Coursera's introduction into General Relativity

I was lured recently to a Coursera's course in General Relativity. It is a course hosted by Emil Akhmedov, professor in the Higher School of Economics.

Do you think too coursera's logo looks like some kind of brand for a dark society?

Not gonna lie, it's not easy but at least it's formal enough to make me re-learn tensor handling in a way Susskind can not. Not his goal anyway. Here is the web in case you want to check it out.

The speed of gravity

The internet is full of cats and porn. No breaking news, here. There are a lot of other places with interesting information, though. One of them is the askscience of reddit. If you follow the link, you'll find this one too.

The question stated is: if gravity is transferred at $c$ (speed of light), and Sun is 8 minutes away at $c$, regarding the fact the Sun moves, does it mean Earth is attracted to the position the Sun had 8 minutes ago?

Video is from a guy named DjSadhu. Music is cool, if you ask me.

Spacetime as a consequence of something else: ER=EPR.

There's so much I don't know about GR that talking about interpretations and physics trends is a dangerous matter, specially if my words are intended to be, you know, taken seriously. That's the reason I try to focus in learning instead of trying to develop things on my own. And I don't always succeed anyway, so look at that.

Nevertheless sometimes comes the time when you find something interesting and you have to share it and... Well, hell, this is my blog, right?

I have found recently the ER=EPR conjecture and I have tried to learn a bit about it. Shall we talk about it a little? I'll go on with Schwarzschild metric inside the Earth a little bit later. :)

Maxima ctensor overview

I have installed Maxima (finally) and I have been playing with it a little. First part of the post is about a general explanation about what Maxima's ctensor can and can't do. Second and third part will try to explain functionality with a couple of easy examples. I have not tried neither itensor nor atensor; as I stated in the past post, their functionality is not what I need for now.
So far, and summarizing, I must admit Maxima falls short for my intentions. Truth is it's a good tool anyway and it should be easy to adapt to what I want when I had the time to go for it.

Coming back ASAP

Sorry again. A couple of months have passed and I haven't find the time to continue with the posting. I should continue ASAP. I left the work at analyzing Maxima and Cadabra, which I should do in the next days, while I prepare my TOEIC exam -lot of work to do as you can see by my english level.

After that, truth is I don't have a clear idea of what to do next. I had a couple of posts ready about Earth nucleus age, and stuff like that but I'm not sure what should come next. Anyway, I should continue, so in a couple of days a new post will appear, but be patient please.

It's just things are not easy everytime :)

Open Source Computer Algebra Systems.

So, I started to look for computer help. As an open source freak, I have found two different packages available: Maxima and Cadabra. This post is about impressions on both of them. Not sure yet what I am going to use yet, but my hopes go to Maxima in a first approach.

Schwarzschild metric (V).

Did you really think we were done with Schwarzschild metric? We have finally found that...
\begin{equation}
ds^2= \left ( 1- \frac{2GM}{c^2r}\right )c^2dt^2- \frac{dr^2}{\left( 1- \frac{2GM}{c^2r}\right)}-r^2(d\theta^2+sin^2\theta d\varphi^2)
\end{equation}
... so... Good for us.

But what does that mean? Why is it important? Why is it useful?

Schwarzschild metric (IV)

This Schwarzchild Schwarzschild topic is getting out of hand!

Just two more posts. In this one we are going to find the final form of the metric. Next one is about interpretations, conclusions and such, which turns out is a lot more faster than deriving the metric itself...

So, here we go again... Math ahead!

Schwarzschild metric (III): the search for simplification

Stubborn as I am, I'm not going to continue with the demonstration demostration until I completely understand why $C=D=1$. I'm not saying it's wrong. Just saying I don't understand why it's right... Yet.
Spoiler: I have found a way of understanding it. Not sure is correct, but it's good for me at the moment. Yes. I do believe too that I have overthought this too much.

Betrand Russell advises.

It really has not much to do with GR, but in the moment I saw the video, I knew I had to post it.

Schwarzschild metric (II)

Are you ready?
We were supposed to end up with the demonstration, but it actually is pretty damn long, so in this post I'm going to just obtain the equations as functions of A, B, C and D. And I'll let the final steps of the demonstration for the next post. So two points already pendant:

1. Why metric tensor has only diagonal components? That is, why there are no cross-terms?
2. Applying $g_{\mu\nu}$ into the $R_{\mu\nu}$ equation.

Warning! Math ahead!

New look!

I'm finishing Schwarzschild post II: the Schwarzs strikes back. For the moment, as you can see, several look changes have been made to the blog. For the sake of clarity.

Maybe I add several changes more in the future. Maybe the font is a little bit small...

Edit: It turns out Mathjax doesn't work well with my android phone... Working on it! :)

Edit2 (05/16/16): It turns out I'm a moron and I didn't activate the custom layout in mobile needed for Mathjax to perform its magic. Several issues regarding line breaking in formulae still remain, so I suggest to use your mobile in landscape mode to fully enjoy the math for now :)

Schwarzschild metric (I)

Maybe I should start explaining what a metric is. Or curvature. Or the energy-stress tensor...

Nahhh... :)

Let's run before walk with an example. It's funnier this way.

Legend says Karl Schwarzschild was a german artillery officer during WWI, but the truth is he was a physicist and astronomer before that. He died several months after finding the first analytical solution to EFE's, of an illness called Pemphigus (click at your own risk). You have read right: Schwarzchild did find an analytical solution to Einstein Field Equations. Exactly. Analytically. Manly.

Truth be said, he did it for the "simplified" case of a spherical static object, but it's a good aproximation, specially for people like me who need to learn. Are you ready? I prevent you I have (finally) learned how to use latex in blogger...

Coming back soon!

Two years and a half. 

Two years. And. A. Half.

Lot of things have happened since the last post. Long story short, I have a little bit of free time again, so I'm preparing new posts in the point where I left.

Next stop, Schwartzschild metric.

See you soon!

EFE -for beginners!

A short post again, sorry.

Looking for info into old posts, youtube has pointed out a nice video from user DrPhysicsA which seems... shiny!

Old fashinated english accent (I think it's English, pretty sure not American nor Australian, nor Irish... what else could it be?) for 2 hours of EFE fun!

Frequency in general relativity.

Ok. It's clear what's the role of frequency in special relativity. Frequency does not change essentially from its origin in classical mechanics. Frequency has units of inverse of time, which means it's something with no dimensions that is measured per unit of time. In case of frequency that 'something with no dimension' is a repetition.

Quite clear, right? Better explanations here, and don't forget to donate. I will when I get paid for what I do, which has not come yet.

Back to the subject, in special relativity time changes and so does frequency. If there's a time dilation according to velocity value, frequency changes as time does. With an inverse proportionality relation, of course. 

In general relativity, all references I've found so far point out frequency is the same. They talk about red-shifting and the same time dilation issues special relativity does.

But, how come?

We have said before time is just another dimension in general relativity (a special one because our perception, but just another dimension). Several questions arise, then:

1. Are there spatial frequencies the same as a temporal one? If so, what's their use?
2. What does it make time so special? Why are we so used to talk about temporal frequency as frequency?

Easy answer for a physicist, but you have to remember I'm not one them.

In any case, what's the point of this? Well, that picture of the response to a step input in a second order system has made me think about frequency and an old friend/foe concept known as resonance. Which I'll go deeper later. Not now.

The Cavendish experiment

Disappointed with this blog? Yeah, sorry again. Too much time between posts and I'm not getting very deep in any subject I deal with. That's because it requires time, and that's a thing I didn't nor I have at the moment.

So, let's talk about a funny experiment finally I've found in the internets. It's called Cavendish experiment and you can get good historical explanations of it here and here.

The experiment seems easy to follow. In Newtonian terms, two masses attract each other.

Proportionally to the masses, inverse proportionality to distance squared. But proportional doesn't mean equal. There's a constant in somewhere you have to add to make things right. Unit conversion stuff. Kinky stuff. If you want to measure how much does it weight Earth (is there anything British-er than asking yourself that kind of questions? ) you need to get that constant first.

But, how can you measure the gravitational constant without knowing Earth mass? And harder still, you are living on Earth. In case you figure out a way of measuring without taking into account Earth gravitational pull, doesn't it still f*ck with your measurements?

Before explaining something you probably already know I need to make clear why am I interested in Cavendish experiment:

1. It's an "easy" experiment which allows us to understand gravity in human size scales.
2. What is its relationship with space-time? Before you say human scales are Newtonian, which is almost truth for every almost example you can find, just keep on reading.

Coordinate systems (I)

There are yet a lot of things to say about manifolds but maybe is a good moment to talk about coordinate systems. Manifolds can have coordinate systems. If not, how the hell it is supposed we can find something in a manifold? We need a point of reference and a way of establishing relationships between dimensions.

Why coordinate systems are important?

Coordinate systems are a way of knowing where things are and/or when they are happening.

If you are familiar with linear algebra (if not, sorry I can't help you) you already know in order to distinguish an element, for instance a vector in a vector space, the element has a set of coordinates that define that particular element. A fast explanation about this: each coordinate corresponds to a dimension and is a number which tell us how far should we go along that dimension in order to "find" the element.

 

A typical coordinate system and a point represented in coordinates, from here. In Linear Algebra, 3 coordinates are enough... And a reasonable way of describing physics. In order to arrive to point (x,y,z) you have to walk exactly x,y and z in the respective coordinates which, to make things easier, represent only one dimension at a time.

If you start walking from the origin of the system the number of units, along one dimension, the coordinates tell you how much walking do you have left in order to arrive to your destination.

When I studied Linear Algebra, only orthonormal basis were approached. Orthonormal means:

• Each element in the basis is orthogonal (which means is linearly independent from the other elements in the basis, which means it can not be expressed as a combination of coordinates from the other elements in the basis, which means roughly, each coordinate represents only a dimension). Orthogonality is typically used to express perpendicularity.
• Each element is also normalized -which means is a unitary- vector, or versor.

Orthonormal basis are the easier basis we can use -and they're quite useful for physics to a quite high level. Easy to grasp, easy to operate. Cool.

The problem comes when we are talking about manifolds which represent curved space-time: generally speaking it's better idea to use another kind of coordinate systems. And when that happens, sh*t gets real.

What the heck is a manifold.

After the failure in describing geodesics from Schwarzschild solution I've decided to come back to the origins. Starting from the basics. Manifolds. Yeah. Manifolds rock!

General Relativity -uppercase means it's important- is defined in a 4 dimensional pseudo Riemannian manifold or more specifically, a Lorentzian manifold.

So, before continuing, what is a manifold?

Some people could say mathematicians are weird. I share this appreciation with a deep and sincerely respect (sometimes a little bit of fear) for mathematicians. Level of mental abstraction in Math is something out of my grasp, so I can only understand it by drama.

That means, in 4 steps:

1. I've got a problem I need to solve but nothing I understand so far helps, so...
2. I find someone's work who already has solved the issue and I try to grasp the Math behind his/her solution...
3. I believe I have understood the Math, so I try to solve the problem again.
4. I come back to the first point if the problem is not solved yet. More than once, 'cause I can be a real donkey sometimes...

Quite algorithmic. You have to remember I'm an engineer...

So instead of explaining what is a manifold, I'm going to explain what I understand a manifold is...

Coming back soon!

So... 2013... Nice!

Hey, don't look at me that way. It's been a couple of hard months. More than a couple, I get it, but I needed to focus in real life problems, like ending a master and looking for a job. Which I already have... For the moment. The job, I mean. I'm still working in the master.

It's hard, OK? Come to Madrid, they said. You'll have fun, they said... :)

Anyway, thanks for the waiting. I'm returning to the subject as soon as I start to re-read all I have posted before. Last thing I tried was obtaining a parametric form for an easy geodesic, and I failed so hard it's been 7 months since the last entry. I have pending a couple of book reviews and lot of extra work, but I'll do my best.

See you in a couple of months!

Edit: It's been so long I almost forget how to post. Sorry for the mistakes!