How measurements are made is one of the most fundamental concepts in physics. Understanding how an observer makes measurements of motion is of paramount importance in determining how that observer interprets events.

A typical thought experiment used to explain time dilation and length contraction uses the two postulates of relativity, two observers, A and B, and a light clock, which is a light source and a vertical rod with a mirror attached at each end. The two observers are in (inertial) motion, and each observer is unaware of his own (inertial) motion. Observer A is the one with the light clock. So while the two observers are in motion relative to each other, observer A and the light clock are stationary relative to each other, and observer B and the light clock are in motion relative to each other.

The two observers are facing each other. So while each observer is unaware of his own motion, each observer sees the other observer moving from his left to his right (at a constant speed v).

Observer A (the one with the light clock) sees observer B moving from left to right, as the the flash of light (of the light clock) travels a vertical path along the vertical rod of the light clock, such that the distance between the rod and the flash of light, called the "bewteen distance" does not change.

Observer B (the one without the light clock) sees observer A (with light clock) moving from left to right, as the the flash of light (of the light clock) travels a vertical path along the vertical rod of the light clock, such that the distance between the rod and the flash of light, called the "bewteen distance" does not change.

So both observers see the flash of light travel vertically along the rod (of the light clock) without the "between distance" changing.

Of course, observer B sees something extra that observer A does not see because observer A and the light clock were stationary relative to each other while observer B and the light clock were not stationary relative to each other. Observer B sees that while the flash of light progress vertically along the (vertical) rod of the light clock (without the "between distance" changing), the (vertical) rod of the light clock progress horizontally from his left to his right (at a constant speed v).

1 is the light source, 2 is the bottom mirror, 3 is the vertical rod
4 is the top mirror, 5 is the flash of light, 6 is the "between distance"

Usually, (in the typical thought experiment for time dilation and length contraction) a diagonal line is drawn, connecting the flashes of light.

Then a line of reasoning is stated, such as the following. Observer B thinks the flash of light had a "farther distance" to travel, and since by the second postulate of relativity, he would measure the speed of the flash of light to be c, he reasons that time is running slow for observer A (as viewed from observer B's frame of reference).

The line of reasoning continues to a mathematical explanation, such as the following.

The symbol t rpresents the time it takes the flash of light to travel from the bottom to the top of the light clock according to observer A (that is, in the frame of reference that is stationary relative to the light clock). Since the speed of the flash of light is c, and the time is t, it follows that the vertical measurement is ct. The symbol t' represents the time it takes the flash of light to travel from the bottom to the top of the light clock according to observer B (that is, in the frame of reference that is not stationary relative to the light clock). Since the speed of the flash of light is c, and the time is t', it follows that the diagonal measurement is ct'. According to observer B, during the time t' the light clock travels horizontally (from his left to his right) at a constant speed v. It follows that the horizontal measurement is vt'.

Then the Pythagorean theorem is applied, and from there the time dilation equation is derived.

From the above we can define three (3) systems of measure.

The Exclusion System Of Measure
The exclusion system of measure excludes the motion of the source (i.e. the light clock).

Observer A uses the exclusion system of measure. Since observer A and the light clock are stationary relative to each other, observer A does not see the motion observer B observes the source to be in relative to him.

The vertical path traveled by the flash of light along the vertical rod of the light clock without a change in the "between distance" represents an exclusion measurement made by observer A. This exclusion measurement is a vector velocity of a single body (i.e. the flash of light).

The Inclusion System Of Measure
The inclucion system of measure includes the motion of the source (i.e. the light clock), and contains the exclusion system of measure within it.

Observer B uses the inclucion system of measure. Since observer B and the light clock are not stationary relative to each other, observer B sees the flash of light progress vertically along the vertical rod of the light clock without the "between distance" changing (an exclusion measurement) while the vertical rod of the light clock progress horizontally from his left to his right at a constant speed v (an inclusion measurement).

So observer B has made two measurements. One of them is an exclusion measurement (the motion of the flash of light), and one of them is an inclucion measurement (the motion of the source).

The Babin Conclusion
Not only do the two measurements made by observer B accurately and completly describe what observer B saw, but they also satisify the two postulates of relativity; thus, logically, there is no need for the line of reasoning that results in the drawing of a diagonal line (where the flash of light had a "farther distance" to travel).

The Mixed System Of Measure
Whereas the inclusion system contained the exclusion system within it, the mixed system of measure combines the exclusion system and the inclusion system, resulting in a measurement that does not represent a vector velocity.

The diagonal line represents a mixed measurement and does not represent a vector velocity of any body.

The Pythagorean Rule
To correctly apply the Pythagorean theorem all three side of the right triangle must represent the same type of measurement.

Since the two sides each represent a vector velocity while the hypotenuse is not a vector velocity, we do not have a right triangle for which all three sides represent the same type of measure.

For relativity theory, side A of the right triangle (which is the exclusion measurement made by observer A) is denoted by /\t and hypotenuse C (which is the mixed measurement made by observer B) is denoted by /\t' The ratio C/A is called the gamma factor and is denoted by y. So we have

Using one second for the time measurement represented by side A, it will take C/A times one second for the flash of light to travel the "farther distance" represented by hypotenuse C. Because C is the hypotenuse of a right triangle, C is always greater than the adjacent side A, so it follows that the ratio C/A is always greater than one.

By using the Pythagorean theorem (which, of course, we should not be doing) we can obtain the inverse of the gamma factor (which is A/C) expressed as the ratio of side B to the hypotenuse C (which is a distance ratio), where side B represents the distance traveled by observer A, and C represents the farther distance the flash of light had to travel (with these motions occurring in the same time interval).

The speeds (using v to denote the speed observer B sees observer A to be traveling, and using c to denote the speed of the flash of light) should have the same ratio as the distances.

Since A/C is the inverse of the gamma factor.

Giving

However, since we were using a right triangle for which all three sides did not represent the same type of measure, we should not have applied the Pythagorean theorem, and, thereby, should not have calculated the gamma factor in the above manner. (In addition, by the Babin Conslusion, the line of reasoning using the diagonal, i.e. the hypotenuse, for the "farther distance" is not necessary.)

The Taylor Rule
Observers must get different measurements while measuring the same thing using the same system of measure in order for time dilation and length contraction to be valid conclusions.

So we need to combine the exclusion and inclusion measurements without a vectorial addition of the speed of light and the speed of the observer, such that the diagonal (i.e. hypotenuse) drops out of the interpretation.

The solution is simple and obvious; however, justifying the solution is neither simple or obvious, and we should not expect it to be because relativity theory goes against the Babin Conclusion (by using the diagonal) and the Pythagorean Rule (using the diagonal as a vector velocity).

Because it is simple, we will do the solution first, and then we will justify it. Of course, in the solution you will realize we are using a measurement for observer B instead a measurement for observer A, and you will think that doing this is wrong, but it will be shown to be correct; however, we can only do one thing at a time so we start with the solution.

The diagonal (i.e. hypotenuse) is a mixed measurement and does not represent a vector velocity.

A mixed measurement combines the exclusion measurement and the inclusion measurement (but does not result in a vector velocity).

Each observer is unaware of his own motion, so V = 0 for him.

Of course, since v = 0, adding it to something is in effect the same as not adding it.

So we can combine the exclusion and inclusion measurements without a vectorial addition of the speed of light and the speed of the observer, such that the diagonal (i.e. hypotenuse) drops out of the interpretation.

The mixed measurement (the diagonal, i.e. hypotenuse) is the exclusion and inclusion measurements combined.

Since V = 0 we get

Since the diagonal (i.e. hypotenuse) was c + V, and since we ended up with just c, the diagonal (i.e. hypotenuse) has dropped out of the interpretation.

This means we are now in compliance with Babin Conclusion and the Pythagorean Rule, as we should have been all along. The consenquence is that the transformation equation is nothing more than a conversion equation for converting from one system of measure (the mixed system) to another system of measure (the exclusion system): there is no actual time dilation or length contraction.

Again, by the The Taylor Rule, Observers must get different measurements while measuring the same thing using the same system of measure in order for time dilation and length contraction to be valid conclusions.

First, the diagonal (i.e. the hypotenuse) was not brought into the interpretation by the Paritas Hypothesis. For the Paritas Hypothesis, by the Babin Conclusion, there was no need for the diagonal (i.e. the hypotenuse). It was relativity theory that brought the diagonal (i.e. the hypotenuse) into the interpretation. So any blame falls on relativity theory, not the Paritas Hypothesis.

Second, since observer A and the light clock are in the same frame of reference, there are only three bodies of concern: observer A (with light clock), observer B, and the flash of light.

Third, the vertical line (i.e. side A of the right triangle) is the motion of the flash of light, and the horizontal line (i.e. side B of the right triangle) is the motion observer B sees observer A (with light clock) to be in.

Fourth, the diagonal (i.e. the hypotenuse) is a mixed measurement (i.e. two measurements combined) and does not represent a vector velocity.

Fifth, relativity theory, by the second postulate, excludes the combined measurements from being the flash of light and observer A (with light clock) --- so any blame falls on relativity theory, not the Paritas Hypothesis.

Thus, there is only one combination left: the flash of light and observer B. According to observer B, his motion during the (thought) experiment was V = 0.

The Schwalm Switch
The switch from an inertial observer not accounting for his or her own inertial motion to that inertial observer accounting for his or her own inertial motion.

It was relativity theory, not the Paritas Hypothesis, that put us in the situation of using the Schwalm Switch. The end result being that the diagonal (i.e.the hypotenuse) drops out of the interpretation, and that the transformation equation is nothing more than a conversion equation for converting from one system of measure (the mixed system) to another system of measure (the exclusion system): there is no actual time dilation or length contraction.

We should be able to use the systems of measure to explain the Michelson-Morley experiment, the case of fast moving muons, and the case of a proton accelerated in a super conductor.

The detector (of the Michelson-Morley apparatus) used an exclusion system of measure. the detector's system of measure excludes the motion of the apparatus (and excludes the motion of any body not at rest with respect to the apparatus).

For the detector there is no direction of motion of the apparatus, and no direction that can be defined as "in the apparatus' direction of motion," and no direction that can be defined as "against the apparatus' direction of motion," and there is no direction that can be defined as "perpendicular to the apparatus' direction of motion."

The beam of light which according to Michelson and Morley was traveling in and then against the apparatus' direction of motion, was, according to the detector, doing no such thing: it was traveling in a horizontal direction with respect to the detector. Also, the beam of light which according to Michelson and Morley was traveling out and back perpendicular to the apparatus' direction of motion, was, according to the detector, doing no such thing: it was traveling in a vertical direction with respect to the detector. So the two beams of light which were set to travel along perpendicular paths of equal distance arrived at the detector at the same time because the detector used the exclusion system of measure. No time dilation or length contraction required to explain the null result of the Michelson-Morley experiment

Is the point of creation of a muon way up in the upper atmosphere? The answer is yes. Is the lifetime of a muon 1.5 microseconds? The answer is yes. Are muons found near the surface of the earth? The answer is yes. Before going on, I want to point out that the measurement from the surface of the earth up to the point where a muon is created is an exclusion measurement. Also, the measurement of 1.5 microseconds for the muon life time is an exclusion measure.

Now, as long as we use these two exclusion measurements time dilation and length contraction represent a valid explanation for why the muon reaches the surface of the earth. But the scientists did NOT (!) use these two exclusion measurements. What the scientists did is they used the 1.5 microsecond exclusion measurement for the muon life time, and the travel time from point of creation to surface of the earth, which is a mixed measurement, since the scientist used two different system of measure, instead of using the same system of measure for both measurements, time dilation and length contraction are not valid conclusions.

When there is no external influence acting on a particle, such as a proton, the particle moves along a straight line. Let’s say we have a transparent tube which we can see through, and that this transparent tube in no way interferes with a proton traveling a straight line as it passes through the transparent tube.

Look at the solid line with slash marks that runs horizontally along the center of the transparent tube, going from the back end of the transparent tube (through the center of the proton) to the front end of the transparent tube; it represents three distance measurements: one of these distance measurements (which is made from the back end of the transparent tube to the front end of the transparent tube) is not a mixed measurement, and the other two of these distance measurements (one made in relation to the proton and the back end of the transparent tube, and one made in relation to the proton and the front end of the transparent tube) are mixed measurements. We can add together the two mixed measurements to find the mixed measurement length of the transparent tube, which is greatly smaller than the non-mixed measurement for the length of the transparent tube.

Look at the solid line with slash marks that runs vertically from the top wall of the transparent tube (through the center of the proton) to the bottom wall of the transparent tube; it represents three distance measurements: one of these distance measurements (which is made from the top wall of the transparent tube to the bottom wall of the transparent tube) is not a mixed measurement, and the other two of these distance measurements (one made in relation to the proton and the top wall of the transparent tube, and one made in relation to the proton and the bottom wall of the transparent tube) are mixed measurements. We can add together the two mixed measurements to find the mixed measurement for the width of the transparent tube, which is greatly smaller than the non-mixed measurement for the width of the transparent tube.

Now let’s let the transparent tube be ring shaped. Because a proton will travel a curved line in a magnetic field, we will use a magnetic field to prevent the proton from traveling a straight line.The strength of the magnetic field is one of the quantities which determines the extent to which the proton’s direction of motion is changed from the straight line it would travel were there no magnetic field acting on the proton.

Of course, there are two measurements of the magnetic field strength: one is not a mixed measurement and one is a mixed measurement. The mixed measurement of the magnetic field strength is made in relation to the proton: it is greatly smaller (or weaker than) the non-mixed measurement of the magnet field strength, and the mixed measurement for the magnetic field strength becomes smaller the faster the proton moves in the ring, even when the non-mixed measurement for the magnetic field strength does not change, which means the faster the proton moves with respect to the ring shaped, transparent tube the stronger we must make the magnetic field strength: the need for a strong magnetic field is not due to an increase in the proton’s mass.

There is no relativistic mass increase: only the strength of the magnetic field (in relation to the proton) and the speed of the proton (with respect to the ring shaped, transparent tube) have changed —- the mass of the proton has not changed from before when it was traveling slower with respect to the ring shaped, transparent tube.