A Einstein Field Equations: Spherical Symmetry Solution

[PAllen]

[Moderator's Note: Thread spin off due to topic and level change.]

For a spherically symmetric solution, if SET components were written in terms a single one of 4 coordinates, in a way plausible for a radial coordinate, the I believe solving the EFE would require spherical symmetry of the metric up to possible implausible noise terms (producing some Weyl curvature).

 

[PeterDonis]

if SET components were written in terms a single one of 4 coordinates, in a way plausible for a radial coordinate, then I believe solving the EFE would require spherical symmetry of the metric

You couldn't tell if it were truly a radial coordinate without knowing the metric. Plenty of functions of one variable might look "plausible for a radial coordinate" without actually requiring spherical symmetry (one obvious alternative is just axial symmetry, i.e., cylindrical instead of spherical).

up to possible implausible noise terms (producing some Weyl curvature).

You seem to be implying that perfect spherical symmetry rules out Weyl curvature. That's obviously false since Schwarzschild spacetime has nonzero Weyl curvature.

 

[PAllen]

You couldn't tell if it were truly a radial coordinate without knowing the metric. Plenty of functions of one variable might look "plausible for a radial coordinate" without actually requiring spherical symmetry (one obvious alternative is just axial symmetry, i.e., cylindrical instead of spherical).

That could be ruled out by coordinate conditions, not metric conditions per se. Though, I see, that they would only have effect when resolving the metric.

You seem to be implying that perfect spherical symmetry rules out Weyl curvature. That's obviously false since Schwarzschild spacetime has nonzero Weyl curvature.

I guess I wasn’t clear. What I meant was that a metric that produces asymmetric Weyl curvature while producing spherically symmetric Ricci curvature would be allowed by the EFE. My suspicion is that this would happen due extra metric terms that look artificial and could be readily removed.

[edit: I wonder if imposing spherical symmetry via coordinate definition on both the Weyl tensor and the Ricci tensor (or Einstein tensor), would force spherical symmetry on the metric]

 

[PAllen]

It occurs to me, on review of material in Synge, 1960, that on the assumption of spherical symmetry, one can specify two arbitrary functions for just two components of the Einstein tensor (rr and tt), and all other components of the Einstein tensor and the metric follow. Thus, if you simply assume spherical symmetry for the spacetime, you can just pick two functions for two components of the SET, and derive everything else. They can even be time dependent functions.

 

[PeterDonis]

if you simply assume spherical symmetry for the spacetime, you can just pick two functions for two components of the SET, and derive everything else.

Not in the general case, no. The most general spherically symmetric Einstein tensor (and hence SET) has three independent diagonal components, not two; the tangential components have to be equal, but they do not have to be the same as the radial one. In order to get the number of independent diagonal components down to two, you need to assume something like a perfect fluid (isotropy of pressure).

Also, as has already been noted, just specifying these three independent functions for the three independent diagonal components of the SET does not specify a unique solution. I believe the minimum number of independent functions required for a unique solution is five: two metric functions (how exactly they appear in the metric can vary, but a common way of specifying them is to specify $g_{tt}$ and the "mass function" $m$, which makes $g_{rr}$ take the form $1 / (1 - 2m / r)$ in Schwarzschild coordinates) and the three SET functions described above (which can be thought of as energy density, radial pressure, and tangential pressure). These functions are related by three differential equations, arising from the Einstein Field Equation, but those equations do not, I believe, reduce the freedom to specify functions any further in the general case (i.e., nothing else specified by spherical symmetry, no specification of perfect fluid, etc.).

I believe MTW has a discussion of this but I can't find it at present.

 

[PAllen]

Not in the general case, no. The most general spherically symmetric Einstein tensor (and hence SET) has three independent diagonal components, not two; the tangential components have to be equal, but they do not have to be the same as the radial one. In order to get the number of independent diagonal components down to two, you need to assume something like a perfect fluid (isotropy of pressure).

Also, as has already been noted, just specifying these three independent functions for the three independent diagonal components of the SET does not specify a unique solution. I believe the minimum number of independent functions required for a unique solution is five: two metric functions (how exactly they appear in the metric can vary, but a common way of specifying them is to specify $g_{tt}$ and the "mass function" $m$, which makes $g_{rr}$ take the form $1 / (1 - 2m / r)$ in Schwarzschild coordinates) and the three SET functions described above (which can be thought of as energy density, radial pressure, and tangential pressure). These functions are related by three differential equations, arising from the Einstein Field Equation, but those equations do not, I believe, reduce the freedom to specify functions any further in the general case (i.e., nothing else specified by spherical symmetry, no specification of perfect fluid, etc.).

I believe MTW has a discussion of this but I can't find it at present.

You can eliminate one of the three metric functions by assuming a specific type of coordinate system. This does not reduce the generality of physics representable. Synge then derives explicit equations relating the two remaining metric functions to just two components of the Einstein tensor. Either can be determined from the other. He also derives the rest of the Einstein tensor from the two chosen arbitrarily. All of these functions may allow time as well as radial coordinate. This is one area where I find MTW seriously deficient. Not only does it discuss spherical symmetry with less completeness than Synge, it has no general discussion of axial symmetry at all. Wald, on the other hand, has the same material on axial symmetry as Synge, expressed in more modern formalism.

 

[PeterDonis]

You can eliminate one of the three metric functions by assuming a specific type of coordinate system.

Yes, I know, that's why I said you only have two independent functions in the metric. You have three in the SET (actually possibly four, see below on the nonzero off diagonal term).

Synge then derives explicit equations relating the two remaining metric functions to just two components of the Einstein tensor.

I'm not familiar with Synge's derivation, but as I've already said, I'm not sure how there could be only two independent components of the Einstein tensor, since tangential and radial pressures do not have to be the same.

Also, now I come to think of it, if $dm / dt$ is nonzero, I'm not even sure the Einstein tensor has to be diagonal. The basic metric ansatz would be something like:

$$
ds^2 = - J(r, t) dt^2 + \frac{dr^2}{1 - \frac{2 m(r, t)}{r}} + r^2 d\Omega^2
$$

Plugging this into Maxima gives a nonzero off diagonal component $G_{tr} = G_{rt}$ for the Einstein tensor.

 

[PAllen]

Yes, I know, that's why I said you only have two independent functions in the metric. You have three in the SET.I'm not familiar with Synge's derivation, but as I've already said, I'm not sure how there could be only two independent components of the Einstein tensor, since tangential and radial pressures do not have to be the same.

Also, now I come to think of it, if $dm / dt$ is nonzero, I'm not even sure the Einstein tensor has to be diagonal. The basic metric ansatz would be something like:

$$
ds^2 = - J(r, t) dt^2 + \frac{dr^2}{1 - \frac{2 m(r, t)}{r}} + r^2 d\Omega^2
$$

Plugging this into Maxima gives a nonzero off diagonal component $G_{tr} = G_{rt}$ for the Einstein tensor.

The key is that these components are not independent. Synge derives formulas for the off diagonal components of the Einstein tensor in terms of just two of the diagonal components. There is also a possible theta,theta and phi,phi component of the Einstein tensor( these are equal to each other) that is derivable from the rr and tt components. So it remains true that you can choose just these two SET components, assume spherical symmetry, and derive everything else.

I thought, at some point, you had acquired a copy of Synge. The relevant pages are 270 to 274. I would be loathe to present 4 pages of terse derivation here.

[edit: you really do want to read from page 270, even though the main derivation is on 273, because he introduces a very odd notation for partial derivatives, used just in this section; the motivation is to save some ink]

 

[PeterDonis]

I thought, at some point, you had acquired a copy of Synge.

I think I have one somewhere, but it's not handy at the moment. If I can find it I'll take a look.

 

[PeterDonis]

Synge derives formulas for the off diagonal components of the Einstein tensor in terms of just two of the diagonal components. There is also a possible theta,theta and phi,phi component of the Einstein tensor( these are equal to each other) that is derivable from the rr and tt components.

Is Synge using the covariant divergence equations? I had not taken those into account in my earlier posts, but yes, those are additional constraints that can reduce the number of independent functions.

 

[PAllen]

Is Synge using the covariant divergence equations? I had not taken those into account in my earlier posts, but yes, those are additional constraints that can reduce the number of independent functions.

Yes, he uses the covariant divergence equations to get the tangential components in term of all the others (that is theta,theta in terms of rr, tt, and rt). Before this he uses clever integrations to invert rr and tt Einstein tensor components in terms of metric functions (that is, getting the two metric functions in terms of these Einstein components). Then these inversions are used to get the off diagonal Einstein components in terms the diagonal components. Altogether very slick IMO.