Spherically symmetric space is conformally flat

[Q-reeus]

You already found some web pages showing that conformally flat 3D space implies zero cotton page,...

If you are referring to the Wiki article https://en.wikipedia.org/wiki/Conformally_flat_manifold (post #4), yes it specifies that. What it doesn't say there is what you think it implies - that *any* spherically symmetric space is conformally flat.

...and in the first message I have shown that spherical symmetry implies conformal flatness.

That is obviously how you interpret it, but I stick by earlier responses in posts #2, #4, #6, #7 (grrrr... can we not have post numbering incorporated here!?)
Look, you are claiming SM has 3D conformal flatness = zero Cotton tensor. We agree I hope that zero Cotton tensor for 3D case is synonymous with locally Euclidean. Well here's your challenge. In part 3 of article, I gave two specific examples of spatially non-Euclidean character of SM. See if you can refute my more easily determined 2nd example - that the proper measure of differential radial gap between closely spaced concentric great circles (centered about a central point mass) is greater by factor than for matter-free Euclidean case. It has to be so.

There are many pages about Cotton tensor and conformal flatness but not so many about conformal flatness of spherically symmetric space.

And I will guarantee you will never find a page in regular literature equating conformal flatness to *any* spherically symmetric space.

[Q-reeus]

Just to be clear that where I twice wrote "*any* spherically symmetric space" in above post, the meaning was in the general sense i.e. *every*. Apart from the one unambiguous case of spherically symmetric space with zero conformal flatness - empty i.e. matter-free, Yilmaz gravity equivalent to SM is as per article locally Euclidean thus conformally flat.

[Mikko]

There are many pages about Cotton tensor and conformal flatness but not so many about conformal flatness of spherically symmetric space.

And I will guarantee you will never find a page in regular literature equating conformal flatness to *any* spherically symmetric space.

Some pages mention the result without proof or reference.

[Q-reeus]

There are many pages about Cotton tensor and conformal flatness but not so many about conformal flatness of spherically symmetric space.

And I will guarantee you will never find a page in regular literature equating conformal flatness to *any* spherically symmetric space.

Some pages mention the result without proof or reference.

Umm - references please, preferably to freely accessible online material. Or a direct cut-and-paste quote, in full context, from mainstream textbook. I extremely doubt the eventuality for reasons already given - SM is for sure not only 4D conformally non-flat = non-zero Weyl curvature, but spatial part also conformally non-flat - non-zero Cotton curvature. I gave you the physical example demonstrating that to be so.

It's like this: On an ideal spherical non-rotating planet, a being climbs a radial pointing step-ladder X proper meters, where X << R the planet's surface radius (strictly speaking, in GR, R is 'just a parameter' defined by C = 2*pi*R). At that proper elevation X there exists a railing of constant nominal radius R+X - i.e. a great circle. What is the circumference C' there, as determined by the being measuring proper distance along the railing? If you get as first-order in metric approximation, start again. SM is locally spatially non-Euclidean. Correct answer is


And just as not one of the assortment of fools and knaves (not all so categorized) I endured in that linked ATM thread ever took up my challenge to refute my two main findings - disappearing functional dependence on gravitational potential for just one metric component, or conflicting predictions for internal spatial metric, so it it has been here (not that I'm implying folks here are all fools and/or knaves! ).

[Mikko]

And I will guarantee you will never find a page in regular literature equating conformal flatness to *any* spherically symmetric space.

Some pages mention the result without proof or reference.

Umm - references please, preferably to freely accessible online material.

Some typical examples are:
http://arxiv.org/abs/1405.1682
http://www.phy.olemiss.edu/~luca/Topics ... om_3D.html
http://homepage.univie.ac.at/piotr.chru ... Energy.pdf (page 5)
All these mention it as if it were in common textbooks but I don't know any such textbook.

[Q-reeus]

Some typical examples are:
http://arxiv.org/abs/1405.1682
http://www.phy.olemiss.edu/~luca/Topics ... om_3D.html
http://homepage.univie.ac.at/piotr.chru ... Energy.pdf (page 5)
All these mention it as if it were in common textbooks but I don't know any such textbook.

Will have to concede these GR experts are indeed agreeing with your thread title. The last one, whether or not you used it directly as basis for post #1, seems to employ the same reasoning - it's always possible to transform a spherically symmetric metric into an isotropic form. Hmm....

This begs the question - what does 3D conformal flatness, in particular relating to exterior SM or alternative, actually physically mean? My understanding is that it implies a differential excursion from any given radial value yields the first proper Euclidean relation for C' I gave last post. But the actual relation is given by my second expression. A manifestly locally non-Euclidean first order in metric relation. What relation for C' do you find in the scenario I gave? How do you explain it?

[Mikko]

This begs the question - what does 3D conformal flatness, in particular relating to exterior SM or alternative, actually physically mean?

Probably not very much. There are no exactly spherical non-rotating bodies big enough to have significant gravity. In less symmetric situation it is not clear how one should choose the time coordinate. The spactime curvature does not depend on choice of coordinates so it can be considered more important. Spacetime is conformally flat only in some special situations.

[Q-reeus]

This begs the question - what does 3D conformal flatness, in particular relating to exterior SM or alternative, actually physically mean?

Probably not very much. There are no exactly spherical non-rotating bodies big enough to have significant gravity. In less symmetric situation it is not clear how one should choose the time coordinate. The spactime curvature does not depend on choice of coordinates so it can be considered more important. Spacetime is conformally flat only in some special situations.

Which all deftly avoids my last point - SM deviates from locally 3D flatness by a first order in metric term. Asking you again - what value do you obtain for C' as per great circle railing scenario? It should be what I gave in second expression. How do you reconcile it with spatial conformal flatness?