আলোর থেকে দ্রুত যাওয়া সম্ভব না

Introduction

Have you ever wondered what it would feel like if you traveled at the speed of light? Well let's think a bit beyond and ask ourselves what would it feel like if you traveled faster than the speed of light? Or wait...is it even possible to travel at or faster than the speed of light?! In today's exploration, we will answer all three of these questions at the same time by simply answering the last one. We will hit three birds with one stone! (or is it three birds with one bird and that one is a suicide bomber...well whatever)

In this lesson, we will explore not just light, but Albert Einstein's famous equation for mass-energy equivalence ($E = mc^2$), and the stuff in the universe (or multiverse for those of you who are fans of Alexander Vilenkin and Alan Guth) which we know travels faster than the speed of light.

Ok, so before we get into the fun implications of Einstein's mass-energy equivalence, we should first talk about objects in general. We know that all forms of matter has a distinguishing feature called mass. For example, let's take a baseball. An average baseball has a mass of $145$ g or $0.145$ kg. But what about the mass of light? But wait, even before that, isn't light not tangible? I mean, we can see it sure but that doesn't mean that we can hold it in our hands...So how do we measure the mass of light? Or...does light even have a mass?

Well, let's take a look at our friend $E = mc^2$. We know the definitions of each of the constants: $E$ is energy, $m$ is the mass and $c$ is the speed of light. See anything that might help us solve our problem?

Einstein saves the day?

The constant for the speed of light, $c$, tells us that the energy held by some body of matter which has mass is relative to the speed of light. So if we rewrite our equation, we can make a new equation for the relative mass of an object:

$$E = mc^2$$ $$m_{\text{rel}} = \frac{E}{c^2}$$

So then what about the mass of light? Well due to $E = mc^2$, we can understand the fact that if an object moves at the speed of light, then it will practically lose all mass. Instead, that mass is converted to energy! Now to be honest, this previous statement is only $100$% true for a massless particles (such as light). Or to reword this, a massless particle has zero rest mass.

However, this does not make it so that we cannot calculate their relative mass. To do so, we simply use the relative mass formula that we found above. The only difference for light and other members of the "photon family" is that their energy is calculated with the formula $E = hf$ where $E$ is the energy, $h$ is the planck constant, and $f$ is the frequency. So let's try to conceptualize what we just talked about using our baseball again.

Food for thought

Scenario 1: Say that I threw a baseball at $93$ mph or approximately $41.57$ m/s. This is fast (faster than most speed limits on the road, if not all of them) but I can still see and feel the baseball. In particular, I can say with 100% confidence that I cannot feel the fact that the baseball lost any mass. Even if I tried to measure the baseball on a scale, unless the scale can measure to way past the billionth or so, it won't be able to catch the difference at all. This is since the baseball isn't moving anywhere close to the speed of light which is $670616629$ mph or $299792458$ m/s (not to mention, the speed of light squared!) even though the mass of the baseball is fairly close to $0$.

Now consider the next scenario (by the way, I will be using the m/s as the units for the velocity of the object from now on).

Scenario 2: Suppose that I threw the same baseball again at $41.57$ m/s, and as it was traveling the baseball just happened to have encountered a bunch of extremely powerful hurricanes (it seems that I picked the wrong park to play in at the wrong time. Stupid meteorologists :P just kidding) which caused the ball to increase in velocity to the point where when it was released, it was magically able to move at a velocity that was extremely close, let's say $224844344$ m/s which is approximately $75$% of the speed of light, for just an instant. Even before that instant, as the velocity of the baseball kept on increasing, the baseball started to lose mass which was all converted to energy.

Unfortunately, unless we can reverse the conversion of mass to energy (the baseball does not have such a fascinating piece of technology attached to it) the baseball will have "lost" that amount of mass forever (assuming that it hasn't completely converted yet, which is possible).

This also applies to humans too. No matter what kind of super powered suit we make (I apologize to avid fans of Ironman in advance), we cannot survive moving at the speed of light. Therefore, no body of matter, that initially traveled at a speed lower than the speed of light, can maintain its mass when it travels at the speed of matter

Faster and Furious-er?

Ok so great, we know why its impossible to move at the speed of light, and we don't have to wonder about what moves beyond that speed...or do we? There are particles out there that supposedly move faster than the speed of light. We call these particles Tachyons or hypothetical particles that travel faster than the speed of light (note that particles which move slower than the speed of light are called bradyons).So far, we have only discovered one general type of these fascinating particles: Neutrinos. If anything, there is one very interesting fact to know about tachyons, and it is all based off of the following equation for the total energy of a particle:

$$E = \frac{mc^2}{\sqrt{1 - \frac{v^2}{c^2}}}$$

Explanation: This equation shows that the total energy of a particle (bradyon or tachyon) contains a contribution from its rest mass (the "rest mass-energy") and a contribution from its motion, the kinetic energy. When $v$ is larger than $c$, the denominator in the equation for the energy is "imaginary", as the value under the radical is negative. Because the total energy must be real, the numerator must also be imaginary: i.e. the rest mass $m$ must be imaginary, as a pure imaginary number divided by another pure imaginary number is a real number.

Using this equation, we can understand the following about traveling faster than the speed of light. Unlike ordinary particles, the speed of a tachyon increases as its energy decreases. In particular, $E$ approaches zero when $v$ approaches infinity. (For ordinary bradyonic matter, $E$ increases with increasing speed, becoming arbitrarily large as $v$ approaches $c$, the speed of light). Thus, just as bradyons are forbidden to break the light-speed barrier (which we have discussed before), so too are tachyons forbidden from slowing down below $c$, because infinite energy is required to reach the barrier from either above or below.

Another effect of faster-than-light travel was presented by Einstein, Toulman, etc. is that, due to special relativity, any particles that move faster than the speed of light could be used to travel backwards in time. However, this may be going a too far indepth so perhaps this will be a great research topic for you to consider in your spare time!

I hope that this was a really fun lesson! This was a true pleasure for me to write, but I hope that you had as much fun as I did exploring thorugh the realm of speed. Gotta go fast! (but not too fast). See you all next time.

T.Kim

You can't travel faster than light? Really?

How do you explain Cherenkov radiation then?

Light moves slower in different mediums depending on the refractive index. Light has 2 different velocities, the phase velocity and the group velocity. We only need the object to have higher speed than the phase velocity of light and this can be done by changing the medium which light is going through.