# Light-cone coordinate and Lorentz transformation

• Chenkb
In summary, it is not possible to eliminate the quark's transverse momentum without changing the longitudinal momentum fraction of the hadron.
Chenkb
Assume that, in cartesian coordinate, we have a quark with momentum ##k=(k_0,0,k_0sin\theta,k_0\cos\theta)## and a fragmented hadron ##p=(p_0,0,0,p_0)##.
Define, in light-cone coordinate, ##k^+ = k_0 + k_3 = k_0(1+cos\theta)##, and ##p^+ = p_0 + p_3 = 2p_0##.
And the longitudinal momentum fraction of the hadron is ##z^+ = p^+/k^+##.

Obviously, we take the hadron direction as z-axis, and the quark has a transverse momentum ##|\vec{k_\perp}| = k_0sin\theta##.
I wonder whether it is possible to make a suitable Lorentz transformation, so that the quark gets no transverse momentum, and maintain ##z^+## unchanged?

Last edited:
No, it is not possible to make a suitable Lorentz transformation that will make the quark have no transverse momentum and keep the longitudinal momentum fraction of the hadron unchanged. The only way to do this would be to rotate the quark's momentum vector so that the transverse component was aligned with the longitudinal component, but this would change the longitudinal momentum fraction of the hadron since it would reduce the quark's momentum in the z-direction.

## 1. What are light-cone coordinates and how are they related to Lorentz transformations?

Light-cone coordinates are a specific way of representing points in spacetime, where one coordinate represents the time dimension and the other represents the space dimension. They are related to Lorentz transformations, which are mathematical equations that describe how measurements of time and space change between two reference frames that are moving relative to each other. Light-cone coordinates are used to visualize and understand these transformations in special relativity.

## 2. Can you explain the concept of a light-cone in relation to light-cone coordinates?

In special relativity, the path of a beam of light can be represented as a cone in spacetime, with the tip of the cone being the source of the light and the base of the cone representing the path of the light over time. Light-cone coordinates are named after this cone, as they represent points along the cone in spacetime.

## 3. How do Lorentz transformations preserve the speed of light?

In special relativity, the speed of light is considered to be constant and the same for all observers, regardless of their relative motion. Lorentz transformations are designed to preserve this constant speed of light by adjusting measurements of time and space between reference frames. This is what allows the laws of physics to remain the same in all inertial reference frames.

## 4. What is the significance of the Lorentz factor in Lorentz transformations?

The Lorentz factor (γ) is a mathematical term that appears in the equations for Lorentz transformations. It represents the amount of time dilation and length contraction that occurs between two reference frames moving relative to each other. It is important because it allows us to calculate how measurements of time and space must change in order to preserve the constant speed of light.

## 5. How are light-cone coordinates and Lorentz transformations used in physics?

Light-cone coordinates and Lorentz transformations are fundamental concepts in special relativity and are used in many areas of physics, including particle physics, cosmology, and electromagnetism. They are essential for understanding the behavior of objects moving at high speeds and for making accurate predictions about how physical quantities change between different reference frames.

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