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This will be the first of several articles which will provide, for reference, useful equations for static, spherically symmeetric spacetimes. This is a common special case that is studied in General Relativity, and it has the advantage of having a general solution for the Einstein Field equation that can be expressed in closed form equations. This first article will focus on those closed form equations for that solution.
First, we should be clear about the set of spacetimes we are talking about. The two key properties of this set of spacetimes can be defined in coordinate-independent terms as follows:
(1) A static spacetime has a timelike Killing vector field which is hypersurface orthogonal.
(2) A spherically symmetric spacetime has a 3-parameter group of spacelike Killing vector fields that satisfy the properties of the Lie group SO(3), which describes rotations in 3-dimensional space.
As noted, these...
Continue reading...
 
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A question from Reddit
I suspect the question is from a person who does not really have the background knowledge to understand the post. Here is a suggested reply:

The gammas are Christoffel symbols [1]; the three indexes are tensor indexes, not variables. The upper index is not an exponent; it's an upper tensor index. (Strictly speaking the Christoffel symbols are not tensors, but they have tensor indexes and are treated like tensors in formulas.) Like all other quantities in the case under discussion, the Christoffel symbols are functions of only one variable, ##r##, because of the staticity and spherical symmetry.

[1] https://en.wikipedia.org/wiki/Christoffel_symbols
 

Ibix

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Hi Peter. An interesting article - thanks. Are you going to go more into the stress-energy tensor and the constraints on it in this case in later articles?
 
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Are you going to go more into the stress-energy tensor and the constraints on it in this case in later articles?
I wasn't planning to. One further article will give Maxwell's Equations in a static, spherically symmetric spacetime (and how they couple to the EFE via the stress-energy tensor for an EM field), and another will give equations for an object being slowly lowered in a static, spherically symmetric spacetime.

Did you have a particular question about the SET and constraints on it?
 

Ibix

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Did you have a particular question about the SET and constraints on it?
Not really anything specific. The stress-energy tensor seems to me to be something GR texts often seem to assume you understand, somehow. I think I may have missed a lecture somewhere, because it's the bit I feel I have the least grasp of. So I'm always interested in seeing someone talking about it.
 

martinbn

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I bit pedantic but shouldn't the definition of spherically symmetric be phrased differently. The Killing fields generate a Lie algebra isomorphic to ##\mathfrak{so}(3)##, the group of isometries is ##\text{SO}(3)##.
 
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I bit pedantic but shouldn't the definition of spherically symmetric be phrased differently. The Killing fields generate a Lie algebra isomorphic to ##\mathfrak{so}(3)##, the group of isometries is ##\text{SO}(3)##.
I think this might be a bit too much detail; I was trying to avoid going into it by saying the Killing fields "satisfy the properties" of ##\text{SO}(3)## (the Lie algebra would be part of the "properties").
 
There seem to be a few errors in the article. ## \partial_{\theta} ## is NOT a killing vector here. I think this is repeated erroneously a few times. Further, ##J(r)## is defined to be the 'redshift factor'. I believe the correct formulation is that the redshift factor is proportional to ##\sqrt{J(r)}## and not ##J(r)##.

It would also be great if you could illustrate the definition of '##s##' in the TOV equation as this seems to have been introduced without adequate precedent.
 
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##\partial_{\theta}## is NOT a killing vector here.
You're right, I have updated the article to correct this.

I believe the correct formulation is that the redshift factor is proportional to ##\sqrt{J(r)}## and not ##J(r)##.
It is ##\sqrt{J(r)}##, that's correct. (There is no constant of proportionality required.) I've updated the article to fix this as well.

It would also be great if you could illustrate the definition of '##s##' in the TOV equation as this seems to have been introduced without adequate precedent.
I'm not sure what you mean by "inadequate precedent". The TOV equation is derived assuming that the matter present is a perfect fluid; that means all three spatial diagonal components of the SET must be equal. If you drop that assumption, the radial component can be different from the tangential components, but the two tangential components still need to be the same by spherical symmetry. That's why there are just two functions, ##p## and ##s##.
 

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