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Relativity – Relativistic Momentum

We have recently learned that the laws of physics work anywhere you are in the Universe. We also learned that some of these laws change depending on the speed of the object. One of these changes is our understanding of momentum. Conservation of momentum says that two interacting bodies have a constant total momentum. This means no matter the inertial frame, the total momentum of the system remains constant unless there are external forces at play. Suppose we measure the momentum of a system involving a collision. Suppose then that we measure the velocities of the system in a second inertial frame using Newtonian transformations. We would notice that momentum is not conserved. Sometime in the future I will go through the derivation for the relativistic momentum equation, but for now I will just tell it to you. We are going to let m equal the mass of a particle at rest.

\[\vec{p}= \frac{m\vec{v}}{\sqrt{1-v^2/c^2}}\]

As you can see, with small values of v, the value of p becomes closer to the traditional value. We can use this same principle to deduce a value for Newton’s Second Law in an inertial frame.

\[\vec{F} = m \vec{a}\]

\[\vec{F} = \frac{\text{d}\vec{p}}{\text{d}t}\]

\[\vec{F} =\frac{\text{d}}{\text{d}t} \frac{m\vec{v}}{\sqrt{1-v^2/c^2}}\]

There are some consequences to these equations that I should point out. Since momentum is no longer proportional to the velocity of an object because of the inertial frame, force is no longer proportional to the change in acceleration. Because of this, constant force in an inertial frame does not cause an object to move with a constant acceleration. An object moving faster and faster towards the speed of light is continually decreasing in acceleration. This should make sense because of the objects inability to surpass c. If you keep pushing on an object with a constant force, it will eventually slow down its acceleration.  

\[F = \frac{m}{(1-v^2/c^2)^{3/2}}a\]

\[a = \frac{F}{m}(1 - \frac{v^2}{c^2})^{3/2}\]

This is just further proof that the speed of light is the ultimate barrier for a massive object, and that no matter how great the force, this barrier can never be passed. There is another consequence to the magic of relativity. An objects mass is dependent on the speed of which it is traveling. Let m be the mass of the object at rest.