “use” of time reduces the availability of space, and vice versa? Time is open-ended. Space is open-ended on a cosmological scale (assuming an ever-expanding universe).

According to general relativity, spacetime is curved; that is, both time and space are curved. Not that they can really be separated, but physicists have developed a dodge. There are time-like objects, whose journey through spacetime can be described in terms of time values. There are space-like objects, whose journey through spacetime can be describe in terms of spatial values. But unless there is an object that truly moves only through time — an object at the non-existent center of the universe — this is a meaningless distinction. To put it another way, if there is spacetime rather than space and time, there can’t be a separate thing called time, curved or not.

But there is a separate thing called time. A stationary human being knows that time is passing because he knows that he has consecutive thoughts. He also knows that he is continuing to breath, that his heart is continuing to beat, and so on. But the sum of these events, when reviewed in the observer’s mind, don’t reveal a curvature of time, merely its continuation, or — more to the point — the continuation of the observer and the objects around him. Time doesn’t curve, go in straight lines, or do loop-the-loops. It just flows. The idea of curved time is a patently nonsensical mathematical abstraction.

Phipps (2006,2012) argues at length — and with seeming authority — that STR is fundamentally flawed because it is based on Maxwell’s equations rather than what Phipps calls a neo-Hertzian alternative. This alternative is based on the field equations of Heinrich Hertz.  I begin with Phipps (1993):

In the last chapter of [Electric Waves], which appeared in 1892, [Hertz] treated the “electrodynamics of moving bodies” by an original set of equations comprising what we would call today an “invariant covering theory” of Maxwell’s equations for vacuum electrodynamics. The new equations differed from Maxwell’s through the inclusion of an extra velocity-dimensioned parameter, the components of which Hertz designated (α, ß, γ). The presence of this extra velocity parameter spoiled the space- time symmetry of Maxwell’s equations, but caused them to become rigorously invariant under the Galilean transformation of coordinates….

[Hertz’s] theory did not become physics, because physics is never equations alone but equations plus physi-cal interpretation. As often as not, and certainly in this case, interpretation proves the stumbling block. On the side of interpretation Hertz made a fatally bad guess….

… Maxwell, Hertz, and most other late nineteenth- century physicists were fixated on ether… so, when Hertz saw a new velocity parameter unavoidably emerging from his invariant mathematics, he automatically identified it with ether velocity….

… [I]t was not Hertz’s mathematics that was empirically discredited, but the combination of that and an obviously (in the modem view) unsound physical interpretation. An identical interpretational mistake (of staking all the physics on an ether mechanism) was made by Maxwell.

History, ever the joker, forgave Maxwell’s errant physics and preserved – indeed, virtually sanctified – his mathematics (a noninvariant special case of the Hertz equations … hence in formal terms a comparatively degraded breed of mathematics)….

Phipps (2006, 2012) concludes that time-dilation is a fact. But it is related to the energy state of a body, and it is asymmetrical (i.e., absolute); it is not the symmetrical time-dilation of STR. Further, he concludes that length-contraction is not a fact; length is invariant with frame of reference.


Online courses in special relativity

Lecture 1 of “Special Relativity,” Stanford University

All lectures of “Special Relativity,” Khan Academy

All lectures of “Understanding Einstein: The Special Theory of Relativity,” Standford University

Selected books and articles about special relativity

Barnett, Lincoln. The Universe and Dr. Einstein. New York: Time Incorporated, 1962.

Bondi, Hermann. Relativity and Common Sense: A New Approach to Einstein. New York: Doubleday & Company, 1946.

Buenker, Robert J. “Commentary on the Work of Thomas E. Phipps, Jr. (1925-2016).” 2016.

Einstein, Albert. “On the Electrodynamics of Moving Bodies.” Annalen der Physik, 322 (10), 891–921 (1905).

———. Relativity: The Special and General Theory. New York: Henry Holt, 1920.

Epstein, Lewis Carroll. Relativity Visualized. San Francisco: Insight Press, 2000.

von Mettenheim, Christoph. Popper versus Einstein. Heidelberg: Mohr Siebeck, 1998.

———. Einstein, Popper and the Crisis of Theoretical Physics (Introduction: The Issue at Stake). Hamburg: Tredition GmhH, 2015.

Noyes, H. Pierre. “Preface to Heretical Verities [by Thomas E. Phipps Jr.].” Stanford: Stanford Linear Accelerator Center, Stanford University, June 1986.

Phipps, Thomas E. Jr. “On Hertz’s Invariant Form of Maxwell’s Equations.” Physics Essays, Vol. 6, No. 2 (1993).

———. Old Physics for New: A Worldview Alternative to Einstein’s Relativity Theory. Montreal: Apeiron, first edition, 2006.

———. Old Physics for New: A Worldview Alternative to Einstein’s Relativity Theory. Montreal: Apeiron, second edition, 2012 (The late Dr. Phipps — Ph.D. in nuclear physics, Harvard University, 1950 — styled himself a dissident from STR, for reasons that he spells out carefully and exhaustively in the book.)

Rudolf v. B. Rucker. Geometry, Relativity, and the Fourth Dimension. New York: Dover Publications, 1977.

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