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## Notions of simultaneity in strongly curved spacetime

 Quote by pervect 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).

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 Quote by zonde 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).
Which is equivalent to having a concept of "now": "static" means you have a family of "surfaces of constant time" that completely cover the region of spacetime you are interested in, and those surfaces define a concept of "now". And judging which clock is running faster means counting how many ticks of each clock there are between two particular surfaces of constant time, i.e., between two particular "nows"; the clock which has more ticks between the first "now" and the second "now" is the one that is running faster. If you don't have a family of "now" surfaces, you can't make the comparison.
 Recognitions: Gold Member I am interested only in two worldlines and relative rates of proper time along them. Try to draw spacetime diagram. You just project one worldline on other using identical null geodesics. There is no need for concept of "now".

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 Quote by zonde You just project one worldline on other using identical null geodesics.
And what makes two null geodesics "identical"? Such a concept only works in a static spacetime, which, as I said, is equivalent to having a concept of "now". In other words, when you project one worldline on another using null geodesics, and then correct for light travel time, the set of events you define as "now" will be the same as the set of events that are in a surface of constant time as I defined them.

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 Quote by zonde 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.
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.

 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.
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. The process of converting the Lorentz interval into time and space was described previously.

 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 GR, all we require is that every point have some unique way of identifying it via 4 coordinates. This defines a coordinate basis at every point in your space-time. The metric coefficients, expressed in this coordinate basis , tells you how the possibly curved 4-d geometry gives you distances in that particular labelling system.

The metric IS the space-time map, as described by Misner:

http://arxiv.org/abs/gr-qc/9508043

 one divides the theoretical landscape into two categories. One category is the mathematical/conceptual model of whatever is happening that merits our attention. The other category is measuring instruments and the data tables they provide. .... What is the conceptual model? It is built from Einstein’s General Relativity which asserts that spacetime is curved. This means that there is no precise intuitive significance for time and position. [Think of a Caesarian general hoping to locate an outpost. Would he understand that 600 miles North of Rome and 600 miles West could be a different spot depending on whether one measured North before West or visa versa?] But one can draw a spacetime map and give unambiguous interpretations. ... eq 1 $d\tau^2 = [1 + 2(V − \Phi_0)/c^2]dt^2 − [1 − 2V/c^2](dx^2 + dy^2 + dz^2)/c^2$ Equation (1) defines not only the gravitational field that is assumed, but also the coordinate system in which it is presented. There is no other source of information about the coordinates apart from the expression for the metric. It is also not possible to define the coordinate system unambiguously in any way that does not require a unique expression for the metric. In most cases where the coordinates are chosen for computational convenience, the expression for the metric is the most efficient way to communicate clearly the choice of coordinates that is being made. Mere words such as “Earth Centered Inertial coordinates” are ambiguous unless by convention they are understood to designate a particular expression for the metric, such as equation (1).

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 Quote by zonde 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).
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.

Use of the static frame's defintion of "now" also becomes problematical when one wants to examine events at or inside the event horizon, because static observers (and their static frame) no longer exist there.

So people who reloy on the static observer's notion of "now" tend to get confused by trying to apply it as if it existed in regions where it doesn't. As a result we get these long, meandering threads.

So short summary:

Use of the static observers "now" in the external region of a black hole is fine. Trying to apply it to the event horizon or inside a black hole just doesn't work. It also doesn't work if you want to consider moving observers, such as those external to the event horizon who are falling in, if they are moving at relativistic velocities.

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 Quote by PeterDonis And what makes two null geodesics "identical"? Such a concept only works in a static spacetime, which, as I said, is equivalent to having a concept of "now".
This is interesting statement and it is directly related to topic of this thread so it requires attention. As pervect has made the same statement I will write replay to both of you.

 Quote by PeterDonis In other words, when you project one worldline on another using null geodesics, and then correct for light travel time, the set of events you define as "now" will be the same as the set of events that are in a surface of constant time as I defined them.
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.

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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.

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.

 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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