Understanding the phrase "simultaneity convention"

[PeterDonis]

I think he intended to allow a foliation such that 9:00 am Eastern is simultaneous with 9:00 am Central.

Perhaps so, but in any event that is not the standard interpretation of time zones. Nobody actually treats 9 am Eastern and 9 am Central as simultaneous. Everybody understands that the latter is an hour later than the former. So if he is claiming that his proposal is just like the standard usage of time zones, I don't think that claim is valid.

 

[Dale]

that is not the standard interpretation of time zones. Nobody actually treats 9 am Eastern and 9 am Central as simultaneous

I agree.

 

[Ibix]

I think that is not his intention. I think he intended to allow a foliation such that 9:00 am Eastern is simultaneous with 9:00 am Central. It is an odd approach, but I think that is exactly what he is proposing.

Is that allowed? The two geographical regions share an edge where the time coordinate is either ill-defined or arbitrarily chosen to be one or the other. Doesn't that make the coordinates either ill-defined or defined on closed regions?

Edit: you could define a finite but narrow transition region between the zones, of course.

 

[cianfa72]

That convention is the "natural" one for Painleve observers, i.e., observers free-falling into the hole from rest at infinity, to use; the surfaces of constant $T$ are everywhere orthogonal to the worldlines of Painleve observers (which of course are timelike). But those worldlines are not integral curves of the Killing vector field $\partial_T$.

Ok, so the Killing vector field $\partial_T$ is spacelike inside the horizon, Painleve observer's timelike curves are not integral curves of $\partial_t$, however they are orthogonal to the spacelike hypersurfaces of constant $T$.

 

[Dale]

Is that allowed? The two geographical regions share an edge where the time coordinate is either ill-defined or arbitrarily chosen to be one or the other. Doesn't that make the coordinates either ill-defined or defined on closed regions?

Edit: you could define a finite but narrow transition region between the zones, of course.

That is a good question. Carroll introduces the requirement that the chart be $C^\infty$. But then he almost immediately backs off on the requirement. So I am not sure what subsequent theorems rely on the continuity and how far they really rely on it. And he does use polar and spherical coordinates later. So I am just not sure how much that requirement can be relaxed.

 

[cianfa72]

Carroll introduces the requirement that the chart be $C^\infty$. But then he almost immediately backs off on the requirement. So I am not sure what subsequent theorems rely on the continuity and how far they really rely on it. And he does use polar and spherical coordinates later. So I am just not sure how much that requirement can be relaxed.

Sorry, which specific chapters/sections of Carroll lectures/book are your referring to ?

 

[Ibix]

That is a good question. Carroll introduces the requirement that the chart be $C^\infty$. But then he almost immediately backs off on the requirement. So I am not sure what subsequent theorems rely on the continuity and how far they really rely on it. And he does use polar and spherical coordinates later. So I am just not sure how much that requirement can be relaxed.

I think it's more a problem with the discontinuity than the differentiability. If coordinates aren't defined on the borders between time zones then you actually have 24 non-overlapping coordinate patches. What does it even mean for two things to be similtaneous if you can't draw a line of constant coordinate time from one to the other?

That isn't really a problem with polar coordinates because there's only a half-infinite line that's problematic and you can always go around it. But in the time zones case I think you end up having to appeal to some kind of global coordinate system, or at least another set of patches that do cover the joins. That doesn't feel like "a" coordinate system so much as several partially overlapping ones.

 

[Dale]

Sorry, which specific chapters/sections of Carroll lectures/book are your referring to ?

It is the first couple of pages of chapter 2 in the Lecture Notes on General Relativity

 

[Dale]

That isn't really a problem with polar coordinates because there's only a half-infinite line that's problematic and you can always go around it. But in the time zones case I think you end up having to appeal to some kind of global coordinate system, or at least another set of patches that do cover the joins. That doesn't feel like "a" coordinate system so much as several partially overlapping ones.

I don't see a difference between the two. I think the problem is the same, but again, this is not my preferred definition but it is in the literature. I am disinclined to argue either for or against it. Unfortunately, I don't know an actual reference that gives a better definition.

To me, this is a definition, so an author can choose a different definition. The real problem in my opinion is what @PeterDonis mentioned: it does not capture the idea that people usually mean by the word. So using this definition will cause confusion at some points, even if it is mathematically OK.

I think that a time coordinate should both be timelike everywhere and foliate the spacetime into spacelike simultaneity surfaces. Any other definition will not be what people generally mean.

 

[cianfa72]

It is the first couple of pages of chapter 2 in the Lecture Notes on General Relativity

I took a look there. Indeed Carroll claims for example that Mercator map doesn't include both North, South pole and International Date Line. As @Ibix said one needs a set of coordinate patches (an atlas) since the domain and the target of each coordinate patch/map must be an open set.

 

[PeterDonis]

the Killing vector field $\partial_T$ is spacelike inside the horizon

Yes.

Painleve observer's timelike curves are not integral curves of $\partial_t$, however they are orthogonal to the spacelike hypersurfaces of constant $T$.

Yes. (But note that it should be $\partial_T$ in the quote above, since I was using $T$ for the Painleve time coordinate.)

 

[PeterDonis]

I think that a time coordinate should both be timelike everywhere and foliate the spacetime into spacelike simultaneity surfaces. Any other definition will not be what people generally mean.

While this is true, it is also true that there will be cases where doing this will end up with something that isn't "what people generally mean" about that specific spacetime.

For example, if you want a chart with these properties on Schwarzschild spacetime including the black hole region, you will need to use something like the Kruskal chart. But the integral curves of that chart's time coordinate aren't the worldlines of any observer that would seem natural, and the foliation surfaces defined by that chart don't correspond to any foliation that would seem natural. The "natural" chart on Schwarzschild spacetime is the Schwarzschild chart, which fails both requirements at and inside the horizon.

 

[Dale]

For example, if you want a chart with these properties on Schwarzschild spacetime including the black hole region, you will need to use something like the Kruskal chart. But the integral curves of that chart's time coordinate aren't the worldlines of any observer that would seem natural, and the foliation surfaces defined by that chart don't correspond to any foliation that would seem natural. The "natural" chart on Schwarzschild spacetime is the Schwarzschild chart, which fails both requirements at and inside the horizon.

I think it is better to simply admit that.

We have a "natural" chart, and we have a common understanding of the requirements for a "time coordinate". The chart that is both natural has the common "time coordinate" is limited to outside the horizon. There are charts that have a common "time coordinates" throughout the manifold. There are charts that are "natural" throughout the manifold. It is only the combination of both features that becomes incompatible at or below the horizon.

I would rather not change the meaning of a "time coordinate" to capture this. After all, not all charts are required to have a time coordinate. And simply using the variable $t$ doesn't mean that the thing represented by the variable must be time.

But again, I don't know of a source that uses the definition I would prefer.

 

[cianfa72]

Since we're talking about coordinates, I'd ask for a clarification about the Schwarzschild metric in Schwarzschild coordinates. At the horizon $r=r_s$ the metric component $g_{rr}$ blows up while $g_{tt}$ goes to zero. My impression is that such coordinate singularity is alike what happens in polar coordinates along the negative (left) half-line starting from a fixed point O in Euclidean plane. In the target codomain of polar map $(r, \theta)$ points along the segment $r=0$ and along the lines $\theta = - \pi, \theta = \pi$ are actually excluded. This way polar map satisfies Carroll's definition of chart since it is a one-to-one map between open sets (and the composition with the identity map is a diffeomorphism where it is defined).

Having said that, is $r=r_s$ in Schwarzschild coordinates a coordinate singularity alike to the above?

 

[PeterDonis]

My impression is that such coordinate singularity is alike what happens in polar coordinates

Not really. 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.

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.

 

[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/