Both Newtonian physics and special relativity conclude that the total kinetic momentum is a conservative quantity for any system containing n moving objects if the system is isolated from any extraneous force. Mathematically this is expressed as

where n can be any positive integer,

stands for the kinetic momentum of the i-th object,

stands as a constant, both in direction and magnitude,

and the bars serve to denote a vectors.

Both theories also mathematically demonstrate that the change of kinetic momentum against time would result in force. Therefore, with being a constant, we have

where stands for the resultant force.

In Newtonian physics, if n=2, Eq.9-1 would leave us with

where m's stand for the quantities of mass,

a's stand for acceleration.

Since relativity considers mass m as being a speed depending quantity, the equality of Eq.9-2 can only hold for some time instant; the varying andforce the resultant values on both sides of the equation out of balance all the time except at some particular moment. Of course, then, with this argument, relativity would think that it can toss Newton's second law and third law out of the window at no time.

In relativity, the clock attached to each moving object would be observed as functioning differently by someone not moving with this object, depending upon the speed of each moving object. The most convenient method for an observer to keep track of the movement of all the objects involved in the observation is to transform all time readings from all the moving clocks into proper time, the time that is recorded by a single clock that is stationary to his laboratory frame. As far as special relativity is concerned, Equation 9-1 could be developed as

Because of the relationship (See notes below)   found in special relativity, for each elementary term in Equation 9-4, according to special relativity, we can have,

where is the moving length of the i-th object that the stationary frame can cover by the relative movement during the

proper time interval recorded by the observer on the stationary frame,

represents the corresponding rest length of the moving length of the i-th object. (The total rest length of the i-th object may be much longer than , but we are only interested in the secment of that is marked by a certain coordinate of the stationary frame during ).

Furthermore, for the scaler quantity m of the i-th object, we have

,

where is the rest mass of the i-th object,

is the relativistic mass of the i-th object.

But is also found in special relativity, so we can have ,  and Eq.9-4 leads us to

Because is a constant, we then further have

When n=2, we would naturally have

or, eventually,

By comparing Eq.9-3 and Eq.9-5, we can immediately realize that special relativity leads itself to an identical equation that is a mathematical expression of Newton's third law. In other words, special relativity, with its mathematical outcomes, leads to the "confirmation" of Newton's third law.

"Confirmation" is concluded.
 

Regarding expression A), 

On page 67, "......the situation is illustrated in Fig.2-4. The passenger, who has a wrist watch, say, sees the light ray follow a strictly vertical path (Fig. 2-4a) from A to B to C (The coordinate system in which A, B, and C are marked is attached to a train, and is not moving with respect to the passenger, so the distance AB or BC represent the rest length to the passenger, noted by this author) and times the event by his clock(watch). This is a proper time interval,......"

Regarding expression B),

On page 69, "The ground observer measures the length to be L (So, this distance represents the rest length to the ground observer, noted by this author) and claims that the passenger covered this distance in a time (where v is the speed, noted by this authoor). This time, , is a non proper time, ..."

Regarding expression C),

On the same page, "The passenger, however, observes...... and......measures a proper-time interval , .... The passenger claims that the platform moves (So, the platform represents a moving length to this passenger, noted by this author) ...... Hence, the length of the platform to him is L' (moving length) = ......"

Does relativity restrict itself to these speed expressions in its mathematical works? Not at all ! For example, whenever the speed of light is involved, relativity just puts up . Of course, then, without any exception, when other values of speed, such as 0.6 C, 0.1 C, are involved, is the expression for relativity to use. In other words, relativity has a fourth expression of speed, which is simply

So, we can easily tell that relativity has at least 4 mathematical expressions for the same physical concept: speed.

We all know that, before relativity appeared, the form had long been exclusively used to express speed and was unambiguously adopted in all speed calculations by Newtonian mechanics, in which the impact of speed on the nature of length and time is unfound.
 
 

A Preceding Question

If one is asked to compare a 7 meter rod moving at speed of 0.34C with a 5 meter rod moving at speed of 0.31C, can he tell which rod will be measured as being longer?

Answer from Newtonian physics: The 7 meter rod is longer, regardless.

Answer from special relativity: No conclusion can be made until the measurements are converted so that they can be compared with a physical measuring system that is stationary to an observer. The conversion is done according to the formula

where is the moving length of the rod being measured,

is the rest length of the same rod,

      v is the speed of the rod, and

     C is the speed of light.

That the unit of length for can not be simply termed as "meter" is obvious because the body it represents is moving. This is to mean that the moving rod with rest length of  7 meters can not be simply called a rod measured as 7 meters; otherwise the impact of speed on the measurement of length is omitted and special relativity will no longer distinguish itself from Newtonian physics. Therefore, the unit of length in relativity pertinent to has to be termed as moving meters at a certain speed. In order to achieve coherence of units for comparison, the conversion, taking the 7 meter rod as an example, must be done in the following manner:

Both the word "moving" and the speed specification on the left side of the equation have formed an indispensable part of the unit of length for the moving rod, otherwise we would have 7 meters = This is obviously unacceptable.

However, for simplicity, if we are dealing with only one speed, we can denote our unit of length in our conversion as shown in the following manner: 7 moving meters =, leaving the speed element as an understandable hidden factor on the left side of the equation. This method of unit denotation would have trouble if our concern were for more than one speed, because 7 moving meters at speed of 0.34C is definitely different from 7 moving meters at speed of 0.35C according to special relativity. By the same token, in relativity, when we have a time quantity, we must qualify the quantity with a description of its being proper or non proper, for example, one proper second, three non proper seconds (at a certain speed) etc., instead of simply one second, or three seconds.

Mathematical Performance Resulting in 0.6=0.48

When a certain object is said to be traveling with speed 0.6C, the versatility of speed expressions, as allowed by special relativity and shown in the Note above, permits us to have several mathematical expressions such as :

Since is legitimate in special relativity, then on the right side of the above equation we have v=0.48C . However, the left side of the same equation is v=0.6C by initial condition, therefore, we have

"Proof" is concluded.

You have also been shown how to acquire 1=0.2=0 with relativity in the symposium paper MATHEMATICAL INVALIDITY OF RELATIVITY . Now,  if you want to, I am sure you know how to formulate 3=5=2 , with relativity, of course.  Do you still feel perplexed with the twin paradox? I think your time is more valuable than looking for the "truth" of this mystery.

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