## Sunday, April 26, 2009

### Nanosuit superhumans

In the future, a device can be worn as a belt and on activation, small nanobots will come out of the device to create a suit that covers the user’s body with specialised carbon nanotubes. This suit can be bullet proof, light and be equipped with weapons. In comparison with the world of fiction, it is the nano version of Iron Man! In short, nanotechnology allows the technical possibility of Superhumans to exist in our world. These superhumans can be a one man tank, with the strength to carry heavy things as the exoskeleton allows it, high energy lasers that can be placed on the arms, jets that be placed at the bottom of the shoes, and anything else that Iron man can do.

If the suit is in the hands of the “right” people, superheroes can easily exist in this world. However, the impact of the existence of Nano Iron-man in the world may be varied. The world might explode in laughter at the person with the nanosuit acting like a superhero. The other possible response is that the world takes the concept of superhero seriously. From this point on, we only have the comics to guide us. And almost all the comics treat the coming of Superheroes as someone to respect and to hail as some saviour. However, the opposite may be true. Superpowers like America may be worried that Superhumans harbour too much power that it would threaten the superhuman to stop and hand over the suit.

Other than having people with “righteousness” having the suit, there can be greedy and ambitious people that can get a hold on this much power. These are the supervillians in the comic books. These people can leave a very negative impression on nanotechnology and possibly trigger wars like the war on terrorism. In this scenario, if a superhero comes and successfully save the day from the supervillian, then people who laugh at the notion of superhero may start to take them seriously. Most probably a team of supercops will be formed and controlled by a certain country. This can also trigger the possession of the suits by countries and start the whole arms race again, only this time; it is to collect the biggest army of Supersoldiers.

Finally, if the suit is available commercially without the weapons, it can be the ultimate protection suit, transforming the human race to a race of superhumans. The function of cops and soldiers are reduced, and murder cases become rare.

In conclusion, the possibilities allowed by nanotechnology in this form are huge and may generate many possible responses from the society.

## Friday, April 24, 2009

### Investigation of the existance of closed timelike curves and the feasibility of time travel

by Han Weiding, Jani Hariom Kirit, Ng Xin Zhao for the SP2171 Discovering Science SPS module.

## Thursday, April 23, 2009

### Fatal Deviation: Bleach versus OnePiece

## Monday, April 13, 2009

### How best to make the prospect of nuclear power in Southeast Asia come true

The first problem is that of nuclear waste disposal. Nuclear fission produces radioactive wastes that have half lives up to thousands of years, making it hard to store them properly. However, the wastes from nuclear fusion would only have half lives of about 50 years and thus it only needs to be stored for a shorter amount of time before being released. The second problem of nuclear proliferation also disappears as Plutonium-239 is not produced as waste in nuclear fusion reactors. The third problem of danger due to accidents is also minimized as runaway nuclear fusion is impossible due to the delicate nature of its process.

Thus it is obvious that the advantages of nuclear fusion reactor would remove all considerations of implementing nuclear power in anyone’s mind. Just one major problem with fusion is that currently it has negative efficiency. Research on nuclear fusion is been actively done to overcome this problem. However various difficulties are being encountered with some estimate of a possible nuclear fusion reactor only at the year 2050. However, looking at the desperate need of a cheap and non-polluting source of energy, the speaker should also encourage the audience to contribute to the feasibility of nuclear fusion as soon as possible.

The physics major students can contribute by increasing the number of talents working on that field, the sociology students can help research on how best to overcome the psychological difficulties of the public in accepting nuclear energy, the economics students can help plan how best to implement the nuclear energy in the near future. The main question unanswered is how best to make the prospect of nuclear power in Southeast Asia come true.

## Wednesday, April 08, 2009

### Lesson from History

The culture of civilisations at any time should be distinct enough to enable the thinkers and scientists of that time to think in very different ways. From there, the different view points on the same topics can be produced and when a meeting of culture happens, the ideas can be combined, argued, and compared to advance the best idea, theory or even create a better one with the ideas available.

I feel that the speaker did not make the connection of this to what is happening now.

With the advent of the internet and the trend towards globalisation, many cultures are meeting at the same time, thus producing a lot of output in terms of the advancement in sciences. Universities tend to hold international conferences and dialogues for researches, this way, a lot of ideas are communicated freely and progress are made fast. The surges of ideas exchanged and the increased advancement of science can be clearly seen now as in the case of the Large Hadron Collider where physicist from many countries come together to discover the fundamental theory of the universe.

However, as globalisation progresses, the cultures of the world are slowly unifying. Even though there will always be individual thinkers, it might be harder to think of the something that others had not already thought of as everyone tends to share the same unshakable unspoken assumptions. They simply cannot think out of the box. Then the progress of science would slow down considerably, and science might have to rely on the occasional genius like Einstein to break our everyday notion of the absoluteness of some principles like absolute time.

This implies that maybe it might not be very good for scientific advance that the world is heading towards globalisation. But certainly, more research needs to be done in this respect.

This research is important to clarify these questions. Does world peace or world government come at a price of scientific advancement? And if so is it worth it? Furthermore, will the momentum of science be stopped eventually and forced to turn back? Will the events similar to Asimov’s Foundation series in which a great empire collapsed due to lack of advancements of science happen? Is it inevitable or can we somehow avoid it?

Of course this whole speculation is based on the assumption that science is advanced based on different viewpoints of scientist. This is indirectly implied from the topic of the speaker, “The Dialogue of Civilizations in the Birth of Modern Science” and is also open to research.

## Monday, April 06, 2009

### Books for QM?

Thank you.

http://www.physics.nus.edu.sg/~phyebg/index.php?page=LQM

Here I believe the three books in this website are useful. After you finished them off, you'll be as knowledgable in QM as most physics undergrad in their 4th year. Well to tell the truth, you'll find them boring and totally unrelated to what you read on those popular books. The maths will be unfamilar to you and hard too unless you learn linear algebra.

And for your Pre-QM stuffs, I would recommend "30 years that shook physics" by George Gamov as a begining popular text (since you're fond of them) to get you familar with the history of QM. Then I can only say, read form 6 text book, the part that you'll know is important from your QM history background. Then read a University Physics Textbook. These kind of text books are thick and only usable for 1st years. However, to save some money, don't buy the thick University textbook, go for modern physics, the front few pages describes relativity and the rest are QM and it's implications.

So now I have provided some guidelines and key words. To show your initiative to learn, it's up to you to be resourceful enough to google them and find those books. And I think there are ebooks too, so try looking for the free books online before spending most of your savings buying a book.

And don't give up too soon if you find that you're stuck. Go slower and give yourself time to learn. Hope to see you in this field in the future. Meanwhile keep me posted if you got any difficulty.

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### Finally Report done.

An Investigation of the Existence of Closed Timelike Curves and the Feasibility of Time Travel.

Man I'm not going to do this (delay till last week/day) again.

Been awake for 21 hours now, hopefully I can get some 2 hours of sleep.

## Sunday, April 05, 2009

### How to score full As in STPM

Disclaimer: even if you somehow managed to follow all these advices, there is no guarantee of 4 flat. More importantly is you do your best, then whatever results you get, be contented.

- If you don't mind having only 4 subjects, don't take 5. It's a risk. But if you're as hard headed as me or already taken it, then do your best. But don't worry anyway, even if you miss out on 1 subject out of 5, you still can get 4 flat, unless the subject is PA. Like what I missed.
- Don't take Further Maths. It's a hard to score subject, and too little people in Malaysia take it, mostly the ones who consider themselves the smartest in their school or one of the smartest. So I need not say more, but if this is your 5th subject, then concentrate on the other subjects(to get 4 flat) if you find that you won't get A in this, but at least make some effort not to fail it. However if you're also one of those who consider yourself a maths genius, be my guest, take it. Here's the blog to read if you do take it: http://furthermathematicst.blogspot.com/
- Don't play the fool. Yes you can be active in CCA(co-curicular activities), but don't spend all day not studying. Ya, the most basic way of scoring, study more hours per day! Do your homework in school, during recess, whenever you can, do extra exersices, ask your teacher things you don't understand..... do I need to continue? Of course, if you're like me, who thinks that homework is a waste of time and don't do the extra self exercise, and going online everyday, then...umm.... make sure you score in your monthly tests.
- Don't spend all day looking for job/uni/doing extra stuffs for your uni applications. Ya, devote your life to studying. If you think form 6 is a store house for studying, you aint seen Uni yet. But of course this is a unreasonable thing to ask of you.

What for you want to get 4 flat? to get into uni of course. But if you aim for local uni, just do your CCA, if you aim for MIT, just do your essay now, if you aim for NUS, then....umm..... get good result. If you aim for outer univer, Ivy leagues, then find a job, do extra stuff, join competition in thing you are good in until the highest level you can. National, International..... train hard. Get king scout, get black belt........ all these.... - Don't spend unnessary time having a boyfriend/girlfriend. Most likely you two are not going to the same Uni or same course. ( If you are, then congrats, you're an ideal pair) However if you already got one.... then manage your time well. Oh and be prepared for break ups, especially around the time of exams. So one way to avoid this is to agree to break up now and reunite after exam (warning: possible passion reduction after long time no see each other.)....... Ok perhaps it's not nice to tell people to break up. But anyway, you're lucky to have one.
- Don't make an enemy in classes. you're going to spend around 30 hours per week in class, studying. So make sure you don't have someone you dislike in class. It's bad for your mental balance. and you might not absorb what the teacher is saying. However if you already have one, agree to make peace. And see a psychologist if you still can't stand his sight. Don't be too alarmed to do that, I did it too believe it or not.
- Go for some International Olympiads. Search the net for them and go through my blog for them. The contacts should be within your reach if you're serious enough to google it. Oh no.... this has nothing to do with scoring full As. Ok so don't go (to get more study time). But be aware that you're missing a very beautiful experience.
- On study places, if you're in Malacca, go to Tzu Chi, at RM5, you can sit there all day doing your work and enjoying your tea. No food there through, so be prepared. Or more popularly, study at your home, you room or anywhere you can sit for hours on end and keep on having the mood to study.
- On studying with whom.... I'll say your friends of course. Go for some group study. That is a bunch of friends studying at a certain place where everyone is comfortable and can study continously for at least 5 hours. But if you're not with your friends, then study alone.

It's not a crime, unlike shopping alone, which I don't think is a crime too. Oh ya, choose your friends properly, those who talk non-stop when in a group are to be avoided. Ideally it should be everyone doing their own work and anyone got a question can rise it up and the whole group or whoever's listening can think can contribute to the discussion. 10 minutes later, it should be back to individual studies. - Read some books on how to study. It's plentiful and contains all the nessary and basics skills on studying. I'm getting lazy here trying to explain the basics to you, but just that if you don't know them, then you shouldn't be here(taking STPM that is).
- Don't waste time, do some reading in your car, bus, train...etc. And ok Now I can't take it........ Just use common sense and study. Ok, good luck!

### Holiday mood

## Saturday, April 04, 2009

### Time Management

living which includes eating, drinking, toilet, bathing in another section

The situation is that I studied for 73 hours last week and it impressed J, so she asked me how I did it. I told her that I keep a notebook to put some strokes everytime I do something for 15 minutes per stroke. There are 6 sections in my timetable and the first one was sleeping, the second, living, that is eating, bathing, toilet, brushing teeth. and the third is:

##### 03:56Ng

travelling, that means walking to class, anytime you're in a car or bus, or simply exercising in a third one

##### 03:58Ng

Lectures including studying, homework anything that you do in academic, including studying with friends, doing report, project, lab, online assignment, tutorial, you get the idea

then CCA, for co-curicular activities, that means the time you need to commit to do CCA

##### J:

WAT IS CCA?

OOO..

co curricular activities

##### 04:00Ng

and Playing at last, which includes everything else, watching anime, waste time talking to friends, go shopping, colour your hair, lol, and party!

lol

put whatever you need to reduce in playing

then divide your time into stokes, last time I used 24 minutes as 1 stroke

and then 1 day got 60 stroke to do

##### J

BUT.. I SERIOUSLY WONDER Y U CAN SLEEP ONI 4HOURS A DAY?? IF U R SLEEPY N TIRED, U HAVE NO MOOD TO STUDY OSO

HMMM..

HMMM..

##### 04:02Ng

well, I did had a few 6 hours, and one 17 hours sleep during last weekend

##### J

OOO..

##### 04:03Ng

oh ya, divide the number of strokes accordingly

then bring along the note book and a pencil whenever you go

and record them as often as you can remember what you did.

I'm using 15 minutes per stroke now

##### J

OOO..

OK

##### 04:04Ng

and devoting 55 strokes to lectures

per day

##### J

I WIL TRY TO USE UR METHOD

##### 04:04Ng

ok yu

ty

tell me if it works

##### J

HAHA

##### 04:05Ng

oh ya the main thing is to be flexible

##### J

OK SURE

=)

THC A LOT

##### 04:05Ng

start small then slowly increase the number of lecture hours

welcome

and do some analysis every week

see what is obtructing you from your goal

##### J

OK

=)

I WILL TRY TO DO TAT.. =)

##### 04:07Ng

ty

lol

I developed this system for PMR

and used it for STPM too.The thing about it is that I managed to watch movies, animes, play games, go to buddhist activity and learn martial arts despite studying 73 hours per week, so she wanted to learn it. And I hope this helps for those desparate ones out there.

### Realised, Awaken, and Understood.

Then I've gone through the 2 Physics and scapegoated the maths together, not giving my best in them, but doing good in the exams. And so I only got 3 As for them and no A+. Moreover, I don't think I can recall what they taught me very well now. My additude was terrible in the sense that I didn't took much of anything in Sem 1 seriously. I was more of a bitter guy for missing out on a lot of oppotunities (scholarship, MIT, advanced placement, gold medal) before coming here. And so I didn't really think I accomplished anything significant during sem1.

Come sem2, I realised I don't wanna repeat my mistakes and I should do something more than just study or else I'll regret it as one sem loss. However, almost the same additude on the 2 physics and one maths carried themselves over here. In addition, I had one English module that carries no MC! But I planned properly and didn't take too heavy a module from USP this time. In the course of time, I made a timetable for sem 2 to make sure that I catch up to the others in my weekly homework and performance. And I noticed that I actually spend a lot of time in the 2 level 1 physics modules, due to Lab and Mastering physics mostly. So they are quite heavy modules. So I did had some more bitterness on having to take them. I did my Mastering during lectures and take my liberty to skip them to do other modules and attend other talks.

All this lasted until Prof. Yeo Yi had pointed out at the end of the module that "there are a lot of smart people, but it is the smart people that has the additude will make it." Humility is the key. He said that Asian people going to the US had some advantage towards their peers in the first year. However, at 3rd year, the very same people who were behind them caught up and was better than them now. What happened was that the things taught too early will fix the taught of the students. But then their peers learned the basics of physics with an open mind, so they think deeper, and ask more questions. And what happened in year 2 is that the physics students cannot see the physics in the maths, so basic understanding of physics is important. He told us to try to see physics with an open mind, try to understand the structure of it.

I was quite sure that this partly if not mostly is directed to me. It got me thinking, should I take 8 and a half module next sem? How can I have the right additude while doing that much things? I tried to listen to maths class, but got asleep soon after. It was a pattern, a habit born of the first impression that the module is nothing to me. So I was quite depressed on Thursday, on the subject of taking more modules or having the right additude. The friends I have consulted told me that what's the point of taking that many module if what I gain from it is nothing? or I don't fully immerse in the module.

Just now, talking to Hariom, I commended him as having a dissatisfaction on understanding and wants to understand as much as he can on each physics modules. And so after that I realised that's what I lack. the urge to understand deeper, the dissatisfaction with what is presented and asking for more. I knew then what Prof. Edward wanted me to learn: to throw off the mental mechnical way I do physics by questioning why, why whY down to the last why. I realised that the IPhO is nothing really to be proud of, there's no understanding there, we are nothing compared to the physicists. And so this lesson is learnt. And it is definately worth all those hours of the 4 physics modules for me to realise it. At least at this point in time. But I am quite afraid that I need the mental tool for IPhO yet to serve me in the coming exam. No worries, I know now how to procede. And I thank all the Profs and friends that helped me realised this.

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## Friday, April 03, 2009

### Finally I've seen the light!

## Thursday, April 02, 2009

### General Relativity Introduction Part 1

It's in Latex code, so just download Latex then copy and paste, and voila, you get to see my work. and above is the picture, just convert it to eps and save as LC.eps in the same folder as the tex file.

\documentclass[a4paper,12pt]{article}

\renewcommand{\baselinestretch}{1.5}

\usepackage{amsmath}

\usepackage{graphicx}

\setlength\topmargin{-0.75in}

\setlength\headheight{0in}

\setlength\headsep{0in}

\setlength\textheight{10in}

\setlength\textwidth{6.5in}

\setlength\oddsidemargin{0in}

\setlength\evensidemargin{0in}

\setlength\parindent{0in}

\setlength\parskip{0in}

\title{Introduction to General Relativity}

\author{Ng Xin Zhao}

\newcommand{\eq}{\begin{equation}}

\newcommand{\qe}{\end{equation}}

\newcommand{\tyh}{\begin{align}}

\newcommand{\hyt}{\end{align}}

\begin {document}

\maketitle

\section{Background}

The Special Theory of Relativity describes physical laws based on inertial reference frames, and successfully merged the consistency of the speed of light found by Maxwell with Newton's law of motion; however Newton's law of gravitation still conflicts with the consistency of the speed of light. This problem is solved by the General Theory of Relativity. The theory is based on the equivalence principle, that is the observation that the gravitational mass of an object is the same as the inertia mass of the object. Thus there is no distinction between locally gravitational frames with accelerated frames. The Theory of General Relativity states that gravity is nothing but curvature of four dimensional spacetime and the distribution of mass-energy bends spacetime, thus producing the familiar gravity. The sections below highlight the components of the General Relativity Theory and the physical meaning of them.

\subsection{Tensors}

Since the General Theory of Relativity is formulated to describe the laws of physics in the same form in all frames, it must be coordinate independent. Not only that, the theory should be able to describe how the change of the observer's frame affects how the observer looks at a physical phenomena. In the mathematical sense, the transformation of coordinates must be defined. And the mathematics for this kind of requirements exists in the form of Tensors. Therefore, the General Theory of Relativity must be expressed in the Tensor form. A Tensor then consists of the following properties:

\paragraph{}

It is a array of numbers in many dimensions relative to a choice of basis chosen. Here basis means a coordinate basis. However, tensors are geometrical in nature and thus are independent of any coordinate system that describes it.

\paragraph{}

A tensor has ranks; a tensor of rank 0 is a scalar, or a single number. A tensor of rank 1 is a vector or an array of numbers. A tensor of rank 2 is a matrix, or a 2 dimensional array of numbers. The definition extends this to any number of dimensions.

\paragraph{}

A tensor is usually denoted with subscripts and superscripts. The sum of subscripts and superscripts is the rank of a tensor. For example, $T^{\mu}$ is a rank one tensor and $T$ is a rank 0 tensor. A tensor with rank $n+s$ can be denoted as $T^{\mu_1 \mu_2 ...\mu_n}_{\nu_1 \nu_2 ...\nu_s}$.

\paragraph{}

A tensor is defined by the coordinate transformation

\eq

T^{\mu\prime_1...\mu\prime_n}_{\nu\prime_1...\nu\prime_s}=X^{\mu\prime_1}_{\mu_1}....X^{\mu\prime_n}_{\mu_n}X^{\nu_1}_{\nu\prime_1}...X^{\nu_s}_{\nu\prime_s}T^{\mu_1...\mu_n}_{\nu_1...\nu_s},

\qe

where

\eq

X^{\nu\prime}_{\nu}=\frac{\partial x^{\nu\prime}}{\partial x^{\nu}}.

\qe

\paragraph{Remarks on notation}

The notation used in General Relativity follows the Einstein notation where $\lambda^{\mu}\lambda_{\mu}$denotes a summation on $\mu$ for all the values of $\mu$. Or more simply,

\eq \lambda^{\mu}\lambda_{\mu}=\lambda^1_1+\lambda^2_2+\lambda^3_3+\lambda^4_4,

\qe

if $\lambda$ is the value 1 to 4.

Partial derivatives can be written as

\eq

\frac{\partial x^\alpha}{\partial x^\beta}=\partial_{\beta} x^{\alpha}=x^{\alpha}_{,\beta}.

\qe

The basis can be written in short notations of $x^\mu$ where $\mu$ range from 1 to 4 in 4 dimension spacetime. Thus,

\eq x^0=t, \hspace{3mm} x^1=r, \hspace{3mm} x^2=\phi, \hspace{3mm} x^3=z, \qe in cylindrical coordinates.

\section{Einstein's Equation}

\begin{align}

G_{\mu\nu} +\Lambda g_{\mu\nu} =R_{\mu\nu} -\frac{1}{2}Rg_{\mu\nu}=\frac{8\pi G}{c^2}T_{\mu\nu},

\end{align}

The Einstein Equation above is written in Tensorial form and it just equates the curvature of spacetime with the mass-energy that causes it. The terms are:

\begin{itemize}

\item $G_{\mu \nu}$, which represents the curvature of spacetime,

\item $T_{\mu\nu}$, the Stress Energy Tensor which describes completely the distribution of mass, energy and stress or pressure,

\item $\Lambda$, an extra term introduced to represent negative gravitation or more positive gravitation. Historically Einstein introduced it as a repellent force to balance against gravity to maintain a static Universe. Currently it is believed that this value for our Universe is small but non-zero, the force that is responsible for the acceleration of expansion of the Universe. It is called Dark Energy too due to ignorance of its nature.

\item $R_{\mu\nu}$ is called the Ricci Tensor, it tells how much the spacetime differs from Euclidean space, the traditional notion of space before Special Relativity.

\item $R$ is the Ricci Scalar, it denotes a single real value to every point of spacetime. If the value is positive, the volume of a object is smaller at that point in spacetime compared to the volume at Euclidean space and vice-versa.

\item $g_{\mu\nu}$, is called the metric tensor, it captures the geometric and casual structure of spacetime.

\end{itemize}

\subsection{The Metric Tensor, $g_{\mu\nu}$}

\paragraph{Euclidean Space} is just three dimensional space, when there is a rod with a length $l$, the length can be found in terms of a coordinate by Pythagorean theorem, that is $l^2=x^2+y^2+z^2$, and the infinitesimal form is $dl^2=dx^2+dy^2+dz^2$. In another coordinate basis, the length $dl^2$ remains the same that is, $dl^2=dx^2+dy^2+dz^2=dx\prime^2+dy\prime^2+dz\prime^2$.

\paragraph{Minkowski Spacetime} is the flat spacetime of Special Relativity where time is treated as another dimension and therefore the line element $ds^2$ is modified to include the time axis.

\eq ds^2=-dt^2+dx^2+dy^2+dz^2=-dt\prime^2+dx\prime^2+dy\prime^2+dz\prime^2. \qe

$ds^2$ is an invariant in spacetime, it does not change with the change of basis.

The time axis is negative to show that it is a special dimension.

\begin{figure}[ht]

\begin{center}

\includegraphics[scale=0.5]{LC.eps}

\label{Light Cones}

\vspace{-8cm}

\caption{Light Cone}

\end{center}

\end{figure}

The figure \ref{Light Cones}, is drawn in Minkowski spacetime. The horizonal plate is two of the three space dimension and the vertical axis is the time dimension. One more space dimension is compressed to draw the time axis. The cones are light paths tracing from the origin of the axis to both the posive time or future and the negative time or the past.

\paragraph{The Metric Tensor} is a rank 2 symmetric tensor that describes the geometry of spacetime. It can act on two vectors to produce a scalar number. The notation can be written as

\eq

g_{\mu \nu}[x^\mu ,x ^\nu]=g_{\mu \nu}x^\mu x ^\nu=c,

\qe

where $c$ is a scalar. In fact, when the metric tensor interacts with the infinitesimal change in the elements of the basis,$dx^\mu$, the scalar produced is called the line element, $ds^2$.

\eq

ds^2=g_{\mu \nu}dx^\mu dx ^\nu.

\qe

The metric tensor for Minkowski spacetime is then

\begin{eqnarray}

g_{\mu\nu}=

\left(

\begin{array}{cccc}

-1 & 0 & 0& 0 \\

0 & 1 & 0 & 0\\

0 & 0 & 1 & 0\\

0 & 0 & 0 & 1

\end{array}

\right).

\end{eqnarray}

Thus it can be seen that the metric tensor represents the form of the line element which in turns tells the properties of the spacetime in question.

The equation $ds^2=g_{\mu \nu}dx^\mu dx ^\nu$ arises because not all spacetime is flat, so curved spacetime can be described by just changing the metric tensor.

\paragraph{Metric signature} used here is $(-,+,+,+)$ and is just a convention for writing a metric. It comes from the signs used in Minkowski metric. The opposite signature of $(+,-,-,-)$ can be used but the whole system must be consistent in using just one signature.

\subsection{Casual Structure}

\paragraph{Manifold} is a mathematical space that resembles Euclidean space at a small enough scale. In General Relativity, a manifold,$M$ is then a curved spacetime that can be approximated to Minkowski spacetime locally. Here, locally is a term for a small enough scale so that the approximation holds and globally is used for big scales where the approximation fails.

\paragraph{Tangent Vector}

\end {document}