Conditions for spacetime to have flat spatial slices

[Passionflower]

Here is your original posting (I boldfaced the relevant parts):

No, as I noted elsewhere in the post, it's actually a 2-surface when the angular coordinates are taken into account. Also, I was *not* saying that any traveler crossing the horizon passes through this point; as I noted further on, all the actual physics at the horizon is on the "future horizon" null line that runs at 45 degrees up and to the right from the center point in the Kruskal diagram. That's where worldlines crossing the horizon go, and they can cross at anyone of an infinite number of different events.

The "3-volume" spanned by r = 2M, t = minus infinity to plus infinity, theta = 0 to pi, phi = 0 to 2 pi. Since the metric coefficient $g_{tt}$ is zero at r = 2M, the integral corresponding to this 3-volume vanishes, indicating that what looks like a 3-volume in Schwarzschild coordinates is actually, at most, a 2-surface. (We can verify that it is, in fact, a 2-surface and not something with even fewer dimensions by, for example, integrating over the full range of angular coordinates at the "point" at the center of the Kruskal diagram, which gives the nonzero area of the horizon.)

In both parts you say it is actually a 2-surface and not a 3-surface seems indicated by Schw. coordinates. But one posting ago you agreed it is actually a 3-surface? I must be seriously mistaken.

 

[PeterDonis]

In both parts you say it is actually a 2-surface and not a 3-surface seems indicated by Schw. coordinates. But one posting ago you agreed it is actually a 3-surface? I must be seriously mistaken.

You are. Here's the full relevant part of my post #125, which originally prompted your question (but you only quoted part of it in your question):

For example, in Schwarzschild coordinates, there appears to be an entire infinite line at the horizon, r = 2M, t = minus infinity to plus infinity, that actually, physically, is just a point, as you can see by transforming to Kruskal coordinates, where that entire line becomes the single point at the center of the diagram. (Here I've been ignoring the angular coordinates; when we put them back in, the "point" is actually a 2-surface.) This transformation also maps the "point" at t = infinity in Schwarzschild coordinates to an entire null line (the 45-degree line between regions I and II in the diagram with a yellow background on the Wikipedia page here: http://en.wikipedia.org/wiki/Kruskal–Szekeres_coordinates); this null line, the "future horizon", is where all the interesting physics at the horizon actually happens, and it is "invisible" in Schwarzschild coordinates, which often leads to confusion if those coordinates are taken too literally.

I've stated from the very beginning that "point" and "2-surface" refer only to the "apparent line" from t = - infinity to infinity in *Schwarzschild* coordinates, which is physically just a point (or 2-surface if angular coordinates are included) and appears that way at the center of the Kruskal diagram, and that there *is* a "real" line (or 3-volume if angular coordinates are included) at the horizon that is covered by Kruskal coordinates but which does *not* appear in Schwarzschild coordinates.

 

[Passionflower]

which is physically just a point

Didn't you just agree that, for instance when two observers passing it after each other, it is a line?

 

[PeterDonis]

Didn't you just agree that, for instance when two observers passing it after each other, it is a line?

No. Go back and read my posts again, carefully, and see exactly what I've referred to as a "point" and what I've referred to as a "line", and look at a Kruskal diagram and see how the two relate to each other. Then fill in a couple of timelike worldlines on the diagram, crossing the future horizon line at different events. Where are those events placed, in relation to the point at the center? Do they pass through that point?

 

[Passionflower]

No.

Ok, so you are saying it is a point?

So then I ask you again, observer A passes this point before observer B, clearly their worldlines do not cross there, how do you explain that if there is no line but a point?

 

[PeterDonis]

Ok, so you are saying it is a point?

So then I ask you again, observer A passes this point before observer B, clearly their worldlines do not cross there, how do you explain that if there is no line but a point?

Did you do what I asked in my last post, looking at the Kruskal diagram? You will find that you can draw two timelike worldlines that cross the future horizon line at two different events (points), but neither of those events are the center point of the diagram (the "point" which appears as a line in Schwarzschild coordinates). Please do that before asking this question again.

 

[Passionflower]

Did you do what I asked in my last post, looking at the Kruskal diagram? You will find that you can draw two timelike worldlines that cross the future horizon line at two different events (points), but neither of those events are the center point of the diagram (the "point" which appears as a line in Schwarzschild coordinates). Please do that before asking this question again.

I am not talking about coordinate charts but about physics. An observer A passes the EH at the same spatial location before an observer B, what does that imply?

Too bad Peter, I think you are trying to avoid my questions.

 

[PeterDonis]

I am not talking about coordinate charts but about physics. An observer A passes the EH at the same spatial location before an observer B, what does that imply?

Um, that spacetime has a time dimension? I'm not trying to avoid anything, I just don't understand what point you're trying to make. I've already agreed that observer A and observer B, in your example just above, would cross the horizon (the line U = 0, V > 0 that goes up and to the right at a 45 degree angle from the center of the Kruskal diagram) at two different events. Neither of those events is the point at the center of the diagram.

If it's the bit about "the same spatial location" that's bothering you, I've also already said that the "future horizon" line I described has r = 2M all along it. I gave the function relating r to the Kruskal U and V that makes that clear. It's true that on the Kruskal diagram, the two events where A and B cross the horizon have different *Kruskal* coordinates (different V, or different Kruskal spatial coordinate X or R, if you use those coordinates instead of U and V to label events in the diagram). That's because the Kruskal chart is a different chart that labels events differently. What's the problem?

 

[Passionflower]

What's the problem?

Do you think r=rs, phi=0, theta=0 is a line or a point in spacetime?

 

[PeterDonis]

Do you think r=rs, phi=0, theta=0 is a line or a point in spacetime?

You said you weren't talking about coordinate charts, but about physics. Physically, I've agreed several times that there is a line at the horizon (for a given angular direction in space), so that different infalling observers (or ingoing light rays) can cross the horizon at different events. Whether or not that line is covered by a given coordinate chart depends on the chart, so if you want an answer to your question exactly as you posed it, you'll need to specify which chart the coordinates you gave relate to.

 

[Passionflower]

You said you weren't talking about coordinate charts, but about physics. Physically, I've agreed several times that there is a line at the horizon (for a given angular direction in space), so that different infalling observers (or ingoing light rays) can cross the horizon at different events. Whether or not that line is covered by a given coordinate chart depends on the chart, so if you want an answer to your question exactly as you posed it, you'll need to specify which chart the coordinates you gave relate to.

I am talking about this statement:

For example, I asserted just now that the apparent "line" at the horizon in Schwarzschild coordinates is actually just a point--or, if we include the angular coordinates, what appears to be a 3-surface is actually just a 2-surface. How do I know this is right?

So with actually you mean 'coordinates' and not 'physically'?

But I am confused by this statement:

For example, in Schwarzschild coordinates, there appears to be an entire infinite line at the horizon, r = 2M, t = minus infinity to plus infinity, that actually, physically, is just a point

Doesn't this say the opposite to what we now agree on?

 

[PeterDonis]

So with actually you mean 'coordinates' and not 'physically'?

No, the "actually" there meant "physically", because, as I've noted several times, the actual, physical line at the horizon is not covered by the Schwarzschild chart. Let me go ahead and answer the question you posed in your last post but one, exactly as you posed it, in terms of the Painleve and Schwarzschild charts, since both use the r coordinate directly.

In terms of the Painleve chart, the answer to your question exactly as you posed it is "a line". What appears as a line in this chart, the line theta = phi = constant, r = 2M, T = minus infinity to plus infinity (I'll use capital T for Painleve time to avoid confusion with Schwarzschild time t), is actually, physically, a line--it's the "future horizon" line we've been talking about, which different infalling observers can cross at different events.

In terms of the Schwarzschild chart, however, the answer to your question exactly as you posed it is "a point". What *appears* as a line in this chart, the line theta = phi = constant, r = 2M, t = minus infinity to plus infinity (small t this time), is actually, physically, just a single point, *not* a line.

George Jones and I had an exchange about the relationship between the Painleve chart and the Schwarzschild chart earlier in this thread: his posts #80 and #121 have good information (and a helpful diagram in the latter post).

https://www.physicsforums.com/showpost.php?p=2988151&postcount=80

https://www.physicsforums.com/showpost.php?p=3000266&postcount=121

 

[Passionflower]

In terms of the Schwarzschild chart, however, the answer to your question exactly as you posed it is "a point". What *appears* as a line in this chart, the line theta = phi = constant, r = 2M, t = minus infinity to plus infinity (small t this time), is actually, physically, just a single point, *not* a line.

Well there is the rub, you speak about Schw. coordinates and seems to assign physical attributes to it. The line physically exists in spacetime but Schw. coordinates do not cover this line, that does not mean it is not physically there is just means that the chart has limitations.

 

[PeterDonis]

The line physically exists in spacetime but Schw. coordinates do not cover this line, that does not mean it is not physically there is just means that the chart has limitations.

Which is exactly what I've been saying all along. That was my whole point in bringing up the example in the first place.

 

[PeterDonis]

But I am confused by this statement:

For example, in Schwarzschild coordinates, there appears to be an entire infinite line at the horizon, r = 2M, t = minus infinity to plus infinity, that actually, physically, is just a point.

Doesn't this say the opposite to what we now agree on?

I saw this edit to your post #151 just now--it must have appeared while I was editing one of mine. No, it doesn't say the opposite. I've stated what the physical point and line are several times, but maybe a quick summary will help:

The Physical Line: The "future horizon" line. This appears as:

-- A vertical line in the Painleve chart (r = 2M, T = minus infinity to infinity)

-- A 45 degree line up and to the right in the Kruskal chart (U = 0, V > 0).

-- Does *not* appear in the Schwarzschild chart (it's shoved up to "plus infinity", off the chart).

The Physical Point: This appears as:

-- A vertical line in the Schwarzschild chart (r = 2M, t = minus infinity to infinity).

-- The "center point" in the Kruskal chart (U = 0, V = 0).

-- Does *not* appear in the Painleve chart (it's shoved down to "minus infinity", off the chart).

 

[Passionflower]

I saw this edit to your post #151 just now--it must have appeared while I was editing one of mine. No, it doesn't say the opposite. I've stated what the physical point and line are several times, but maybe a quick summary will help:

The Physical Line: The "future horizon" line. This appears as:

-- A vertical line in the Painleve chart (r = 2M, T = minus infinity to infinity)

-- A 45 degree line up and to the right in the Kruskal chart (U = 0, V > 0).

-- Does *not* appear in the Schwarzschild chart (it's shoved up to "plus infinity", off the chart).

The Physical Point: This appears as:

-- A vertical line in the Schwarzschild chart (r = 2M, t = minus infinity to infinity).

-- The "center point" in the Kruskal chart (U = 0, V = 0).

-- Does *not* appear in the Painleve chart (it's shoved down to "minus infinity", off the chart).

Ok, I see, and now understand what the confusion was. Looks like we agree. :)

 

[JDoolin]

There are two variables in the Friedmann Walker Diagram, the horizontal variable is a "space-like" variable, and the vertical is a "time-like" variable. To map to the Comoving Particle Diagram, I'm not sure exactly how it is done, but I think the vertical "time component" is just mapped straight over, while the horizontal "space component" is some form of velocity * distance. I may be wrong, but I *think* it is the integral of the changing scale factor with respect to the "cosmological time."

The horizontal variable is SPACE, and the vertical variable is TIME; some kind of "Absolute" or "cosmological" time, which really doesn't exist in the realm of Special Relativity.

What the Milne model does is treats the horizontal variable in the FWD as sort of a "rapidity-space" The mapping from the FWD to the MMD assumes that the meaning of the FWD "space-like" variable is distance = rapidity * proper time. For a set of particles all coming from an event (0,0), giving the rapidity and proper time for a particle uniquely defines its position in space and time. To map from the FWD to the MMD, you are simply mapping: (d'=rapidity*proper time,t'=proper time) to (d=space, t=time).

To map from the FWD to the CPD, you are mapping (d'="Stretchy" Velocity * Cosmological Time, t'=Cosmological Time) to (d=Space,t=Cosmological Time).

I'll see if I can express this as mathematically and unambiguously as I can, so that if I'm wrong it can be corrected.

$$\begin{matrix} FWD \mapsto CPD \text{ as }(d\int a(\tau)d\tau,\tau)\mapsto(d,\tau) \\ d=Proper Distance = Cosmological Distance \\ \tau=Proper Time=CosmologicalTime \\ a(\tau)=ScaleFactor \end{matrix}$$​

On the other hand, the Milne mapping looks like this:

$$\begin{matrix} FWD \mapsto MMD \text{ as }(\varphi \cdot\tau,\tau)\mapsto(v \cdot t,t) \\ \varphi=rapidity \\ \tau=proper time \\ v = velocity \\ t = time \end{matrix}$$​

As you can see, the Milne mapping is linear; there's no changing scale factor. The relation between rapidity and velocity and distance, time, and proper time is the same as is usually given in Special Relativity.

Rapidities between -infinity and +infinity map to velocities between -c and +c. So the horizontal plane (representing infinite rapidity) in the Friedmann Walker Diagram maps to the light-cone in the Milne Minkowski Diagram.

There is a very clear difference between the two mappings, but I am still uncertain of the mapping from the FWD to the MMD. I used an integration where I think it may have been unnecessary, but I'm pretty sure the scale factor is invoked in the mapping from FWD to MMD.

$$\begin{matrix} FWD \mapsto CPD \text{ as }(d a(\tau)\tau,\tau)\mapsto(d,\tau) \\ d=Proper Distance = Cosmological Distance \\ \tau=Proper Time=CosmologicalTime \\ a(\tau)=ScaleFactor \end{matrix}$$​

Is there anyone who can verify or correct this mapping?

 

[JDoolin]

Ok, I see, and now understand what the confusion was. Looks like we agree. :)

Alright, that was a nice discussion, but over my head, I'm afraid. But regardless of the complexity of the mathematics, there are some things that are fundamental.

The issue is whether there is a difference between a single event, and an infinite number of events occurring in an environment which has been scaled to zero.

If you represent anything in polar coordinates, then you can represent the origin by any number of possibilities: (0,0 radians) (0,1 radian) (0,2 radians) etc. It is the only point on the circle that is non-uniquely defined in polar coordinates. Yet, obviously, there is only one point. Yet it would not take too much work to distort that coordinate system so that that point turned into a horizontal line, and the concentric circles became horizontal lines, and the radial lines became vertical.

Now, you could map this coordinate system using a scale factor a(r)=r, which would tell you the ACTUAL arc-length of each segment.

You can have an infinite number of lines coming out of that point, but all of those lines meet at the event, this is ONE EVENT. But if you try to say those lines don't meet at that point; it is just the scale factor reduces to zero, making them look like they meet, then you're talking about an infinite number of events.

I feel like Peter is trying to have it both ways, claiming that with the singularity, that the Big Bang is, at the same time, one event, and an infinite number of events. You're violating a most basic law of logic, because you can't have something be both true, and untrue at the same time.

The two models resolve the mystery in two different ways. The Milne Model says there is one event with many particles coming out. The Standard Model says there are many events, reduced in scale to zero, so that it only LOOKS like one event.

Peter is saying there are examples of this happening all the time in General Relativity where single events become multiple events or vice versa, but you have to do more than handwaving to convince me that you have successfully violated the law of the excluded middle, and proven it mathematically.

 

[PeterDonis]

You can have an infinite number of lines coming out of that point, but all of those lines meet at the event, this is ONE EVENT. But if you try to say those lines don't meet at that point; it is just the scale factor reduces to zero, making them look like they meet, then you're talking about an infinite number of events.

I feel like Peter is trying to have it both ways, claiming that with the singularity, that the Big Bang is, at the same time, one event, and an infinite number of events. You're violating a most basic law of logic, because you can't have something be both true, and untrue at the same time.

The two models resolve the mystery in two different ways. The Milne Model says there is one event with many particles coming out. The Standard Model says there are many events, reduced in scale to zero, so that it only LOOKS like one event.

No; you're misunderstanding the standard FRW model. I've been saying all along that, physically, the initial singularity in the FRW models is *one event*. I've also been saying all along that in the "conformal" diagram, that one event *appears* to be a line, but physically, it's still just a point (one event). I've never claimed anything else. Please read carefully what I've posted; I've tried to be careful about making these distinctions, between the actual, physical invariant objects (points, lines, etc.) and the *appearances* in various coordinate systems, which may not reflect the actual, physical reality.

You seem to believe that somehow, because the FRW model achieves the initial singularity by reducing a scale factor to zero, the intial singularity is really an infinite number of events. That's not correct. I showed earlier how you can tell that it's just one event: look at the spatial "volume element" at a constant time t, which is just the square root of the product of the spatial metric coefficients. (As I cautioned before, this looks this simple only because the FRW metric is diagonal; in a non-diagonal metric things are more complicated.) Since *all three* of the spatial metric coefficients are multiplied by the scale factor a(t), if a(t) goes to zero, the volume element vanishes identically, and this happens whether we try to compute a 3-dimensional volume, a 2-dimensional surface area, or even a 1-dimensional length. That means that, physically, the "volume" t = 0 (the initial singularity) is actually just a single point, with zero dimensions, regardless of how it *appears* in some coordinate system.

Peter is saying there are examples of this happening all the time in General Relativity where single events become multiple events or vice versa, but you have to do more than handwaving to convince me that you have successfully violated the law of the excluded middle, and proven it mathematically.

Once again, I've *never said* that single events actually, physically, become multiple events. Obviously that's absurd. I've *always said* that some transformations can make single events *appear* to be lines instead of points, or conversely, it can make actual, physical lines (or surfaces or volumes) effectively "invisible" because of some strangeness in a particular coordinate system. But obviously you can't change the actual, physical nature of an event (a point), or a line, or a surface, or a volume, by changing coordinates. I've said that before too, in almost exactly those words. Once again, please read carefully what I've posted.