# Notions of simultaneity in strongly curved spacetime

by PAllen
Tags: curved, notions, simultaneity, spacetime, strongly
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 Quote by harrylin In fact that chart is an equation. You next suggest that another equation has more physical content than the equation which was derived from it, using reasonable physical assumptions.
No, I don't. I suggest that the first chart/equation (exterior Schwarzschild) does not cover a particular portion of the spacetime that the second chart/equation (Kruskal) does.

However, underlying all of this is just one equation, the EFE. That equation is what's really at issue here. See below.

 Quote by harrylin If I correctly understood the explanations, those equations lead to white holes when blindly followed through without physical concerns; following your arguments, white holes are "really GR".
You didn't correctly understand the explanations. The EFE leads to white holes only if we assume the spacetime is vacuum everywhere (and spherically symmetric, but that's a minor point for this discussion). Nobody thinks that assumption is physically reasonable. If the spacetime is not vacuum everywhere--for example, if there is collapsing matter present--then the EFE does *not* predict white holes. So white holes are part of the set of all possible mathematical solutions of the EFE, but they are not part of the set of physically reasonable solutions of the EFE.

Just an "equation" isn't enough; you have to add constraints--initial/boundary conditions--to get a particular solution. Which solution of the equation you get--i.e., which spacetime geometry models the physical situation you're interested in--depends on the constraints.

 Quote by harrylin However, and as we discussed earlier, we agree that considerations of what makes physical sense should play a role. Most theories do contain more than mere equations, and GR is no exception. For me a theory consists of its physical foundations (both those postulated and those clearly mentioned).
Of course. See above.

 Quote by harrylin Those foundations are all kept except if they lead to contradictions; and an equation is only physically valid insofar as those physical foundations are not violated. An obvious one is the Einstein equivalence principle, but there are also the physical reality of gravitational fields and the requirement that theorems must describe the relation between measurable bodies and clocks.
Sure.

 Quote by harrylin Consequently I expect that Einstein would reject Peter's argument and say that Peter denies the physical reality of gravitational fields.
Einstein *did* reject arguments of this type. Einstein was wrong.

 Quote by harrylin Einstein would argue that the clock's worldline never really stops, but does not reach beyond 42 due to the physical reality of the gravitational field.
What is "the gravitational field"? What mathematical object in the theory does it correspond to? Before we can even evaluate this claim, we have to know what it refers to. But let's try it with some examples:

(1) The "gravitational field" is the metric. The metric (the coordinate-free geometric object, not its expression in particular coordinates) is perfectly finite and continuous at the horizon, and for reasons that both PAllen and I have explained, it can't "just stop" at the horizon without violating the EFE.

(2) The "gravitational field" is the Riemann curvature tensor. Like the metric, this is perfectly finite and continuous at the horizon.

(3) The "gravitational field" is the proper acceleration experienced by a "hovering" observer (an observer who stays at the same radius and does not move at all in a tangential direction). This *does* increase without bound as you get closer and closer to the horizon. However, there is *no* "hovering" observer *at* the horizon, because the horizon is a null surface: i.e., a line with constant r = 2M and constant theta, phi is not a timelike line; it's a null line (the path of a light ray--a radially outgoing light ray). So there is no observer who experiences infinite proper acceleration, and this definition of "gravitational field" simply doesn't apply at or inside the horizon.

As far as I can see, the only possible basis you could have for claiming that "the physical reality of the gravitational field" means that the clock's worldline stops as tau->42, would be #3. However, #3 doesn't apply to infalling observers; it only applies to accelerated, "hovering" observers. Infalling observers don't feel any acceleration, so there's nothing stopping them from falling through the horizon. The "gravitational field" in the sense of #3 is simply not felt by them at all.

Note that in all these cases, the physical "field" has to correspond to something invariant in the mathematical model, *not* something that only exists in a particular coordinate chart. That is something Einstein would have *agreed* with. Note also that none of the definitions of "gravitational field" I gave above used Schwarzschild coordinate time, or the fact that t->infinity as you approach the horizon. Einstein simply didn't understand that claims about t->infinity as you approach the horizon were claims about something that only exists in a particular coordinate chart.

 Quote by harrylin I think that Kruskal's white holes and his inside solution for black holes are not compatible with Einstein's GR.
See above. You are equivocating on different meanings of "Einstein's GR". White holes are mathematically compatible, but not physically reasonable. Black hole interiors are both mathematically compatible *and* physically reasonable.

 Quote by harrylin And don't you think that Einstein would have exclaimed the same, if it was really a "gross violation of the principle of equivalence"? Instead he commented that "there arises the question whether it is possible to build up a field containing such singularities". (E. 1939.)
As I've said before, Einstein's paper only considered the stationary case--i.e., he only considered systems of matter in stable equilibrium. All his paper proves is that *if* a system has a radius less than 9/8 of the Schwarzschild radius corresponding to its mass, the matter can't be in stable equilibrium. A collapsing object that forms a black hole meets this criterion: the collapsing matter is not in stable equilibrium. So Einstein's conclusion doesn't apply to it.
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 Quote by zonde I found your example. I just have to think what I have to say about it. It is just mock transformation when you undo all the consequences of transformation using transformed metric.
I only have time for one quick comment:

But that is the way all coordinate transforms work in differential geometry! That is the whole point! The the metric is transformed along with coordinates so all geometric properties (lengths, angle, intervals, etc.) are preserved as mathematical identities.
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 Quote by harrylin According to the Schwartzschild equations as used by Einstein and Oppenheimer, that clock will only reach the horizon (and indicate 3:00pm) at t=∞. In common physics that means "never"; however some smart person (in fact, who?) invented a different interpretation!
That is not true, the clock will pass the horizon, and its proper time will go after 3:00pm.
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 Quote by PeterDonis [..] Einstein *did* reject arguments of this type. Einstein was wrong.
Just give me the physics paper that proves that your philosophy is right, and Einstein's was wrong.
 What is "the gravitational field"? [..]
Perhaps your beef with Einstein could be summarized as follows:

Peter: What is "the gravitational field"? It is not a real mathematical object
Einstein: What is a "region of spacetime"? It is not a real physical object.

In my experience it can be interesting to poll opinions, and to inform onlookers about different points of view; however discussions of that type are useless.
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 Quote by martinbn That is not true, the clock will pass the horizon, and its proper time will go after 3:00pm.
Not in "Schwartzschild" time t of the Schwartzschild equation. Did you calculate it?
Once more: According to the Schwartzschild equations as used by Einstein and Oppenheimer, that clock will only reach the horizon (and indicate 3:00pm) at t=∞. In common physics that means "never"; however some smart person (in fact, who?) invented a different interpretation. In that different interpretation, which I still don't fully understand, the clock will pass the horizon despite Schwartzschild's t=∞.

For details, see the ongoing discussion: http://www.physicsforums.com/showthread.php?t=651362
incl. an extract of Oppenheimer-Snyder: http://www.physicsforums.com/showpos...5&postcount=50
P: 329
 Quote by harrylin Not in "Schwartzschild" time t of the Schwartzschild equation. Did you calculate it? For details, see the ongoing discussion: http://www.physicsforums.com/showthread.php?t=651362
This makes no sense. The clock shows its proper time, nothing else, and it will not stop when it reaches 3:00pm.
P: 3,179
 Quote by martinbn This makes no sense. The clock shows its proper time, nothing else, and it will not stop when it reaches 3:00pm.
A so-called "asymptotic observer" predicts that it will slow down so much that it will not reach 3:00pm before the end of this universe. However, a "Kruskal observer" says that that is true from the viewpoint of the asymptotic observer but predicts that the clock will nevertheless continue to tick beyond 3:00pm.

PeterDonis and PAllen say that these predictions do not contradict each other. And that still makes no sense to me, despite lengthy efforts of them to explain this to me.
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 Quote by harrylin And a quick comment on that quick comment: I don't see the qualitative difference with "The light speed limit doesn't exist; the tachyon space works just fine, it is the same solution. You just have to deal with technical problems to get through c".
It does require closer inspection to see if the apparent singularity in the equations of motion is removable or not.

What do I mean by a removable singularity?

[quote]
http://en.wikipedia.org/w/index.php?...ldid=507006469

 n complex analysis, a removable singularity (sometimes called a cosmetic singularity) of a holomorphic function is a point at which the function is undefined, but it is possible to define the function at that point in such a way that the function is regular in a neighbourhood of that point. For instance, the function $$f(z) = \frac{\sin z}{z}$$ has a singularity at z = 0. This singularity can be removed by defining f(0) := 1, which is the limit of f as z tends to 0. The resulting function is holomorphic.
It's been known for a very long time that in the black hole case that the singularity is removable.

IT does takes a bit of work to decide if the apparent singularity is the result of a poor coordinate choice , or is an inherent feature of the equations.

It might be helpful to give a quick example of how this happens. Consider the equations for spatial geodesics on the surface of the Earth. (Why geodesics? Because that's how GR determines equation of motion. So this is an easy-to-understand application of the issues involved in finding geodesics).

If you let lattitude be represented by $\psi$ and longitude by $\phi$, then you can write the metrc $ds^2 = R^2 (d \psi^2 + cos^2(\psi) d\phi^2)$ and come up with the equations for the geodesic (which we know SHOULD be a great circle) for $\psi(t)$ and $\phi(t)$

$$\frac{d^2 \psi}{dt^2} + \frac{1}{2} \sin 2 \psi \left( \frac{d \phi}{dt} \right)^2$$$$\frac{d^2 \phi}{dt^2} - 2 \tan \psi \left(\frac{d\phi}{dt}\right) \left( \frac{d\psi}{dt} \right) = 0$$

Now, one solution of these equations is $\phi$ = constant. It makes both equations zero. It's also half of a great circle. But, if we look more closely, we see that we have a term of the form 0*infinity in the second equation as we approach the north pole, because of the presence of $\tan \psi$ when $\psi$ reaches 90 degrees.

THis apparent singularity is mathematical, not physical. If you're drawing a great circle around a sphere, there's no physical reason to stop at the north pole.

Of course we already know what the answer is - we need to join two half circles together. In particular, we know we need to splice together two half circles, 180 degrees apart in lattitude, though as far as I know all the solution techniques (change of variable, etc) are equivalent to not using lattitude and longitude coordinates at the north pole, because the coordinates are ill-behaved there.

The same is in the black hole case, though to justify it you need to either do the math yourself, or read a textbook where someone else has. Note that you probably won't find this sort of thing in papers so much, it's assumed everyone knows it in the literature. Where you're more likely to find an explanation in a textbook or lecture notes.

Which brings me to the next point.

We don't have textooks online, but we've got several good sets of lecture notes.

What does Carroll's lecture notes have to say on the topic?
He defines the geodesic equation of motion - they're pretty complex looking, and I wouldn't be surprised if you didn't want to solve them yourself. But what does Caroll have to say about solving them?

I'll give you a link http://preposterousuniverse.com/grno...otes-seven.pdf, and a page reference (pg 182) in that link.

Then I'll give you some question

1) Does Carroll support your thesis? Or does he disagree with it?
2) What do other textbooks and online lecture notes have to say?

And for my own information
3) Do you think you know the difference between "absolute time" and "non-absolute time"
4) Do you think your argument about "time slowing down at the event horizon" depends on the existence of "absolute" time?
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 Quote by harrylin A so-called "asymptotic observer" predicts that it will slow down so much that it will not reach 3:00pm before the end of this universe. However, a "Kruskal observer" says that that is true from the viewpoint of the asymptotic observer but predicts that the clock will nevertheless continue to tick beyond 3:00pm. PeterDonis and PAllen say that these predictions do not contradict each other. And that still makes no sense to me, despite lengthy efforts of them to explain this to me.
Here's an analogy that may help it make a bit more sense.
Consider an ordinary boring constant-velocity special relativity problem: You are rest and you watch me passing by at some reasonable fraction of the speed of light, so you observe that my clock is ticking more slowly than yours. If the universe has a a finite age, it is certainly possible for me to observe a time on my clock that you will claim will never be reached - all that necessary is that:
1) I get to read my clock on my worldline before it terminates at the end of the universe.
2) Your worldline terminates at the end of the universe before it intersects the line of (your) simultaneity through the event of me reading my clock.

But, you will say, I'm cheating by introducing this arbitrary "end of the universe" to cut off your worldline (actually, you introduced it - I'm just abusing it )before it can intersect the relevant line of simultaneity. If I didn't do that, then no matter how much of my time passes before I read my clock, you'd be able to extend your worldline to intersect the line of simultaneity. That is true enough, but then again the entire concept of "line of simultaneity" only really makes sense in flat space.

The bit about a "Kruskal observer" is a red herring. The geometry around a static non-charged non-rotating mass is the Schwarzchild geometry, no matter what coordinates we use, and the only meaningful notion of time that we have is proper time along a time-like worldline. The Kruskal coordinates allow us to calculate the proper time along the infalling clock's worldline as it crosses the Schwarzchild radius, whereas the the Schwarzchild coordinates (as opposed to geometry) do not. So it's not that the "Schwarzchild observer" and the "Kruskal observer" are producing conflicting observations, it is that the Kruskal coordinates are producing a prediction for the infalling observer's worldline and the Schwarzchild coordinates are not.
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 Quote by PAllen I only have time for one quick comment: But that is the way all coordinate transforms work in differential geometry! That is the whole point! The the metric is transformed along with coordinates so all geometric properties (lengths, angle, intervals, etc.) are preserved as mathematical identities.
Okay, I have kind of working hypothesis about how this works.
We have global coordinate system where we know how to get from one place to another i.e. it provides connection, but this global coordinate system does not tell anything useful about distances and angles and such. And then we have another coordinate system that tells us distances and angles but it works only locally, meaning that if we have two adjacent patches with local coordinate systems we don't know how to glue them together.
So we take take global coordinate system with metric that will give us geometric values in accord with local coordinate systems.

Something like that. Only I don't know how to check if this is right.
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 Quote by PeterDonis Since you're so insistent on doing calculations in Schwarzschild coordinates, try this one: write down the equation defining the proper time of an object freely falling radially inward from a finite radius r = R > 2M, to radius r = 2M. Write it so that the proper time is a function of r only (this is straightforward because it's easy to derive an equation relating r and the Schwarzschild coordinate time t, so you can eliminate t from the equation). This equation will be a definite integral of some function of r, from r = R to r = 2M. Evaluate the integral; you will see that it gives a finite answer. Therefore, the proper time elapsed for an infalling object is finite, even according to Schwarzschild coordinates. .
 Quote by Austin0 Correct me if I am wrong but it appears to me that the integration of proper falling time does not have a finite value..
 Quote by PeterDonis Yes, it appears that way, if you just try to intuitively guess the answer without deriving it. But when you actually derive it, you find that it *does* give a finite answer, despite your intuition.
 Quote by Austin0 It asymptotically approaches a finite limit.

 Quote by PeterDonis This is equivalent to saying the proper time integral *does* have a finite value. If you try to evaluate the integral in the most "naively obvious" way in Schwarzschild coordinates, you have to take a limit as r -> 2m, since the metric is singular at r = 2m; but the limit, when you take it, is finite..
From the statement the limit "does" have a finite value can I assume you are basing this on a mathematical theorem "proving" that such limits at 0 or infinity resolve to definite values??? While I understand the truth of such a theorem within the tautological structure of mathematics and also it's practical truth as far as, for most applications in the real world, the difference becomes vanishingly small (effectively vanishes) this does not imply that it necessarily has physical truth.

Example: Unbounded coordinate acceleration of a system under constant proper acceleration as t ---->∞

Mathematically you can say this resolves to c but in this universe as we know it or believe it to be, this is not the case.

What you are doing here seems to me to be equivalent to integrating proper time of such a system to the limit as v --->c to derive a finite value. Thus demonstrating that such a system could reach c in finite time even if it never happens according to external clocks..

The analogy is particularly apt as by assuming the free faller reaches the horizon this is also equivalent to reaching c relative to the distant static observer yes??

What difference do you see between the two cases????

In both cases it is equivalent to directly assuming reaching c or the horizon independent of determining whether they could actually arrive there. And then determining a temporal value for your assumption. Just MHO

 Quote by PeterDonis However, even if you insist on doing the integral in Schwarzschild coordinates, you can still write it in a way that doesn't even require taking a limit; as I said in the previous post you quoted, you can eliminate the t coordinate altogether and obtain an integrand that is solely a function of r and is nonsingular at r = 2m, so you can evaluate the integral directly. .
The comments above apply to any method of integration but if freefall proper time is derived from the metric how does the additional dilation factor from velocity enter into this integration??
If you are directly integrating the metric without reference to coordinate time isn't this actually integrating an infinitesimal series of static clocks between infinity and 2M???

It seems to me that either the Schwarzschild metric accurately corresponds to reality outside the hole or it doesn't. But the idea that its perfectly true up to some indeterminate pathological point "somewhere" in the vicinity of the horizon seems very shakey.
Actually the idea of a horizon as a third sector of reality between inside and outside seems like a pure abstraction. Is there a surface between air and water?
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 Quote by pervect And for my own information 3) Do you think you know the difference between "absolute time" and "non-absolute time" 4) Do you think your argument about "time slowing down at the event horizon" depends on the existence of "absolute" time?
Does the returning twins age difference depend on a concept of absolute time????

What if the traveling twin hangs out close to the horizon for a time before traveling back to his distant outside brother. Does his younger age indicate time slowing down at the horizon?
Does it depend on an absolute time??? Is it a coordinate effect???
P: 2,365
 Quote by Austin0 It seems to me that either the Schwarzschild metric accurately corresponds to reality outside the hole or it doesn't. But the idea that its perfectly true up to some indeterminate pathological point "somewhere" in the vicinity of the horizon seems very shakey.
Be careful with that term "Schwarzchild metric"....

There's the metric that Schwarzchild discovered as a solution of the Einstein field equations. It corresponds to reality (assuming spherical symmetry, no charge, no rotation, static - the conditions under which the SW metric is solution of the EFE) inside the event horizon, outside the event horizon, and at the event horizon itself.

Then there are Schwarzchild coordinates, which we often use when we want to write that metric down in a particular coordinate system. These coordinates do not work well at the event horizon. That doesn't mean that there's anything wrong there with the spacetime described by the Schwarzchild solution to the EFE; it just means that we should use some other coordinates to describe the metric there.
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 Quote by harrylin Just give me the physics paper that proves that your philosophy is right, and Einstein's was wrong.
How about every paper published on black holes since the 1960's, and every major GR textbook since then?

 Quote by harrylin Perhaps your beef with Einstein could be summarized as follows:
My "beef" isn't with Einstein; last I checked he doesn't post on PF.

 Quote by harrylin Peter: What is "the gravitational field"? It is not a real mathematical object
Huh? I gave several examples of mathematical objects that could be reasonably associated with the term "gravitational field".

 Quote by harrylin Einstein: What is a "region of spacetime"? It is not a real physical object.
Einstein thought spacetime *was* physically real; since a "region" of spacetime is just a portion of it, it should be real as well, since a portion of a real object would also presumably be real.

 Quote by harrylin In my experience it can be interesting to poll opinions, and to inform onlookers about different points of view; however discussions of that type are useless.
I agree, but that's not the discussion we're having. You are stating your understanding of a physical model, and I am saying your understanding is mistaken. You are then quoting Einstein as an authority supporting your understanding, and I am repeating that your understanding is mistaken, and also that, in so far as Einstein's understanding was the same as yours, his was mistaken too.

You might well say that discussions of that type are useless too; I agree to the extent that I think quoting authorities is useless if the objective is to talk about the physics. We should be able to talk about the physics without caring what Einstein, Oppenheimer, Schwarzschild, or anyone else thought; we can talk about the mathematical model and its physical interpretation directly. You're having trouble understanding how the things PAllen and I and others have been saying about the mathematical model can all be consistent with each other; fine, I understand that. But it does no good to quote Einstein or anyone else; either you are able to construct the model yourself, or you're not. If you're not, IMO you need to learn how to do so before criticizing it--or else you should be able to show your partial construction of the model and exactly where you are hitting a stumbling block.

It seems to me that your current stumbling block is the fact that t->infinity as tau->42; you appear to think that this requires the infalling object to never reach tau>=42. What is your argument for this? By which I mean, what are the specific logical steps that get you from "t->infinity as tau->42" to "tau can't be >=42", and what assumptions do they depend on? I know it seems obvious to you, but it's not obvious to me, because I have a consistent mathematical model that shows how tau>=42 is possible despite the fact that t->infinity as tau->42. So one or the other of us must have a mistaken assumption somewhere. Let's see if we can find it.

If it will help, I can post *my* logical argument; but that will have to wait for a separate post.
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 Quote by harrylin A so-called "asymptotic observer" predicts that it will slow down so much that it will not reach 3:00pm before the end of this universe.
That's *not* what the asymptotic observer predicts. What he predicts is that he will never see a light signal from the infalling object that says "my clock reads 3:00 pm", and light signals saying "my clock reads 2:59 pm", "my clock reads 2:59:30", "my clock reads 2:59:45", etc., etc. will reach him at times on his clock (the asymptotic observer's clock) that increase without bound.

The asymptotic observer may try to *interpret* this prediction as showing that the infalling observer's clock will slow down so much that it will not reach 3:00 pm before the end of this universe. But that interpretation depends on additional assumptions, such as the adoption of a particular simultaneity convention for distant events. As PAllen has pointed out repeatedly, simultaneity conventions are just that: conventions. They can't be used as the basis for making direct physical claims like those you are trying to make.

 Quote by harrylin However, a "Kruskal observer" says that that is true from the viewpoint of the asymptotic observer but predicts that the clock will nevertheless continue to tick beyond 3:00pm.
No, a "Kruskal observer" says that the asymptotic observer is claiming too much (see above).

Btw, all this talk about different "observers" making different predictions is mistaken as well. Predictions of physical observables are the same regardless of which coordinate chart you adopt. Also, which coordinate chart you adopt is not dictated by which worldline in spacetime you follow; there is nothing preventing the "asymptotic observer" from adopting Kruskal coordinates to do calculations.
P: 3,179
 Quote by PeterDonis That's *not* what the asymptotic observer predicts. [..] all this talk about different "observers" making different predictions is mistaken [..]
As I said, I will get to the bottom of this in the appropriate thread for a detailed discussion of Oppenheimer-Snyder.
I let myself be held up by the continuing conversation in this thread. Consequently I will not anymore reply in this thread until that is done. http://www.physicsforums.com/showthr...=651362&page=6

PS (in contradiction to my remark above - but I won't add another post for the time being!):
 Quote by PeterDonis [..] My "beef" isn't with Einstein [..]
 Quote by PeterDonis Einstein *did* reject arguments of this type. Einstein was wrong.
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 Quote by zonde Okay, I have kind of working hypothesis about how this works. We have global coordinate system where we know how to get from one place to another i.e. it provides connection, but this global coordinate system does not tell anything useful about distances and angles and such.
It does, but not directly. The easiest way to get this information out of the global coordinates is to transform them so that locally they DO directly tell us about angles and distances in the manner in which we are used to.

 And then we have another coordinate system that tells us distances and angles but it works only locally, meaning that if we have two adjacent patches with local coordinate systems we don't know how to glue them together.
I don't view it as a matter of gluing, but I suppose if you are thinking of trying to glue together all the local maps you can think of it this way.

Consider the problem of making a map of the earth. The issue is that the Earth's surface is curved, and our paper is not.

If we do a straightforwards projection, we can make a map that is "to scale" near any particular point we choose. (The further away we are from the point, the more distorted the map gets).

Occasioanlly you'll see maps like this - looking up the topic for definitess, I find Goode homolosine projection :
http://en.wikipedia.org/w/index.php?...ldid=508879282

So to summarize, using the example of the Earth's curved surface as a model for the similar problem of making maps of curved space-time.

Global coordinate information (lattitude and longitude in our example) does exist and does provide information on distances and angles, but the information requires decoding.

We can map the surface of the Earth in a variety of ways, but while we can't make the resulting map projections appear to be in one piece and drawn to scale on a flat piece of paper.