Albert Einstein's
Special Theory of Relativity

§ I. Introduction

In 1905, Albert Einstein's special theory of relativity was published under the title: "On the Electrodynamics of Moving Bodies." (1) In it Einstein reasoned from two postulates to several necessary conclusions.

In this study, Einstein's argument is presented on two levels, or as having two structures: one simple and one complex. Both must be recognized in order to understand the following discussion. The argument's simple level, or structure, consists of an assertion that "A = B", a second assertion that "A = C", and the necessary conclusion that "A = B = C". The argument's complex level, or structure, is revealed when A, B, and C are defined as ratios where "A = D/E", "B = F/G", and "C = H/I".

Let no one be intimidated by this study's topic or the number of its symbols and pictures. An adequate understanding of this presentation, and the special theory of relativity, can be achieved with no more than a basic understanding of algebra in general and the Pythagorean Theorem in particular.

§ II. The First Postulate

Einstein's first postulate presents the Principle of Relativity. This principle originated in the observation that events occurring in a vehicle seem to occur in much the same way whether the vehicle is at rest or moving along a straight-line path at some constant ratio of distance to time. In Einstein's words,

the phenomena of electrodynamics as well as of mechanics possess no properties corresponding to the idea of absolute rest. [Thus] ... 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 hereinafter be called the "Principle of Relativity") to the status of a postulate .... (2)

The laws by which the states of physical systems undergo change are not affected, whether these changes of state be referred to the one or the other of two systems of co-ordinates in uniform translatory motion. (3)

This postulate tells us that, if a scientific law (L) describing an event in a physical system is valid, it will be equally valid whether the system is a stationary system (S) or a moving system (M), as long as M's motion is along a straight-line path through S at some constant ratio of distance (d_S) to time (t_S). Briefly, "L_S = L_M" as long as M's d_S/t_S is constant.

§ III. The Second Postulate

Einstein's second postulate expresses his law of the constant velocity of light. In Einstein's words,

[we] 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. (4)

Any ray of light moves in the 'stationary' system of coordinates with the determined velocity c, whether the ray be emitted by a stationary or by a moving body. (5)

According to this postulate, a little bit of light generated by a source in a stationary system (S) always travels at the same velocity (C_S) relative to S. By definition, this velocity will always be the same constant ratio of distance in S (D_S) to time in S (T_S). In summary, "D_S/T_S = C_S".

§ IV. The Postulates Combined

What, then, must we believe if we believe that both postulates are true simultaneously?

According to the first postulate, if one of two physical systems is stationary and the other is traveling at some constant velocity relative to the first, then a scientific law which is valid in one system will be valid in the other. Given the second postulate's assertion that the law of light's constant velocity is valid in a stationary physical system, if both postulates are true, then the first postulate is tantamount to an assertion that the law of light's constant velocity is also valid in a moving physical system traveling at some constant ratio of d_S/t_S relative to the stationary system. In other words, C_S = D_S/T_S = D_M/T_M = C_M.

Necessarily,

To visualize these statements, imagine M traveling along a straight-line path through S at some velocity d_S/t_S.

Figure 1.

Next, imagine that, when M is in Position 1, two light sources, one in S only and the other in M, simultaneously generate little bits of light that travel in their separate physical systems while M travels to Position 3 in S. According to the first postulate, if the only significant difference between S and M is M’s motion relative to S at some constant ratio of d_S/t_S, then the two lights, one in S only and the other in M, will travel at equal velocities in and relative to their respective systems, as shown in Figure 2.

Figure 2.

Thus, Figure 2 portrays the the Principle of Relativity which implicitly asserts that "C_S = C_M" and Einstein's law of light's constant velocity which asserts that "D_S/T_S = C_S".

This, however, is not all there is to Einstein's special theory of relativity. In fact, Figure 2 is incomplete with regard to the second postulate, which addresses the motion of any light in S, including one that travels in M and relative to both M and S simultaneously. Thus, the figure above is incomplete because it identifies only one light’s motion relative to S, although two lights have moved relative to S in the scenario given. To complete the figure, we must map the second light’s motion relative to S while it travels in M, which we can do by drawing an arrow from M’s light source when M is in Position 1 to the light’s location in M when M is in Position 3, as shown in Figure 3.

Figure 3.

Because K_S, as defined, is a light's velocity relative to S and because the second postulate states that any light travels at the constant velocity C_S when it travels in S, we realize that the second postulate's definiton of C_S implicitly asserts that "C_S = K_S". Necessarily,

If we assume that all motions in the scenario underlying Figure 3 are concurrent, then "τ_S = T_S" and "τ_S/T_S = 1". Therefore, δ_S must equal D_S and δ_S/D_S must equal "1" for the assertion "C_S = K_S" to be true. The fact that the vertical arrow on the left and the diagonal arrow in the middle of Figure 3 are not equal in length implies that "C_S = K_S" is not true if we assume that "τ_S = T_S". But if we make the two arrows equal in length, we obtain Figure 4.

Figure 4.

A problem presents itself, as follows:

• Figure 3 reflects the second postulate's assertion that light's velocity relative to
  a stationary physical system is a constant (D_S/T_S = C_S) and the first postulate's assertion that "C_M = C_S", but the figure does not reflect the second postulate's
  implicit assertion that "K_S = C_S".

• Figure 4, on the other hand, reflects the second postulate's assertion that light's
  velocity relative to a stationary physical system is a constant (D_S/T_S = C_S)
  and the second postulate's implicit assertion that "K_S = C_S", but the figure does
  not appear to reflect the first postulate's implicit assertion that "C_M = C_S".

• But, if both postulates are true simultaneously, then:
  C_S = C_M = K_S

  or, in terms of the ratios of distance to time represented by each velocity,

  D_S/T_S = D_M/T_M = δ_S/τ_S

  which means that:

  • If "T_S = T_M = τ_S" is true, then "D_S = D_M = δ_S" must be true;
  • If "T_S = T_M = τ_S" is false, then "D_S = D_M = δ_S" must be false;
  • If "D_S = D_M = δ_S" is true, then "T_S = T_M = τ_S" must be true; and
  • If "D_S = D_M = δ_S" is false, then "T_S = T_M = τ_S" must be false.

• Because D_S and δ_S are unequal when D_S and D_M are equal, and D_S and D_M are unequal when D_S and δ_S are equal, "T_S = T_M = τ_S" must be false.

• Therefore,
  • Either Einstein's postulates are irreconcilable and his special theory
    of relativity is false,

  • Or the motions in the scenario underlying Figures 3 and 4 are not as
    concurrent as we might otherwise have imagined them to be.

Einstein resolved the problem and reconciled his otherwise irreconcilable postulates by assuming that "D_S/δ_S = T_S/τ_S =1" and identifying the mathematical relationships between δ_S and D_M and between T_M and τ_S that would preserve his theory. The task was a simple one. To see how he did it, let us begin with the statement which summarizes the special theory of relativity:

Given the definitions of these velocities established above, this statement translates into:

• Theoretically, assuming that "T_S = τ_S" allows us to imagine that "D_S = δ_S"
  and to assert that "C_S = K_S".

• If "C_S = C_M = K_S" must be true for the special theory to hold true, then
  "D_M/T_M = δ_S/τ_S" must true for the theory to hold true.

• If "D_M/T_M = δ_S/τ_S" must be true, "D_M/δ_S = T_M/τ_S" must be true.

• If "D_M/δ_S = T_M/τ_S" must be true, then determining the mathematical relationship between D_M and δ_S will tell us what the mathematical relationship between T_M
  and τ_S must be in order for the special theory of relativity to hold true.

The task is, then, to determine the mathematical relationship between D_M and δ_S.

§ V. The Mathematical Relationship between D_M and δ_S

To determine the mathematical relationship between D_M and δ_S, we begin with a stripped-down and enlarged version of Figure 4.

Figure 5.

Note that Figure 5 represents the motion of M through S, the motion of a little bit of light in and relative to M while M is traveling through S, and the motion of the same little bit of light as it moves relative to S. Note that the paths of these three "concurrent" motions form a right triangle. Thus, the relationships between d_S, D_M, and δ_S are analogous to the relationships between the two legs and hypotenuse of a right triangle.

Figure 6.

Knowing this allows us to use the Pythagorean Theorem to determine the relationships between d_S, D_M, and δ_S. According to the Pythagorean Theorem, "(Leg A's length x Leg A's length) + (Leg B's length x Leg B's length) = (The length of the Hypotenuse x The length of the Hypotenuse)", or "(Leg A²) + (Leg B²) = (Hypotenuse²)". The relationship between d_S, D_M, and δ_S, then, must be:

We now solve for D_M.

Thus, the mathematical relationship between D_M and δ_S is:

§ VI. The Postulates Reconciled

Because the ratio between T_M and τ_S must equal the ratio between D_M and δ_S in order to make C_M equal K_S,

When both statements are combined, we obtain:

which reduces to "D_M/T_M = (δ_S/τ_S)(1)" or "C_M = K_S".

Because we began our reconciliation of Einstein's postulates with the assumption, in § IV, that:

we realize that,

We now know the mathematical relationships between D_M, δ_S, D_S, T_M, τ_S, and T_S that must be true in order to reconcile Einstein's postulates and maintain the desired equality of C_M, K_S, and C_S.

§ VII. Remarkable Consequences

As shown, Einstein's first and second postulates reconcile when:

• D_S/δ_S = T_S/τ_S =1;
• D_M/T_M = (δ_S) Ö [1 - (d_S²/δ_S²)] / (τ_S)Ö[1 - (d_S²/δ_S²)]; and
• D_M/T_M = (D_S) Ö [1 - (d_S²/δ_S²)] / (T_S)Ö[1 - (d_S²/δ_S²)].

Theoretically,

• If d_S (the distance M travels in the time period "t_S") is less than δ_S (the distance traveled by the little bit of light generated in M when the light's velocity is measured relative to S), then D_M and T_M will each be greater than "0" (zero) but less than δ_S (and D_S) and τ_S (and T_S), respectively. When this happens, the width of M will actually be less than it would have been if it had been traveling at a lesser velocity. Otherwise an occupant of M would discover the light's failure to reach its expected goal timely, which would contradict the Principle of Relativity.

• If d_S equals δ_S, which occurs if M travels at the same velocity in S that the little bit of light generated in M is moving relative to S, then D_S and T_S will each be "0" (zero). In other words, if M travels at the velocity of light, time and space inside of M will cease to exist as we recognize them.

• These two consequences support an assertion that M and its contents will shrivel if M accelerates while a light shines in it.

• The density of M and its contents increase when their volumes shrink.

• If d_S is greater than δ_S, which occurs when M travels through S faster than a little bit of light in M moves relative to S, then D_S and T_S will each be a negative number. This mathematical possibility has inspired much fiction regarding the notion of "time travel."

Einstein acknowledged the first three posibilities when he wrote:

Thus, whereas the Y and Z dimensions of the sphere (and therefore of every
rigid body of no matter what form) do not appear modified by the motion, the
X dimension appears shortened in the ratio 1:Ö(1 - v²/c²), i.e. the greater the value of v, the greater the shortening. For v = c, all moving objects--viewed
from the "stationary" system--shrivel into plain figures. (6) (7)

Although the fourth possibility above attributes M's increase in density to a decrease in M's volume, Einstein appears to have promoted the notion that M's mass increases when M accelerates.

Einstein precluded the fourth possibility listed above, involving negative values for D_M and T_M, when he wrote:

For velocities greater than that of light our deliberations become meaningless; we shall, however, find in what follows, that the velocity of light in our theory plays the part, physically, of an infinitely great velocity." (8)

And

Velocities greater than that of light have ... no possibility of existence.(9)

The special theory of relativity generates other conclusions, but these few are sufficient to give the reader a taste of the beliefs which Einstein's theory obliges us to accept.

§ VIII. Commentary

As shown, Einstein's special theory of relativity obliges us to believe that:

where (r) is the "relativistic" factor Ö[1 - (d_S²/δ_S²)]. (10)

The notion that " 'T_S = T_M = τ_S' must be false if 'D_S = D_M = δ_S is false' " is clearly consistent with the goal of maintaining the assertion that "C_S = C_M = K_S"; however, the notion that "T_M = τ_S(r) = T_S(r)" is troublesome, because it is so strikingly counterintuitive.

The notion that M shrivels when a bit of light travels through M while M accelerates is also troublesome because it raises questions regarding the cause-and-effect relationship between light’s motion in M and the shriveling that Einstein says M and its contents must undergo.

Even more troubling is the theoretical obligation to believe that light’s velocity is some form of “universal speed limit” when the claim sounds in the “right-triangle” relationships between d_S, D_M, and δ_S that exist only when a light travels in M. After all, if there is no light in M, there seems to be no basis for the claim.

Most troublesome of all, however, is Einstein’s assertion--inherent in the claim that "C_M = K_S"--that a light can move at the same velocity relative to S that it travels in and relative to M while M is traveling through S. With Figure 7 in mind, this claim says that a single light’s concurrent motions along the vertical path in M and the diagonal path in S take place at the same velocity.

Figure 7.

Although the language of mathematics permits this possibility, reality forbids it.

The objection may be raised that this study inaccurately imagines M to be a physical system where Einstein merely imagined a moving coordinate system unassociated with any actual physical environment. The presence of the Principle of Relativity in Einstein's theory belies this objection. If the theory does not need two distinctly separate physical environments, one "at rest" and one in motion relative to each other, to make sense, then the Principle of Relativity is irrelevant and misleading.

In summary, Einstein’s special theory of relativity is counterintuitive and untenable, despite experimental proofs offered in support of it. (11) Given Einstein's assertion that his special theory departs from “classical mechanics” through the second postulate, not through the Principle of Relativity, it is reasonable to believe that, if Einstein’s theory is untenable, it is because his second postulate is false. (12)

NOTES:

(1) Einstein, Albert. (1905). “On the Electrodynamics of Moving Bodies,” [Annalen Der Physik, 17, 1905]. English-language publication in The Principle of Relatiity: A Collection of Original Papers On The Special And General Theory Of Relativity. [Dover Publications Edition, 1952]. Translation by W. Perrett and G. B. Jeffrey.

(2) Ibid. Pages 37-38.

(3) Ibid. Page 41.

(4) Ibid. Page 38.

(5) Ibid. Page 41.

(6) Ibid. Page 48.

(7) The word “plain” in the quote should be “plane,” as in “two-dimensional.” Einstein spoke of the change in the X dimension only because that was, in his theory, the dimension parallel to the path of the light traveling in and relative to M. Because the light’s path, addressed in Einstein’s theory, was not parallel to the Y or Z dimensions, those two dimensions were not imagined subject to the shriveling effect that Einstein spoke of in the quote above. It should be emphasized that Einstein’s words make it clear that he conceived this shriveling effect to be real, not illusory.

(8) Einstein, A. (1905). Page 48.

(9) Ibid. Pages 63-64.

(10) Readers who are familiar with the Lorentzian Transformations may be interested in seeing what the transformations look like when translated into the terminology used in this study. The transformations are commonly presented as follows:

and

These equations translate into:

and

Recognizing that "t_S = τ_S", we can reduce last two equations to:

and

When both are combined and the relativistic factors are canceled, we are left with:

(11) To be valid, an experimental proof of the special theory should take place in S and measure a light’s velocity relative to S before the light has left M. Albert Michelson’s measures of light’s velocity, on the other hand, confirm light’s constant velocity in and relative to one physical system only, not two. In the terms used in this study, Michelson’s experimental measures support the hypothesis that “C_M = C_M".

(12) Einstein, Albert. (1916). “The Foundation of the General Theory of Relativity,” (Annalen Der Physik, 49, 1916.) Cf. The Principle of Relativity: A Collection of Original Papers On The Special And General Theory Of Relativity. Page 111. [Dover Publications Edition, 1952].

© 2002, Terry M. Sampson, All rights reserved.

Those who would instruct me may contact me at:

Anti-relativity@worldnet.att.net

Acknowledgements

My thanks

To Angelica, whose critiques inspired significant changes;

To Jane, for her encouragement;

To all who listened, over the years, in good humor and disbelief;

To Dover Publications for permission to reprint; and most of all

To my wife, who let me "play".