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Welcome to our Physics lesson on Mass in Relativistic Events, this is the first lesson of our suite of physics lessons covering the topic of Relativistic Dynamics. Mass, Impulse and Energy in Relativity, you can find links to the other lessons within this tutorial and access additional physics learning resources below this lesson.
Let's start by analyzing the mass of objects - a parameter that characterizes the inertia of objects in Newtonian system. In the (classical) Newtonian system, mass is considered as absolute (unchangeable) in all inertial systems of reference.
Let's suppose we have a particle having a Newtonian mass m which moves at constant velocity v in a fixed inertial system S. We analyze this motion in another inertial system S'. From previous articles, it is known that when S' moves at high velocities comparable to that of light relative to S, the velocity of particle in S' is different from that measured in S. However, we have only dealt with changes in velocity - a kinematic parameter; nothing has been said about any change in mass so far. In other words, is it possible to accept the idea of an absolute (unchangeable) mass, independent from the system of reference we choose? Our experience with spacetime related quantities urge us to reject this claim.
From previous articles, it is known that the speed of light c is unreachable by material particles. A material particle can only get closer to the value of c but is can never equal or overcome it. In the actual stage of learning, we already consider the Newtonian system as outdated in relativistic events but we still accept the Newton's First Law of Motion as absolute. In the practical sense, this law is also valid in relativistic events, as we known that a free particle is at rest or in uniform motion in all inertial systems.
What about the Newton's Second Law of Motion? Doubting about the concept of absolute mass means putting in discussion the truthfulness of Newton's Second Law. This law in the form F⃗ = m ∙ a⃗ is valid in inertial systems but always in the framework of absolute time and for a fixed mass. We will try to prove this law is valid in relativistic events as well.
Let's suppose a constant force F is exerted on a particle in the X-direction of the inertial system S. At t = 0, the particle was at rest in the origin O of the system. Since there is a constant force is acting on the particle, there is a constant acceleration a⃗ as well, based on the Newton's Second Law of Motion. In the classical approach, the constant acceleration brings a constant increase in velocity by a · t, which means that theoretically, the velocity can take very large values depending on the duration of event. These values can overcome the limit velocity of light, which results in a paradox - now it is proven that nothing can overcome the value of light speed c.
Since this cannot occur, we believe something has happened to the inertia of particle. When v → c, the particle resists more and more to any change in velocity. This means its mass increases with the increase in its velocity. If this situation is handled mathematically, we will see that when v → c, then a → 0. Eventually, since F is constant, then m → ∞ because only when we multiply zero by infinity we have any chance to get a fix number (from theory of limits explained in mathematics).
Let's enforce this reasoning using another approach. We have explained that Newton's Second Law of Motion is also expressed in terms of impulse and time as
where p⃗ = m ∙ v⃗ is the instantaneous momentum of particle (impulse is a change in momentum). When F⃗ is constant, the above rate is constant too. This means the particle gains a constant impulse regardless the velocity in reality increases less and less when approaching the value of c. This can occur only when mass increases more and more. In other words, in relativistic events mass does not remain constant but increases with the increase in velocity according the function m = m(v⃗) such that m → ∞ when v → c.
You have reached the end of Physics lesson 18.6.1 Mass in Relativistic Events. There are 3 lessons in this physics tutorial covering Relativistic Dynamics. Mass, Impulse and Energy in Relativity, you can access all the lessons from this tutorial below.
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