One new reality check on “The Relativistic Mass” concept

Remember that when we used to study in high school, some of our friends used to meet and discuss topics related to physics. Among those discussions, one thing would always come up, that in future we will know Einstein’s theory of relativity in detail.  We know some things about the length contraction, time dilation and an idea that states, mass increases with the increase in speed (Velocity). In simple words an in reality, the concept of Relativistic mass; this particular idea of Relativistic mass(M) was always fascinating.

Building up the background

Today also, some experts from physics understand this and believe that the real fact is that mass is a function of velocity. Thus, faster the object moves, the greater is its mass.  Surprisingly, using this idea, they want to explain why no object can move faster than the speed of light. They argue that for this infinite energy is required and mostly refer to the formula:

        ———————Equation (1)  

Asking someone who has some knowledge of mathematics, he would say that the mass(M) will increase if the velocity (v) is increased.  That is, for most people this formulation is meaningful and true, which even nature accepts.

In science, nothing can be accepted quickly. It is necessary to strengthen the criterion, whether it is the criterion of logic or experiment.  So now we must know whether mass is really just a function of motion?  I propose that we can find a solution to this task with the help of logic, irrefutable logic!

Seeking help from the history

If we look at the history, we will find that the origin of this sutra was initiated by Albert Einstein to keep the concept of momentum alive.  When he was writing special theory of relativity in detail, he understood that the expression of conventional linear momentum given by Descartes as p=mv, is an utter failure and false when the object is moving at high speed.  As a result, the law of momentum conservation also becomes untrue. To fix this issue, the great physicist imagined that momentum should also be a function of a certain quantity. This might be depending on the speed of the object itself.  When the equations were solved; famous Lorenz Factor equation was obtained. Now the formula for linear momentum modified to:

———————–Equation (2)

We call this formula Relativistic linear momentum.

This is the most accurate momentum formula yet and it also holds some physical meaning. If we want, we can find something more accurate than this, but before that we will have to find some flaws in the sutra of this linear momentum. 

Coming back to the originality to reality

Let us return to our old question where we need to know whether mass is a function of motion or not. To achieve this, we must look back carefully at the formula for this new momentum, and we will find that the numerator of the represents the formula of the old momentum p=mv. This means that if we identify it by following two ways. In reality, it would be:

   ——————–Equation (3a.)

Or

 ———————Equation (3b.)

And again, we want to simplify it, we can write:

————————Equation (4a.)

Or

———————-Equation (4b.)

But this time, remember what is this mass M, and what is the velocity η ? Both situations have their own names in the world of relativity. The first case is called Relativistic mass (equation 3a) and the second case is the proper velocity (equation 3b). We can write our new formula as the old formula if we choose either one of these two cases. (equations 4a. and 4b.)

Understanding with a simple example

These situations can be better explained by a simple case – ‘the likes and dislikes of Mohan and Sohan’. 

If Mohan says I like the first case 4a, that’s how it should be written, then it’s against Sohan. He will then say, I can write the same formula using the second case 4b, so that original form of expression can be achieved.

Moreover, if he gets a little more aggressive, he will also say that this is the only true way to express it.

Let they continue their quarrel, meanwhile we continue to understand that this new sutra can be written in two ways.

If our aim is to write it in original compact form, then both the conditions are the same and the (equation 3) have no earthly and physical meaning. That is, they are called mathematical structures and these mathematical structures have no meaning in physics but are just abstract concepts. 

The juggling of equations starts here…

However, one can get a qualitative description of these mathematical structures like for the relativistic mass M, one can say that it increases with the increase in the velocity, but keep in mind that this thing will not be applicable in reality to the original concept of mass(m).

Similarly, one can also understand the meaning of proper velocity in following ways:

Let simple velocity u of particle in inertial frame S as

    (considering only x-coordinate)

  is the change in displacement from point A to B along  in  time period under inertial reference frame S.

Also, there exist another inertial frame S’, which has frame velocity  with respect to our old S frame. We will imagine our S’ in such a way that old frame (S), these two events might be occurring at same point so that time interval between the event in S’ will be a proper time  as,

 ———————–Equation (5a.)

It is necessary to note down in reality here that, this condition can only be fulfilled if we understand that (i.e. magnitude of particle and frame velocity are equal)

Thus, rewriting Equation (5a.) we get

   ——————–Equation (5b.)

It is interesting here that proper velocity is being defined as,

    —————————-Equation (6)

I know it is difficult to accept that proper velocity is not actually ‘properly’ defined but we can’t do anything 😊.

Into the final stretch of reality – Just there…

One may also find it that it’s of no use but it has it use in the formation of four-vector for velocity & transformation.

Interested can also show that equation (6) as equivalent to equation (3b).

[Hint: Use equation (6) & equation (5b)].

One can say that proper velocity is something as a ratio of a displacement of particle in S-frame to the change in time when observed in S’-frame.

(Such that uniform velocity of the S’-frame with respect to S is same as the velocity of particle moving within S frame.)

Don’t worry if you are getting confused, it had taken me 3 years to understand difference between S & S^’.

In wholesome we can say that when people wanted to stick to the old definition of momentum then two new concepts:

1. concept of relativistic mass and
2. The concept of proper velocity was observed.

Thus, it’s an arbitrary choice for someone to write down the expression of relativistic linear momentum p either in the term of relativistic mass (M) or proper velocity .

This arbitrary choice of writing expression of linear momentum has overthrown the reality of relativistic mass and make us always to recapitulates that mass (rest mass) is a something that does not depend on velocity of any kind.

Culminations and Codas on reality of Relativistic Mass Concept

Now it will be easy for you to understand that wherever you study that mass is a function of motion is not pretty true. There are no events in nature to prove that the mass of moving object increases if its speed (velocity) is increased. Taking a little more time, I would like to show the following picture in which Albert Einstein himself wrote his thoughts about relativistic mass.

It is clear from this picture that Albert Einstein himself has emphasized that concept of Relative mass should be discarded. It clears the mind in reality that mass should be of two types, but we have not seen it in a fair way to accept such things and also understood (hope so!), Why it is like this?

We should always remember that the mass, which is measure of matter/Energy content, (in any extreme condition of momentum) would not show alternation in its value by any amount. And we earthly being use the small m of the English alphabet to represent the mass (generally)…

At last, let’s take a comic picture from reality for fun…

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