Derivation of mass invariance using Lorentz transformation

[chris_0101]

Homework Statement

As the title suggests, I need help finding resources that clearly shows the step by step process of the derivation of the rest or invariant mass using the Lorentz transformation.

Homework Equations

Energy-momentum relation

The Attempt at a Solution

Not looking for an actual solution (even though that would be a bonus, lol) but I'm looking for a resource that will help me understand this concept. I'm not too picky either, let it be a wiki page to a textbook, I'm just desperate for guidance at this point in time.

thanks, in advance

 

[anathema]

!

The concept of rest or invariant mass is an important one in the field of special relativity. It is defined as the mass of an object when it is at rest, or when its velocity is zero. This is in contrast to the relativistic mass, which takes into account the effects of the object's velocity.

To understand the derivation of the rest mass using the Lorentz transformation, it is important to first understand the energy-momentum relation in special relativity. This relation states that the energy (E) of an object is equal to its mass (m) multiplied by the speed of light squared (c^2).

E = mc^2

Now, let's consider an object with a rest mass m0 and a velocity v. According to the Lorentz transformation, the energy of this object can be expressed as:

E = γm0c^2

where γ is the Lorentz factor, given by:

γ = 1/√(1-v^2/c^2)

Substituting this into the energy-momentum relation, we get:

γm0c^2 = mc^2

Simplifying, we get:

m = γm0

This is the relativistic mass of the object. However, when the velocity v is zero, the Lorentz factor becomes 1, and the relativistic mass reduces to the rest mass:

m0 = m/γ

Therefore, we can see that the rest mass is the mass of an object when its velocity is zero, and it can be derived from the relativistic mass using the Lorentz transformation.

For a more in-depth explanation and step-by-step derivation, I would recommend referring to a textbook on special relativity, such as "Introduction to Special Relativity" by David J. Griffiths or "Special Relativity: A First Encounter" by Domenico Giulini.

I hope this helps in your understanding of the concept. Best of luck with your studies!