# Momentum in special relativity

1. Oct 20, 2015

### Sobhan

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

2. Oct 20, 2015

### Orodruin

Staff Emeritus
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).

3. Oct 20, 2015

### 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?

4. Oct 20, 2015

### Orodruin

Staff Emeritus
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.)

5. Oct 20, 2015

### PWiz

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)

6. Oct 20, 2015

### Orodruin

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

7. Oct 20, 2015

### PWiz

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

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