In Tutorial thirteen we showed that relativity is invalid, because the spherical wave proof failed. The proof’s failure means that Einstein did not demonstrate the compatibility of his two postulates. In fact, the failed proof shows that his two postulates – *the principle of relativity* and *the principle of the constant velocity of light* – are not compatible. Although relativity is a failed theory, any analysis of Einstein’s work is incomplete without an explanation of relativity’s most important concepts: time dilation, length contraction, and the limitation on the speed at which something moves. In this tutorial, we examine how motion would be explained if his two postulates were actually compatible.

Before we look at Einstein’s concepts, we have to explain the difference between *absolute* and *relative* measurements. Imagine a bus traveling at 50km/s. A car is driving next to the bus in the same direction and at the same velocity. With respect to someone standing on the sidewalk, the car is moving at 50km/s. However, from the perspective of a passenger sitting in the bus, the car appears stationary because it is neither pulling away from nor falling behind the bus. From the passenger’s perspective, the car appears to be moving at 0km/s. In general, measurements taken *with respect to* the stationary system are called *absolute measurements*. Measurements made *from the perspective of* a moving system are called *relative measurements*. The idea of using *relative* measurements to explain motion is where *relativity* gets its name.

Once again we introduce an example involving a bus, a jogger, and a street. To this example we now add a stationary observer who is standing on the sidewalk and a moving observer who is in the bus.

An observer is simply someone who performs his or her measurements from the perspective of one–and–only-one system. The first observer, placed on the sidewalk, is called that *stationary observer*. He makes his measurements using a ruler embedded in the street. The second observer, placed on the bus, is called the *moving observer*. He makes his measurements using a ruler embedded in the floor of the bus. It is often easier to describe relative measurement in terms of an observer rather than the more verbose “using the ruler embedded in the…”

With the bus stationary, we place the jogger in motion at velocity *w* and make four important observations:

- The jogger is always able to run from the rear of the bus to its front, and vice versa.
- When the jogger has reached the front of the bus, she has traveled the length
*x’*. - When the jogger has traveled one–half the round–trip distance, she has reached the front of the bus.
- The jogger moves at velocity
*w*.

The *principle of relativity* says that what is observed when the bus is stationary **must be** what is observed when the bus is in motion. The *principle of the constant velocity of light* essentially says that the woman always moves with respect to the stationary system and both observers view her as moving at the same velocity. Remember, Einstein believes that **both postulates must always apply**. We must explore the mathematical and conceptual adjustments Einstein makes to remain aligned with this belief.

Let’s examine what happens when the bus is placed into motion at velocity *v*.

#### Observation 1 – The jogger is always able to run from the rear of the bus to its front, and vice versa

The principle of relativity requires that the jogger must always be able to run from the rear of the bus to its front, since she is able to always do so when the bus is stationary. Notice that the jogger can only run from the rear of the bus to its front when the velocity of the bus is less than the jogger’s velocity. In mathematical terms, *v* < *w *must always be true. This restriction is imposed by Einstein’s postulates, where if this restriction were not present we could have a case where jogger would never reach the front of the bus, violating the principle of relativity. The restriction imposed by this postulate disagrees with reality, where we know that the velocity of a bus in not limited by how fast a person can run. None–the–less, this restriction is required to satisfy both postulates.

When discussing the electromagnetic force, the bus is generalized as a *moving system *and the jogger is replaced by a *ray of light* traveling at *c*. This changes the mathematical relationship to *v* < *c*, and is why relativity theory says that the velocity of the moving system has to be less than the speed of light. It is important to remember that this is an artificial constraint that is specifically required to satisfy both of Einstein’s postulates, which we have already shown are incompatible.

#### Observation 2 – When the jogger has reached the front of the bus, she has traveled length *x’*

From the perspective of the moving observer, the principle of relativity requires that that jogger travel length *x’* to reach the front of bus. As illustrated in Figure B, above, the moving observer uses the ruler on the bottom of the bus to conclude that the jogger has moved from the white triangle (when she began running, as shown in Figure A, above) to the white circle. The distance from the white triangle to the white circle, called the *segment length*, represents the length of the bus, *x’*. The stationary observer uses the street as the ruler and measures the *forward intercept length* as the distance the jogger runs from the black triangle (when she began running, as shown in Figure A, above) to the black circle.

While this explanation seems simple enough, there’s one problem: If the jogger has reached the halfway point from the perspective of the moving system, then she must also be at the halfway point with respect to the stationary system. But this isn’t the case. From the perspective of the moving observer, the jogger has run the length of the bus, *x’*, which is one–half the total distance from his perspective. However, with respect to the stationary observer, the jogger has run the *forward intercept length*, which is longer than one–half her round–trip distance. This is not consistent and would violate the principle of relativity, which brings us to the third observation.

#### Observation 3 – When the jogger has traveled one–half the round–trip distance, she has reached the front of the bus

The principle of relativity requires that what occurs when the system is in motion be the same as when the system is stationary. Observation 2 placed the jogger at the front of the bus. Observation 3 says that when the jogger has traveled length *x’* from the perspective of the moving system, that she must be:

- at the halfway point from the perspective of the moving observer, and
- at the halfway point with respect to the stationary observer.

We know from earlier tutorials that one–half of the jogger’s total distance, with respect to the stationary observer, is the *average intercept length*. To align the diagram with this adjustment, relativity requires that we reposition the jogger to the *average intercept length* rather than at the* forward intercept length. *As shown in Figure C, we position the jogger at the* average intercept length *and position the front of the bus at* x”*. Neither the bus nor the jogger are positioned at the forward intercept.

This conceptual repositioning solves one problem, but creates a new one. Notice that when the jogger has traveled the *average intercept length*, she has not yet reached the front of the bus. So in actuality, she has not traveled the length *x’* from the perspective of the moving observer. This problem is overcome by repositioning the front of the bus to also correspond to the *average intercept length*, as show in Figure D.

Although consistent with Einstein’s postulates, both repositioning ignore two important facts. First, the jogger does not arrive at the front of the bus until she has traveled the *forward intercept length* in the stationary system. Second, it ignores that the front of the bus is actually further on the *x* axis than Einstein states. The bus is actually at *x”*, but Einstein has repositioned it to correspond with the *average intercept length*.

Conceptually, these adjustments mean that the jogger has reached the front of the bus when she has reached the *average intercept length *from the perspective of the moving observer. However, we know that the jogger does not reach the front of the bus until she has traveled the *forward intercept length* with respect to the stationary observer. Once again we have a problem where one observer sees something that differs from what the other sees. How can one observer see something that is symmetrical and the other something that is asymmetrical? Einstein says this is possible and gives it a name: **Simultaneity**.

Mathematically, we can’t simply reposition things and say that’s how the world operates. These adjustments have to be proven, which is why Einstein needs the spherical wave proof. Although we have already shown that the proof fails, we continue to proceed with the understanding that Einstein believes his proof worked. This means he must explain the mathematical relationship between the length the moving observer measures, *x’*, and the length the stationary observer measures, the *average intercept length.* This relationship is defined by the *average intercept length* equation:

*ξ = x’ / (1 – v ^{2} / w^{2} )*

or, when specifically discussing a ray of light traveling through an electromagnetic vacuum:

*ξ = x’ / (1 – v ^{2} / c^{2} )*

The *average intercept length* will always be greater than the *segment length* *x’* when the moving system is in motion. The relationship where the length of the bus (eg, the original *segment length)* *x’ *is always less than the *average intercept length* is what Einstein refers to as **length contraction**.

#### Observation 4 – The jogger moves at velocity *w*

When the bus is stationary, the moving observer determines that the jogger runs at velocity *w*. So to remain aligned with both postulates, he must conclude that the jogger is still running at velocity *w* when the bus is moving at velocity *v*.* *Scientists who argue against relativity theory often correctly state that the jogger is moving at the apparent velocity* w – v. *While true, the use of this expression as the velocity of the jogger would violate the first postulate. To remain internally consistent with the use of relative measurements and align with his postulates, Einstein accepts that both observers believe that the jogger is moving at velocity* w*. In such a case, the moving observer would need to treat *v* as if it were zero. In fact, this is why many scientist argue that experiments like the Michelson–Morley experiment must produce a 0km/s result. Regardless of whether you believe the assumption is correct, you must accept it as Einstein’s approach.

If, from the perspective of the moving observer, the jogger has traveled a distance of *x’* and is moving at velocity *w*, then the amount of *time** *required for her to run this distance is simply length divided by velocity. Using variables and expressions that Einstein’s attributes to the moving system:

*τ = ξ / w*

or

*τ* = x’ / (1 – v^{2} / w^{2} ) / w

As discussed above, when discussing the electromagnetic force *w* is replaced by *c* in the equations. When the bus is moving at velocity *v*, the *average intercept time* will always be greater than the amount of time the jogger needs to run the *segment length* *x’* (aka, the *segment time*).* *This relationship where the *average intercept time *is greater than the *segment time* is called: **time dilation**.

It is interesting to consider that we do not use time dilation or length contraction to explain the forward or reflected Doppler shifts (eg, *forward* and *reflected intercepts* in Modern Mechanics). If we don’t use Einstein’s concepts in explaining those terms, then we can reasonably argue that those same terms are not required to explain the average of those equations (eg, the *average intercept length ξ *or the *average intercept time τ*).

#### Summary

Einstein was extremely thoughtful in the development of his work. It is apparent that he accounts for situations that would violate his postulates, which he addresses using concepts and constraints. Conceptually, his theory requires length contraction, time dilation, and simultaneity. You cannot have these terms without relativity theory or vice versa. As a constraint, Einstein’s maximum velocity for a moving system is needed to ensure that everything that can be observed when a system is stationary can also be observed when it is in motion. Unfortunately, Einstein did not fully understand the equations he had found or how they are generalized.

As a recap, the spherical wave proof failed; which means that Einstein was unsuccessful in associating his postulates with one another. So the terms and restrictions associated with Einstein’s work do not apply. Said simply, explaining motion does not require time dilation, length contraction, or simultaneity. In addition, the velocity of a moving system is not theoretically limited. Modern Mechanics provides an intuitive framework for explaining motion that is built upon a well–understood mathematical foundation called geometric transformations. Not only does Modern Mechanics provide equations that perform better than relativity’s equations, it opens the door to the possibility of faster than light interaction and motion. This provides avenues for faster than light communication and travel, as well as possible theoretical explanations for observations like quantum mechanics’ entanglement. Of course, traveling faster than the speed of light would certainly require new engineering and scientific breakthroughs.

*Note: This Tutorial uses Einstein’s equations prior to “substituting x’ with it’s value” in §3 of his 1905 paper. This simplifies our analysis, because it keeps the emphasis on length x’. In addition, our analysis is further simplified by performing the analysis prior to Einstein’s unannounced adjustment where he drops a β term which, as discussed in Tutorial twelve, introduces error not present in the Modern Mechanics equations.*

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