Lorentz transformation lab to CM system

• WarnK
In summary, the Lorentz transformation between the lab and CM systems takes the system to the CM system.
WarnK
Which Lorentz transformation takes the lab system to the CM system?

Lab system: $$p_a = (E^{lab}_a, \vec{p}_a)$$ and $$p_b = (m_b, \vec{0})$$
CM system: $$p_a = (E^{CM}_a, \vec{p})$$ and $$p_b = (E^{CM}_b, -\vec{p})$$

For a binary reaction a+b->c+d, the textbooks I have say quite a lot about the kinematics of such reactions (Mandelstam variables and all that) but how do I go about doing a Lorentz transformation between the lab and CM system? That must be possible to do.

Anyone?

The velocity of CM in the scattering of two particles with rest mass $m_1$ and $m_2$ is given by

$$\vec{u}_{CM}=\frac{\vec{p}_1+\vec{p}_2}{E_1+E_2}\,c^2$$

from which you can find the Lorentz transformation.

So a lorentz transformation from the lab frame to the CM frame would be
$$\left[ \begin{array}{cccc} \gamma & -\beta \gamma & 0 & 0 \\ -\beta \gamma & \gamma & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array} \right]$$
with $$\beta = ||\vec{u_{CM}}||$$ and $$\gamma = (1-\beta^2)^{-1/2}$$?
It doesn't make any sense to me.

It doesn't make any sense to me.

The crusial thing is to make sense in nature, not to any of us.
First system of center of mass for two particles is defined by the requirement, that $(\vec{p}_1+\vec{p}_2)_{CM}=0$. The general Lorentz transormation $(c=1)$ is

$$\Lambda=\begin{pmatrix} \gamma& \gamma\,\vec{\beta} \cr \gamma\,\vec{\beta} & \delta_{\alpha\beta}+\frac{\gamma-1}{\beta^2}\,\beta_\alpha\,\beta_\beta \end{pmatrix}$$

Take now the 4-vector $p^\alpha_1=(E_1,\vec{p}_1)$ and apply $\Lambda$ with $\beta=\frac{\vec{p}_1+\vec{p}_2}{E_1+E_2}$, to get for the 3-momentum $\vec{P}_1,\,\vec{P}_2$ to the CM system:

$$\vec{P}_1=\Lambda\,\vec{p}_1\Rightarrow \vec{P}_1=\gamma\,\left(E_1+\frac{\gamma}{\gamma+1}\,\vec{\beta}\,\vec{p}_1\right)\,\vec{\beta}+\vec{p}_1[/itex] [tex]\vec{P}_2=\Lambda\,\vec{p}_2\Rightarrow \vec{P}_2=\gamma\,\left(E_2+\frac{\gamma}{\gamma+1}\,\vec{\beta}\,\vec{p}_2\right)\,\vec{\beta}+\vec{p}_2[/itex] Adding now the two equations we get [tex]\vec{P}_1+\vec{P}_2=\gamma\,\left(E_1+E_2+\frac{\gamma}{\gamma+1}\,\vec{\beta}\,(\vec{p}_1+\vec{p}_2)\right)\,\vec{\beta}+\left(\vec{p}_1+\vec{p}_2\right)\Rightarrow \vec{P}_1+\vec{P}_2=\gamma\,\left(E_1+E_2+\frac{\gamma}{\gamma+1}\,\vec{\beta}\,(E_1+E_2)\,\vec{\beta}\right)\,\vec{\beta}+\left(E_1+E_2\right)\,\vec{\beta}\Rightarrow$$
$$\vec{P}_1+\vec{P}_2=\gamma\,(E_1+E_2)\,\left(1+\frac{\gamma}{\gamma+1}\,\beta^2\right)\,\vec{\beta}+\left(E_1+E_2\right)\,\vec{\beta}\Rightarrow \vec{P}_1+\vec{P}_2=0$$

Now does it makes sense?

1. What is the purpose of a Lorentz transformation lab?

A Lorentz transformation lab is used to study the effects of special relativity on physical systems. It allows scientists to understand how space and time are affected by the relative motion of an observer and an object.

2. What does the CM system refer to in a Lorentz transformation lab?

The CM (center-of-mass) system is a reference frame in which the total momentum of a system is zero. It is commonly used in physics experiments to simplify calculations and analyze the behavior of particles.

3. How is a Lorentz transformation performed in a lab setting?

A Lorentz transformation is performed by applying mathematical equations that describe the relationship between space and time in special relativity. In a lab, this can be done using various instruments and techniques such as measuring the speed of particles or analyzing data from particle collisions.

4. What are some real-world applications of Lorentz transformations?

Lorentz transformations have many practical applications, such as in particle accelerators, where they are used to calculate the energy and trajectory of particles. They are also important in GPS technology, where they are used to account for the effects of relativity on the timing of signals between satellites and receivers on Earth.

5. Are there any limitations to Lorentz transformations?

While Lorentz transformations accurately describe the behavior of objects at high speeds, they are only applicable in the context of special relativity. They do not account for the effects of gravity, which are described by general relativity. Additionally, Lorentz transformations are only valid for objects moving at constant speeds in a straight line, and do not apply to accelerated motion.

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