Риччи Тензор

[GRstudy]

OK. In which textbook it is discussed?

[Munin]

Misner Thorne Wheeler 'Gravitation'.

Ah sorry! I referred to it as MTU, which is wrong (in Russian Wheeler is written Uiler). Wherever you see MTU, read that as MTW.

[GRstudy]

I found it. What's the page?

-- 02.08.2012, 01:18 --

I looked box 23.2 (every page in that area). I didn't find the clear version of listing of Einstein Tensor's non-zero components. Where is it? I want to find out what they are really bad...

[Munin]

P. 609.

-- 02.08.2012 01:37:21 --

$G_{\hat{0}\hat{0}}$ and $G_{\hat{r}\hat{r}}$ are on pp. 602, 604. Though notice their indexes: they are given in normalised basis, and not in the coordinate basis. That is marked by hats over indexes.

[GRstudy]

Why bases aren't normal? And also, I didn't understand where did exponential growth, e, came from: it is not in metric you gave me. I would like to see components of Einstein Tensor for metric tensor that you gave me.

[Munin]

GRstudy в сообщении #602179 писал(а):

Why bases aren't normal?

Because they are chosen by other criteria - to easen the solving of PDE. That is called separation of variables, and it is a method powerful enough to solve the Einstein's equation for star and for Universe, to solve the Schrodinger's equation for hydrogen atom, to solve Poisson's equation for spherical, linear and planar charges, and so on.

GRstudy в сообщении #602179 писал(а):

And also, I didn't understand where did exponential growth, e, came from: it is not in metric you gave me.

$e$ does not mean exponential growth, it is only a variable substitution: first solving starts with some general metric coefficients

$$ds^2=-g_{tt}\,dt^2+g_{rr}\,dr^2+g_{\theta\theta}\,d\Omega^2,$$

and then these coefficients are renamed as

$$ds^2=-e^{2\Phi}\,dt^2+e^{2\Lambda}\,dr^2+R^2\,d\Omega^2.$$

Then it is said that we have to find not the initial functions $g_{tt},g_{rr},g_{\theta\theta},$ but new functions $\Phi,\Lambda,R$ - that's all. They are chosen as such to make the following calculations and solvings easier, but you won't see that because you don't solve anything. You can just seek for the outcomes: (23.18) with (23.19) and Box 23.2 (2), for example, give

$$\begin{array}{lll}\displaystyle e^{2\Lambda}=\dfrac{1}{1-\tfrac{8\pi}{3}\rho r^2}&\qquad&\text{inside the ball,}\\ \mathstrut&&\\ \displaystyle e^{2\Lambda}=\dfrac{1}{1-\dfrac{\tfrac{8\pi}{3}\rho R^3}{r}}=\dfrac{1}{1-\dfrac{2M}{r}}&\qquad&\text{outside the ball,}\end{array}$$

and (23.26) and Box 23.2 (2) give you $e^\Phi,$ and (23.6) and (23.7) give you $R.$

So, there's nothing obscure in here, you just have to substitute some expressions into others, following the redefinitions in the text. The Real Thing is not in these substitutions, which are mere tools, it is done in chosing the conditions, solving the equations and taking integrals. Those things you prefer to skip, while they are the most important. I cannot compliment your approach.

[GRstudy]

Yeah, I found their solution. I wasn't happy at all They only gave 2 components of the Einstein Tensor; moreover, they assumed c=G=1 which made it less interesting; in addition, I suspect that the solution is for some kind of fluid which makes it even less entertaining. They got too many substitutions and I didn't get what I was expecting to get.

Actually, Einstein Tensor should have many other components. As for Stress-Energy Tensor, the r-r component of it is "p". Actually, r-r component is flux of momentum in x direction.

Are there any other (clear!) examples of Einstein Tensor having non-zero components and Stress-Energy being written out in its full grace? They have 16 components! They solved only 2

[Munin]

GRstudy в сообщении #602525 писал(а):

They only gave 2 components of the Einstein Tensor;

Most other components are zero, and the pressure is the same in all directions.

GRstudy в сообщении #602525 писал(а):

moreover, they assumed c=G=1 which made it less interesting;

:-( That is the 'natural system of units', and it is more interesting this way, making the mathematical structure more clear and essential things more evident.

GRstudy в сообщении #602525 писал(а):

in addition, I suspect that the solution is for some kind of fluid which makes it even less entertaining.

In GR you cannot work without any postulation about the equation of state for the matter. You have to chose either dust or liquid or gas, and the equation of state of liquid is most adequate for such bodies as planets and stars (gas is something else in relativistic sense). That is because in GR not only mass gravitates, but also pressure and tension, and that is caused by the true relativistic union of space and time (pressure and tension are spatial counterparts for energy density).

It disappoints that the most beautiful points and ideas of the elegant theory entertain you the least.

GRstudy в сообщении #602525 писал(а):

They got too many substitutions and I didn't get what I was expecting to get.

That is just trivial and transparent for more experienced readers. Substitutions let you write things compactly, and highlight most important features, or else they are just insignificant.

GRstudy в сообщении #602525 писал(а):

Actually, Einstein Tensor should have many other components.

Only non-zero components are written.

GRstudy в сообщении #602525 писал(а):

As for Stress-Energy Tensor, the r-r component of it is "p". Actually, r-r component is flux of momentum in x direction.

Pressure IS the flux of momentum, BY NATURE.

GRstudy в сообщении #602525 писал(а):

Are there any other (clear!) examples of Einstein Tensor having non-zero components and Stress-Energy being written out in its full grace?

What you call grace I would call untidiness, because the symmetrical tensor (like ET or SET) can always be reduced to its principal axes, where it would have only diagonal elements. And they are energy density and pressure, and the pressure can be same on all directions for isotropic media.

GRstudy в сообщении #602525 писал(а):

They have 16 components! They solved only 2

They didn't 'solve' anything, you again are confusing the solving and the posing of conditions. And posing just two components is enough because that lets solve the entire equation system. By the way, other components are not left unposed, they are just set by symmetry conditions, as zeros or same as given components.