Momentum in special relativity

[Sobhan]

when studying momentum 4 vectors,i encountered the CT momentum which is MC.can some explain where has this come from?

 

[Orodruin]

In special relativity, the 4-momentum is defined as $P = m V$, where $V = dX/d\tau$ is the 4-velocity, $m$ the invariant mass, $\tau$ the proper time, and $X$ the coordinates along the world line. It follows directly that the 0-component of the 4-momentum is given by $m\gamma c$. The classical limit allows the identification of this component with the total energy of the object (as well as of the spatial components with the momentum of the object).

 

[Sobhan]

a question on the energy-momentum triangle:in this triangle one of the sides of it is PC,is this P the momentum in 4 dimensions or 3?

 

[Orodruin]

The energy-momentum triangle is nothing but the norm relation for the 4-momentum. Since the norm of the 4-velocity is 1, the norm of the 4-momentum is always $m^2 c^2$. With the 4-momentum being $P = (E/c,\vec p)$, it follows that $P^2 = E^2/c^2 - \vec p^2 = m^2 c^2$, which may be rewritten as the energy-momentum triangle relation. (So the answer to your question is that the momentum in the triangle is the 3-momentum.)

 

[PWiz]

a question on the energy-momentum triangle:in this triangle one of the sides of it is PC,is this P the momentum in 4 dimensions or 3?

Are you talking about the norm of the four-momentum? If yes, then $\vec p = (\gamma m_0 c, p_x, p_y, p_z)$ and $|\vec p |^2 = \frac{E^2}{c^2} - p^2$ where $p$ is the 3-momentum of the particle and $E$ is the particle's total energy (rest+kinetic) in that particular reference frame. (The norm is invariant in all frames.)

EDIT: Orodruin beat me to it. (No surprises there)

 

[Orodruin]

Are you talking about the norm of the four-momentum? If yes, then $\vec p = (\gamma m_0 c, p_x, p_y, p_z)$

Generally, I would avoid using a vector arrow for 4-vectors and reserve it for 3-vectors. Things can become very confusing otherwise ...

 

[PWiz]

Generally, I would avoid using a vector arrow for 4-vectors and reserve it for 3-vectors. Things can become very confusing otherwise ...

Okay. It's just that when I think of a vector in a SR, it's almost always a four-vector, so I've sort of got into a habit of putting that arrow