Lorentzian Transformations: The Basics Explained Basically

What do we mean by transformations?

How do we know what time it is? Ask someone in your household, and they’ll say something like ‘five in the evening’. But if I asked my relatives in Bangladesh, they’d say it’s around 11 pm. Both are correct, but how can that be when we’ve acquired different measurements for the same thing?

Well, to solve this temporal crisis, we have a third system to compare our measurements to, allowing me to convert my time into Bangladeshi time and vice versa. The Coordinated Universal Time (UTC) gives me an equation, a simple one, but an equation nonetheless which lets me calculate the time anywhere by adding or subtracting a certain value from the time that I observe.

The fundamental idea behind transformations tackles the fact that every measurement we take is not absolute, meaning that they’re all relative to the observer. Whilst London may be 16 miles to my west, it’s about 120 miles south to someone in Brum. But neither distance is incorrect, they’re just measurements taken from different places.

Galilean Transformations

Physicists love arguing about everything. After all, whoever wins ends up with their name inscribed on Wikipedia as ‘one of the greatest physicists in the x-teenth century’, whilst obtaining worldwide recognition and fame.

So, when early physicists realised that the calculated velocity of an object depends on how fast the observer themselves is moving, chaos briefly arose because it implied the laws of motion were wrong as different observers yield different results.

Galileo cracked down on the idea and invented Galilean Relativity. Though somewhat intuitive, it states that the laws of motion are the same in all inertial frames of reference (a frame which has 0 net force acting on it thus is not accelerating). Indirectly, this implies the general idea of relativity. After all, if no physical experiment can distinguish between an object moving at constant velocity and one at constant rest, then the only motion that matters is the relative motion between objects. There is no absolute motion.

Then how do we know how fast another object is moving if we ourselves are moving? Well, the idea of Galilean Relativity was developed further to form the Galilean Transformations, a bunch of equations which allowed you to calculate the position and velocity of another object using the position and velocities of the observer. They’re relatively simple to be fair.

The Downfall of the Galilean Transformations

The sharp eyes amongst us may have picked up on one thing unusual about the Galilean Transformations though. Unusual according to modern science at least. They assume that the time perceived by a stationary observer is the same as the time perceived by a moving observer. Sounds about practical, you might think, but why is this scientifically incorrect?

Well, the Transformations are built on classical Newtonian mechanics, so whilst they may work for common, low-velocity objects, they begin to break down when velocities reach near the speed of light.

Why?

Because of Einstein’s theory of Special Relativity. I don’t have the time (nor ability, yet) to explain why the theory is as it is, but SR concerns the consequences of travelling near the speed of light. Basically, the nearer you are to the speed of light, the more:

- Time dilation (time slows down)

- Length contraction (longer distances become shorter)

- Mass dilation (increase in mass)

So now, once again, physicists have been proved wrong, and were sent down a spiral of conceptual chaos, theoretical turmoil and scientific squabbles. The Galilean Transformations were about to be banished from science, and embarrassingly because of an abstract thought experiment about hypothetically travelling near the speed of light, until another famous physicist came to save the day. Who was it?

Hendrik Lorentz.

Finally, The Lorentzian Transformations

Lorentz was a very efficient man. Instead of going to the effort of creating a whole new set of equations from scratch, he took the near-perfect Galilean Transformations and tweaked them a tiny bit to make them fully perfect.

The equations look largely the same, except the first two have a new λ (gamma) multiplier in front of them.

The λ is the symbol for the Lorentz Factor, which is a quantity used to calculate how much the measurements of time, length and mass are affected by special relativity. It looks like this:

Although it may seem that all Lorentz did was slap on a λ in front of already formed equations and then take the credit for it, the beauty of the Lorentzian Transformations lies in the fact that they work in both classical and relativistic mechanics with no exceptions at all. From these equations, we can calculate the displacement, time and velocity from another object’s frame of reference using the observations from our own frame of reference, just like how you can use GMT to calculate the time for someone else in a different country. Maybe physics, when taught simply, isn’t so difficult at all.