OnlyMe said:
P.S. The Lorentz transformations in SR play a role in being able to project what things look like from differing inertial frames of reference. That is if you measure the impact of a bullet, F = mv in one frame you can figure out what its mass is in its rest frame. It does not say that it gains mass with velocity.
So why does Okun use the Lorentz transform to calculate mass?
Here is the quote from Okun, again...
The Concept of Mass
This choice is caused by the confusing terminology widely used in the popular science literature and in many textbooks. According to this terminology the body at rest has a "proper mass" or "rest mass" m moving with velocity v has "relativistic mass" or "mass" m, given by, $$m = E/c^2 = \frac{m_o}{\sqrt{1-v^2/c^2}}$$
As I will show, this terminology had some historical justification at the beginning of our century, but it has no rational justification today. When doing relativistic physics (and often when teaching relativistic physics), particle physicists use only the term "mass." According to this rational terminology the terms "rest mass" and "relativistic mass" are redundant and misleading. There is only one mass in physics, m, which does not depend on the reference frame. As soon as you reject the "relativistic mass" there is no need to call the other mass the "rest mass" and to mark it with the index O.
In the first part of the quote he identifies, the form of the equation quoted, as
"confusing terminology widely used in the popular science literature and in many textbooks."
He then goes on to explain, in the next sentence, what that form of the equation is describing, which is what creates the confusion. By incorporating an object's (
he uses the word "body"), velocity and mass (rest mass), mass takes on two separate definitions. Okun uses $$m$$, in this equation to indicate "relativistic mass", which he is arguing against, and $$m_o$$ to indicate an object's frame invariant mass, also known historically as, "proper mass" and/or "rest mass". Einstein used $$M$$ for "relativistic mass" and $$m$$ for mass.
Here Okun, is explaining the misuse and origin of the confusion, not his own or Einstein's definition of $$m$$, in the equation, $$E = mc^2$$.
In the final paragraph of the above quote, he sets out his intention to explain why, the concept of relativistic mass should be discarded and only a single definition and term for mass be used.
Later in the paper, Okun does show how $$E = mc^2$$, can be incorporated in a description of the momentum of both a massive object and massless photon. He does use the Lorentz factor in doing so, but I think that Trippy put it a little more clearly in
Post 231.
$$E=mc^2$$ is only true for a stationary object.
For a fast bullet, the correct equation is:
$$E=\sqrt{(mc^2)^2+(pc)^2}$$
Which as you can see for a stationary object (p=0) boils down to:
$$E=mc^2$$
And for a photon boils down to:
$$E=pc$$
While Trippy begins by describing $$E = mc^2$$ as a description of the mass of a stationary object only, explaining, in a later post, that he assumes E to always denote total mass, it is my contention that that where ever it appears in an equation, $$mc^2$$ always denotes the energy (E) associated with the "rest mass" or "proper mass", of an object at rest or in motion. (I believe that this is consistent with both Einstein and what Okun was trying to explain.)
OnlyMe said:
If you look at the equation as presented by Okun, $$m = E/c^2 = \frac{m_o}{\sqrt{1-v^2/c^2}}$$, the first part $$m = E/c^2$$ is the same as $$E = mc^2$$. The last section $$\frac{m_o}{\sqrt{1-v^2/c^2}}$$ then includes a velocity, which then associates it with momentum rather than a direct description of mass.
I mean, if you want momentum instead of "relativistic mass", just use: $$ mv = \frac{m_o v}{\sqrt{1-v^2/c^2}}$$.
P.S. I think you made a mistake with F = mv, too. And your comment about how the Lorentz factor "includes a velocity", has nothing to do with momentum.
The whole point has been to discard the idea of using $$m$$ and $$m_o$$.., to essentially discard the description of momentum as, "relativistic mass" and accept a single definition for mass.
As to the F = mv issue, if you measure the momentum of a bullet from a frame of reference at rest relative to the bullet's velocity, as F (force), that force is the product of the bullet's frame invariant mass and its velocity relative to the frame of reference from which the F (force) is measured. It is the same as p = mv, in this case. Force is a more general term, but it includes the more specific definition consistent with momentum.
Once the concept of mass is accepted as frame invariant, there are some issues that seem to come up relative to the Lorentz factor, inertia and mass, when relativistic velocities are also, involved... An apparent conflict with SR, but this is an issue that cannot really be addressed, until some consensus definition of mass, is established.
