Re: [Phys-l] Relativistic Mass or lack thereof in intro books

[John Denker <jsd@av8n.com>]

On 01/27/2012 01:13 PM, Rauber, Joel wrote:
> I just received a copy of the 3rd edition of Randall Knight's Physics
> for Scientists and Engineers - with modern physics
>
> And was pleased to see that he eschews relativistic mass.

Very nice.
 
> i.e. his momentum for a particle is
>
> P = m *(Delta x over Delta Tau) =gamma*m*u

Hmmmmm.  Let's take apart the double equation on the last line:

  P = m *(Delta x over Delta Tau)               [2]

That is entirely reasonable and conventional.  It is an equation
involving the 4-vector momentum P and the 4-vector position x.
So far, so good.

Meanwhile, I am confused by this part:

   m *(Delta x over Delta Tau) =gamma*m*u        [3]

Conventionally, 
        v = dx_S/t  =  classical 3-vector velocity (in some frame)
             where x_S is the projection of the position vector (x)
             onto the spatial part of the given frame,
             and the denominator involves t, the projection onto
             the time-axis of the given frame.
        u = dx/dτ  =  the 4-vector velocity (in all frames)
             where the denominator involves τ i.e. the proper time

If we take the u in equation [3] to represent the 4-velocity, the
factor of gamma is wrong there.  The usual 4-vector equation is
simply
       P = m u                                   [4]

If we take the u in equation [3] as a typo or as an unconventional
representation for the classical velocity dx/dt, the equation doesn't 
make sense, on dimensional grounds.  That is: it appears to have a 
4-vector on the LHS and a 3-vector on the RHS.  Similarly, the LHS 
is valid in all frames, while the RHS only makes sense in some chosen 
frame.

For more on how to handle this topic correctly, see
 http://www.av8n.com/physics/spacetime-welcome.htm#sec-velocities-momenta