Notions of simultaneity in strongly curved spacetime

[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.

 

[zonde]

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".

 

[zonde]

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.

 

[PeterDonis]

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.

 

[PeterDonis]

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.

 

[zonde]

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.

Hmm, the way you say it gives me feeling that I am just fussing without much reason.

I guess it could be just enough to settle for meaning of "frozen star" as predictions of GR that are testable (falsifiable) by observations from Earth or at least without going on the suicide mission.

 

[harrylin]

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"?

Certainly not: "there is no time for Eve when, in her co-moving inertial reference frame, Adam passes through the horizon. In that sense, Eve could claim that Adam never reaches the horizon as far as she's concerned." It is specified that Eve uses a "co-moving inertial reference frame", which is only justified if she thinks that she is not accelerating. The author suggests further on in his discussion that she should adopt Adam's perception of reality. I agree that that is a more sensible approach, but my opinion is based on the fact that her "gravitational field" looks fictive to me: there is no physical cause that could allow for a difference from SR.

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.

Perhaps we simply misunderstood each other on words; I'm not sure. Probably everyone proposes an asymmetry for such cases.
"They can make the same computations" is very much the "twin paradox". As Einstein explained, a symmetrical interpretation for an asymmetrical physical situation is incompatible with the foundations of GR. In fact, I don't know any theory of physics that violates that principle. In his discussion of the twin paradox (which was, I think, the first time that Einstein gave reason to doubt the reality of his "induced gravitational fields") he details how different the physical interpretation of a gravitational field is from that of acceleration; only the observable phenomena are held to be identical. And that brings me to a related point (I rearrange):

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).

I did not say the contrary - and the point that I tried to make is obscured by your precision. I'll try again. In this illustration it is assumed that Adam uses his newly found rest frame as reference for physical reality. At the moment that Eve starts accelerating away, Adam ascribes the frequency difference that Eve observes to "classical" Doppler; according to him, her rocket has still negligible length contraction so that her clocks go at nearly equal rate. In contrast, Eve claims to be in rest and ascribes the frequency difference to the effect of a gravitational field which makes her clocks go at a different rate. This is just to illustrate how a different interpretation of gravitational fields and acceleration is both necessary and understood.

[..] the reason I brought it up was that it appeared that you were contradicting the equivalence principle). [..] What "inverse reasoning". Spell it out, please.

After some of you brought it up, I elaborated on the equivalence principle because you and several others seem to interpret it as requiring that we can make gravitational fields from matter "vanish" (which is simply wrong), and you seem to deny the physical reality of gravitational fields of matter. You thus claimed that in 1916GR, 'gravitation" can be turned into "acceleration" by changing coordinates'. That is the inverse of the equivalence principle that I have seen proposed in GR (of course, I may have just missed it; if so, please cite it!). GR is based on the assumption of physical reality of gravitational fields and the equivalence between acceleration and a homogeneous gravitational field. What Einstein originally denied was the physical reality of acceleration, which he thought could be "relativised" by pretending that instead a homogeneous gravitational field is induced. If we hold that GR was wrong on that last point, then that merely makes such induced fields "fictive" and acceleration "absolute". Einstein warned for a misconception that you seem to hold, and which I'll now cite it in full:

"From our consideration of the accelerated chest we see that a general theory of relativity must yield important results on the laws of gravitation. In point of fact, the systematic pursuit of the general idea of relativity has supplied the laws satisfied by the gravitational field. Before proceeding farther, however, I must warn the reader against a misconception suggested by these considerations. A gravitational field exists for the man in the chest, despite the fact that there was no such field for the co-ordinate system first chosen. Now we might easily suppose that the existence of a gravitational field is always only an apparent one. We might also think that, regardless of the kind of gravitational field which may be present, we could always choose another reference-body such that no gravitational field exists with reference to it. 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."

[...] Do you really not understand what the physical meaning of "proper time" is? It's at the foundation of the physical interpretation of relativity.

I wonder what you mean with "physical"; certainly nothing measurable! But now that also you talk of "the foundation of the physical interpretation of relativity": I may have overlooked it but I do not find the word "proper" in either "The Foundation of the Generalised Theory of Relativity" or "Relativity: The Special and General Theory". One should expect something that is at the foundation of the physical interpretation of relativity to be easy to find. So: reference please!

Do you think "spacetime" itself is a "mere mathematical term"?

"Mere mathematical" in the sense of Applied Mathematics? Not only I do I think so, GR is based on such thinking:

"The non-mathematician is seized by a mysterious shuddering when he hears of "four-dimensional" things, by a feeling not unlike that awakened by thoughts of the occult. [..] the world of physical phenomena which was briefly called "world" by Minkowski is naturally four dimensional in the space-time sense. For it is composed of individual events, each of which is described by four numbers" - Relativity:The Special and General Theory

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)?

That is completely wrong: he makes no such implicit assumption. Following your misunderstanding, Einstein would have meant with "For v=c all moving objects—viewed from the “stationary” system—shrivel up into plane figures" that it is *possible* for an object to move at c.

[..] I thought that by "Kruskal observer" you meant "someone calculating things using the Kruskal chart". [..] If you mean "coordinate chart", then say "coordinate chart". "Observer" does not mean "coordinate chart". [..]

PAllen introduced the Kruskal chart as giving a picture that differs from the equations of Oppenheimer. However I did not mean "coordinate chart", as I distinguish a chart from the opinion of the user of such a chart - PAllen suggested that the user of a Kruskal chart interprets the inside area as physical reality. Charts do not catch the topic of this thread which concerns human notions. However:

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.

The discussion of this thread appears to be about metaphysics. In order for the mentors not to close it, we should see if there is anything left related to this topic that is either physical in a verifiable sense, or pertaining to official GR theory. But even then, it may be better to start a fresh thread on that, as this thread is getting rather long.

 

[PeterDonis]

It is specified that Eve uses a "co-moving inertial reference frame", which is only justified if she thinks that she is not accelerating.

Eve *is* accelerating; she feels a nonzero acceleration, an accelerometer attached to her reads nonzero, if she stood on a scale it would register weight, etc. There is no way in which she "thinks she is not accelerating".

The term "co-moving inertial reference frame" is more precisely stated as "momentarily co-moving inertial reference frame" (MCIF). You can construct such a frame centered on any event on Eve's worldline; it will be the inertial frame in which Eve is momentarily at rest at the chosen event. This does not mean that Eve is not accelerating; it just means that it is easier to do a lot of the math in an inertial reference frame. The disadvantage of doing this is that, as I said, Eve is only at rest momentarily in any such frame; so if you want to look at the physics along Eve's worldline for any significant period of time, you can't use a single MCIF to do it. So the MCIF isn't a good representation of "Eve's point of view" over any significant length of Eve's time.

An MCIF is also *not* the same as Rindler coordinates; those are non-inertial. The advantage of Rindler coordinates is that they can cover all of Eve's worldline with a single coordinate chart in which Eve is "at rest" (her spatial coordinates in this chart don't change). This makes Rindler coordinates a better candidate for representing "Eve's point of view". But since the coordinates are non-inertial, they behave differently from the inertial coordinates you're used to in SR. For one thing, as has been noted before, Rindler coordinates don't cover the entire spacetime, so if Rindler coordinates describe Eve's "point of view", then as I said before, Eve's point of view is necessarily limited in a way that Adam's is not.

The author suggests further on in his discussion that she should adopt Adam's perception of reality. I agree that that is a more sensible approach, but my opinion is based on the fact that her "gravitational field" looks fictive to me: there is no physical cause that could allow for a difference from SR.

There is no "difference from SR". Rindler coordinates and Adam's coordinates (Minkowski coordinates) both describe the same spacetime; they describe the same geometric object (or at least a portion of it, in the case of Rindler coordinates). That spacetime is flat, so there is no spacetime curvature present. Whether that counts as a "fictive gravitational field", or no "gravitational field" at all, depends on how you define the term "gravitational field". The physics is the same either way.

This brings up a general comment: you are insisting on centering your reasoning around terms like "gravitational field" that are simply not fundamental according to GR. That is why you are having all these problems trying to interpret what's going on. There is no single consistent interpretation of the term "gravitational field" that matches all the physics. There just isn't. To find an interpretation that matches all the physics, you have to give up the term "gravitational field" and change your set of concepts, to include things like "spacetime curvature", "stress-energy", "Einstein Field Equation", etc. instead.

"They can make the same computations" is very much the "twin paradox". As Einstein explained, a symmetrical interpretation for an asymmetrical physical situation is incompatible with the foundations of GR.

How does making the same computations require a "symmetrical interpretation"? In the case of the twin paradox, both twins can compute that the traveling twin will have aged less when they meet up again. In other words, both twins compute an asymmetrical result. The same is true here; both observers (Eve and Adam) compute that Adam's point of view is not limited, while Eve's point of view is. What's the problem?

In this illustration it is assumed that Adam uses his newly found rest frame as reference for physical reality. At the moment that Eve starts accelerating away, Adam ascribes the frequency difference that Eve observes to "classical" Doppler; according to him, her rocket has still negligible length contraction so that her clocks go at nearly equal rate.

I didn't say Adam attributed the frequency difference to "length contraction". I said he attributed it to the fact that the nose of Eve's rocket is moving more slowly than the tail, in Adam's rest frame. A more precise description would actually be pretty much identical to "classical Doppler":

(1) A light beam emitted from the nose of Eve's rocket to the tail will look blueshifted at the tail, because the tail is accelerating towards it (in Adam's rest frame).

(2) A light beam emitted from the tail to the nose will look redshifted at the nose, because the nose is accelerating away from it (in Adam's rest frame).

Nowhere in any of this does "length contraction" appear; the reasoning applies equally well at the moment Adam drops off the rocket and at a later time when the rocket is moving at nearly the speed of light relative to Adam. If you work through the math, the observed blueshift/redshift depends only on Eve's proper acceleration; it does not depend on her instantaneous velocity relative to Adam. So it does not depend on the absolute value of her "length contraction" or "time dilation"; it only depends on the *change* in those values during the time of flight of a light beam across her rocket, which depends on her acceleration.

In contrast, Eve claims to be in rest and ascribes the frequency difference to the effect of a gravitational field which makes her clocks go at a different rate. This is just to illustrate how a different interpretation of gravitational fields and acceleration is both necessary and understood.

Yes, no problem here.

After some of you brought it up, I elaborated on the equivalence principle because you and several others seem to interpret it as requiring that we can make gravitational fields from matter "vanish" (which is simply wrong)

Agreed; the quote you gave from Einstein on this was apposite. I wasn't intending to argue about that.

and you seem to deny the physical reality of gravitational fields of matter.

I certainly didn't intend to deny that matter causes gravity; I was only bringing up issues relative to the term "gravitational field" and what it means. See my comments above.

You thus claimed that in 1916GR, 'gravitation" can be turned into "acceleration" by changing coordinates'.

I suppose I should have included the qualifier "locally", since I was really just trying to affirm the equivalence principle, and the EP only says that you can do this locally (i.e., in a small patch of spacetime centered on a particular chosen event).

That is the inverse of the equivalence principle that I have seen proposed in GR (of course, I may have just missed it; if so, please cite it!).

There are a number of different ways of stating the EP; the Wikipedia page gives a decent overview:

http://en.wikipedia.org/wiki/Equivalence_principle

The key thing I was trying to focus in on is that, in GR, you can always set up a local inertial frame centered on a particular event, in which "the acceleration due to gravity" vanishes. More precisely, you can always set up a local inertial frame centered on a particular event in which the following is true:

(1) The metric in the local inertial frame, at the chosen event, is the Minkowski metric; i.e., it is

ds^2 = - dt^2 + dx^2 + dy^2 + dz^2

where t, x, y, z are the local coordinates in the local inertial frame, whose origin (0, 0, 0, 0) is the chosen event.

(2) The first derivatives of all the metric coefficients are zero at the chosen event; this means that the metric coefficients are the Minkowski ones not just at the chosen event, but in the entire local inertial frame.

The second condition is what ensures that there is no "apparent gravitational field" in the local inertial frame; i.e., that the worldlines of inertial objects (i.e., freely falling objects) are straight lines in the local inertial frame. But this also means that the worldlines of accelerated objects--for example, the worldlines of objects at rest on the surface of the Earth, if we set up a local inertial frame centered on some event on the Earth's surface--are *not* straight lines in the local inertial frame: in fact they are hyperbolas, just like Eve's worldline in Adam's frame. This is the sense in which, locally, we can "make gravity look like acceleration"; we are making objects that are static in the local gravitational field look like accelerated objects in flat spacetime [Edit: and we are also making objects that are freely falling in the local gravitational field, and hence are "accelerating" from the viewpoint of an observer static in the field, look like objects at rest in an inertial frame in flat spacetime.]

But the local inertial frame only covers a small piece of spacetime around the chosen event; how small depends on how curved the spacetime is and how accurate our measurements of tidal gravity are. Spacetime curvature, i.e., tidal gravity, depends on the *second* derivatives of the metric coefficients, and those *cannot* all be set to zero by any choice of coordinates if the spacetime is curved. This is the sense in which we *cannot* "make gravity vanish" by choosing coordinates; the curvature will always be there, as in Einstein's example of being unable to make the gravitational field of the Earth vanish in its entirety by any choice of coordinates.

GR is based on the assumption of physical reality of gravitational fields

I would say it is based on the assumption of the physical reality of *spacetime* as a dynamical object. As I said above, the term "gravitational field" is problematic.

and the equivalence between acceleration and a homogeneous gravitational field.

With the qualifier "locally", and subject to reservations about the term "gravitational field", yes, this is OK.

What Einstein originally denied was the physical reality of acceleration, which he thought could be "relativised" by pretending that instead a homogeneous gravitational field is induced.

This is only true if "acceleration" is interpreted to mean "coordinate acceleration". Einstein never, AFAIK, claimed that *proper* acceleration (i.e., feeling weight, registering nonzero on an accelerometer, etc.) could be relativised. With the proper terminology, Einstein was correct: coordinate acceleration *can* be relativised (again, with the qualifier "locally"), and proper acceleration cannot (which is good since it's a direct observable).

If we hold that GR was wrong on that last point

It wasn't and isn't. See above.

Einstein warned for a misconception that you seem to hold

I don't. See above.

I wonder what you mean with "physical"; certainly nothing measurable!

You don't think proper time is measurable? What do you think your watch measures?

But now that also you talk of "the foundation of the physical interpretation of relativity": I may have overlooked it but I do not find the word "proper" in either "The Foundation of the Generalised Theory of Relativity" or "Relativity: The Special and General Theory".

If you want to learn about how a physical theory actually works, you can't depend on popular books, even if they're written by the person who invented the theory.

One should expect something that is at the foundation of the physical interpretation of relativity to be easy to find. So: reference please!

Try any relativity textbook. MTW talks extensively about proper time. So does Taylor & Wheeler's Spacetime Physics, which may be a better starting point since it is only about the fundamentals of relativity; MTW has a *lot* of other material.

"Mere mathematical" in the sense of Applied Mathematics? Not only I do I think so, GR is based on such thinking:

Einstein used the term "the world of physical phenomena". He wasn't talking about a mathematical abstraction; he was saying that *the real, actual universe* is a four-dimensional thing, which we call "spacetime".

That is completely wrong: he makes no such implicit assumption. Following your misunderstanding, Einstein would have meant with "For v=c all moving objects—viewed from the “stationary” system—shrivel up into plane figures" that it is *possible* for an object to move at c.

I have no idea what you're trying to say here. What you are claiming "follows my misunderstanding" does not follow from what I said at all, as far as I can see.

Perhaps I should belabor this some more, since it is an important point. Here's what you quoted from Einstein: "a clock kept at this place [i.e., at the horizon] would go at rate zero". I can interpret this one of two ways:

(1) Einstein is claiming that a clock can be kept at the horizon, and saying that it would go at rate zero.

(2) Einstein is claiming that a clock *cannot* be kept at the horizon, because if it could, it would go at rate zero, and that doesn't make sense.

Are you interpreting him as saying #1 or #2? If it's #1, the refutation is pretty easy: the clock would have to go at the speed of light, and no clock can do that. So let's look at the other possibility; #2 leads to one of the following:

(2a) A clock can't be kept at the horizon, because if it could, it would go at rate zero, and that doesn't make sense. Therefore the horizon can't exist, and neither can any spacetime inside it.

(2b) A clock can't be kept at the horizon, because if it could, it would go at rate zero, and that doesn't make sense. Therefore the horizon (i.e., a curve of constant r = 2m) can't be a timelike curve, because if it were, a clock could follow it as a worldline. But the horizon could still exist if it were some other type of curve, such as a null curve; and if so, there could also be a region of spacetime inside the horizon, where curves of constant r < 2m are also not timelike.

Einstein, as far as I can tell, believed #2a; but "modern GR" says #2b. I can't tell for sure what Oppenheimer and Snyder thought, since they didn't address the question in their paper; but everyone who has extended their model has come up with #2b as well.

I'll put comments on the "metaphysical" aspects of all this in a separate post.

 

[grav-universe]

(1) Einstein is claiming that a clock can be kept at the horizon, and saying that it would go at rate zero.

(2) Einstein is claiming that a clock *cannot* be kept at the horizon, because if it could, it would go at rate zero, and that doesn't make sense.

Are you interpreting him as saying #1 or #2? If it's #1, the refutation is pretty easy: the clock would have to go at the speed of light, and no clock can do that.

Just thought I'd add my two cents, but if that is your refutation of #1, then we should also add

(3) A clock can't cross the horizon, because if it could, it would have to go at the speed of light, and no clock can do that.


Of course there's still a way out. Due to the extreme gravity at the horizon, all matter might be annihilated so that only massless and timeless particles actually cross, but personally I opt for #2a.

 

[PeterDonis]

(3) A clock can't cross the horizon, because if it could, it would have to go at the speed of light

You stated this wrong. The correct statement is "a clock can't cross the horizon moving outward, because to do so it would have to go at the speed of light." That in no way prevents the clock from crossing the horizon moving inward.

Of course there's still a way out. Due to the extreme gravity at the horizon, all matter might be annihilated so that only massless and timeless particles actually cross, but personally I opt for #2a.

Then you opt incorrectly.

 

[PeterDonis]

I did not mean "coordinate chart", as I distinguish a chart from the opinion of the user of such a chart

I don't get this at all. A coordinate chart is a well-defined mathematical entity, and I understand what it describes: it describes a spacetime, or a portion of one. I don't understand what an "opinion of the user of such a chart" is--at least, not as it relates to any sort of actual physics.

PAllen suggested that the user of a Kruskal chart interprets the inside area as physical reality.

This seems very confused to me. The Kruskal chart, like any chart, maps 4-tuples of numbers to points of a geometric object. If you are trying to say that that, in itself, is "just mathematics", and doesn't necessarily have any physical interpretation, I agree. But such mathematical objects certainly serve as "building blocks" out of which we construct models that *do* have a physical interpretation. For example, we can take a portion of the geometric object described by the Kruskal chart and "glue" it together with another geometric object described by a collapsing FRW chart. The physical interpretation of this model is a spacetime containing a collapsing object such as a star, plus the vacuum region surrounding it.

Of course such a model is idealized; so is every model we use in physics. But its physical interpretation is not a matter of "opinion". Whether or not it's a *valid* model, taking its idealizations into account, is a separate question of what the physical interpretation of the model is.

Charts do not catch the topic of this thread which concerns human notions.

But coordinate charts are how we express the particular human notions that we are talking about. At least, they're a very convenient way of doing so. If you would prefer another way of expressing those notions, fine, please propose one. But you can't just punt on using coordinate charts without giving some other way of making precise, unambiguous statements about the subject under discussion.