World Line and Galilean Transformation
Definition
The line that represents the track of a particle in space and time is called a world line.
Description
Let’s think only about a coordinate system moving at a constant speed in one direction. In the $A$ coordinate system, there is a particle at rest at the origin. The world line of this particle is as follows.
And there is a $A^{\prime}$ coordinate system moving at a speed of $v_{0}$ in the $+x$ direction relative to the $A$ coordinate system.
When observing the motion of the same particle from the $A^{\prime}$ coordinate system, the world line is as follows.
Since the $A^{\prime}$ system moves only in the $x$ direction, the value of $y, z$ does not change. That is, it is as follows.
$$ \begin{align*} t^{\prime}&= t \\ x^{\prime} &= -v_{0}t \\ y^{\prime} &= 0 \\ z^{\prime} &= 0 \end{align*} $$
If one considers the particle to be stationary at an arbitrary point $P(x,y,z,)$, it is as follows.
$$ \begin{align*} t^{\prime} &= t \\ x^{\prime} &= x-v_{0}t \\ y^{\prime} &= y \\ z^{\prime} &= z \end{align*} $$
Then, the following equation holds true between the two coordinate systems in space-time, which is called the Galilean transformation.
$$ \begin{pmatrix} t^{\prime} \\ x^{\prime} \\ y^{\prime} \\ z^{\prime} \end{pmatrix} = \begin{pmatrix} 1 & 0 & 0 & 0 \\ -v_{0} & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix} \begin{pmatrix} t \\ x \\ y \\ z \end{pmatrix} = \begin{pmatrix} t \\ x-v_{0}t \\ y \\ z \end{pmatrix} $$
One can observe the following characteristics of the Galilean transformation:
- Time is absolute and unchanging between the two coordinate systems.
- The speed of the particle differs by the speed difference between the two coordinate systems.
- Furthermore, there is no difference in speed in the direction that the coordinate system does not move. Mathematically, it is expressed as follows. $$ \begin{align*} t^{\prime} &= t \\ v_{x}^{\prime} &= v_{x}-v_{0} \\ v_{y}^{\prime} &= v_{y} \\ v_{z}^{\prime}&= v_{z} \end{align*} $$
See Also
[Lorentz Transformation]
Galilean transformation is a transformation equation that does not consider the effects of relativity. When the speed is not close to the speed of light, this approximation matches reality quite well. However, as it approaches the speed of light, one must consider the effects of relativity, and the expression reflecting this is the Lorentz transformation.