Prequel: Einstein’s Special Theory of Relativity, but Much Simpler?

A man’s travelling with a speed ‘w’, while you’re at rest. Now you start moving at the speed ‘v’. All in vacuum. Now, what will be the speed of that man, as you see moving at that speed ‘v’? Add your speed plus the man’s. Problem’s over. To help you see it more: if you’re not moving at all, that v is just zero. Hence, the speed of that man will be only what he runs at for you, and as you start moving, yet, since this relativity is quite a special one, at a constant speed, you add up to the man’s speed with yours, as you perceive the speed of that man.

Likewise, replace that man with light. When you’re at rest, you see a known value as its speed. Start moving. You perceive the speed of light speeding up? Speeding over ‘c’? Einstein has to take an inhale slowly, thinking, here goes his day rescuing light from getting violated second time (technically, first time, but since, I wrote about general theory first heh and also, you can insert ‘violating the speeding limit joke’). The law of speed of light in vacuum is that it is constant, and so can’t speed up or slow down than that ‘c’.

So, he just had to take those equations (Galilean equations), and had to transform using Lorentz transformation such that in either case (you at rest or moving), it gives only the ‘c’. Such a simple thing been done. But hasn’t anyone said something like ‘Special Theory of Relativity is easy’ ever before? Yeah, he must have thought about that. About what? What in the equations to transform? Simple, as you saw before, the speed it is. Yeah, it is. Speed is dependent on two things: the distance you travel and the time you take to travel that distance.

His famous thought experiment, may I say it is famous, is that: now, you’re back to rest again and on either side of you, lightning strike. Each striking at the same distance from you. Now, coming back in time, you start moving, and towards which lightning you prefer approaching, it will strike first and the other second. Moving clearly does its thing that it makes your time slow. Such a statement which I just said lacks the progress. Let’s make that progress happen.

Measuring some distance when you’re at rest and when you’re moving, well, you may find two different values measured. So do time and distance become shorter or longer depending on whether you’re at rest or moving? Yeah, for you to see a beam of light moving alongside you at that ‘c’ in vacuum, your time has to slow down and/or the distance you’re covering, and by this, you’ll slow down (since speed is just distance covered per unit of time), and sees that as it happens. You’ll take more time to cover a distance which is shrinking from that one you see at rest, in order that you’ll keep your speed that, even if it adds up to the speed of light, it will not even a slightest to the ‘c’.

Yeah, once he got to know that he would transform the Galilean equations into those that shall give ‘c’ in inertial frames irrespective of your state of motion (at rest or in uniform motion), he also got to know that this specific condition needs sacrificing time and distance for the meaning they were before (at that time) so you would continuously be at that speed at which in vacuum you would see the speed of light at just ‘c’ in all inertial frames.

By saying it, he makes the justification to the Lorentz factor. How the factor is used? At rest, you won’t need it. But when you start moving, you want to know how your time or distance change. So simply, multiply by it the Galilean equations. For example, Einstein multiplied the distance x when you are moving expressed in terms of that distance when you’re at rest x’ (or generally in other uniform speed) plus the distance now you’re travelling (which in turn is expressed as your speed times time vt’ ( the time as you see it when you’re at rest or generally in other uniform speed), with this Lorentz factor. He got to know how much distance x you travel, which always comes to be different than one you will find using Galilean equations: that x’ + vt’. Similar process to all other quantities uniformly moving.

Author’s note: I’m aware, originally he considered both that frame of reference which is at rest (or in different uniform speed), and that frame of reference which is in uniform speed at the same time. Then, he would use his relativity. But since, we don’t in everyday life compare two people with two different frames of reference at the same time to understand time dilation or space contraction, I thought of using one person with two different frames of reference at two different times. This makes easier to understand since you are the only subjective.

Lorentz factor’s inclusion also means that no other thing than light itself can travel faster (in vacuum), because doing so gives us an imaginary number in the math, to which we can’t attribute to any real event.

A number greater than ‘c’ yields the ratio a number greater than 1, which when subtracted by that 1, gives a negative number, and the square root in turn gives an imaginary number.

His famous mass-energy equivalence is also derived from this. Not the one you think. But this:

Or simply, in terms of Lorentz factor:

He applied the principle ‘work done = force times distance.’ Force, for newton, is that ‘rate of change of momentum with time.’ Therefore, work done is not ‘force times distance’ if we switch to that integration of ‘distance times the rate of change of momentum.’ But momentum is ‘velocity times mass’ and distance now left with time gets paired up to give velocity. He used that momentum having ‘mv’ to multiply with Lorentz factor, and after using calculus, he derived the relativistic equation.

Yeah he used calculus just to get as simple as that. Well, work done is equal to the change in kinetic energy and not the total energy. So, when he derived this:

That mc² ? It’s the energy you posses when you’re totally at rest compared to the first term where you’ll be moving. When you’re at rest such that the first term loses its Lorentz factor, the equation becomes zero, meaning, at rest, you do not possess the kinetic energy, whereas you will possess the energy as measured when you’re moving minus the energy when you’re at rest as your kinetic energy.

If you kept a close eye to what I said before, you’d have noticed that he used two thought experiments to explain how time is not the same in every frame of reference, and dealt the same fate for distance (which in three dimensions is space), in order he could make sense in the equation he got every time he would multiply with Lorentz factor. Now he had one to explain how Energy and Mass can be of a same thing.

You, completely standing still, emit from your both hands to your left and right, light (the source is you). They’ll be of the same thing as started (he assumed them identical at the outset, so don’t come back at me for this). But when you start move, the frequencies of light change (which he attributes to Relativistic Doppler Effect). As frequency is directly dependent on energy, energy must change too. He measures this change again using Lorentz factor. The change between the energy from when you were at rest and when you were moving gives you a state where you have to decide either your velocity changes or your mass changes. Perfectly similar to how he sacrificed time and distance, but he chose to sacrifice only one thing now: mass.

Since, he yielded velocity as c² and on comparing it with the formula for change in kinetic energy, your velocity becomes constant, and what is now doomed to change is your own mass. This is because the velocity does not change (when you were at rest, you weren’t moving at all, and when you were moving, you have a constant speed v, since we’re strictly talking about inertial frames), yet when we applied the Lorentz factor to see how the energy is changing, you get a completely different value for each circumstance under consideration. Since energy (here particularly kinetic energy) is dependent on only two things and one of them is not changing, mass does it.

The mass itself is converted as the energy which you’re emitting as light. Same thing as how the sun emits light. Thus if you can convert mass into energy without ever needing another thing, then, they’re basically the same thing. The c² is just another factor like Lorentz factor to tell how 1 unit mass equals this much energy (the difference between them is c² is a conversation factor, that is, takes one quantity here mass and converts it into another (energy) totally, whereas the Lorentz factor is a transformation that takes a circumstance describing particular quantity and how that same particular quantity feels like in different one. The former is there to tell that they are same, while the latter is there to tell the difference).

That’s all about Special Theory of Relativity.

If you want the math behind it to see why one led to another, I can help with that too.