Deriving the Lorentz Factor (γ) of Special Relativity
The Lorentz factor is a fundamental component of special relativity, and there’s an elegant and simple way to derive it.
Albert Einstein is, without doubt, one of the most popular physicists of the 20th century. The German-born theorist laid the foundations for modern theoretical physics with his groundbreaking research on fundamental physics. The year 1905 is special in the history of physical science because it is considered Einstein’s annus mirabilis or “miracle year”, in which he published four papers on the photoelectric effect, Brownian motion, special relativity, and mass-energy equivalence respectively, the last of which was the origin of the famous but seldom understood equation E=mc².
How important were these papers to physics? Well, I think it’s safe to say that each one of his papers revolutionized at least one field of physics. The photoelectric effect made physicists reconsider the nature of light, and led to physicist Max Planck proposing the light quanta hypothesis which birthed the field of quantum mechanics, while the theory of Brownian motion finally established that atoms and molecules are a thing and actually exist (scientists were surprisingly skeptical of them at that time). Perhaps his most important contributions, however, came in the forms of special relativity and the mass-energy equivalence, ideas which fundamentally changed our perspective on the nature of reality itself. The impact of his work cannot be understated, and physicists are still busy unraveling its consequences more than 100 years later.
Einstein’s special theory of relativity describes motion at high-speeds, that is, speeds close to the speed of light c, and predicts how fundamental quantities such as mass, energy, distance, and time behaves at such speeds. At the heart of special relativity, lies the Lorentz factor. Named after the Dutch physicist Hendrik Lorentz and denoted by γ, the factor determines how much a system deviates from its predicted classical behavior under relativistic conditions. It is used in operations such as the Lorentz transformation which converts physical quantities when changing from one frame of reference to another, and in the equations that describe the bizarre phenomena of time dilation and length contraction.
There is no hard-set derivation for the gamma factor, as you can arrive at it in a number of different ways. I have seen derivations that involve using everything from the spacetime interval to differential calculus. However, after searching the internet and not a small number of books on relativity, I present in this post what I’d call an “easy” method to derive the Lorentz factor — one which uses only middle-school mathematics like algebra and the Pythagorean theorem. To do this, let us embark on a popular thought experiment — that of the near-lightspeed train.
Suppose there is an imaginary train that travels at relativistic speeds (close to the speed of light). The train contains an observer, a light source, and a reflecting mirror attached to the inside of the train’s wall. Standing outside the train is another observer who can see what’s happening inside.
Turning on the light source would cause light to travel the length of the train, reflect off the mirror on the opposite wall, and return back to the source. So, according to the Stationary Observer who is at rest with respect to the mirror and light source, the path taken by light is that of a straight line:
This means that, if l is the distance from the source of light to the mirror, t is the time taken by the ray of light to travel to the mirror and back, and c is the speed of light, l will equal c times t:
The External Observer standing outside the train, however, has a different take on things. From his perspective, the train and everything inside it is moving so the path taken by the light ray for him won’t appear as a straight line. According to him, light takes a triangular path and comparatively more time to reach the mirror and back:
From the perspective of the External Observer, t’ is the time taken for light to reach the mirror and back, l is the distance from the source of light to the mirror (same as that of the Stationary Observer case), c is the speed of light, and v is the speed of the train. As such, the distance traveled by the light ray and the distance traveled by the train when the light ray reaches the mirror is ct’/2 and vt’/2 respectively, and we see that these two distances along with the perpendicular distance between the mirror and the light source l form a perfect right-triangle, compelling us to apply the Pythagorean theorem:
But we know from equation I that l = ct where t is the time taken for the light ray to travel to the mirror and back, and since here we’re operating under the scenario of the light ray traveling to the mirror which is only half of its total journey, we will instead use the relation l = ct/2. substituting that we get:
Now, one way to define the Lorentz factor γ is:
The ratio between the time interval in the frame of reference with respect to which the object is moving to the time interval in the frame of reference in which the object is at rest.
Therefore, if we rearrange the equation we have in terms of a ratio between t’ and t, we should technically arrive at a valid expression for the γ factor.
And we are done :). As you can see, we needed no more than basic middle school math to arrive at this result, and if this is not an indication of how simple but powerful the theory of Special Relativity is, I don’t know what is.
Thanks for reading and I hope you enjoyed and learned something about Special Relativity from this blog post. I apologize for the LaTeX equations being presented as images. Medium doesn’t have LaTeX support which is a bummer (something Quora blogs had). Coming from Quora, this is my first blog post on Medium and it’s starting to grow on me. A similar post (which I published a year ago) can also be found on my now-defunct Quora blog of the same name: https://qr.ae/TWvAWz.
I plan to continue publishing posts like these regularly and even start a series of posts regarding a certain topic. If there are any topics you want me to write about, please mention in the comments below.
“I… a universe of atoms, an atom in the universe.” — Richard P. Feynman
