# Applying Lorentz transform to current 4 vector

• Finch91
In summary, the problem involves finding the current and charge densities in a moving frame, given an infinite line of charge with a density of lambda per unit length along the z-axis in a stationary frame. The Lorentz transform is used to calculate the current and charge densities in the moving frame, resulting in a current density of (-\gamma\beta c \lambda,0,0,\gamma c \lambda) and a charge density of \gamma c \lambda. However, this solution assumes a uniform charge distribution, whereas the problem specifies a charge distribution concentrated on the z-axis. Therefore, a different mathematical object would need to be used to accurately represent the charge distribution. Additionally, the convention for writing charge and current components may vary.
Finch91

## Homework Statement

Consider an infinite line of charge with density λ per unit length lying along the z-axis. If the line of charge is stationary in frame S, use the Lorentz transform to find the current and charge densities in a frame S' which is moving with velocity v parallel to the z-axis of S.

## Homework Equations

4 current defined as

$$j_\mu = (J_1,J_2,J_3,c\rho)$$

## The Attempt at a Solution

In S I find

$$j_\mu = (0,0,0,c\lambda)$$

then applying the Lorentz transform to find

$$j'_\mu = (-\gamma\beta c \lambda,0,0,\gamma c \lambda)$$

in S'

Is this correct?

Last edited:
Hey, welcome to physicsforums!
It's almost correct. You have the correct answer if the charge was uniform over all space. (since you have put the constant lambda, the charge will spread the same over all space). What the question asked for was a charge distribution that is concentrated on the z axis. So which mathematical object can you use to represent this charge distribution?

Also, usually the charge is written as the zeroth component, and the current as the other 3, but I guess it is only convention, so it doesn't really matter. It just might be confusing when you look at other people's work.

## 1. What is the Lorentz transform?

The Lorentz transform is a mathematical formula used in special relativity to describe how measurements of space and time change between two different frames of reference that are moving at a constant velocity relative to each other.

## 2. How is the Lorentz transform applied to a current 4-vector?

The Lorentz transform is applied to a current 4-vector by using the matrix multiplication of the 4-vector with the Lorentz transformation matrix. This results in a new 4-vector that is transformed to the new frame of reference.

## 3. What is the significance of applying the Lorentz transform to a current 4-vector?

Applying the Lorentz transform to a current 4-vector allows for the calculation of how physical quantities, such as velocity and momentum, change between different frames of reference in special relativity. This is important in understanding the effects of time dilation and length contraction.

## 4. Can the Lorentz transform be applied to any 4-vector?

Yes, the Lorentz transform can be applied to any 4-vector, as long as the vector represents a physical quantity that is affected by changes in space and time between different frames of reference.

## 5. How does the Lorentz transform relate to Einstein's theory of special relativity?

The Lorentz transform is a key component of Einstein's theory of special relativity, which states that the laws of physics should be the same for all observers moving at a constant velocity. The Lorentz transform allows for the transformation of physical quantities between different frames of reference, which is essential in understanding the effects of special relativity.

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