How Do Contradictory Metrics Align with GR Field Equations?

[StateOfTheEqn]

I don't understand what's the significance of that limit
4\pi R^2/4\pi r^2 \rightarrow \infty

You're saying that it has to do with unbounded negative spatial curvature?

Consider the spatial manifold $\mathbb{R}^+\times S^2$. Suppose there are two metric on $\mathbb{R}^+\times S^2$, one Euclidean and the other non-Euclidean. Define $R=Area(S^2)/4\pi$ as the area radius for the non-Euclidean and $r=Area(S^2)/4\pi$ the area radius for the Euclidean. For the Euclidean, $r=\sqrt{x^2+y^2+z^2}$ in Cartesian coordinates centered at the origin. Now, consider how radial lines diverge. The distance between where two radial lines intersect the $\{R\}\times S^2$ sphere is $Rd\theta$ in the non-Euclidean metric and where two radial lines intersect the $\{r\}\times S^2$ sphere is $rd\theta$ in the Euclidean metric. Assume $\{R\}\times S^2$ is the same sphere in $\mathbb{R}^+\times S^2$ as $\{r\}\times S^2$ and $r \neq R$ . We can make the assumption $r \neq R$ because of the different metrics. Otherwise, if the metrics were the same then $r=R$.

If $Rd\theta>rd\theta$ we can say the non-Euclidean space is curved negatively because the radial lines diverge more than their Euclidean counterparts and if $Rd\theta<rd\theta$ we can say the non-Euclidean space is curved positively because the radial lines diverge less.

In the Schwarzschild paper of 1916 he defines $R=(r^3+r_s^3)^{1/3}$ where $r$ is the Euclidean distance from the origin. So, $R>r$ and the space will be negatively curved. The limit mentioned above is the limit in the ratio of areas $4\pi R^2/4\pi r^2=R^2/r^2$ which grows arbitrarily large as $r \rightarrow 0$ and which is a result of the negative curvature growing arbitrarily large near a mass concentrated (theoretically) at a single point.

If you accept that space can be negatively curved in this way by a gravitating body then you can get rid of the irregular sphere at $r_s$ called the event horizon of the Black Hole. Then all space-time around the gravitating body is regular except at $r=0$ which is also where $R=r_s$.

 

[atyy]

A word of caution about taking the 1916 paper as entirely correct http://www.staff.science.uu.nl/~hooft101/lectures/genrel_2010.pdf :

"In his original paper, using a slightly different notation, Karl Schwarzschild replaced (r³-(2M)³)^1/3 by a new coordinate r that vanishes at the horizon, since he insisted that what he saw as a singularity should be at the origin, claiming that only this way the solution becomes "eindeutig" (unique), so that you can calculate phenomena such as the perihelion movement (see Chapter 12) unambiguously. The substitution had to be of this form as he was using the equation that only holds if g = 1 . He did not know that one may choose the coordinates freely, nor that the singularity is not a true singularity at all. This was 1916. The fact that he was the first to get the analytic form, justifies the name Schwarzschild solution."

 

[PeterDonis]

Suppose there are two metric on $\mathbb{R}^+\times S^2$

This is mathematically fine but physically meaningless; physically there can only be one metric. Having two metrics would require the same physical measurements to yield two different results, which is impossible. So your proposal is not relevant to determining the actual physical structure of Schwarzschild spacetime, since the "non-Euclidean" metric is the one we actually physically observe. (For example, for the "Euclidean" metric to be physically relevant, $r$ would have to be the actual physical distance from the origin, but our actual physical measurements say it isn't.)

 

[TrickyDicky]

This is mathematically fine but physically meaningless; physically there can only be one metric. Having two metrics would require the same physical measurements to yield two different results, which is impossible. So your proposal is not relevant to determining the actual physical structure of Schwarzschild spacetime, since the "non-Euclidean" metric is the one we actually physically observe. (For example, for the "Euclidean" metric to be physically relevant, $r$ would have to be the actual physical distance from the origin, but our actual physical measurements say it isn't.)

Peter, I think you are misinterpreting StateOfTheEqn's point. First he is only referring to the spatial part of the spacetime and by saying that certain metrics could apply to it mathematically I don't think he is saying anything about "physically having two metrics". When for instance in the FRW case we consider three possible spatial metrics nobody thinks 3 physical measurements are to be yielded, which is absurd but that only one is eventually right.
IMO the argument StateOfTheEqn clearly is referring to an actual non-euclidean case.

Another plausible interpretation related to the above: let's recall that the way the metrics were represented back then was different to the current way, once again one can think of the way the early cosmological models of Einstein, de Sitter or Friedmann were written in the 1916-1922 period, they were usually obtained by an embedding in a higher dimensional manifold and then parametrizing and constraining it to the desired geometry thru an equation that was then differentiated. In all these cases it is possible to do this because foliation of the 3-hypersurfaces slices is allowed.
To use a trivial example of the way metrics were usually constructed back then thru embeddings in higher dimensional spaces, let's imagine we want to construct a line element for a 3-sphere(S3) by using its embedding in a 4-dimensional Euclidean space: $ds^2 =dx^2+dy^2+dz^2+dw^2$ . We would consider a hypersphere equation on that space as a hypersurface constraint:$x^2+y^2+z^2+w^2 =a^2$, with the paameter a being a euclidean radius in the embeeding 4-dimensional space but actually a radius of curvature in the non-euclidean three dimensional line element obtained just by differentiating, substituting for dw and transforming to spherical coordinates. We get $ds^2 = \frac{a^2}{ a^2−r^2} dr^2+r^2d\theta^2+r^2 sin^2\theta d\phi^2$
Here it is obvious that the case r=a doesn't mean there is a physical singularity there, it is just a sign of the way the line element was constructed by a constraint from a 4-dimensional embedding space to the three dimensional hypersurface.
In this vein the original Schwarzschild line element can be interpreted for its spatial part in a similar way, as valid only for values of R bigger than $r_s$ by construction, and considering $r_s$ as a sectional curvature radius of the spatial part of the manifold. I'm not saying it must be interpreted this way but that it is mathematically possible and nothing physical that I can think of right now goes againt it.

 

[PeterDonis]

First he is only referring to the spatial part of the spacetime

Yes, I understand that, but that doesn't change the fact that he is assuming that $r$ (lower case, using his definition) has a physical meaning that it doesn't actually have. In fact, what he's trying to do with his lower case $r$ basically amounts to re-inventing the Flamm paraboloid, but he's misinterpreting what it says.

I don't think he is saying anything about "physically having two metrics".

He is assigning a physical meaning to the ratio $R^2 / r^2$, which amounts to saying that $r$ has a physical meaning. Perhaps "two metrics" is not the way he would describe what that physical meaning is, but the fact remains that that ratio does *not* correspond to anything physical; see above.

the original Schwarzschild line element can be interpreted for its spatial part in a similar way, as valid only for values of R bigger than $r_s$ by construction, and considering $r_s$ as a sectional curvature radius of the spatial part of the manifold.

But the radius of curvature of the spatial part is *not* $r_s$ (except at the horizon--see below), nor is it $R^2 / r^2$. That's the point. There isn't even a single "radius of curvature of the spatial part" at all, since the curvature is different in the radial and tangential directions (see above); but in so far as we can define a "radius of curvature" in the radial direction, I think the best expression of it is $\sqrt{R^3 / r_s}$ (where I've used upper case $R$ again to make it clear that it's the area radius), which is the square root of the corresponding value of the Riemann curvature tensor. As you can see, at $R = r_s$, this radial radius of curvature is finite--in fact it is $r_s$. So $r_s$ can be thought of as the "radial radius of curvature at the horizon"--but of course that's not at all what StateOfTheEqn is claiming.

 

[TrickyDicky]

But the radius of curvature of the spatial part is *not* $r_s$ (except at the horizon--see below).That's the point. There isn't even a single "radius of curvature of the spatial part" at all, since the curvature is different in the radial and tangential directions (see above); but in so far as we can define a "radius of curvature" in the radial direction, I think the best expression of it is $\sqrt{R^3 / r_s}$ (where I've used upper case $R$ again to make it clear that it's the area radius), which is the square root of the corresponding value of the Riemann curvature tensor. As you can see, at $R = r_s$, this radial radius of curvature is finite--in fact it is $r_s$. So $r_s$ can be thought of as the "radial radius of curvature at the horizon".

Sure, I was referring about the curvature radius at the origin(let's not use the current terminology of horizons since we are talking about the original 1916 paper), the solution is asymptotically flat so the curvature radius will grow asymptotically as R goes to infinity and curvature goes to zero.

My point (as can be seen in the example I used) was that (and remember we are always referring here to the spatial hypersurface slice of the static spacetime) $r_s$ can only be seen as a distance, that is as a radius of a true sphere in the euclidean interpretation, but if we agree that we are dealing with a non-euclidean hypersurface(i.e. that we are using a non-euclidean metric in S2XR), it can only be interpreted as a curvature radius of the hypersurface at points closest to its singular origin, that in the weak field physical interpretation is proportional to the mass introduced as boundary condition. Therefore R can only be > $r_s$ by construction.

 

[PeterDonis]

Therefore R can only be > $r_s$ by construction.

Yes, all this is true of the Flamm paraboloid construction: it's only valid for $R > r_s$. But that's *not* the same as showing that the region $R > r_s$ is the entire spacetime. Schwarzschild (I think, based on what I've read) believed it was; Einstein apparently believed it was too. But the arguments that they thought proved that, did not actually prove it.

In so far as these arguments prove anything, they prove that the region $R > r_s$ is the entire *static* portion of the spacetime; but that's not the same as proving that the entire spacetime must be static. As far as I can tell, Schwarzschild and Einstein simply did not consider the possibility that the spacetime could have an additional region that was not static. (This appears to be a similar error on Einstein's part to the error that led him to miss predicting the expansion of the universe; he wanted a static solution to describe the universe and added the cosmological constant to the EFE to get one, rather than considering the possibility that the universe as a whole was not static.)

One reason, even within the Flamm paraboloid construction, to doubt that the region $R > r_s$ is the entire spacetime, is the fact, which I've mentioned repeatedly, that the physical area of 2-spheres at $R$ does not approach zero as a limit as $R \rightarrow r_s$; it approaches $4 \pi r_s^2$. The fact that $r = 0$ when $R = r_s$ does not change that, nor does it give $r$ a physical meaning; $r$ is just an abstract coordinate in an embedding diagram. I've repeatedly asked StateOfTheEqn to say how he would physically measure $r$ and have received no response.

 

[TrickyDicky]

Yes, all this is true of the Flamm paraboloid construction: it's only valid for $R > r_s$. But that's *not* the same as showing that the region $R > r_s$ is the entire spacetime. Schwarzschild (I think, based on what I've read) believed it was; Einstein apparently believed it was too. But the arguments that they thought proved that, did not actually prove it.

In so far as these arguments prove anything, they prove that the region $R > r_s$ is the entire *static* portion of the spacetime; but that's not the same as proving that the entire spacetime must be static. As far as I can tell, Schwarzschild and Einstein simply did not consider the possibility that the spacetime could have an additional region that was not static.

The Flamm's paraboloid representation of the spatial hypersurface have problems of its own as it tries to ilustrate in two dimensión what is very difficult or impossible to visualize in 3D(S2XR) and therefore misses some subtleties, but I guess you are just using it to refer to the spatial part of the spacetime so we probably agree about the math.
My aim wasn't to show "that the region $R > r_s$ is the entire spacetime" but only that it is a possible interpretation of the line element written by Schwarzschild if one ignores the current context and for instance do not consider the condition of analyticity for all the points of the manifold preventing the analytical extensión or considers the staticity requirement strictly and not replaceable by any orthogonal killing vector field.
You simply put more emphasis on different points from the ones I stress, like you seem more concerned about what such and such proved or didn't prove( always keeping an eye on the mainstream interpretation which is fine) while I prefer to look at the math in a more agnostic way.
Schwarzschild was certainly limited when he wrote the 1916 paper by the fact he knew the EFE only in its incomplete form( previous to Nov.25th 1915).

 

[StateOfTheEqn]

The error in the notion of Black Hole 'event horizons' at r=2GM has been exposed back in 1989. The error began with Hilbert. See the paper Black Holes:The Legacy of Hilbert's Error. See also Schwarzschild's original 1916 paper in English.

We summarize the result of the preceding sections as follows. The [Kruskal-Fronsdal] black hole is the result of a mathematically invalid assumption, explains nothing that is not equally well explained by [the Schwarzschild solution], cannot be generated by any known process, and is physically unreal. Clearly, it is time to relegate it to the same museum that holds the phlogiston theory of heat, the flat earth, and other will-o’-the-wisps of physics.

 

[stevendaryl]

The error in the notion of Black Hole 'event horizons' at r=2GM has been exposed back in 1989. The error began with Hilbert. See the paper Black Holes:The Legacy of Hilbert's Error. See also Schwarzschild's original 1916 paper in English.

I think that paper is wrong, or at best, misleading. The authors write:

Since each of these space-times assigns a different number to the limiting value of a radially approaching test particle’s locally measured acceleration, it is necessary to supplement the historical postulates by one that fixes this limit.

The Schwarzschild geometry is the unique (up to equivalence under coordinate transformations) spherically symmetric solution to the vacuum Einstein field equations. The acceleration of an infalling test particle is not an input to the Schwarzschild geometry, it's an output---it's computable from the Schwarzschild geometry.

Comparing the Kruskal extension to phlogiston and flat-earth is just trolling. No serious researcher would say something like that.

The significance of the Kruskal extension is not that it's a realistic model for the collapse of realistic stars. It's just another, interesting solution to the Einstein field equations. It helps in understanding a theory to have a bag of exact solutions (which are scarce for General Relativity).