Proper Velocity: Intro to Elementary Particles

[hagopbul]

TL;DR Summary

some idea in proper velocity i dont understand

Hello :

as i read during quarantine introduction to elementary particles by griffth i encounter the following paragraph
"When we speak of the "velocity" of a particle (with respect to the labo-
ratory), we mean, of course, the distance it travels (measured in the lab frame)
divided by the time it takes (measured on the lab clock):

v = dx/dt

But in view of what has just been said, it is also useful to introduce the "proper" velocity, zeta, which is the distance traveled (again, measured in the lab frame)
divided by the proper time:

zeta = dx/d(tao) : d(tao) = dt/gamma*

According to equation (3.28), the two velocities are related by a factor of gamma:

zetta = gamma*v"

what i don't understand is how we can use two different quantities to describe velocity the distance is from the lab frame and the time is from particle frame

why zetta isn't written like this

zetta = dx'/d(tao)

 

[PeroK]

what i don't understand is how we can use two different quantities to describe velocity the distance is from the lab frame and the time is from particle frame

You have two measures of time you can use: the coordinate time of the laboratory ($t$) and the proper time of the particle ($\tau$). Any quantity can be differentiated with respect to either of those two time variables.

In SR generally this leads to having two versions of many quantities: three-velocity and four-velocity; three-momentum and four-momentum; three-force and four-force. It turns out that both are useful concepts, so something like three-velocity doesn't get entirely replaced with four-velocity. Both are useful.

Note also that the particle's proper time is essentially an alternative way to parameterise motion of the particle. You can look at it like this as well. You either have $(x(t), y(t), z(t))$ or $(x(\tau), y(\tau), z(\tau))$ as the parametrisation of the particle's path.

 

[vela]

But in view of what has just been said, it is also useful to introduce the "proper" velocity, zeta, which is the distance traveled (again, measured in the lab frame)
divided by the proper time:

zeta = dx/d(tao) : d(tao) = dt/gamma*

Griffiths actually used the Greek letter eta, $\eta$, not zeta, $\zeta$, and the proper time is represented by tau, $\tau$, not tao.

what i don't understand is how we can use two different quantities to describe velocity the distance is from the lab frame and the time is from particle frame.

I think the point you're overlooking is that $d\tau$ is an invariant. It's not so much that it's $dt'$ in the particle's rest frame, but it's the invariant quantity that all observers can calculate from their own measurements.

why zetta isn't written like this

zetta = dx'/d(tao)

That would just be the spatial components of the four-velocity in the S' frame, right? If you're an observer at rest in S, why would you want to use that? And if S' is the particle's rest frame, it would be 0.

 

[robphy]

The key idea is to describe the "separation between inertial worldlines meeting at a common event".

Analogous to Euclidean geometry,
the separation of worldlines can be described by

• the velocity (slope) ( the ratio of the legs of the 4-velocity, $v=\frac{\Delta x}{\Delta t}=\frac{opp}{adj}=\tanh\theta$ , which is not additive since $v_{AC}=\frac{(v_{AB}+v_{BC})}{1+v_{AB}v_{BC}}$ )
• the rapidity ( Minkowski-angle ) $\theta=\mbox{arctanh}(v)=\mbox{arctanh}(\frac{\Delta x}{\Delta t})$ which is an additive quantity ($\theta_{AC}=\theta_{AB}+\theta_{BC}$),
• the celerity or "proper velocity" (spatial-component of the [unit] 4-velocity $\gamma v=\frac{\Delta x}{\Delta \tau}=\frac{opp}{hyp}=\sinh\theta$), which is also not additive since $(cel)_{AC}=(cel)_{AB}\gamma_{BC}+\gamma_{AB}(cel)_{BC}$
  (I think proper-velocity is a potentially confusing term. Unlike "proper time" and "proper acceleration" which are invariants, "proper velocity" is not an invariant.)

(Incidentally, in the Galilean limit, these three quantities coincide.
So, the fundamental "additivity of angles" implies --only in the Galilean limit-- the "additivity of velocities".
Unfortunately, our common sense mistakenly regards "additivity of velocities" as fundamental, which impedes our intuition for special relativity.)

 

[anuttarasammyak]

what i don't understand is how we can use two different quantities to describe velocity the distance is from the lab frame and the time is from particle frame
why zetta isn't written like this
zetta = dx'/d(tao)

The reason why we introduce 4-velocity in lieu of usual 3D velocity is that
TOR requires 4 component vector to describe the motion and such introduced velocity is convenient to describe energy or momentum.

Say the start point of particle motion $(ct_s,x_s,y_s,z_s)$,
The goal point of particle motion $(ct_g,x_g,y_g,z_g)$, so the difference $(ct_g-ct_s,x_g-x_s,y_g-y_s,z_g-z_s)$

The same difference in the particle's rest frame
Difference $(c\tau_g-c\tau_s,0,0,0)$

These two are connected with the relation :
$$(c\tau_g-c\tau_s)^2 = (ct_g-ct_s)^2-(x_g-x_s)^2-(y_g-y_s)^2-(z_g-z_s)^2$$

So 4-vector
$$(\frac{ct_g-ct_s}{c\tau_g-c\tau_s},\frac{x_g-x_s}{c\tau_g-c\tau_s},\frac{y_g-y_s}{c\tau_g-c\tau_s},\frac{z_g-z_s}{c\tau_g-c\tau_s})=(u^0,u^1,u^2,u^3)$$
satisfies
$$(u^0)^2-(u^1)^2-(u^2)^2-(u^3)^2=1$$
normalized, in TOR sense, to 1 and its spatial direction is same with familiar 3D velocity.

Making use of it 4-momentum (energy/c and momentum) is expressed simply by $p^i=mcu^i$ which is similar to usual non relativistic relation $p=mv$.