Notions of simultaneity in strongly curved spacetime

[PeterDonis]

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.

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.

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.

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.

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.

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.

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.

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.

 

[PAllen]

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.

 

[martinbn]

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.

 

[harrylin]

[..] 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.

 

[harrylin]

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: https://www.physicsforums.com/showthread.php?t=651362
incl. an extract of Oppenheimer-Snyder: https://www.physicsforums.com/showpost.php?p=4162425&postcount=50

 

[martinbn]

Not in "Schwartzschild" time t of the Schwartzschild equation. Did you calculate it?
For details, see the ongoing discussion: https://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.

 

[harrylin]

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.

 

[pervect]

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?

http://en.wikipedia.org/w/index.php?title=Removable_singularity&oldid=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)

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

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/grnotes/grnotes-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?

 

[Nugatory]

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 Schwarzschild 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 Schwarzschild radius, whereas the the Schwarzschild coordinates (as opposed to geometry) do not. So it's not that the "Schwarzschild 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 Schwarzschild coordinates are not.

 

[zonde]

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.

 

[Austin0]

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.

.

Correct me if I am wrong but it appears to me that the integration of proper falling time does not have a finite value..

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.

It asymptotically approaches a finite limit.

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

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?

 

[Austin0]

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?

 

[Nugatory]

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 "Schwarzschild metric"...

There's the metric that Schwarzschild 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 Schwarzschild 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 Schwarzschild solution to the EFE; it just means that we should use some other coordinates to describe the metric there.

 

[PeterDonis]

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?

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.

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".

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.

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.

 

[PeterDonis]

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.

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.

 

[harrylin]

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. https://www.physicsforums.com/showthread.php?t=651362&page=6

PS (in contradiction to my remark above - but I won't add another post for the time being!):

[..] My "beef" isn't with Einstein [..]

Your memory is short :

Einstein *did* reject arguments of this type. Einstein was wrong.

 

[pervect]

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?title=Goode_homolosine_projection&oldid=508879282So 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.

 

[pervect]

How about every paper published on black holes since the 1960's, and every major GR textbook since then?

You might want to "tweak" Harry on whether or not he bothered to look at Caroll's online lecture notes about this topic. Specifically, I'd like to know if he _really_ thinks that Caroll's written views support his thesis.

He doesn't appear to have responded to my question on the point when I asked. Perhpas he just missed it.

http://preposterousuniverse.com/grnotes/grnotes-seven.pdf around pg 182. Perhaps I should quote it, but I'm hoping to try and motivate people to look up references.

 

[atyy]

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.

Greg Egan gives a similar situation in special relativity. http://gregegan.customer.netspace.net.au/SCIENCE/Rindler/RindlerHorizon.html (See the section "free fall")

 

[zonde]

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.

Question is about role of metric.
And as I understand metric gives easier way to get distances out of global coordinates. There is no need to do any transformation. And distance is between two points and you might not be able to transform coordinates so that neighbourhood of both endpoints can be considered flat.
This might be different about angles.

And another part of the question was about role of coordinate system if it does not provide distance information. And the answer seems to be that it provides correct proportions between distances in local neighbourhood so that we know what is connected to what.

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?title=Goode_homolosine_projection&oldid=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.

But with the Earth map it is clear why we can't do that - surface of Earth and surface of flat piece of paper are different in 3D. But globe is not very handy for carrying around so we use flat piece of paper instead.

But what about GR maps? What is the correct embedding? Is it related to extra dimension or distortion of measurement system?

 

[zonde]

Specifically, I'd like to know if he _really_ thinks that Caroll's written views support his thesis.

He doesn't appear to have responded to my question on the point when I asked. Perhpas he just missed it.

http://preposterousuniverse.com/grnotes/grnotes-seven.pdf around pg 182. Perhaps I should quote it, but I'm hoping to try and motivate people to look up references.

These Caroll's views seems like a start of long discussion. Do you want to start one?

 

[pervect]

These Caroll's views seems like a start of long discussion. Do you want to start one?

Who me? Perish the thought. I think we can settle for "Yes, Caroll disagrees with me" or "No, when Caroll says

Thus a light ray which approaches r = 2GM never seems to get there, at least in this
coordinate system; instead it seems to asymptote to this radius.
As we will see, this is an illusion, and the light ray (or a massive particle) actually has no
trouble reaching r = 2GM. But anobserver far awaywouldnever be able to tell. Ifwe stayed
outside while an intrepid observational general relativist dove into the black hole, sending
back signals all the time, we would simply see the signals reach us more and more slowly. This should be clear from the pictures, and is confirmed by our computation of &)1/&)2 when we discussed the gravitational redshift (7.61). As infalling astronauts approach r = 2GM, any fixed interval &)1 of their proper time corresponds to a longer and longer interval &)2 from our point of view. This continues forever; we would never see the astronaut cross r = 2GM, we would just see them move more and more slowly (and become redder and redder, almost as if they were embarrassed to have done something as stupid as diving into a black hole).

The fact that we never see the infalling astronauts reach r = 2GM is a meaningful
statement, but the fact that their trajectory in the t-r plane never reaches there is not. It
is highly dependent on our coordinate system, and we would like to ask a more coordinateindependent question (such as, do the astronauts reach this radius in a finite amount of their proper time?). The best way to do this is to change coordinates to a system which is better behaved at r = 2GM. There does exist a set of such coordinates, which we now set out to find.

that's just what I've been saying all along... :-)

I'm open to short, focused discussions as my time and interest permit, of course.

 

[DrGreg]

But with the Earth map it is clear why we can't do that - surface of Earth and surface of flat piece of paper are different in 3D. But globe is not very handy for carrying around so we use flat piece of paper instead.

But what about GR maps? What is the correct embedding? Is it related to extra dimension or distortion of measurement system?

In theory you could embed in extra dimensions. But you don't need an embedding at all. All you need is a map and the correct formula (i.e. the metric) for converting map-distance to real-distance/time.

 

[zonde]

I'm open to short, focused discussions as my time and interest permit, of course.

Is white hole and black hole the same thing or two different things?

 

[zonde]

In theory you could embed in extra dimensions. But you don't need an embedding at all. All you need is a map and the correct formula (i.e. the metric) for converting map-distance to real-distance/time.

Do we need a map? As I perceive it, this map is measurement system distortion type embedding. If you say we need a map I say this means we need embedding.

 

[DrGreg]

Do we need a map? As I perceive it, this map is measurement system distortion type embedding. If you say we need a map I say this means we need embedding.

In this analogy, the map is the coordinate system. Or, to be more precise, it's a diagram drawn using a particular coordinate system. If you draw a diagram using Schwarzschild coordinates, the diagram is the "map" of part of the spacetime around a black hole or spherically symmetric mass. If you draw a diagram using Kruskal coordinates, the diagram is a different "map" of part of the spacetime around a black hole or spherically symmetric mass.

Why do you need to know about an embedding? The map with its metric has all the information you need.

If it helps you understand the concept, you can certainly consider that the embedding exists (as a mathematical construct). It's just that there's no need to calculate what it is.

 

[Nugatory]

If you say we need a map I say this means we need embedding.

An embedding is most useful as an aid to visualizing curvature - provided that there are no more than three dimensions involved, so that we can visualize it.

But embedding is not necessary. Given enough time and sufficiently accurate distance and angle measuring instruments, I could construct a complete description of the two-dimensional surface of the earth, one that would allow me to calculate the distance between any two points and the angles between any two lines on that surface. And I could do all this while working only with two dimensions, never using any third dimension and certainly not embedding my two-dimensional surface into a third dimension.

 

[zonde]

DrGreg and Nugatory,
When you speak about embedding you mean curvature in extra dimension. But I don't mean that. Have you heard about Einstein's marble table analogy?

EDIT: Thought that rather well known example would be variable coordinate speed of light type embedding. Using variable coordinate speed of light type we can embed curved spacetime within Euclidean spacetime using isotropic coordinates.

 

[pervect]

Does the returning twins age difference depend on a concept of absolute time?

No

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?

One of the lessons one should learn from SR before GR is that there isn't a universal concept of "now", and that hence the problem of determining which of two spatially separated clocks is faster or slower is in general ambiguous. For in order to compare two clocks, one first needs a concept of "now" to do the comparison.

Hence the title of this thread - "notions of simultaneity in strongly curved space-time".

The notion of time dilation can (and IMO should) be understood as comparing proper time (the time measured by a clock) to coordinate time. So time dilation, understood in this manner, obviously becomes a coordinate dependent notion.

Within the framework of a system of "static observers", the notion that time slows down works pretty well, and one might forget for a moment (if one's learned it in the first place) that simultaneity is relative. But when one broadens one'sr class of observers to include non-static observers such as infalling ones, the idea that "time slows down" becomes an obstacle to understanding, just as it does in special relativity with the twin paradox.

 

[zonde]

One of the lessons one should learn from SR before GR is that there isn't a universal concept of "now", and that hence the problem of determining which of two spatially separated clocks is faster or slower is in general ambiguous. For in order to compare two clocks, one first needs a concept of "now" to do the comparison.

Hey, this is not true. You don't need concept of "now" to determine which clock is faster. You just have to have concept of static position in center of mass reference frame i.e. you just have to have some static background against which you can define static position (for example, planet surface).