I did not intend this posting for didactic analysis; but rather was putting it as a hyper-quick posting (in terms of my time) and intended it to be merely a quick informational posting. This means I did not bother to define terms; thus leaving potential for confusion.
The author makes clear that u is the usual 3-velocity in some frame (actually one component of the 3-velocity where the spatial axis is aligned with the direction of the 3-velocity; meaning we are effectively dealing with a spatially one dimensional situation and a 1+1 dimensional space-time). The equation is therefore writing a relationship for one component (a spatial component) of the 4-momentum. IMO, it is a scalar equation and not susceptible to criticism on dimensional grounds. (see the textbook for other assumptions)
I agree with JD, that my posting gives the appearance of such confusion, since I didn't write a careful posting clearly defining the symbols.
| -----Original Message-----
| From: firstname.lastname@example.org [mailto:phys-l-
| email@example.com] On Behalf Of John Denker
| Sent: Friday, January 27, 2012 4:57 PM
| To: Forum for Physics Educators
| Subject: Re: [Phys-l] Relativistic Mass or lack thereof in intro books
| 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) 
| 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 
| 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  to represent the 4-velocity, the
| factor of gamma is wrong there. The usual 4-vector equation is
| P = m u 
| If we take the u in equation  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
| For more on how to handle this topic correctly, see
| Forum for Physics Educators