The Lorentz Factor

Hello!

This is an advanced topic so if you don't know much math, I would recommend to read it later.

Let us suppose you are in a spaceship (that would already be an incredible experience) but make the velocity of the spaceship increase until it reaches the speed of light. What do you think will happen?

What if I tell you that at those speeds, time literally slows down! No, not an illusion, it literally slows down! Well, maybe not for you but for everyone not in the spaceship, it looks like time for you has slowed down. Now you might be wondering how and why, and that is what special relativity explains us. This phenomenon is known as time dilation and is directly linked to today's topic, the Lorentz Factor.

The Lorentz Factor is a quantity describing the magnitude of how time changes for an object with different velocities with respect to an inertial observer. It is usually shown as:

\(\gamma=\frac{1}{\sqrt{1-\beta^2}}\)

where \(\beta\) is the ratio of the velocity of the object with respect to an inertial observer to the speed of light \(c\). It is represented as \(\beta = \frac{v}{c}\) where \(v\) is the instantaneous velocity of the object. This equation can be also be generalized as:

\(T_{0}=\frac{T}{\sqrt{1-\beta^2}}\)

where \(T\) is the time that passes for a clock in an inertial field and \(T_{0}\) is the time that passes for a clock in a instantaneous velocity \(v\) observed by a person in an inertial field. If \(v\) reaches \(c\), the answer will be \(\infty\) because the limit of the following function \( \frac{1}{1-\beta^2}\) as \(v \rightarrow c\) equals \(\infty\) when \(\beta=1\) and above the speed of light, the results will be in the form of complex numbers which cannot be defined as a specific velocity in our real world.

Graph created in desmos.com/calculator

This shows the value of the Lorentz factor as the velocity reaches c.

The Lorentz Factor is extremely important also to find the position of an object because in very high speeds, the relative position of an object changes too! We can find out that using the Lorentz Transformation Matrix which is as follows:

\(\begin{bmatrix} ct' \\ x' \\ y' \\ z' \end{bmatrix} = \begin{bmatrix} \gamma &-\beta\gamma &0  &0 \\-\beta\gamma &\gamma  &0  &0 \\0 &0  &1  &0 \\0 &0  &0  &1 \end{bmatrix}\begin{bmatrix} ct \\ x \\ y \\ z \end{bmatrix}\)

where \(ct\) is the time, \(x\) is the x-position, \(y\) is the y-position, \(z\) is the z-position (all for inertial frames of reference) and the corresponding variables like \(ct'\) shows the values for the corresponding qualities when in an defined velocity. \(\gamma\) is the Lorentz Factor and \(\beta\) is the ratio we mentioned earlier. It gives us an accurate description of objects in a moving frame of reference from our frame.

These transformations are useful when dealing with very high speed elementary particles such as electrons. CRT screens often are designed to deal with relativistic effects and even the Global Positioning System (GPS) has to deal with time dilation to give an accurate result. Lorentz Transformation Matrix is especially useful in astrophysics to understand more about the phenomena happening in space and will ultimately give a deeper understanding of our universe.

I know this is quite complicated, but honestly, what do you expect from special relativity?

Thanks for reading!