Complex Results of Time dilation & Tachyons - Part 3 in 26th Dimension Series

Part 3 - Interesting cases of time-dilation and time at v=c 

“Time is what happens when nothing else happens.” - Richard Feynman 

Lets now see, some of the interesting things related to Time dilation, and after that we will discuss some interesting paradoxes related to it.

The above picture is one of the visualization of faster than light travel. Tachyons are hypothetical particles that are supposed to travel faster than light. We will discuss them soon, but first we need to have some other things to discuss.

So, for this article, recall that train scenario, and when the train is stopped, then the path of the ball is the same for an observer on the ground and also for the observer on the train that is not in motion.

Now, don’t be clever and say you are observing it while moving, here, we are considering an observer on ground to be in rest. The terms rest and motion can be very tricky sometimes, but that will be a topic of another article, not this one.

I assume you know rectangular components, in which we break(or decompose) a force into into two components, called x-component and y-component. To avoid any confusions, check out the drawing below:

For further explanation of this decomposition of force, just Google the term “decomposition of force into its rectangular components”, and after that you may proceed.

 

So, coming back to the train analogy. We will assume the following scenarios:

1.       Train is at rest, so the velocity will be zero.

2.       Train is moving at some velocity v1.

3.       Train is moving at speed of light (don’t worry, imagination is not a crime).

 

For the train at rest, we see that there won’t be any time dilation, in other words, the component of Force along x-axis, which is also called horizontal component, will be zero. Check out the next scenario to get a better hold of what is being said in this scenario.

 

Now, for the second scenario, train is moving with different velocities, v_1, v_2, v_3, so we get a path traced by the ball as follows, where v_3 is greater than v_2, and v_2 is greater than v_1.

Here we observe that, the yellow and grey line, which represent the x and y components respectively. The component in the direction of y (or negative y-axis to be precise) seems constant, and it’s just the x-component that keeps on increasing.

 

Moving forwards toward our final scenario, where the speed of our train is almost equal to speed of light (just saying, so, cool down). In our previous scenario, we have seen that the horizontal component keeps on increasing with the increasing speed of the train, and now we assumed that speed of the train is almost equal to the speed of light. So, horizontal component will be maximum as we have reached the ultimate speed limit of the cosmos. So, it will take infinite amount of time for the ball to reach the ground, and then coming back to its initial position is not an option right now. So, it will trace an unending straight line.

Lets check out this thing with our derived formula, (the time-dilation formula that we derived the other day).

So, here, take a look at that formula again in case you forgot it

Now, for the velocity of the train equal to that of speed of light, that fraction gets equal to 1, and then under the square root we get 0, so time is stopped.

Now, once again, consider v to be 0, then we get t=t_0, that means time dilation won’t occur if both observers are at rest with respect to each other, as we stated before.

 

Now, lets play around with the formula and check what happens when the speed of the train is k times the speed of light, where k can be 1,2,3,4, …., or any other integer.

Substituting v=kc, we get the equation as under:

And then, after simplification, we get:

Lets suppose k to be 2, 3, 4, and then we will try to generalize it if possible.

We got the pattern, but complex numbers? So does this say anything about the arrow of time and its reverse direction? Or time travel? Let us keep that to us for now.

Using out previous analogy, we saw that time starts to slow down whenever speed of a body increases. And at higher velocities, like speed of light, time can be negligible or in other words, it seems to have stopped. And using some daily life examples, we will try to understand the behavior of time when velocity is greater than speed of light. Well, we can’t see even if we are able to move faster than the speed of light, because light is what helps us see things.

We will discuss what is known as Intermediate Value Theorem to some of you, if you don’t know what it is, don’t worry, we are actually using an example here.

Throw a ball into the sky and then most probably it will come back to you. So it was thrown in , let’s say, positive Y-direction and then it went higher and higher and then at one point, it came at rest, and then changed its direction to the opposite one , i.e. the negative Y-direction.

 

Now, back to time-dilation, we see that time started slowing down and then for the speed near speed of light, it almost stopped. If we compare it to the analogy of a ball thrown upward, or using the IVT(Intermediate Value Theorem), there are three possible cases:

1.       Time goes back towards dilation

2.       Time anti-dilation?

3.       Reversing the arrow of time, or time-travel?

 

We will start the next article using these three cases, and then we will discuss them.