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



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: phys-l-bounces@carnot.physics.buffalo.edu [mailto:phys-l-
| bounces@carnot.physics.buffalo.edu] 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)               [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
| _______________________________________________
| Forum for Physics Educators
| Phys-l@carnot.physics.buffalo.edu
| https://carnot.physics.buffalo.edu/mailman/listinfo/phys-l
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