science stuff

[demagogue]

I know we have some scientifically inclined members. I had a science question, but if other people wanted to talk about other science stuff here too, feel free.

 

so I'm watching a lecture on General Relativity. Can any science people help me understand something?

 

I got contravarient transformations (upper indicy) transform opposite to the coordinate change. Easy, rotate the coords &, e.g., the velocity or position appears to rotate the other way (same with translation, or expansion, etc). But how does a covarient transform (lower indicy) play out? It's supposed to transform along with the coords. They gave an example of a gradient, but if I imagine you rotate the coords, the gradient again rotates the opposite way, right? or if you spread the coords out, the gradient just shrinks, so how is it covarient? It also looks contravarient to me. another way they explained it was covarient was a projection on the (new?) axes (and something about the normals of planes), but I had no concept how a transformation in coords can looks like a projection or a normal. does anybody get what I'm missing?

[Obsttorte]

the covariance refers to the way the linear transformation (for example a gradient) appears after the transformation. If you transform a vector v it looks like

v_new=A*v

where A is your transformation. On the other hand if you transform a scalar-valued linear function L (like a gradient for example) it looks like this

L_new=L*A

So the difference is from which side the transformation gets applied.

 

I hope that helps.

 

EDIT: I've corrected my post.

 

EDIT2: Referring to the gradient example you've mentioned you have to think of it as a function, not as a vector. So it is something that gets applied to vectors. This is a huge difference.

[lost_soul]

The tablet I ordered has an accelerometer in it. Which directions can this thing detect? I've seen it rotated left and right, but can I also pitch it forward and back? I ask because it does not support multitouch and I am going to load Doom on it.

 

http://www.youtube.com/watch?v=KZVgKu6v808

 

Also I wonder how relative positioning affects this. If I'm sitting in a chair and the device is not at a perfect 90 degree angle (up/down), how does it know when I want to move forward/back and where the "still zone" should be? Or do I have to hold it at 90 degrees to not move in game?

Edited March 13, 2013 by lost_soul

[demagogue]

I hope that helps.

 

Thanks. Yes, I could see the math saying "v_new=A*v" (for contravarient) vs. "L_new=L*A" (for covarient), and for contravarient it was easy to imagine. But I think I needed an image in my mind of what a function in space is & how it transforms. do you mind spelling it out a bit more like I'm a dunce?

 

I imagine a gradient like a more or less smooth build-up/down (of whatever stuff it is) in a direction at each point. Before I was thinking of it like a given direction(s) before the transform, like building-up "southwards" at X point, which would also transform contravariently.

 

But it looks like you're saying I really need to think of it as just the function itself, right?, not like an actual position or vector "out there", but an abstract function you'd, e.g., apply de novo to a vector *after* the transformation. so you'd have some simple function that drops a simple linear gradient on a field of vectors, but now you're in a moving or curved frame, you'd have to transform that function so when you applied it to vectors in the new coord's, it'd look like it should? Is that right?

 

But (if so) I'm still wanting a concrete image of such a transform following "L_new=L*A". Like if you stretch out the coords, does a gradient function really stretch out too (as in varying "with" the coord change, in tandem, covarient)? Just seem like then you have the same result, the gradient looks the same whether you're, e.g., moving or not. -- or is that the desired result? I didn't think so. Maybe I should just ask how does a gradient look after some basic transforms, then I could better imagine how the math got it there. I need a mental image like that. I think my understanding of "varying with the coord change" isn't right.

 

Thanks for the help by the way. I know I must be asking such newb questions.

[Obsttorte]

I guess the problem why you mix this up is that gradients of scalar-valued functions are represented by vectors. So your first thought is that a transformation to it should apply in the same way as to a vector. The difference is however that a vector means a certain point in your space (vector space, the area that is created by the base). The point stays at the same position when switching to another base (like for example via rotating) but its representation in the new base is different.

 

The function (gradient in this case) however takes a vector and creates a scalar out of it (sorry for the strange description, I'm not very familiar with the english terms). So in the end the image of the vector shouldn't change after the transformation, as all it does is to change the representation of space and function, but not the function itself.

 

This leads to the above formulas.

 

Another thing to keep in mind is, that while the vector representing a point is a column vector, the vector representing the gradient is a row vector. This little difference seems a bit unimportant for non-mathematicians, but it isn't.

 

If you look at the term for the gradient you get

L_new=L*A=(A^T*L^T)^T

A^T is the transposed matrix of A (so every entry mirrored at the main diagonal). The transposition of a row vector turns it into a column vector and vice versa. The last transposition turns the result of the matrix-vector-product into a row vector again.

 

What you can basically see here is, that while the vector representing a point in space gets applied to A, the function gets applied to A^T. In case of a rotation this is the inverse of A. This means that it is a function that does exactly the opposite of what A itself does. So if A rotates the vector by 30 degrees into one direction, A^T rotates it 30 degrees into the other direction.

 

You can think of it as following: If the result of a*v (a the gradient, v the vector-representation of a point in space) should stay the same regardless of coordinate transformation, then a_new must compensate the changes leading to v_new under the specific transformation. This applies not only to rotation, but also to compression.

 

I'm not sure if this makes things clearer or more blur. It's a little bit hard to really understand were your understanding ends.

[Diego]

Maybe a step back to the math behind vector spaces and dual spaces would help. I remember a good introduction covering these aspects in the first chapters of Robert Wald's General Relavity.

Basically you first define a vector space and then a space of linear functions acting on the elements of that vector space, called its duel space. A contravariant vector is an element of the vector space first defined and the covariant vectors are linear functions from the dual space. However, this dual space is also a vector space and, furthermore, for every element of the original vector space (the contravariant vectors) there is one and only one linear function in this dual space. These are two identical vector spaces, that's why in GR is convenient to use a notation that differentiates them simply by the index position.

I hope that complements what obsttorte said about the action of an operator and helps you understand this stuff

[Obsttorte]

However, this dual space is also a vector space and, furthermore, for every element of the original vector space (the contravariant vectors) there is one and only one linear function in this dual space.

I guess what you've meant is that every linear function on the vector space can be represented by the scalar product with a specific vector, which is uniquely defined. (Riesz). I may add that this only applies in Hilbert rooms. In general vector spaces this is not right.

 

Another point to mention is that the terms covariant and contravariant does always rely to specific functions. In this case coordinate transformations. So the elements of the vector space are contravariant in respective to coordinate transformations. And the linear functions (the elements of the dual space as you said correctly) are covariant in respective to coordinate transformations. It's like with the term invariant, which does also rely to specific functions applied to certain elements of certain rooms/spaces/sets or whatever.

[Diego]

Well, it's not automatic, the dual space V* can be a vector space itself if one defines sum and multiplication by scalars. While the field of V can be real or complex numbers, the field of V* is V. I don't know how this behaves in different vector spaces but I totally believe you but assuming he's interested in GR I don't know how interesting those other cases are.

Edited March 14, 2013 by Diego

[Obsttorte]

Actually I was more referring to the second part of your sentence.

 

But I guess our little mathematical excourse doesn't make things much easier for demagogue.

[demagogue]

To give context, I'm following YT lectures by susskind because I like how he explains things, and it's at my level in the sense it's just over my horizon; he doesn't venture too far out. But in his lecture on this (

), there was an equipment failure & some of the video cut out at 55:02 (I don't know for how long)... Unfortunately at one of the absolute worst places, introducing the indices (and frustrating because he keeps referring back to it; "remember when I said this..."), and I want to know what I missed. I mean he introduced these two new concepts, upper & lower indices, and then the video cuts out, so I didn't fully catch what these 2 things even are; well I missed what the lower was... I just see him using them later & see the math without knowing what they are or are doing.

 

I did up to like 3rd or 4th year calculus in undergrad, so roughly get partial derivatives, but it was years ago, and playing with vectors is new.

 

I think it's good to step back to basics of the math first too & feel out how it's applying to different objects as I go.

 

so I found this explanation online: http://www.farmingda...pdf/Ch04Rel.pdf

 

Its examples are just translating a vector to a skewed coord frame with components contravariently & covariently, which here just seem to mean "using paralells to the new axes" (contra-) vs. "using right angles to the new axes" (co-). If that's all they mean, that's not too hard to get. But I fear there's more to it. I am still going through susskind's lectures. do you think this tutorial give the same math I need to get GR indices, or at least a starting point (is it using the concepts contraviently & covariently the same way?), or is there still some missing?

 

I'll see if I can find a good textbook too.

 

Edit: this might explain some of my confusion because he was speaking as if contra- & co- were for different objects, when the example above show anything you can describe in one you can translate into the other. so I think what he meant was each is more fit to different objects. Translating a vector was more natural with contravar components (parallels), even though you can do it with covar ones, and I think a gradient (function) is more natural with covar components even though you can do it with contravar ones, but I haven't seen that yet.

 

as an aside, I also need to figure out how the matrix description fits in. It seems relates the (new?) axes to one another for each combination, what's more or less parallel (1), perpendicular (0), or in between, I think.

[Obsttorte]

Regarding the pdf you've posted. The terms contravaiant and covariant are used in a different way here and have, as far as I can tell, nothing to do with your first post.

 

Again, contravariance and covariance are just some properties under a certain function. This is quite similar to invariance as I meantioned earlier.

 

Let's start with the latter so you get what I mean.

 

Invariance means that a certain characteristic of an object does not change under a specific function. You are maybe familiar with the determinant of a matrix. This characteristic is invariant under coordinate transformations. So if I change from one coordinate system to another one the determinant of any matrix does not change at all.

 

Covariance now means that a certain characteristic changes in exactly the same way as it does with some reference object. In our case, the reference object is our coordinate system. The function is, again, a coordinate transformation, and the object is a linear function (for example a gradient). So if a transformation for example rotates the coordinate system in a specific direction, the linear function (or more precisely it's representation as a vector) gets rotated in exactly the same way.

 

Contravariance now means that a certain object is changed in exactly the opposite way as our reference object. In our case, the coordinate transformation rotates the coordinate system in a specific direction, and a vector of the underlying vector space (or more precisely it's representation as a coordinate vector) gets rotated in the opposite direction.

[Diego]

Too long. Did not read. Spam bot?

 

edit: I mean the deleted spam, not Obsttorte's post

Edited April 2, 2013 by Diego

[demagogue]

Definitely spam.

 

Edit: Lol, the deleted advertising spam, not the very helpful & generous math explanations.

[simplen00b]

Definitely spam.

And the scary thing is, presumably it must work - i.e. enough people actually click on the links and then 'buy' something to make it worthwhile - ???

 

Which frankly makes even less sense than what you guys were talking about beforehand....

[demagogue]

I really wanted a qualitative understanding of GR, just enough of the math (for now) to understand what the equations are roughly doing qualitatively ... what things are out there, what space looks like, what's getting summed where, and what all the symbols & indicies in an equation are roughly doing.

 

Explanations at this level are helpful -- it's answer to my original question makes it really concrete & easier to see what it means -- but too bad it's incomplete.

[lost_soul]

http://www.youtube.com/watch?v=gQ3dT7xcMgU

 

Spooky stuff... This video makes you think about things you otherwise wouldn't. For example, some of this waste does not decay for over 25,000 years, so they're stashing it away under ground. Problem is, what happens if people uncover it later? How do you leave a good sign that is guaranteed to let future generations know not to mess with this stuff? What if said future generations can't even read our current language?

 

Some of the funnier comments I've seen are about just launching this stuff out into space, or to the sun. I had that idea when I was little, but what happens when a rocket carrying this waste eventually blows up while leaving the planet? Answer: major clusterfuck of epic proportions.

Edited April 9, 2013 by lost_soul

[Sotha]

Attention, local geologists!

 

Are you (Mg,Fe)7Si8O22(OH)2?

 

Sometimes science conceals great mysteries and arcane messages.

http://imgur.com/gallery/RTZCF

[demagogue]

Classic. I took geology in college, and I even wrote a song about all the rocks we studied called "Rocks in a box". This rock would definitely have a welcome place in my song.

[stumpy]

nuclear waste is usally hidden in places where the background radiation is highter than the waste. and the waste is usally the waste that you would get in any office, but this waste came from a nuclear power station or reprocessing plant.

[demagogue]

I'm an environmental lawyer, so this kind of thing is my bread & butter.

[lost_soul]

Its time for lost_soul's random weird science question of the week. We all know that radioactivity is frequently described in popular media as glowing green. This is NOT true. Radioactivity does not glow by itsself, but it can make other things glow... e.g. phosfers. That was how those old watches were made. The radioactivity excited the phosfers, which made the light.

 

Now the question. I've read about underground facilities where they hold nuclear waste. I was wondering if they even need to have conventional AC power down there for lighting? In theory, the high levels of radioactivity can cause fluorescent tubes to light, right? Again, we're just talking about exciting phosfers in the tubes so... The single reason I can think of that this will not work is if all ambient radiation that is in the underground facility is unable to pass through the glass of the fluorescent tube.

[Sotha]

My understanding is that the waste is packed into lead (or other metal) capsules which should prevent radiation from leaking outside. Nobody could work in areas like that if the ambient rads were too high.

 

I mean, so high that the fluorescent tubes would get on.

[demagogue]

There are stories of the trees glowing around Chernobyl in the weeks after the incident, but short of that I wouldn't expect much power from disposal vats either. It's also probably lower level than you'd think since it's spent fuel, not like enriched fuel.

 

On the other hand, I was watching Lewin's MIT lectures on electro-mag & did you know you can light a florescent tube light by sticking it upright in the ground under high volt powerlines, because the potential varies so much from one end to the other it induces a current.

[Diego]

I don't know how they do it either but I believe sotha is correct. To be able to use phosphorous as a light source means there has to be a radiation leak. I don't think they use lead though, as it can be expensive, steel and concrete are sufficient.

[Obsttorte]

But this would provide a cheap warning system, though. If one of the containers is leaking, the phosphor glows.

 

So you see a wonderful green light before the hurting begins