This post was co-written by Monica Ekal along with Arnav Nair. Last week Monica pitched this idea and we thought it would be great to make a compilation of paradoxes in a post. So this is the result. Lets get started.
Consider that the Greek war hero Achilles races a tortoise. Our hero is compassionate and decides to give the tortoise a head start. Who would win the race? Easy, you say. Achilles!
Erm…no. Not as the Greek philosopher Zeno imagined it, at least.
When Achilles starts the race, the tortoise has already covered some distance. Now, by the time Achilles gets to the point that the tortoise previously was, the tortoise has covered some more distance. “Okay”, says Achilles and bridges that gap too. By then, the tortoise has gone a little further…See where this is heading? The tortoise would always lead this race, even if it is by an infinitesimally small distance. HOW???
One reason this paradox is misleading is due to thinking in terms of discrete steps. Of course Achilles would rush past the tortoise in a matter of seconds in continuous time! This is like walking from a point A to point B while taking steps that are half as long as the previous one each time. (Apart from this being humanely impossible after the point of a few millimeters,) At some point of time there would hardly any perceivable distance left between the person and point B. Can we then say that he has reached point B?
An alternative explanation to this is the case of convergent series. Like cutting a pie into half, then splitting the remaining half into quarters, splitting the quarters into one-eights and so on. We’re splitting something finite like distance into infinite pieces. We know that the series converges from the fact that the pie was whole before we started cutting it in this bizarrely obsessive way. The sum of the infinite series 1/2 +1/4 +1/8 +1/16 +….. is, in fact, 1. So, in this race of infinite steps, at some point Achilles and the tortoise are bound to meet. And yet, that point, my friend, is infinitely far.
(Under special conditions, this paradox can also be divergent. See http://www.slate.com/articles/health_and_science/science/2014/03/zeno_s_paradox_how_to_explain_the_solution_to_achilles_and_the_tortoise.html for more)
Another explanation for this thought experiment is the endlessness of the sequence. With each step, the lead that the tortoise will have over Achilles goes on decreasing. It decreases to the extent that it is almost zero. This article about the paradox is detailed, but worth the read!
So, no matter what, Achilles will overtake the tortoise at some point of time. May be if we can even the odds by giving Achilles some more handicap. May be if we ask him to wear high heeled shoes during the race. Could it be his Achilles’ heel?
Imagine you have created a time machine. Now you go back in time and have this crazy idea – Let me kill my grandfather. So you aim the gun at your two year old grandfather and pull the trigger. So, your grandfather’s dead, which means he will never have a chance to create your father, which means you were not born, which means you did not create the time machine, which means you don’t go back in time and you don’t kill your grandfather and your grandfather lives and … Honestly, at what point in this sentence did your head start paining? This is the classic grandfather paradox.
So do you exist or not? What does it mean? One simple explanation is travelling back in time is not permitted by the laws of physics. If this hypothesis is correct, then life would be much simpler. You can’t go back in time and cannot cause any of such crazy nonsense.
But think about what Einstein has taught us, life is not simple. Of course we cannot say that this hypothesis is false, but we should search for other explanations in case this doesn’t turn out to be true.
Another explanation which would help us sleep at night is, the moment you do this crazy nonsense, the universe splits into two strands, an alternate timeline (Damn you Barry Allen). One is the one in which you are alive and the second one in which your grandfather’s dead at 2 and you don’t even exist.
That’s cool; I go in the past from one timeline and create an alternate universe. I could live with that. However, there could be a third possibility.
Now comes the crazy one for the crazy scientist in you, who decided to ruin his/her existence. The universe is in a superposed state i.e. Schrodinger’s Cat’s Grandfather State. Such kind of weird superposition thingies happen all the time at small, teeny tiny quantum level. Could this happen at big, macroscopic level? There’s only one way to find out. (Statutory Warning: The author of this article does not encourage murder. Please find an alternative way to test this hypothesis). If true, your grandfather is both simultaneously alive and dead; and you simultaneously exist and do not. Both the histories run in parallel.
For more on this check out the video made by minutephysics on this topic: https://www.youtube.com/watch?v=XayNKY944lY
Okay, if you need to take an aspirin, go ahead, because the next paradox is even crazier.
Ah, Einstein, making life miserable since 1905. 1905, is regarded as the miracle year or “Annus Mirabilis ” of Einstein, as he published four amazing papers which revolutionized physics. One of these papers was the special theory of relativity (Not the Title of the Paper). We’ll not go into detail of relativity, but we’ll look at one aspect of it – The Time Dilation a.k.a. that weird time stuff in the movie “Interstellar”.
Einstein derived mathematically that as a person moves faster relative to observer, that person’s clock ticks slower than the one with the observer. However, if we look from the moving person’s point of view, the observer is moving and that person is at rest. So according to the observer, the moving person’s clock is ticking slower whereas from the moving person’s point of view, the observer’s clock ticks slower.
How can it be that two people see two different things? Who is correct? Answer: They both are. The order of events and the times depend upon which frame of reference you are present in. This is called “Relativity of Simultaneity.” Guys, we have not entered into paradox yet. Keep the aspirin handy in case you haven’t taken it already.
Okay so if we accept, although it is hard to accept even if it is true, that time is reference frame dependent, imagine identical twins, say Albert and Rupert Einstein, both standing near the launch site from where one of the twin, say Rupert, is going to start its interstellar journey to Proxima Centauri.
Not a huge distance, just a four light year trip, eight for a return trip. The twins are currently 30 years old. Rupert embarks upon this journey at almost the speed of light. He reaches a planet of our neighbour star system, collects some samples and comes back to earth again with near light speed. Since Rupert was travelling at almost the speed of light, Albert sees that the clock of Rupert slows down during the journey and he notices that the astronaut had hardly aged at all whereas Albert himself aged 8 years and had grey hair and a cholesterol problem. So far, so good. But from Rupert’s point of view, Earth itself is moving away from him at the speed of light as they reach the planet. And then Earth moves towards him at the speed of light as they head back home, near light speed to be precise. So according to him, Albert should age slower than him. But that doesn’t happen. Even he finds out that he is the younger twin. How is that possible? A TWIN PARADOX.
Fortunately, this paradox is resolved by a simple logic. Let’s take a look at the nature of two reference frames. Albert is in what’s called an inertial reference frame. Any reference frame which is not accelerating is called an inertial reference frame. Of course it is not a perfect inertial frame, if you take into consideration the rotation and the revolution of Earth, you will get some acceleration, but the acceleration is relatively small and can be neglected. But in case of Rupert, the frame which he is in first accelerates to attain speed of light and then decelerates to zero as it reaches the planet. And again on the journey back, we see the same.
Of course there is a period where Rupert’s frame is moving with a constant speed and is inertial, at that time dilation takes place as prescribed by Dr. Einstein and Rupert does see the clock of Albert ticking slower than his. But during the acceleration something different happens.
Remember when I told you that according to special theory of relativity, faster you go slower the time ticks. Well, it’s called “special” for a reason. It is applicable for only a special case of inertial frame of references. In order to study the accelerating frame of references as well, we need to go to more general theory – The General Theory of Relativity. If the movie interstellar has taught us anything, it is that more the gravitational pull, slower our clocks tick. And in general theory we have something called an equivalence principal which states that there is no difference between acceleration due to gravity and the acceleration in general, somewhat like that. So during the time of acceleration and deceleration of Rupert’s frame of reference, Rupert’s time runs slower, way slower and accounts for the solution to the paradox.
Another minutephysics video explains this beautifully, check that out using the following link: https://www.youtube.com/watch?v=0iJZ_QGMLD0
If you need a prequel to this video, which talks about the tools needed to solve this paradox, follow this link: https://www.youtube.com/watch?v=Bg9MVRQYmBQ
The Birthday Paradox
This paradox says that in a group of 23 people, there is a 50% chance that two of them will share the same birthday. This seems ridiculous at first, because there are 365 days in a year! Well, this is exactly why it is called a paradox. Lets get to the math to prove this-
Lets look at the probability that two people do NOT share a birthday. This means that the second person out of the two should have his/her birthday on one of the other 364 days of the year, making the probability 365/365 * 364/365 = 0.9972. This means that two people will not share birthdays 99.72% of the time.
As the number of people go on increasing, the probability that they do not share a birthday goes on decreasing.
For 3 people, the chance that they will not have the same birthday is 365/365 * 364/365 *363/365 or 0.9917, for 4 people, 0.9863, for 10 people, 0.8830, for 15, 0.7471 it and so on..
For 23 people, this probability comes down to 0.4927, which means there is a more than 50% (50.73% to be precise chance that any two people in the group share the same birthday. As people in the group go on increasing, the chances will increase even further… This implies that in a group of 57 people, there will be approximately 99% chance that birthdays will be shared! A great and in-depth article on this …is https://betterexplained.com/articles/understanding-the-birthday-paradox/
The Infinite Hotel Paradox
Another paradox/thought experiment on just how mind-boggling infinity is. Consider that there exists a hotel with infinite rooms where each room has a guest in it. Now, a guest turns up looking for a room. Can he be accommodated? Yes! Each guest is asked to move to room n+1 , thus freeing up the first room in the series of an infinite rooms. Next, infinite guests show up, all clamoring for rooms. It is possible to accommodate them too. How, you ask? Well, we need to free up infinitely many rooms. So each guest is asked to move to a room with a number that is twice of his current room. Let’s see if you’re following here. If there are 10 rooms, and guests in 1-10 move to rooms 2, 4, 6…20, we free up 10 rooms, right? Simple. So if we send the infinite number of guests to rooms up to 2 x infinity (= infinity???) we still free up infinite rooms. Err… But the best is yet to come. What if, infinitely many coaches with infinitely many passengers show up? We would want to vacate infinity x infinity (= infinity???) number of rooms. We will let you think. (Hint: The prime rule is to avoid double bookings)
Also, this video is a delight! https://www.youtube.com/watch?v=Uj3_KqkI9Zo
This Wikipedia post elaborates further https://en.wikipedia.org/wiki/Hilbert’s_paradox_of_the_Grand_Hotel
PS: In case you didn’t watch the video, the infinity here is countable infinity (which means only positive real numbers…No complex or negative numbered rooms, phew!)
Hope you have enjoyed reading this post. Let us know your feedback or if we made any boo boo, in the comments.