I Understanding the phrase "simultaneity convention"

[cianfa72]

In polar coordinates the metric coefficient $g_{rr}$ vanishes at $r = 0$. That means the metric has a vanishing determinant, but it doesn't make it undefined.

At $r=0$, however, the polar coordinates/map is not defined.

In Schwarzschild coordinates, the metric coefficient $g_{rr}$ is undefined at $r = r_s$ (zero in the denominator). That's not the same thing as the above.

I believe the point is the following: is $r = r_s$ actually part of the Schwarzschild map/coordinates ? If not it is more o less alike polar coordinates for Euclidean plane, I believe.

 

[PeterDonis]

At $r=0$, however, the polar coordinates/map is not defined.

If you insist on a one-to-one mapping of both coordinates, yes, that's true, because there is not a unique value of $\theta$ assigned to the point $r = 0$.

That doesn't change the fact that the coordinate behavior in the metric (line element) is not the same for this case as for Schwarzschild coordinates at $r = r_s$.

I believe the point is the following: is $r = r_s$ actually part of the Schwarzschild map/coordinates ?

Strictly speaking, no, it's not. There are several ways to look at why:

(1) No finite value of $t$ can be assigned at $r = r_s$.

(2) In the underlying spacetime geometry, the locus $r = r_s$ is more than one 2-sphere, but no other coordinate in Schwarzschild coordinates can distinguish the different 2-spheres at $r = r_s$; the natural candidate to do that would be the $t$ coordinate, but it doesn't work.

(3) Surfaces of constant $t$ in Schwarzschild coordinates intersect at $r = r_s$.

If not it is more o less alike polar coordinates for Euclidean plane, I believe.

Not really, since the reasons are quite different. See above.

 

[cianfa72]

(3) Surfaces of constant $t$ in Schwarzschild coordinates intersect at $r = r_s$.

You mean all the loci of $t=const$ in spacetime intersect each other in a set that is characterized by $r=r_s$, right?

 

[PeterDonis]

You mean all the loci of $t=const$ in spacetime intersect each other in a set that is characterized by $r=r_s$, right?

Not in a "set"--in a point of $t$, $r$ space. (Or a single 2-sphere in the full spacetime.)

 

[cianfa72]

Not in a "set"--in a point of $t$, $r$ space. (Or a single 2-sphere in the full spacetime.)

Ah ok, so actually this intersection point/event in 2D Schwarzschild spacetime model (or the single 2-sphere in full spacetime) is characterized by $(r_s, t)$ where $t$ takes uncountable many values.

Therefore, since they intersect, hypersurfaces of constant Schwarzschild coordinate time $t$ do not foliate the entire Schwarzschild spacetime.

 

[PeterDonis]

so actually this intersection point/event in 2D Schwarzschild spacetime model (or the single 2-sphere in full spacetime) is characterized by $(r_s, t)$ where $t$ takes uncountable many values.

It does not have well-defined Schwarzschild coordinates at all. $r = r_s$ is on the horizon, so we cannot even say that it has "uncountable many" finite values of $t$. Read item (1) of my post #122 again.

since they intersect, hypersurfaces of constant Schwarzschild coordinate time $t$ do not foliate the entire Schwarzschild spacetime.

This is correct. But note also that we cannot even say that all such surfaces are spacelike.

 

[cianfa72]

It does not have well-defined Schwarzschild coordinates at all. $r = r_s$ is on the horizon, so we cannot even say that it has "uncountable many" finite values of $t$. Read item (1) of my post #122 again.

I believe it is much easier to look at/grasp such Schwarzschild coordinate singularities in Kruskal–Szekeres (KS) coordinates $(T,X)$.

Schwarzschild coordinate singularities are actually two folded: all spacetime hypersurfaces that map to $t=c$ for any constant $c$ intersect at a single point/event (or 2-sphere in full spacetime) that has KS coordinates $(T = 0, K = 0)$ -- it is alike $\theta = c$ polar coordinate lines for the euclidean plane that all intersect at the common origin O.

The other kind of Schwarzschild coordinate singularity is due to the fact that the set of events/points (or set of 2-spheres) that make up the horizon, all share the same values ($r=r_s, t= \pm \infty$). There is no "equivalent behaviour" for polar coodinates in the plane, though.

 

[PeterDonis]

I believe it is much easier to look at/grasp such Schwarzschild coordinate singularities in Kruskal–Szekeres (KS) coordinates .

Yes, as I said in this Insights article:

https://www.physicsforums.com/insights/schwarzschild-geometry-part-2/