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Black Holes- Part 2

Hi Guys,

As mentioned before, i will be writing about black holes this week.

The previous post from this series was done by my co-writer, Jackson, who mysteriously disappeared after posting it. Don’t worry, he’s still alive and well helping me solve my darn math and physics problems.

Okay so on to black holes.

In the previous post, Jackson covered the formation and structure of black holes.

To make up for the previous post’s lack of graphics, here’s a helpful diagram of how a black hole looks like:

black hole

Yes. It is black. Or rather, dark. As mentioned before, the gravity from the region beyond the event horizon is so strong that even light cannot escape. As such, it is well, dark.

Some of you might be thinking: “So as time passes, all the matter in the universe will end up in black holes. Since nothing can escape a black hole, whatever goes in, doesn’t come out. So we’ll all end up as black holes one day!”

If you seriously thought that, then you were thinking what i thought when i first learned about black holes.

However, that (in reality) is incorrect.

That brings me to what this post is about. This post is about the thing that kills black holes. How all the black holes will meet their end.It’s a story about the weirdness of the universe, and also how cool it can be.

This post is about Hawking radiation.

So what is Hawking radiation?

Simply put, Hawking radiation is radiation that is emitted by black holes. Over time, the black hole loses mass as it emits more Hawking radiation.

According to Einstein’s famous equation, E=mc^2, the energy from the radiation emitted is related to the mass loss by a factor of c^2 as such, more energy lost, more mass lost.

And what happens after the black hole loses all its mass? It basically disappears. It evaporates.

That’s also why scientists can make micro black holes in particle accelerators without getting the entire earth sucked into oblivion. The micro black holes evaporate in an extremely short span of time, thus we are all safe!

Also, it gets its name from the famous physicist Stephen Hawking (the cool guy who helped us understand black holes a lot more and is also famous from the awesome move Theory of Everything)

stephen_hawking-starchild

At this point you’re probably wondering how can Hawking radiation escape from black holes when basically nothing can escape from black holes?

Well, to answer that, we need to look into quantum physics, where things coming from nothing and teleportation make it possible for the Hawking radiation escape.

The first theory of how Hawking radiation escapes from black holes is by the separation of particle-antiparticle pairs that form in the vacuum of space.

So some of you might be familiar with Heisenberg’s Uncertainty Principle. Most know it for its ‘position-momentum’ uncertainty version. The theory states that the more certain an observer is of an object’s position, the less certain he will be about the object’s momentum. However, this uncertainty is extremely minute and undetectable by most of us in our daily lives. But it is indeed experimentally proven and consistent with many other theories and experimental results.

Apart from ‘position-momentum’ uncertainty, there is also ‘energy-time’ uncertainty. And due to this, in an extremely short span of time, there will be an extremely huge energy uncertainty, causing the energy to possibly go high enough that random particles may be generated. Once again, Einstein’s equation: E=mc^2 comes into play here as mass in the particles is a result of the energy uncertainty. However, as we all know, energy has to be conserved (or else evil scientists would have created an infinite energy producing machine and taken over the world with crazy weapons or maybe just selling us power at ridiculous prices) as such, the particles soon disappear as they are annihilated by the antiparticle that forms with it. As such, the universe remains in equilibrium and evil scientists don’t destroy the Earth.

This also occurs at the edges of the black holes. At the event horizon.

Sometimes, a pair might form with one particle forming within the event horizon with the other particle forming outside.

antiparticle-particle event horizon

As such the antiparticle will fall into the black hole while the particle goes out into the unexplored wilderness that is the universe. Since the antiparticle must fall in as it can’t escape, all pairs that form this way will probably end up releasing a particle outside the black hole. Thus, we observe it as Hawking radiation.

The second way Hawking radiation could possibly form is by quantum tunnelling.

This effect occurs when a particle bypasses an energy barrier without reaching the required energy. For example, certain chemical reactions require a certain temperature (which is proportionate to the average energy of the particles) for the reaction to occur. As such, quantum tunnelling would be when a particle in the solution reacts without reaching the required energy.

This is possible in black holes too. The energy barrier is the event horizon, the particle is a photon.

Regarding how this effect actually works, it relies on the fact that particles can be described as waves (something that is pretty complex and something that I’m leaving out in this post).

The idea of Hawking radiation is extremely important as we can learn a lot from observing Hawking radiation. Since it’s basically the only thing that we can see coming out of a black hole (actually not exactly but hey who cares) it serves an important role in our understanding of these immensely powerful balls of uncertainty.

There still is much more to cover on black holes. Being such an extreme object, it provides insight into what happens under extreme conditions.

Well, that’s it for this fortnight’s post, look forward to the next one!

 

Thanks for reading!

Clyde Lhui 🙂

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Explanation on Notation of Summations And Products

I’ve been wanting to make this post for a long time, since it is severely limiting what I can post on this blog.

These symbols (∑ and ∏) might look very daunting at first but I assure you, even a toddler can understand what it means.

“Hey Fredrick, wanna try a new game?”

 

Game Poster 2.png

Game PosterI spent far too long on those posters. Thanks to Echonox and Cotchy for the design. Posters done using Scrub Microsoft PowerPoint 2012.

Let’s first start with the summation: ∑ | Sigma notation

Delete.png

The above symbol basically means “To sum” or “Add everything together”. You guys know what is addition right? 1+1=3 ( ͡° ͜ʖ ͡°)? No? Ok nvm.

But sum what with what? Well, let me introduce the dummy variable:

Delete.png

The dummy variable is basically an instruction of how to sum. For the example on the left, “n” is the dummy variable.

“a” is the starting point

“b” is the end point

and “f(n)” is the function you want to sum over.

What does this all mean?

Well, basically:

Delete.png

That’s it! Notice how “a” is the starting point and “b” is the end point?

Let’s take a look at some examples:

 

Delete

Delete

Delete

For the third example, note that “x” is the dummy variable and not y.


Now, what happens if you are walking along a beautiful path on a bright sunny day when suddenly a burly man pulls out a butterfly knife with a pearl-embedded, elephant tusk handle and put it against your throat. Adrenaline rushes through your veins as your limbs go numb with fear while your eyes watch the empty space ahead, futile in making sense of your awaiting doom. With a deep croak, your adversary spat

“We meet again. Now let’s talk.”

He pulls out a slip of paper. The paper cried as it flapped against the wind.

“Take a look. Take a close look at this. Explain.”

The slip of paper makes it’s way to inches from your face. Close up, you piece together the various symbols on the paper and break out in cold sweat. On the paper, largely printed, you saw:

Delete.png

Now what the hell is that “infinity” symbol doing there? Julian your master failed to cover this in his lectures, and you know you are doomed, for you don’t know what it means.


You don’t want to end up in the situation above do you? So listen up!

The infinity symbol up there means that you sum to infinity. You don’t stop summing, you just add and add and add and add…….. You are adding an infinite number of terms.

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You don’t stop!

You might be asking now, wouldn’t the sum diverge to infinity? In other words:

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Well the answer is: Yes, most of the time. MOST of the time.

You see, for some sums, it doesn’t approach infinity. For instance, you can proof that

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Try it on your calculator! Or… research on Infinite Geometric Progressions… or not.

When it comes to infinite sums, there can be some quite unexpected results. For instance:

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And yes, the π is indeed the one we are familiar with ( π=3.1415…)

To learn more about it, you can research on the Basel Problem, or the Riemann Zeta Function.


There are other types of summations, such as cyclic sum, symmetric sum and sum-over-n-that-satisfies-some-conditions (I’m don’t know the actual name for this type of sum, hopefully somebody can help fill me up on that). Anyway, I’m not going to go through those sums cause I’m lazy. Let’s move on to ∏ (product):

Delete

∏ is essentially the same as ∑, just that instead of summing, you multiply!

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Now let’s take a look at random examples:

Delete

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For the second example, note that “n” is the dummy variable and not “x”.

Similar to infinite sums, there are also infinite products, where you multiply FOREVER! HAHAHA!


Now, what happens if you are walking along a beautiful path on a bright sunny day when suddenly a burly man pulls out a butterfly knife with a pearl-embedded, elephant tusk handle and put it against your throat. Adrenaline rushes through your veins as your limbs go numb with fear while your eyes watch the empty space ahead, futile in making sense of your awaiting doom. With a deep croak, your adversary spat

“We meet again. Now let’s talk.”

He pulls out a slip of paper. The paper cried as it flapped against the wind.

“Take a look. Take a close look at this. Tell me. Tell me the exact value.”

The slip of paper makes it’s way to inches from your face. Close up, you piece together the various symbols on the paper and break out in cold sweat. On the paper, largely printed, you saw:

Delete

You faint like a Victorian lady, for you do not know.


Well, for this case, I’ll probably let you die, because I’m too lazy to give you a solution to the above problem. (Also because I’m not sure (Also because I don’t know)). However, I can give you the answer though:

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Basically, my point is, just like an infinite sum, an infinite product can also yield interesting results.


Challenge: Using the above result regarding the infinite product, proof that:

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If the LHS and the RHS exists. Hence, proof that:

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First part of the challenge question can be solved with Sec 4 syllabus while the second part involves A-level syllabus. However, the second part is easier as compared to the first.


I hope that now you know what ∏ and ∑ means! Just don’t go around doing stuff like this:

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Yes Mr.Philosopher, I see you just considered the first term

Or this:

 

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Same guy as above. Yep, I agree 1=1

Thank you for your kind attention

 

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Accountability-Time for more blogging

Hi guys!

So I just ended my mid-year exams (or MYEs as we like to call them) today and I have decided to start blogging once more! Yes, from now on I’ll be posting (or rather will be trying my best to post) regularly. Well in intervals of once a fortnight! (for those who have absolutely no idea as to what a fortnight is, it’s basically 2 weeks).

“Why is this post called Accountability?”

Yes I hear you dear reader.

So recently (while I was on my mega blogging hiatus) I have been doing a bunch of stuff. One of these things is, you guessed it, WATCHING YOUTUBE VIDEOS!!!

So while I was watching YouTube videos, I stumbled upon this channel called College Info Geek . He’s a dude called Thomas Frank and he makes these videos about productivity and study tips. There’s a video on his channel about how he was able to wake up at 6am every day:

So what I’m doing is basically what he’s doing just that I’m using the system with a couple of modifications.

So what am I going to actually do?

First, I’m posting this post which is basically a commitment. Every person who reads my blog will now expect me to post something 2 weeks later. And if I do end up not posting anything for 2 weeks, people gonna be angry.

So what if people are like:

Then won’t I just end up slacking off?

Welcome to part 2 of the system.

I made a deal with my friend. So if he sees that I don’t post anything for 2 week, I’ll buy him a drink and after 3 weeks I buy him a meal. It basically escalates to the point where I owe him a Starbucks drink if I don’t post for 5 weeks.

So yeah, accountability.

Well I actually have quite a lot of topics to discuss. Like I said I have been doing a lot of stuff while I wasn’t blogging.

From meeting Nobel laureates to playing piano and even watching anime.

Lots of anime.

Expect a lot more in the future. The next post will probably be something about black holes since it has been quite a while since I last did something on that topic.

Well, look out for the next post within the next 2 weeks!

 

 

Thanks for reading!

Clyde Lhui

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Quick Post

Today, I found an interesting identity. First, let the number of ways of partitioning a set of n elements into m nonempty sets be S(n,m). Also, let

Eqn 1

This can be seen either as the Taylor Series of f(x) or the generating function of B_n

The identity would be:

Eqn 2.png

A special case of this identity would be:

eqn 3.png

Where e is euler’s constant or e=2.71828182846…

It looks really complicated with all the notation but in essence, the identity can be easily understood. I will explain the summation notation in a future post regarding Analytic Number Theory.

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Physics Notes: Turning Effects of Forces

Hi Guys,

As some of you might know i have been working on some notes on Turning Effects of Forces. These are meant to summarise the CHS notes that have been given.

I make these notes to help people score better (at least i hope i do help). If you find these notes helpful, leaving a comment on this post or a like on this post or saying thanks IRL (in real life, if you still haven’t gotten used to internet slang) would be SUPER DUPER AWESOME. It’s pretty tough making notes especially when i can’t use any diagrams i find online and i have to draw out everything myself.

Since I’m on this topic, a teacher once told me that if things come too easy, people may take them for granted. He said this after going through a similar situation where he made a bunch of notes that took him several months of hard work to make and yet no one appreciated them. (I apologise for the bad grammar here but hey with exams in full swing, I’m pretty tired so gimme a break XD). Well i don’t know if my notes are appreciated so i can only hope that they are. Many friends have suggested selling the notes and they say that i would earn a bunch of cash. I don’t know if that’s viable but that isn’t happening in the near future. My ideology tells me that if i sell the notes, less people will be able to get them so they’ll lose their effectiveness.

Sorry for making you read this long post which was written extremely incoherently.

The link for the notes is here: Turning Effects of Forces- vetted

If you ever have any queries, please leave a comment, fill out the contact form or contact me personally.

 

Good luck for your exams!

Clyde Lhui 🙂