Notions of simultaneity in strongly curved spacetime

[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).

 

[PeterDonis]

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.

 

[zonde]

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".

 

[PeterDonis]

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.

 

[pervect]

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 let's 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).

 

[pervect]

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.

 

[zonde]

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.

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.

 

[PeterDonis]

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.

 

[zonde]

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 let's 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.

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.

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.

 

[zonde]

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?

 

[PeterDonis]

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.

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.)

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.

 

[pervect]

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.

 

[PeterDonis]

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.

 

[zonde]

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.

 

[PeterDonis]

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.

 

[harrylin]

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:

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

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. [..]

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 realize 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/Rela...nces_from_the_General_Principle_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/Rela...ument_for_the_General_Postulate_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/physics/Relativity/SR/TwinParadox/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 realize 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.

 

[harrylin]

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/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?

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".

 

[PeterDonis]

According to Eve's view of reality (I suddenly realize 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"?

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.

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.

"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).

Well, you still don't seem to realize that logically exactly the *inverse* reasoning applies to the case of a black hole.

What "inverse reasoning". Spell it out, please.

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.

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"?

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).

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.

 

[PeterDonis]

"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)?

 

[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. [...]

[..] 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?

 

[PeterDonis]

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.

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.