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Re: What hath Einstein Writ?



Forwarded from John Howell, please direct responses directly to him
if you are using provate email.

Dan M

Re: What hath Einstein Writ?

Joel Rauber wrote:

Hmm? Well let's see. What is it we need mass for? We need a proportionality
constant between momentum and v/Sqrt(1-v^2) {units of c=1}. I rather guess
that rest mass will suffice quite well and we don't need a concept like
relativistic mass.



My point: momentum is

m_o*v/Sqrt(1-v^2)

One has two options, you may associate the ugly looking Sqrt term with m_o
and invent the concept relativistic mass. The main justification being that
you may then write formulas for momentum that "look" like the old Newtonian
formulas, i.e. m*v. (The price paid is losing a nice invariant, the rest
mass.)

The other choice is to simply realize that the "look" of the correct special
relativistic formula for momentum does not have to be the Newtonian "look"
and one may simply realize that the proper way to write momentum is

m*v/Sqrt(1-v^2) where m is the invariant rest mass.

Neither choice is wrong! You can develope a fully consistant and correct
special relativistic kinematics either way; but one does see relativistic
mass less and less these days. . .



There's actually another way to think of momentum. In Newtonian physics is was
mass * v where v = dx/dt.

It's sometimes useful to define a new "relativisitic" kind of velocity, as

V_rel = dx/d_tau (where tau is the proper time).

{This, of course, is the spacial component of the velocity 4-vector.}

Then - to follow Joel's teminology - the "ugly" sqrt is absorbed
into the definition of velocity. With this notation, the
relativistic momentum is

p = mass * V_rel

The (total) energy is likewise proportional to the "time" component
of the 4-velocity, so the proportionality that Barlow Newbolt desired
is also in evidence.

John Howell
Earlham College, Department of Physics