Notions of simultaneity in strongly curved spacetime

by PAllen
Tags: curved, notions, simultaneity, spacetime, strongly
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 Quote by zonde You don't need to correct for light travel time as this does not change result. You are subtracting the same value from starting point and ending point so the difference between starting point and ending point stays the same no matter what correction you make. But of course static spacetime (spacetime with static curvature) is needed for this to work.
Yes, exactly; "you are subtracting the same value from starting point and ending point" is only true in a static spacetime. More precisely, it is only true in a static spacetime *region*; there are spacetimes (such as Schwarzschild spacetime) which are static in one region (outside the horizon) but not static in another region (inside the horizon). Your definition of "which clock runs faster" only works in the static region of such spacetimes.

You are correct that, strictly speaking, your definition of "which clock runs faster" does not "require" a concept of "now"; you are basically using null curves as references, whereas the other definition of "which clock runs faster" uses spacelike surfaces of constant time, i.e., "now" surfaces, as references. But the difference is really immaterial: both definitions only work in static spacetime regions, so they both cover exactly the same set of cases; and one can always translate freely between them, so there is no reason other than personal preference for choosing one over the other.
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 Quote by pervect The metric gives you the Lorentz interval between any pair of points in space-time that are sufficiently close together. You can use this information to get distances, as long as you define exactly your notion of simultaneity. This definition of simultaneity defines how you split the Lorentz interval, which is a space-time interval and independent of the observer, into a part that's purely space-like (this depends on the observer) and a part that's purely time-like (which also depends on the observer). This is the domain of SR, and its my impression that a lot of people get lost at this point. Once you've managed the notion of simultaneity, you can slice 4-d space-time into a bunch of 3-d hypersurfaces of simultaneity. The distance then becomes defined in the usual way one defines distance on a possibly curved manifold. You can use the 4-d techniques to find the Lorentz interval between any two nearby points on hypersurface, and because you've defined the time difference to be zero you know that this Lorentz interval gives you the proper distance between the nearby points. So you've got an "induced metric" that lets you find the distance between any two nearby points on the hypersurface. Given the infinite set of distances between all nearby points, you can find the curve of lowest distance connecting your two points, and call this the distance.
Sorry, with distances I meant spacetime distances not space distances.

 Quote by pervect All the coordinate system needs to do is to assign all points in space-time a unique label that identifies it. That's pretty much it. Once you've defined your labeling system, the metric provides the mecchanism for finding the Lorentz interval between points.
Hmm, you need numbers. Just labels won't work.

 Quote by pervect The metric IS the space-time map, as described by Misner: http://arxiv.org/abs/gr-qc/9508043
The statement sounds like: function defines it's arguments. But this just does not sound right.
But he explains what he means with additional statements and it requires a bit of thinking over.
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 Quote by pervect The static frame DOES provide a unique defintion of "now" - in the region external to the black hole at least. Use of the static frame's defintion of "now" is fine as long as none of your observers are moving. When you start to have moving observers (such as the ones falling into a black hole), the moving observers will have a different defintion of "now" than the static frame has.
Have you anything to say about SC coordinates vs GP coordinates?
To me it seems that they have different "now" and that is the main difference between them.

GP is based on time of moving observers but coordinate orgin is the same as for stationary observer and radial distance too is from SC coordinates.

PeterDonis: you made the same (or very similar) statement. What do you think about "now" of SC vs "now" of GP coordinates?
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 Quote by zonde PeterDonis: you made the same (or very similar) statement. What do you think about "now" of SC vs "now" of GP coordinates?
The SC coordinate chart does have a different set of "now" surfaces--surfaces of constant coordinate time--than the GP coordinate chart does. The GP surfaces are "tilted", so to speak, compared to the SC surfaces, because the GP surfaces are orthogonal to the worldlines of infalling observers, while the SC surfaces are orthogonal to the worldlines of "hovering" observers.

 Quote by zonde GP is based on time of moving observers
Yes, in the sense that the GP surfaces of constant time are orthogonal to the worldlines of infalling observers, so GP coordinate time is the same as proper time for those observers. However, the infalling observers do not stay at the same spatial coordinates in the GP chart; curves of constant r (and theta, phi if we include the angular coordinates) in the GP chart are the worldlines of "hovering" observers, just as they are in the SC chart. (Note, though, that that doesn't mean the r coordinate in the GP chart is exactly the same in all respects as the r coordinate in the SC chart--see below.)

 Quote by zonde radial distance too is from SC coordinates.
No, "radial distance" is *not* the same in GP coordinates as in SC coordinates. What is the same is the labeling of 2-spheres by the radial *coordinate* r--in both charts, r is defined such that the physical area of a 2-sphere labeled by r is 4 pi r^2. But the radial distance between the same pair of 2-spheres is different in GP coordinates than in SC coordinates; that's obvious just from looking at the coefficient of dr^2 in the line element (it's 1 in GP coordinates, but it's 1/(1 - 2m/r) in SC coordinates). That's because radial distance is evaluated in a surface of constant coordinate time, and as I said above, the two charts use different sets of surfaces of constant time.
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 Quote by zonde Have you anything to say about SC coordinates vs GP coordinates? To me it seems that they have different "now" and that is the main difference between them.
I don't think I've said much about them.

Offhand, I don't see any problem with your statement about the main difference between GP coordinates and SC coordinates being the assignment of the time coordinate. Perhaps problems with it will show up later, but at the moment I think it's OK.

GP coordinates are sort of a hybrid coordinate system, they've got the time coordinates of the infalling observers mixed with the space coordinates of the static observers. But they're mathematically pretty convenient to use for many purposes.
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 Quote by pervect GP coordinates are sort of a hybrid coordinate system, they've got the time coordinates of the infalling observers mixed with the space coordinates of the static observers.
I would add a caution about interpreting this statement, though; as I pointed out in my last post, even though the spatial coordinates assigned to events are the same in both charts, the relationship between radial coordinate differentials and radial distances is different in the two charts.
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 Quote by pervect Offhand, I don't see any problem with your statement about the main difference between GP coordinates and SC coordinates being the assignment of the time coordinate.
But you said: The static frame DOES provide a unique defintion of "now"
So where is the catch? We have two coordinate systems with different "now", object with static spatial coordinates in one coordinate system has static spatial coordinates in other coordinate system as well.
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 Quote by zonde We have two coordinate systems with different "now", object with static spatial coordinates in one coordinate system has static spatial coordinates in other coordinate system as well.
But in the static coordinate system (SC coordinates), the metric is diagonal; that means the surfaces of constant SC time are orthogonal to the worldlines of objects with static spatial coordinates. And *that* means the definition of "now" given by SC coordinates is the *same* as the definition of "now" given by the local inertial frames along the worldlines of objects with static spatial coordinates.

In the non-static coordinate system (GP coordinates), the metric is not diagonal; there is a dt dr "cross term" in the line element. That means the surfaces of constant GP time are *not* orthogonal to the worldlines of objects with static spatial coordinates. And that means the definition of "now" given by GP coordinates is *different* than the definition of "now" given by the local inertial frames along the worldlines of objects with static spatial coordinates.

So the sense in which the definition of "now" given by static (SC) coordinates is "unique" is that it is the only one that matches up with the definition of "now" in the local inertial frames of static observers.
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Now that my urgent questions concerning Oppenheimer-Snyder having been answered (thanks Peter), I'm returning to this thread. Atyy gave here an interesting link on which I already commented there. Retake:
 Quote by atyy Greg Egan gives a similar situation in special relativity. http://gregegan.customer.netspace.ne...erHorizon.html (See the section "free fall")
Quote by harrylin
Atyy gave a for me useful reference about a nearly equivalent system with accelerating rockets [..]. The interesting phrase for me is:

"Eve could claim that Adam never reaches the horizon as far as she's concerned. However, not only is it clear that Adam really does cross the horizon".

I agree with that, but it appears for different reasons than some others.

In fact, according to 1916 GR, Eve's point of view is equally valid as that of Adam; according to that, acceleration and gravitation are just as "relative" as velocity, and their coordinate systems are valid GR systems.
However, the interpretation of what "really" happens is very different, even qualitatively; and in modern GR many people reject "induced gravitation" and agree that we can discern the difference between gravitation and acceleration.

We thus distinguish in that example that Eve's acceleration is real, and that her gravitational field is only apparent because the effect is not caused by the nearby presence of matter. For that reason I think that we should prefer Adam's interpretation. Similarly, in case of a real gravitational field that we ascribe to the presence of matter, it is Eve's interpretation that we should prefer. [..]
 Quote by PeterDonis For the region of spacetime that both coordinate systems cover, yes, this is true. However, if Adam's coordinate system covers a portion of spacetime that Eve's does not (in the scenario on Egan's web page, Adam's coordinates cover the entire spacetime, but Eve's only cover the wedge to the right of the horizon), then Eve's "point of view" will be limited in a way that Adam's is not.
According to Eve's view of reality (I suddenly realise that "perspective" can be misleading) her view is not limited at all.
I wonder if you mean that a symmetrical interpretation can be valid. That can't be correct: Eve is the one who fires the rocket engines and feels a force, in contrast to Adam. Compare https://en.wikisource.org/wiki/Relat..._of_Relativity
 References, please? In "modern GR", people recognize that the word "gravitation" can refer to multiple things. If it refers to "acceleration due to gravity", then "modern GR" agrees with "1916 GR" that "gravitation" can be turned into "acceleration" by changing coordinates, so both are "relative" in that sense.
That is the exact contrary - Einstein mentioned in his 1911 paper and in both his 1916 papers that not all gravitational fields can be turned into acceleration by changing coordinates, because only homogeneous fields can be made to vanish. See for example:
"This is by no means true for all gravitational fields, but only for those of quite special form. It is, for instance, impossible to choose a body of reference such that, as judged from it, the gravitational field of the earth (in its entirety) vanishes."
[..]
Even though by no means all gravitational fields can be produced in this way [= from acceleration], yet we may entertain the hope that the general law of gravitation will be derivable from such gravitational fields of a special kind. "
- starting from section 20 of: https://en.wikisource.org/wiki/Relat..._of_Relativity

And a modern point of view (for there is by far no unity):
"A gravitational field due to matter exhibits itself as curvature in spacetime. [..] modern usage demotes the uniform "gravitational" field back to its old status as a pseudo-field. "
- http://math.ucr.edu/home/baez/physic...x/twin_gr.html
 you still don't appear to realize that exactly the *same* reasoning applies to the case of a black hole.
Well, you still don't seem to realise that logically exactly the *inverse* reasoning applies to the case of a black hole. Perhaps we won't be able to convince each other, due to incompatible bases of reasoning. And as Wheeler noticed, we can never verify it so that this is in fact personal opinions and philosophy...
 In the Adam-Eve scenario, Eve can easily compute that the proper time along Adam's worldline [..] region of spacetime [..]
Sorry, once more: those are for me mere mathematical terms. Their physical meaning depends on their physical application:
 If Eve were hovering above a black hole, and Adam stepped off the ship and fell in, *exactly* the same reasoning would apply. [..]
According to Adam, clocks at different locations in Eve's accelerating rocket tick at nearly the same rate (small difference, only due to Lorentz contraction) and you hold that Adam should follow exactly the same reasoning for a gravitational field - correct?
In contrast, according to Einstein, clocks in a gravitational field go at different rates - much more different than what he should conclude according to you.
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 Quote by pervect It does require closer inspection to see if the apparent singularity in the equations of motion is removable or not. [..]
In fact, I don't think that that is really an issue; I found that the real issue is interpretation (and thus metaphysics) - not math. Thanks anyway - your explanation could be useful for others.
 [..] 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.
I'm not up to the math (tensors are just not my thing), and by chance the only textbook on GR that I have in my possession dates from before black holes.
 [..] 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?
I looked it up (interesting, thanks!) and I note that he has a different opinion of reality than I have. In my experience, only opinions about verifiable facts can be argued in a convincing way for those who are of a contrary opinion. Do you disagree?
 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?
I know and can explain the term "absolute time". I never heard of "non-absolute time", but logically it should be expected to mean the same as "relative time". And I don't think that my reasons for "time slowing down before the event horizon" require the existence of "absolute" time, already for the simple reason that Einstein did not believe in absolute time but had no issue with Schwartzschild's solution on the essential point that, as he put it, "a clock kept at this place would go at the rate zero".
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 Quote by harrylin According to Eve's view of reality (I suddenly realise that "perspective" can be misleading) her view is not limited at all.
Are you including the events that Eve calculates must exist, but can't receive light signals from (i.e,. events behind the Rindler horizon), in her "view of reality"?

 Quote by harrylin I wonder if you mean that a symmetrical interpretation can be valid. That can't be correct: Eve is the one who fires the rocket engines and feels a force, in contrast to Adam.
You're correct that Eve and Adam are in physically different states of motion. I'm not sure how that impacts their ability to have a "symmetrical interpretation". Both can make the same computations.

 Quote by harrylin That is the exact contrary - Einstein mentioned in his 1911 paper and in both his 1916 papers that not all gravitational fields can be turned into acceleration by changing coordinates, because only homogeneous fields can be made to vanish.
 Quote by harrylin "A gravitational field due to matter exhibits itself as curvature in spacetime. [..] modern usage demotes the uniform "gravitational" field back to its old status as a pseudo-field."
These quotes are from popular presentations, and it doesn't appear to me that you fully understand the actual theory underlying them; or at any rate you are leaving out important context. I'm not sure it's worth trying to disentangle all that, because in your response to the exchange between me and Mike Holland in the other thread you said (or appeared to say) that you did not intend to question the equivalence principle; and as long as you accept the equivalence principle, I don't think we need to pursue this sub-thread about what "gravitational field" means further (since the reason I brought it up was that it appeared that you were contradicting the equivalence principle).

 Quote by harrylin Well, you still don't seem to realise that logically exactly the *inverse* reasoning applies to the case of a black hole.
What "inverse reasoning". Spell it out, please.

 Quote by harrylin Perhaps we won't be able to convince each other, due to incompatible bases of reasoning.
I don't think the bases of our reasoning are incompatible; I just think you are reasoning incorrectly from our common bases. For an example, see below.

 Quote by harrylin Sorry, once more: those are for me mere mathematical terms. Their physical meaning depends on their physical application
Which I have described already. Do you really not understand what the physical meaning of "proper time" is? It's at the foundation of the physical interpretation of relativity.

"Region of spacetime" I can see being a bit more difficult because it's not a standard term; but its physical interpretation is no more difficult than the interpretation of the term "spacetime" itself, and you don't seem to have any problem with that. Or do you? Do you think "spacetime" itself is a "mere mathematical term"?

 Quote by harrylin According to Adam, clocks at different locations in Eve's accelerating rocket tick at nearly the same rate (small difference, only due to Lorentz contraction)
No, according to Adam, clocks at different locations in Eve's accelerating rocket are moving at different speeds. The clock at the nose of Eve's rocket is moving more slowly, according to Adam, than the clock at the tail of the rocket; so the clock at the nose will be ticking faster, according to Adam, than the clock at the tail (slower motion = less time dilation).

 Quote by harrylin and you hold that Adam should follow exactly the same reasoning for a gravitational field - correct?
Yes, the reasoning is "the same", but it's the correct reasoning I just gave, not the incorrect reasoning you gave: the clock at the nose is "higher up" in the gravitational field, so it runs faster.
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 Quote by harrylin "a clock kept at this place would go at the rate zero".
A quick comment: do you see how this statement of Einstein's makes an implicit assumption that it is *possible* for a clock to be "kept at this place" (i.e., at the horizon). Have you considered what happens if that assumption is false--i.e., if a clock *cannot* be "kept" at the horizon (because it would have to move at the speed of light to do so, and no clock can move at the speed of light)?
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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. [...]
 Quote by PeterDonis [..] 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.
I did not pretend that all predictions are for verifiable to us; and you made a good case that these different interpretations cannot be tested by experiment. Note that this is very different from SR's "relativity of simultaneity", which relate to mutually verifiable events that different systems of observation agree on as possibly going to take place.
 No, a "Kruskal observer" says that the asymptotic observer is claiming too much (see above).
If so, then there are some others here who make unwarranted claims about what Kruskal says.
 [..] 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.
That is merely a mutual misunderstanding of terms: I mean with "asymptotic observer" a coordinate system, corresponding to what you call the "outside map". If that is confusing for you then I will try to use another term - perhaps "SC observer" will do?
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 Quote by harrylin If so, then there are some others here who make unwarranted claims about what Kruskal says.
Kruskal himself, or a "Kruskal observer"? If you intended both of these terms to refer to the actual physicist/mathematician, then I misinterpreted what you were saying; I thought that by "Kruskal observer" you meant "someone calculating things using the Kruskal chart". Kruskal himself did not do all the calculations that can be done with that chart, nor did he claim it was the only valid one.

 Quote by harrylin That is merely a mutual misunderstanding of terms: I mean with "asymptotic observer" a coordinate system, corresponding to what you call the "outside map". If that is confusing for you then I will try to use another term - perhaps "SC observer" will do?
If you mean "coordinate chart", then say "coordinate chart". "Observer" does not mean "coordinate chart".

Of course, if you start saying "coordinate chart" when that's what you mean, it will become more evident that many of the things you are saying are dependent on which chart you use, meaning that they're not statements about actual physics, just about coordinate charts.
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 Quote by PeterDonis But in the static coordinate system (SC coordinates), the metric is diagonal; that means the surfaces of constant SC time are orthogonal to the worldlines of objects with static spatial coordinates. And *that* means the definition of "now" given by SC coordinates is the *same* as the definition of "now" given by the local inertial frames along the worldlines of objects with static spatial coordinates. In the non-static coordinate system (GP coordinates), the metric is not diagonal; there is a dt dr "cross term" in the line element. That means the surfaces of constant GP time are *not* orthogonal to the worldlines of objects with static spatial coordinates. And that means the definition of "now" given by GP coordinates is *different* than the definition of "now" given by the local inertial frames along the worldlines of objects with static spatial coordinates. So the sense in which the definition of "now" given by static (SC) coordinates is "unique" is that it is the only one that matches up with the definition of "now" in the local inertial frames of static observers.
So you are saying that the catch is that GP coordinates are non-static. But I don't see anything non-static about them. Slices of "now" are identical as we go along time coordinate.

I think that the catch is that in GP coordinates time coordinate is not orthogonal to space (radial) coordinate and in that sense they are not "right".
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 Quote by harrylin In my experience, only opinions about verifiable facts can be argued in a convincing way for those who are of a contrary opinion. Do you disagree?
Made me think once more about such question: what are observable differences between "frozen star" and "black hole" for distant observer? And I think that there are none.
But then there is an argument that "black hole" is somehow more correct extrapolation of known physical laws beyond limits of testability than "frozen star". This type of discussion is just empty.

Reminds me of Feynman's comments on the field of gravity, back in the 60's in his private letter to his wife.
 I am learning nothing. Because there are no experiments this field is not an active one, so few of the best men are doing work in it. The result is that there are hosts of dopes here and it is not good for my blood pressure: such inane things are said and seriously discussed here that I get into arguments outside the formal sessions (say, at lunch) whenever anyone asks me a question or starts to tell me about his "work". The "work" is always: (1) completely un-understandable, (2) vague and indefinite, (3) something correct that is obvious and self evident, but a worked out by a long and difficult analysis, and presented as an important discovery, or, a (4) claim based on the stupidity of the author that some obvious and correct fact, accepted and checked for years, is, in fact, false (these are the worst: no argument will convince the idiot), (5) an attempt to do something probably impossible, but certainly of no utility, which it is finally revealed at the end, fails (dessert arrives and is eaten), or (6) just plain wrong. There is great deal of "activity in the field" these days, but this "activity" is mainly in showing that the previous "activity" of somebody else resulted in an error or in nothing useful or in nothing promising. It is like a lot of worms trying to get out of a bottle by crawling all over each other. It is not that the subject is hard; it is that the good men are occupied elsewhere. Remind me not to come to any more gravity conferences!
There has to be something to discuss that is within limits of testability.
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 Quote by zonde So you are saying that the catch is that GP coordinates are non-static. But I don't see anything non-static about them. Slices of "now" are identical as we go along time coordinate.
That's not enough for coordinates to be static. The slices of "now" also have to be orthogonal to the integral curves of the time coordinate, and as you observe, they're not.
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 Quote by zonde Made me think once more about such question: what are observable differences between "frozen star" and "black hole" for distant observer? And I think that there are none.
It depends on what you mean by "frozen star". Most people, when they use that term, really mean the *same* thing as other people mean by "black hole". In other words, they are using exactly the same spacetime and exactly the same solution of the EFE--their model of the physics is the same. They are just interpreting it differently. But since they're using the same model of the physics, they will make the same predictions for all observables. The difference is just a matter of interpretation.

(IMO, even the "difference in interpretation" is somewhat strained, since the "frozen star" people agree that an object falling into the hole/frozen star/whatever it is will experience only a finite amount of proper time to the horizon. That means that if we use coordinates that are not singular at the horizon, such as GP coordinates, we can assign a *finite* time to the event of any infalling object crossing the horizon. But I don't think we'll get any further with that discussion here.)

It might be possible to come up with a *different* model of a "frozen star", one which used a *different* spacetime and a *different* solution of the EFE, which actually made different physical predictions about what we would see because it was using a different model of the physics. But I've never seen one.

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