# 4-momentum and relative velocity

• scottJH
In summary, the speed of the particle relative to the observer can be determined by solving the equation ##P_p.P_o = m_pm_o\gamma(-c^2)## for the relative velocity ##v##.

## Homework Statement

In a particular frame of reference a particle with 4-momentum ##P_p## is observed by an observer moving with 4-momentum ##P_o##. Derive an expression for the speed of the particle relative to the observer in terms of the invariant ##P_p.P_o##

I am completely stuck on this question. Any help would be appreciated.

## The Attempt at a Solution

I used the equations

##P_p = m_p\gamma(u_p, ic)##

and

##P_o = m_o\gamma(u_o, ic)##

I tried to obtain ##P_p.P_o## in the rest frame of the observer and obtained

##P_p.P_o = m_pm_o\gamma(-c^2)## where ##\gamma## is in terms of ##u_p##

This is because ##\gamma## in terms of ##u_o## is equal to 1 in the rest frame of the observer

I'm pretty sure that my answer is incorrect. However even if it is correct I am unsure how I would be able to determine the relative velocity from it.

scottJH said:
I tried to obtain ##P_p.P_o## in the rest frame of the observer and obtained

##P_p.P_o = m_pm_o\gamma(-c^2)## where ##\gamma## is in terms of ##u_p##

This is because ##\gamma## in terms of ##u_o## is equal to 1 in the rest frame of the observer

The ##\gamma## here is based on the relative velocity between the observer and the particle (which in the rest frame of the observer happens to be the same as the velocity of the particle in that frame). Since you have expressed the invariant in terms of invariants (##m_p##, ##m_o## and ##c##) and the relative velocity ##v## (through ##\gamma##) you can solve this equation to find the relative velocity.

## 1. What is 4-momentum?

4-momentum is a mathematical concept used in the field of relativity to describe the motion of an object. It combines the object's energy and momentum into a four-dimensional vector, with the three spatial components representing momentum and the time component representing energy.

## 2. How is 4-momentum related to relative velocity?

4-momentum is directly related to relative velocity because it takes into account both the object's speed and direction of motion. As an object's relative velocity changes, so does its 4-momentum vector.

## 3. Can 4-momentum be used in non-relativistic situations?

No, 4-momentum is a concept that only applies in the context of special relativity. In non-relativistic situations, where speeds are much slower than the speed of light, the simpler concept of momentum is used instead.

## 4. How does 4-momentum affect the mass of an object?

According to Einstein's famous equation, E=mc^2, energy and mass are equivalent. 4-momentum takes into account both energy and momentum, so it also affects the mass of an object. As an object's energy or momentum changes, its mass will also change accordingly.

## 5. Can 4-momentum be used to describe interactions between particles?

Yes, 4-momentum is commonly used in particle physics to describe the interactions between subatomic particles. It allows scientists to accurately calculate the energy and momentum transfer during collisions and other interactions.