Basic matrix representation question

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Discussion Overview

The discussion revolves around the matrix representation of the Schwarzschild metric in spherical polar coordinates, particularly focusing on the implications of the absence of infinitesimal cross terms and the appropriate labeling of matrix components. Participants also inquire about the metric inside a rigid gravitating body and the Oppenheimer-Volkoff equation.

Discussion Character

  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant questions whether the absence of infinitesimal cross terms in the Schwarzschild metric implies that the non-diagonal entries in the matrix representation are zero.
  • Another participant confirms that it is correct to take the coefficients of dr², dθ², etc., from the Schwarzschild spacetime interval and use them as diagonal components of the metric tensor in spherical coordinates.
  • A participant expresses a desire for references regarding the metric inside a rigid gravitating body, suggesting that the Schwarzschild metric is only valid for r > R.
  • Several references are provided for further reading on the interior solutions of the Schwarzschild metric and related topics.
  • Concerns are raised about the Oppenheimer-Volkoff equation being based on an incorrect Einstein tensor, leading to potential inaccuracies in solutions derived from it.
  • Participants discuss the implications of using different forms of the Einstein tensor and the relationship between pressure and density in the context of hydrostatic equilibrium.

Areas of Agreement / Disagreement

There is some agreement on the interpretation of the Schwarzschild metric in spherical coordinates, but there are also disagreements regarding the validity of the Oppenheimer-Volkoff equation and its implications for solutions in general relativity. The discussion remains unresolved on the correctness of certain references and equations.

Contextual Notes

Participants note the limitations of the Schwarzschild metric's applicability and the potential errors in the Oppenheimer-Volkoff equation based on differing interpretations of the Einstein tensor.

Who May Find This Useful

This discussion may be useful for students and researchers interested in general relativity, particularly those exploring the Schwarzschild metric, interior solutions of gravitating bodies, and the mathematical foundations of general relativity.

FunkyDwarf
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Hi guys,

Just brushing up on my GR for a project and i have a silly question:

For the spherical polar representation of the schwarzschild metric, the fact that there are no infintesimal-cross terms implies that the non-diagonal entries in the matrix representation are zero, correct? I guess what I am asking is i know what the matrix looks like in cartesians for minkowski space, but obviously it doesn't look the same in spherical polars. Thus, can i simply take the coefficients of dr^2, dtheta^2 etc from the schwarzschild spacetime interval and plug them in as the diagonal components of g-mu,nu and label the columns/rows as spherical coordinates rather than cartesians?

Hope that made sense :S

Cheers
-G

EDIT: Also, can somebody provide a reference (i had a go at looking but couldn't find anything substantial) to show the metric inside a rigid gravitating body (Shwarz metric only valid r>R of course) or do you have to solve for it directly from the Einstein field equations? Surely this has been done before? Metric inside a static, constant density body?
 
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Brilliant! Cheers. I hate reading stuff on google books so i went and grabbed it out the library, seems a very well laid out book in relativity in general, very useful!
 
FunkyDwarf said:
Hi guys,

Just brushing up on my GR for a project and i have a silly question:

For the spherical polar representation of the schwarzschild metric, the fact that there are no infintesimal-cross terms implies that the non-diagonal entries in the matrix representation are zero, correct? I guess what I am asking is i know what the matrix looks like in cartesians for minkowski space, but obviously it doesn't look the same in spherical polars. Thus, can i simply take the coefficients of dr^2, dtheta^2 etc from the schwarzschild spacetime interval and plug them in as the diagonal components of g-mu,nu and label the columns/rows as spherical coordinates rather than cartesians?

Hope that made sense :S

Cheers
-G

EDIT: Also, can somebody provide a reference (i had a go at looking but couldn't find anything substantial) to show the metric inside a rigid gravitating body (Shwarz metric only valid r>R of course) or do you have to solve for it directly from the Einstein field equations? Surely this has been done before? Metric inside a static, constant density body?

Can you clarify the first question and say what exactly you want? I can guess one senario: You might be looking for the Cartesian form of Schwarzschild metric which is presented in polar coordinates as

ds^2=(1-2m/r)dt^2-(1-2m/r)^{-1}dr^2-r^2({d\theta}^2+\sin^2(\theta)d{\phi}^2).

So if you want to transfer this setup into the Cartesian coordinates by just labeling the rows and columns of the metric tensor of the SM as t,x,y,z, I must say it is incorrect! If this is what you mean, then I would go into the details. But if not, then adding a little bit more clear insight into the matter would be helpful for us to realize what the problem is!

For the second question, see

1- RELATIYITY THERMODYNAMICS AND COSMOLOGY by RICHARD C. TOLMAN, pp 245-247 (discussion of interior solution of Schwarzschild metric).

2- Interior Solutions for Reissner-Nordstrom Field by N.O. Santos, Progress of Theoretical Physics, 1980, Vol. 64, No. 6, pp. 2021-2028. Find this article http://adsabs.harvard.edu/abs/1980PThPh..64.2021S" (discussion of interior solution of Reissner-Nordstrom metric in which the author obtains some interior solutions for a charged perfect fluid sphere for which |q|>m. The solutions of Reissner-Nordstrom field are discussed for the case |q|<m in almost every textbook about GR.)

3- EXACT SOLUTIONS AND SCALAR FIELDS IN GRAVITY RECENT DEVELOPMENTS by ALFREDO MACIAS, JORGE L. CERVANTES-COTA and CLAUS LÄMMERZAHL.

4- GENERAL RELATIVITY by Robert M. Wald, pp 125-135.

AB
 
Last edited by a moderator:
Oppenheimer-Volkoff equation...


I have a theoretical problem with reference 2:
A first course in general relativity By Bernard F. Schutz

The interior structure of the star: (page 258 - eq. 10.39)

Oppenheimer-Volkoff equation (O-V):
\frac{dP}{dr} = - \frac{(\rho + P)(m + 4 \pi r^3 P)}{r(r - 2m)}

According to reference 1, the Oppenheimer-Volkoff equation is based upon an incorrect Einstein tensor.

Most of the equation solutions based upon the (O-V) equation listed on pg. 262 are probably also incorrect.
[/Color]
Reference:
https://www.physicsforums.com/showthread.php?t=372432"
http://books.google.com/books?id=qhDFuWbLlgQC&lpg=PP1&dq=schutz&pg=PA258#v=onepage&q=&f=false"
 
Last edited by a moderator:


Orion1 said:
I have a theoretical problem with reference 2:
A first course in general relativity By Bernard F. Schutz

The interior structure of the star: (page 258 - eq. 10.39)

Oppenheimer-Volkoff equation (O-V):
\frac{dP}{dr} = - \frac{(\rho + P)(m + 4 \pi r^3 P)}{r(r - 2m)}

According to reference 1, the Oppenheimer-Volkoff equation is based upon an incorrect Einstein tensor.

Most of the equation solutions based upon the (O-V) equation listed on pg. 262 are probably also incorrect.
[/Color]
Reference:
https://www.physicsforums.com/showthread.php?t=372432"
http://books.google.com/books?id=qhDFuWbLlgQC&lpg=PP1&dq=schutz&pg=PA258#v=onepage&q=&f=false"

Thanks goodness, Schutz recognizes the correct components of Einstein tensor, as is clear from the Eqs. (10.14) to (10.17). But guess what and think how he gets the equation (10.30) from the same corrected things we have https://www.physicsforums.com/showpost.php?p=2549441&postcount=6"?!

Our {\nu}&#039; based on the correct G_{11} is

{\nu}&#039; = \frac{1}{r} \left( - \frac{8 \pi G T_{11} r^2}{c^4} + e^{\lambda} - 1 \right).

While his is the same as ours calculated according to the correct G_{11} if and only if

T_{11}=P(r)e^{\lambda}. (1)

Now let's denote the wrong G_{11} by G^W_{11} and Schutz's T_{11} by T^S_{11} thus keeping our own information as before. If again we set the Einstein equation for G_{11}, using (1) we would shockingly lead to

8\pi GT_{11}/c^4= G^W_{11}, (2)

where T_{11}= P(r). The whole thing is now clear: If we suppose the original OV equation that can be obtained from (2) is correct, then the differential equation of state for hydrostatic equilibrium wouldn't have a form like

\frac{dP(r)}{dr} = - \frac{(T_{11} + T_{00})}{2}{\nu}&#039;. (3)

Rather it must be of the form

\frac{dP(r)}{dr} = - \frac{(\rho (r) + P(r))}{2}{\nu}&#039;.

You can see that due to this reason, Schutz does not use (3) because he is smart enough to not make a colossal mathematical mistake. But I have to say that this kind of tricky way of salvaging OV equation doesn't seem rational but it is definitely true because T^S_{11} is the hydrostatic pressure around the Schwartzchild field while P(r) in the equation of state for hydrostatic equilibrium is the pressure of fluid itself!

AB
 
Last edited by a moderator:
So if you want to transfer this setup into the Cartesian coordinates by just labeling the rows and columns of the metric tensor of the SM as t,x,y,z, I must say it is incorrect! If this is what you mean, then I would go into the details. But if not, then adding a little bit more clear insight into the matter would be helpful for us to realize what the problem is!
No, i most definitely want to use polar coordinates so i can make use of spherical symmetry.
 
FunkyDwarf said:
No, i most definitely want to use polar coordinates so i can make use of spherical symmetry.

I think I got you now! So

Thus, can i simply take the coefficients of dr^2, dtheta^2 etc from the schwarzschild spacetime interval and plug them in as the diagonal components of g-mu,nu and label the columns/rows as spherical coordinates rather than cartesians?

Yes.

AB
 
  • #10
FunkyDwarf said:
Awesome, thanks! Next question :D

Can you recommend any books that address the klein gordon equation in the presence of a curved metric as seen here:

http://en.wikipedia.org/wiki/Klein–Gordon_equation#Gravitational_interaction

cheers
-G

That is a simple calculation which can be done here! I've not found any books giving the details of it because it is not of any interest! But If you are eager to know how they get it, I'm ready to show it in an exclusive thread, not in here!

AB
 
  • #11
The calculation to include a curved metric into a quantum mechanical equation is simple? Yes, please show me :)
 

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