Риччи Тензор

[epros]

GRstudy в сообщении #598939 писал(а):

What I need is a clear concise explanation.

Consider how the gradient of free fall acceleration near the Earth works: Tidal forces compress a body in tangential directions and stretch it in radial direction. But they don't create a PRESSURE on a body on average! That's because the Ricci Tensor is equal to zero.

[GRstudy]

кстати плотност земли будет

$$\rho(R)=\rho_{0} e^{-cR}$$

I think density increases as you move towards the center.

[GRstudy]

http://upload.wikimedia.org/wikipedia/c ... ential.jpg

Which function corresponds to this picture? (in terms of z(x,y)?)

[Munin]

zask в сообщении #598978 писал(а):

С чего народ перешел на английский? По-моему г-н Зубелевич просто попытался пошутить.

Официальные языки форума - русский и английский. Так что, почему бы и нет? Мне, например, такой практики в языке не хватало.

GRstudy
I hope my clumsy English is not too bad?

GRstudy в сообщении #598998 писал(а):

I have followed your instructions and I put them into a visual representation below

Yes, that's right, though i was thinking of a surface with a positive curvature, a 'hill', thus the word 'cap' to describe the idea.

GRstudy в сообщении #598998 писал(а):

Basically, the divergence, the measure how the field spreads out, of a gravitational acceleration, g, is - 4 pi G rho.

$$\nabla g = - 4 \pi G \rho$$

, where both g and rho are functions of x,y,z
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я забыл сказать что

$$F=ma=m\frac{d^2x}{dt^2}=m\nabla \varphi$$

Actually this is standardly defined as

$$m\mathbf{g}=\mathbf{F}=-\nabla U=-m\,\nabla\phi,$$

so two minuses cancel out and the result is

$$\nabla^2\phi=4\pi G\,\rho_{\text{gr}},$$

which I call gravitational Poisson's equation. It differs in sign from the more familiar electrostatic Poisson's equation,

$$\nabla^2\phi=-4\pi k\,\rho_{\text{el}},$$

though this is a matter of simple redefinition, mathematically unimportant, and generally for differential equations it is more natural to be written with the right hand part to be the same sign as the left hand part.

GRstudy в сообщении #599017 писал(а):

кстати плотност земли будет

$$\rho(R)=\rho_{0} e^{-cR}$$

I think density increases as you move towards the center.

Actually, no, not that function. Yes, density increases, but the exact funcion is more complicated, it is made by the different geological materials that make the Earth's body, in varying conditions (temperature and pressure), and it is not known in much detail. Though the general picture is clear, from the research of seismic waves propagation, and some simple model assumptions, and you can look it by Google images 'Earth density profile'.

GRstudy в сообщении #599032 писал(а):

http://upload.wikimedia.org/wikipedia/commons/d/d9/GravityPotential.jpg

Which function corresponds to this picture? (in terms of z(x,y)?)

Most probably, it is a ball with constant density, in an empty space (called in the theory of fields vacuum).

[GRstudy]

Actually, I would define Poisson's equation in two different ways:

$$\nabla g=-4 \pi G \rho$$

given that

$$g=\nabla \Phi$$

, we can conclude that

$$\nabla^{2} \Phi=-4\pi G \rho$$

This is a differential equation and that's why we don't actually put values into it, right? We take potential as a function of two variable and density as a function of radius.

Цитата:

Most probably, it is a ball with constant density, in an empty space (called in the theory of fields vacuum).

I actually wanted to have equation like z(x,y)=2x+y, for example. And we would put that into 3d grapher and get something that looks similar to a picture that I linked.

I think that we can find an approximation of a density function of the Earth. Because if we don't, then Poisson's equation would only be nothing more than a purely theoretical insight.

Your English is great!

Thanks!

-- 25.07.2012, 15:55 --

One more thing, why Ricci Tensor outside the Earth is zero? Because Earth's gravity is weak?

[Munin]

GRstudy в сообщении #599071 писал(а):

This is a differential equation and that's why we don't actually put values into it, right?

Are you not familiar with differential equations? This makes things even harder. I suggest you reading some DE textbook and get used to basic examples and problems. Maybe, a separate textbook about PDE after that.

GRstudy в сообщении #599071 писал(а):

I actually wanted to have equation like z(x,y)=2x+y, for example. And we would put that into 3d grapher and get something that looks similar to a picture that I linked.

Let us call the height coordinate $\phi,$ not $z$ here, because $z$ needs to be a third spatial coordinate ignored in this picture (which shows a section by the plane $z=0$) but absolutely necessary for the differential equation itself. Then, the 'equation' of this surface is something like that:

$$\phi(x,y)=\left\{\begin{array}{lll} -C+\dfrac{\rho}{6}(x^2+y^2)&\qquad&\text{inside the circle \(x^2+y^2=R^2\), i. e. \(x^2+y^2\leq R^2\)}\\ \mathstrut\\ -\dfrac{M}{\sqrt{x^2+y^2}}&\qquad&\text{outside that circle, i. e. \(x^2+y^2\geq R^2\),} \end{array}\right.$$

where the constants $C,M$ and $\rho$ are bound by conditions

$$M=\dfrac{4\pi R^3\rho}{3},\qquad \phi_{\text{inside}}(R)=\phi_{\text{outside}}(R)\quad\text{(which fixes \(C\) with respect to \(M\))}.$$

This 'equation' you can give to a grapher (for a more natural picture you could turn it into polar coordinates). This solution solves the equation

$$\nabla^2\phi(x,y,z)=4\pi\,\rho(x,y,z),$$

with the right hand part

$$\rho(x,y,z)=\left\{\begin{array}{lll} \rho&\qquad&\text{inside the sphere \(x^2+y^2+z^2=R^2\)}\\ \mathstrut\\ 0&\qquad&\text{outside that sphere,} \end{array}\right.$$

notice that the equation and the density function are 3-dimensional. More precisely, i've given not the full solution formula, but only what it looks on the $z=0$ plane. The full 3-d solution is

$$\phi(x,y,z)=\left\{\begin{array}{lll} -C+\dfrac{\rho}{6}(x^2+y^2+z^2)&\qquad&\text{inside the sphere \(x^2+y^2+z^2=R^2\)}\\ \mathstrut\\ -\dfrac{M}{\sqrt{x^2+y^2+z^2}}&\qquad&\text{outside that sphere,} \end{array}\right.$$

and you can check it by hand. The idea how this solution has been found is much more complex, and better be studied by PDE textbooks (or video lectures if you prefer).

-- 25.07.2012 18:55:17 --

GRstudy в сообщении #599071 писал(а):

One more thing, why Ricci Tensor outside the Earth is zero? Because Earth's gravity is weak?

This does not have anything with the strength or weakness, this is the difference of nature of the gravitational field shape inside and outside the Earth. As you've been told, tidal forces inside the Earth tend to compress a test sphere (a spherical surface covered by test points) towards its center, because there is some attractive matter inside every small volume. And outside the Earth, there is no matter inside the test sphere, and it is pulled by tidal forces with conservation of its volume. You can think of the Ricci tensor as of showing that change of volume.

In the Newtonian theory, in the terms of the Poisson's equation, this is shown exactly by the $-\nabla^2\phi$ part, because tidal forces are calculated by taking derivative of the force field, that is, the tensor product $\nabla\otimes\mathbf{g},$ and taking the average by all directions means contracion of that tensor to give a dot product $\nabla\cdot\mathbf{g}.$ And we have seen that by Poisson's equation, $-\nabla^2\phi=0$ in the vacuum. Inside the Earth, it is negative, and the change of volume is also negative.

[GRstudy]

I am going to learn Differential Equations because I have no idea of what they really are. Don't be too harsh, I am a high school student (я школьник).

[Munin]

Ah. I feel sorry.

GR is just a number of levels higher than where you are today. There is a large bunch of subjects needed to be studied before GR can be grasped. The short list is like such (you may know some of this, then start from the following steps):

Calculus line:
1c. Basic calculus - taking derivatives and integrals.
1c -> 2c. Ordinary differential equations.
1c,1g -> 3c. Calculus of several variables.
2c,3c -> 4c. Partial differential equations.
3c -> 5c. Variational calculus (of infinitely many variables).

Geometry line:
1g. Vector algebra.
1g,3c -> 2g. Tensor algebra and calculus.
3g. Either Lobachesvky geometry or (3c ->) external differential geometry - some preparatory step for 4g.
3c,2g,3g -> 4g. Internal (riemannian) differential geometry.

Physics line:
1g -> 1p. Electricity and magnetism.
3c,1p -> 2p. Electrodynamics - Maxwell's theory.
3c,2g,2p -> 3p. Special relativity, and SR electrodynamics (same physics, new language).
5c,2p,3p -> 4p. Lagrangian and hamiltonian formalism (even new language) - in mechanics, electrodynamics, SR.
2c,4c,2p,3p -> 5p. Mathematical physics - a bunch of physical phenomena and their mathematical desciptions, as solutions of basic equations.

Usually a textbook cover one to several points of this list. And a lecture course the same. I can suggest some textbooks, if you ask, though not for every topic.

What concerns me is that you've said you're too busy to read a 500+ pages textbook. I'm afraid all these subjects take more than 1000+ or 1500+ pages total. Though of course they'll be much easier (and faster) to read and comprehend when started from the beginning, and not from some near-the-end point.

Sorry to disappoint you.

[GRstudy]

Thank you very much for your help!

I fully understood where I was standing. I have just graduated from a high school and I am going to get my Bachelor's degree in the US.

I was only doing General Relativity for fun while I am waiting departure.

I used MIT Open Course Ware for Calculus. And furthermore, my final 11th grade Algebra included basic differentiation and integration.

I think that required Math courses are the following:

1) Single Variable Calculus
2) Multivariable Calculus
3) Differential Equations
4) Linear Algebra

I am an Engineering major so basically DE and LA are not required for me.

As for Physics, I need to take 2 semesters of them: Mechanics and Electricity/Magnetism. I don't even have a GR course in my university--only SR. It makes me think that GR is greatly undervalued for undergrads. And moreover, I met some physics grad students who don't even know what a Ricci or Riemann Tensor is. This topic is very special. Where did you learn it? I mean, what's your Background?

Because I am so interested in physics, I am thinking of actually doing a Minor in Physics/Math. I don't think that B.S in Engineering and PhD in Physics would be a bad idea.

[Munin]

GRstudy в сообщении #599391 писал(а):

I am an Engineering major so basically DE and LA are not required for me.

Sounds strange, I am sure DE are of super importance for engineering. Though I'm Russian and know nothing of the US standards.

GRstudy в сообщении #599391 писал(а):

I think that required Math courses are the following:

1) Single Variable Calculus
2) Multivariable Calculus
3) Differential Equations
4) Linear Algebra

Yes. For the Calculus of Variations, only a slight aquaintance can be sufficient, since not much work is done with its techniques, only concepts are used. And this aquaintance can be earned by the Physics textbooks covering Lagrangian mechanics.

As for Riemannian Geometry, you'll have to study it by yourself. Though GR courses like that by Susskind, are almost self-sufficient.

GRstudy в сообщении #599391 писал(а):

As for Physics, I need to take 2 semesters of them: Mechanics and Electricity/Magnetism. I don't even have a GR course in my university--only SR. It makes me think that GR is greatly undervalued for undergrads. And moreover, I met some physics grad students who don't even know what a Ricci or Riemann Tensor is. This topic is very special.

GR is generally thought to be slightly off the main line of physics education. Its domain is too narrow, and it is too mathemathcal and alien by ideas, not much of use for those who will not choose it as their main area. GR is more important for theoretical physicists who are few, it is closer to the ideas of quantum fields and gauge fields, rather than to something utilitarian like plasma physics, solid state physics, microscopic physics and such. And GR is important for astrophysicists and cosmologists, who are also not quite physicists.

GRstudy в сообщении #599391 писал(а):

Where did you learn it? I mean, what's your Background?

I learned it by myself, by books like
Landau and Lifshitz 'Theory of Fields',
Weinberg 'Gravitation and Cosmology',
Misner, Thorne and Wheeler 'Gravitation' (was read last but better should have been first)
'Feynman's Lectures on Gravitation'
It is not too hard, when you learned SR, and know field physics, and ready for the world view shock :-) though my first and main shock was the SR itself (not saying about quantum physics). I knew SR and electrodynamics by my college physics.

GRstudy в сообщении #599391 писал(а):

Because I am so interested in physics, I am thinking of actually doing a Minor in Physics/Math.

I'm very happy to hear that. And may Feynman bless you!

[casualvisitor]

Munin в сообщении #599412 писал(а):

It is not too hard

Well, ... it is not. Still, studying your 'three lines short list' would take years of dedicated work (and THOUSANDS of pages of serious and slow reading).

GRstudy писал(а):

I don't think that B.S in Engineering and PhD in Physics would be a bad idea.

I suspect that there are at least two better ideas:
B.S in Physics and PhD in Physics
or
B.S in Engineering and PhD in Engineering.

If you do plan to become a physicist, make sure that studying Engineering would not be waste of time. Contact people at uni and ask them if your idea of getting B.S in Engineering first is good. They probably have roadmaps to all degrees, lists of courses with topics, descriptions, etc. It would be even better if you manage to talk to a professor, say, lecturing GR. As to what a physicist needs to REALLY know, pay attention on what Munin says. Munin has offered giving you a detailed advice on textbooks, do not neglect this offer. And compare content (not just ToC) of the textbooks he would mention with the uni curriculum.

If you choose to become an engineer, then your interest to physics will remain a hobby. Normally hobbyists are satisfied with popular texts (videos) on the subject ( ... but they rarely ask questions about Ricci tensor ...). In this case, I am sure, Munin would be able to suggest some really good reading too (and he has already mentioned a few titles, pay attention).

You have an important decision to make. Think hard and good luck!

(Оффтоп)

Munin,
1. Насколько я знаком с западным образованием, они часто (всегда?) дают первокурсникам одни и те же по содержанию курсы, независимо от специальности; например, ВСЕМ студентам читают один и тот же Calculus. Поэтому "idea of getting B.S in Engineering first", возможно, не так плоха, как нам может казаться. Но и вряд ли оптимальна.
2. Этот парень, если внемлет Вашим советам, книги будет, скорее всего, покупать ... . Западные люди часто принимают во внимание только тот совет, который имеет форму: "If you have to buy just one book then ..."

[Munin]

casualvisitor в сообщении #599465 писал(а):

Well, ... it is not. Still, studying your 'three lines short list' would take years of dedicated work (and THOUSANDS of pages of serious and slow reading).

I would not make such a gloomy picture of it. Of course, it is vast and takes much seriousness, but in the same time it is joy for someone who likes physics and maths. The key to walking down the path to the end is to take fun of every step on it. Those steps have to be taken not just stages of something big, but fun by themselves, giving new objects to play with, new ideas to think of, new light and perspective of the world and familiar things, new answers and new riddles.

Fortunately I was led to the end by several questions shaped in my youth, 'what is a black hole' and 'what are quarks', though sometimes I did not see the road towards them from where I was, and I stopped for a while. This lack of roadmaps was one of my biggest obstacles, and so I don't hesitate to give a full scheme to anyone who shows even a slight interest. Maybe that's not the best thing I can do, telling a person how long is the way he has to go, maybe that sounds scary, and he wants to lose heart seeing it. But I promise, that this road not in the least goes down the dry desert! It is strewn with excellent treasures and it goes through beautiful gardens with wonderful fruits. You will never get tired of walking and never regret you started this road. And the distant goal won't float away like Fata Morgana.

casualvisitor в сообщении #599465 писал(а):

Поэтому "idea of getting B.S in Engineering first", возможно, не так плоха, как нам может казаться. Но и вряд ли оптимальна.

А я её и не критиковал, кажется. It's always the person's choice what to do with his life, and if he feels B. S. in Engineering fits him best - let it be that. It's always OK to keep such interests as GR on the level of hobbies. And it's harder to find a place in life for a scientist that for an engineer.

casualvisitor в сообщении #599465 писал(а):

2. Этот парень, если внемлет Вашим советам, книги будет, скорее всего, покупать ... . Западные люди часто принимают во внимание только тот совет, который имеет форму: "If you have to buy just one book then ..."

I'm not sure that's the worst thing to do. In the West, paper books are much easier to find, and cheaper, so gathering one's own collection would possibly be a good idea. Nevertheless, one should always consider other ways, like reading textbooks in a library or in a bookstore (some give such an option), making copies of the most important pages. Here in Russia it is OK to download tons of books for free into one's own PC, ignoring the rights of their owners, but I just don't know if it's OK for students in US, if this is widespread, if this is gravely unethical or only slightly, and if a teacher could advise such a thing to a student.

Though of course I won't keep secret a link as http://bib.tiera.ru/ where all referenced books can be found, at least in Russian.

For just one book, of course, it's not possible to learn from. Even if consider the all-in-one compendium 'The Road to Reality', I don't think it is enough. You always have to read many books, you have to keep them at hand, you have always to check with them, because they give the same things from different aspects, and you never know which one will confuse and which one will help you. Be prepared to that. Also it's helpful to ask teachers, to take lectures, to watch video lecture courses on YouTube, which are plenty, and so on. Sometimes even online tutorials on the level of Wikipedia can be good. Let your sources be many.

[GRstudy]

To all above posters:

I fully understand that B.S in Engineering is not going to be a good idea if I would eventually choose to get PhD in physics. However, tuition and expenses of getting a 4 year degree in the US are extremely high for a mediocre income family in Post-Soviet country where I live. There weren't scholarships available for Science majors so I choose Engineering because it is the closest that I can get to Science. And by the way, I would love to be scientist, but 30k salary is not something that I expect from life.

I absolutely love doing Engineering because you can see physics applied in real life situations. Furthermore, as it was above noted, that all Science and Engineering majors have almost the same first 2 years of educations. Eventually, if I will turn out to be such a big enthusiast of physics, I can Minor in Physics. But the problem is that there are many topics that I don't really enjoy in Physics so I guess I will stay only as a Physics amateur. However, engineering and physics don't go far apart.

I appreciate all the suggestions that were made by above posters.

Also, Ricci tensor is not something that only physicist can be interested in. Yes, it is a rare topic; Yes, it is mostly used by "high-profile" cosmologists; but it doesn't mean that a physics enthusiast like me can't be fascinated by it. On the contrary, I think that Einstein Field Equations, which contain Ricci tensors, are a perfect way to appreciate an amazing beauty of higher math. And what is more marvelous, is that no one can explain in clear way what it is exactly.

As for resources that are available, I would like to point out Walter Lewin's Lectures on Physics (8.01, 8.02, 8.03). They are all available on YouTube.

KhanAcademy is also a great way of learning Calculus, DE, and LA.

P.S И так просто для информации, никакой я не "западный". Я учился на русском языке. Просто я отвык от русской клавиатуры на компе--that's all.

[Munin]

GRstudy в сообщении #599803 писал(а):

But the problem is that there are many topics that I don't really enjoy in Physics

I've experienced many cases when some topic seemed to be boring and unintelligible from distance, but proved to be interesting after a close acquaintance. So let you take this state of affairs as transient, and not think it cannot change later.

GRstudy в сообщении #599803 писал(а):

Also, Ricci tensor is not something that only physicist can be interested in.

Differential geometry is an independent area, a branch of mathematics, and it is very fascinating by itself. As usual, physics finds a use for only a small part of the mathematical theory, and there are many ideas that can be studied without any relation to GR.

GRstudy в сообщении #599803 писал(а):

And what is more marvelous, is that no one can explain in clear way what it is exactly.

Oh no, don't perceive the physics that way! It is only a matter of fund of knowledge needed to comprehend it, and no vagueness will remain when you are ready to learn it. It is only a statement that one thing is equal to another, and those things are just bunches of numbers, with exact receipts how to calculate them. What these bunches are, why those numbers are taken, and not some others, is also explained at some point, and if you don't like the explanations, you are free to try and build your own theory of gravitation, though many restrictions on such a theory have to be studied as well.

GRstudy в сообщении #599803 писал(а):

As for resources that are available, I would like to point out Walter Lewin's Lectures on Physics (8.01, 8.02, 8.03). They are all available on YouTube.

Great, I see MIT has a lot of courses there.
Also, I heard, Stanford's courses are available, Susskind lectures being one of them.

But those particular courses (8.01, 8.02, 8.03) seem to me insufficient in one important way. In theoretical physics, it is very important to know and understand the Lagrangian and Hamiltonian mechanics. They are not just a weird look at the mechanics, they are the language in which most other parts of theoretical physics is narrated, and possibly the language in which the Nature itself is written. The word 'mechanics' should not confuse you, because the very same formulas and principles are used for many other topics, almost all topics: electomagnetism, optics, waves in media, atoms, subatomic particles and forces, gravitation. Only heat and statistical physics stay aside.

And those versions of mechanics are very rarely taught to the wide audience. They are not needed for engineers, they are not needed for many non-theoretical physicists. Feynman's lectures don't teach them (though mention them). Many other courses don't too. So a special attention has to be paid to not get deprived of this subject. Maybe, the best way is to read Landau and Lifshitz's 'Mechanics' in parralel with those introductory courses. I don't know its counterpart amid Western textbooks.

[GRstudy]

I actually watched all General Relativity lessons (total 12 lessons) by Susskind. He does a great job generally; however, his lectures don't have a single example of concepts that he explains.

Also, in Poisson's equation, if density is constant, then what's phi as a function of position?