# Conditions for spacetime to have flat spatial slices

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.

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?

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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

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.

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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.

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But I am confused by this statement:

PeterDonis said:
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).

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
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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
Gold Member
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.

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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.