# Are standard and isotropic forms of Schwarzschild metric truly equivalent?

My admittedly rudimentary understanding of physical meaning of conformal flatness - as pertaining to a stationary observer exterior to a spherically symmetric static gravitating mass $M$:

Locally Euclidean, in that the proper differential volume $dV$ between two concentric spherical surfaces centered about M, having differential proper radial spacing $dR$, is given by the Euclidean formula for enclosed volume $dV$ = $AdR$, where $A$ can be taken as the mean value of the two proper surface areas, and we take the limit as $dR\rightarrow0$. Correct?

And that in GR, SM (standard Schwarzschild metric) is not conformally flat since there $dR$ is greater by factor $\sqrt{-g_{rr}}$, as can be determined by inspection of the line element - e.g.:Wikipedia - archived revision

That is, $dV = \sqrt{-g_{rr}}AdR$ for SM, $> dV$ (Euclidean) - for a given proper areal difference between shells.

It's evident by inspection of the line element for ISM: here, for the equivalent differentially separated concentric shells arrangement as above, $dV = AdR$ asymptotically applies as per Euclidean formula? In other words, by virtue of it's construction as spatially isotropic, ISM necessarily claims a conformally flat metric, in-principle measurably physically distinct from that of SM? How then is it that the two are claimed to be physically equivalent?

Also is there a precise technical term and definition specifying departure from conformal flatness here? While the above concentric spherical shells situation is the one I was introduced to, there is surely no reason preventing it being dimensionally reduced to one of proper radial spacing between concentric great circles, and in fact then further reduced to an arbitrarily small local sector cut from such concentric circles. Meaning it must be an in-principle locally observable quantity rather than only determinable globally?

• A metric is conformally flat iff its Weyl tensor vanishes, so the Weyl tensor is what measures conformal non-flatness. Jun 18, 2014 at 8:51
• OK now how does that relate to the case of proper volume between concentric spherical surfaces as computed for SM vs ISM? Jun 18, 2014 at 9:08
• the spatial component of the schwarzschild metric is conformally flat in the isotropic coordinate system, hence the name. With regards to volume elements, it is just a matter of keeping track of the Jacobian. Jun 18, 2014 at 11:27
• Not up with bit about keeping track of the Jacobian. But given agreement with my point ISM isotropy ≡ conformal flatness, and SM = conformal non-flatness, how can that by definition the two describe physically different metrics be avoided? Hence the long held position ISM is merely a coordinate transformation of SM is wrong surely. Jun 18, 2014 at 12:04