Are flowing space models compatible with GR?

[harrylin]

In reverse order, by SC coordinates I mean the Schwarzschild form of metric [..]

Thanks for the clarification!

On the 1393 Einstein paper, I cannot read more of it because your link only allows reading one page. [..] The Oppenheimer-Snyder solution [..] Exactly what to make of this, and whether the result was in any way general, took time to work out. [..] every major feature of the modern view of black holes in classical GR was present and computable in this solution. For example, right from their abstract: "The total time of collapse for an observer comoving with the stellar matter is finite, and for this idealized case and typical stellar masses, of the order of a day"

Probably you need an institutional license for both papers (alternative is the library). And I already summarized the gist of both. Once more: their solution is that what for a non-infalling observer (a so-called "distant" observer) takes infinite time, corresponds to very little proper time for an infalling observer. Einstein concluded that "the Schwartzschild singularities don't exist in physical reality". I have no problem with those conclusions at all; the issue here came from our opinion that Hamilton's model disagrees with such Schwartzschild-based results, which I assume to be compatible with GR.
In this thread we are scrutinizing Hamilton's model, so I won't discuss more comments on the context of this thread in this thread.

 

[PAllen]

Once more: their solution is that what for a non-infalling observer (a so-called "distant" observer) takes infinite time, corresponds to very little proper time for an infalling observer. Einstein concluded that "the Schwartzschild singularities don't exist in physical reality". I have no problem with those conclusions at all; the issue here came from our opinion that Hamilton's model disagrees with such Schwartzschild-based results, which I assume to be compatible with GR.
In this thread we are scrutinizing Hamilton's model, so I won't discuss more comments on the context of this thread in this thread.

Ok, on this core issue, there is no disagreement between Hamilton's model and Schwarzschild conclusions:

- a key point is that your infinite time, for the umpteenth time, is coordinate time, as recognized right at the beginning of Einstein's 1939 paper. Coordinate time is not a physical observervable.

- Both SC metric and Hamilton river model agree that a free faller crosses horizon in finite physical time (time on their clock).

- Both SC metric and Hamilton river model agree that no light, signal, or causal influence can propagate from the horizon or inside to an external observer (while causal influences can freely propagate in the other direction - distant to interior).

It is impossible for them to disagree on any invariant, and all observables are invariants.

 

[Mentz114]

... and all observables are invariants.

We can measure components of tensors ( like x,y,z velocities and tidal forces) which are not invariant but frame dependent. Of course you could mean something different by 'observable'.

 

[PAllen]

We can measure components of tensors ( like x,y,z velocities and tidal forces) which are not invariant but frame dependent. Of course you could mean something different by 'observable'.

This was discussed at length in some threads by Ben Niehoff. A measurement of tensor components is a really a contraction of the tensor with the frame basis of specified world lines (of the instruments). It is thus invariant. In any coordinates or even no coordinates, once you have specified the instruments (thus physically chosen basis), you get the same results for the computation.

 

[Mentz114]

This was discussed at length in some threads by Ben Niehoff. A measurement of tensor components is a really a contraction of the tensor with the frame basis of specified world lines (of the instruments). It is thus invariant. In any coordinates or even no coordinates, once you have specified the instruments (thus physically chosen basis), you get the same results for the computation.

Basis dependent invariants. I can see what you mean. But the result still depends on the choice of instrument basis so it's not invariant in the sense that a scalar like charge (say) is.

Anyhow, thanks for the clarification of 'observable'.

 

[PAllen]

Basis dependent invariants. I can see what you mean. But the result still depends on the choice of instrument basis so it's not invariant in the sense that a scalar like charge (say) is.

Anyhow, thanks for the clarification of 'observable'.

Yes, not like a scalar field. However, a contraction produces a scalar invariant, and a physical measurement in GR is modeled as one or more contractions. This is really just the mathematical implementation of the required goal that changing coordinates you compute in must not change the result of any measurement. Also, that all observers see a given instrument reading the same way (No case of: I see it reading an B field of strength 3 along its axis; you see it reading no B-field. You may know that your instrument will read an E field and no B field, but you don't see my instrument reading any differently than I see it).

 

[Mentz114]

Yes, not like a scalar field. However, a contraction produces a scalar invariant, and a physical measurement in GR is modeled as one or more contractions. This is really just the mathematical implementation of the required goal that changing coordinates you compute in must not change the result of any measurement. Also, that all observers see a given instrument reading the same way (No case of: I see it reading an B field of strength 3 along its axis; you see it reading no E-field. You may know that your instrument will read an E field and no B field, but you don't see my instrument reading any differently than I see it).

Oh I absolutely understand that everyone must agree on what a certain instrument reads.
But different observers instruments may disgree.

 

[PAllen]

Oh I absolutely understand that everyone must agree on what a certain instrument reads.
But different observers instruments may disgree.

Right. And each instrument's measurements may be formulated as contractions of a tensor field and vectors derived from the instrument world line(s), producing one or more scalars. This is what guarantees that different observers and coordinate systems may explain a given instrument's reading differently, but the result will never difffer.

That is, I may say: your instrument measures a B field because it is moving relative to the E field; you say: there is a B field that my instrument (at rest) measures. Mathemetically, the measurement in either frame is characterized by contraction of a tensor field and vectors derived from the instrument.

 

[Mentz114]

Right. And each instrument's measurements may be formulated as contractions of a tensor field and vectors derived from the instrument world line(s), producing one or more scalars. This is what guarantees that different observers and coordinate systems may explain a given instrument's reading differently, but the result will never difffer.

That is, I may say: your instrument measures a B field because it is moving relative to the E field; you say: there is a B field that my instrument (at rest) measures. Mathemetically, the measurement in either frame is characterized by contraction of a tensor field and vectors derived from the instrument.

Everything after the word 'right' is a repetition of what I've already agreed. I'm not trying to argue or disagree.

 

[harrylin]

Here are a few things that may need more elaboration. PAllen, sorry if I did not make the subject matter clear enough for you:

"In this model, space flows like a river through a flat background, while objects move through the river according to the rules of special relativity. " - Hamilton

We are here not discussing a precise transformation equation (which is necessarily only valid for a particular case) but a physical model, which pervect presumed to be a serious one.
Equations fail with a little change of situation, while a physical model accommodates itself to the changed situation. Zonde understood this subtle point of the topic, see my post #29 where I explained this, and his post #38.

A correct, functional physical model does not require tinkering with mapping, although one can make of course maps from it. Which brings me to the next point:

I am not sure you recognize that Hamilton's rive model is a specialized interpretation, of two special case GR geometries. Use of this model is not part of any general understanding of BH, EH, etc. The general modern consensus comes from studying such solutions in coordinate independent ways, and from the global methods developed by Hawking, Penrose, and others. It is not derived or understood by most using Hamilton's river model.
[..]
There are two aspects to the principle of equivalence:

- that acceleration via applied force can be treated (almost) as a gravitational field
- that free fall can be treated (almost) as at rest.

The river model that you think is so anathema is simply defining a 'river' as a particular family of free falling frames. It is absolutely consistent with the second flavor of the principle of equivalence above.

I already stated, without really explaining, that Hamilton puts the equivalence principle on its head. I have been giving examples of how Hamilton's model is inconsistent with any GR reference system - including free fall - from post #29 onward. I'll try to explain it better by building on my earlier illustration in post #56.

First, let's recall GR's equivalence principle; certainly the free fall case is valid and standard use of GR. And no Einstein free fall reference system will break either the law of inertia nor the law of local constancy of the velocity of light.

My illustration of yesterday does not sufficiently "catch" what is going on in Hamilton's model. It comes closer if we take a second Mercator projection, put that one its head, and stick it to the first one, with the North poles against each other. Now the description looks like something much more fancy, as the traveller from the equator to the North pole and on to the equator on the other side does not need to make bends anymore. Of course the resulting patchwork is still not a globe and neither is it really a Mercator projection anymore. And it still has a velocity discontinuity at the North pole, but some people might not notice.
A true Mercator projection is centred around the area of interest, and admits that it provides a distorted perspective except very close to the centre. It is similar with the equivalence principle, which knows no flowing space. I gave the example of a hole in the Earth, or in any heavy piece of matter: Hamilton's model is in several ways a fake Einstein free fall frame, and it breaks GR's laws of inertia and local speed of light. This is why I concluded that not only in spirit but also in action, the model does not explain GR; it is its antithesis. It is not compatible with GR in that sense, even though it gives for a number of cases the right answers.

 

[PAllen]

Here are a few things that may need more elaboration. PAllen, sorry if I did not make the subject matter clear enough for you:

"In this model, space flows like a river through a flat background, while objects move through the river according to the rules of special relativity. " - Hamilton

We are here not discussing a precise transformation equation (which is necessarily only valid for a particular case) but a physical model, which pervect presumed to be a serious one.
Equations fail with a little change of situation, while a physical model accommodates itself to the changed situation. Zonde understood this subtle point of the topic, see my post #29 where I explained this, and his post #38.

In this case, the whole model is special case. Hamilton not only admits this, but derives it in has paper. Every part of:

"In this model, space flows like a river through a flat background, while objects move through the river according to the rules of special relativity.

applies only to two special geometries. If you want to be pejorative, the whole river model is a trick for understanding perfectly ideal black holes (or regions of spacetime that match a portion of such - e.g. outside the earth, but not on or in the earth) - and for nothing else in GR. This is why few authors on relativity besides Hamilton (maybe Visser, on occasion?) bother with it.

 

[PAllen]

My illustration of yesterday does not sufficiently "catch" what is going on in Hamilton's model. It comes closer if we take a second Mercator projection, put that one its head, and stick it to the first one, with the North poles against each other. Now the description looks like something much more fancy, as the traveller from the equator to the North pole and on to the equator on the other side does not need to make bends anymore. Of course the resulting patchwork is still not a globe and neither is it really a Mercator projection anymore. And it still has a velocity discontinuity at the North pole, but some people might not notice.
A true Mercator projection is centred around the area of interest, and admits that it provides a distorted perspective except very close to the centre. It is similar with the equivalence principle, which knows no flowing space. I gave the example of a hole in the Earth, or in any heavy piece of matter: Hamilton's model is in several ways a fake Einstein free fall frame, and it breaks GR's laws of inertia and local speed of light. This is why I concluded that not only in spirit but also in action, the model does not explain GR; it is its antithesis. It is not compatible with GR in that sense, even though it gives for a number of cases the right answers.

Most of what you say genuinely makes no sense to me, specifically, what it has to do with the river model.

The river model does not have any violation of local speed of light or inertia that I can see. Light moves through the river at all times with local speed of c relative to the co-moving observer at that point. The river itself, which is a non-observable abstraction, moves at speed > c relative to infinity; but > c coordinate speeds are actually quite common in GR.

I definitely don't see where inertia is violated since SR applies in comoving frame; and each comoving frame is a 'free fall from infinity' frame.

The real break down of the river model is that it wants to describe the motion of the river against a flat background, governed by Newtonian laws (then apply SR at a local frame at each point in the river, and boosts based on the difference in river velocity between points). This conceit can only be made to match GR for the very special geometries described in the paper. [As admitted and proven in Hamilton's paper.]

 

[zonde]

In GR, Einstein felt its biggest contribution was general covariance, which despite controversy about how much it means, meant that all coordinate systems are equal. Einstein viewed the preference for inertial frames in SR a fundamental weakness of the theory. There is no other possible interpretation of Einstein's writing on this.

The reference I made about "don't bring Einstein into it" was a preference for SC coordinates. I stand by the view that Einstein would have considered such a preference an abomination.

In GR, there is no such thing as a global frame even for an inertial body. For a hovering body in SC geometry, there isn't even an inertial local frame, because such a body is not inertial. However, if you want to consider local inertial frames, there is an unambiguous answer that is coordinate independent (because local frames are just a matter of the local basis on a world line) - an inertial frame crosses the event horizon in finite time in that frame, and continues to the singularity. This was proved by Robertson in the early 1940s.

Your view of coordinate system features is quite wrong. GP coordinates represent a collection of free fall frames which is the GR analog of rest frames. SC coordinates represent frames of non-inertial observers, with proper acceleration approaching infinite for near horizon.

I am not going to respond to all your statements in this post as it will lead us away from topic at hand. So just one thing now.

When we speak about gravity of gravitating body we always view it in the rest frame of gravitating body.

Do you agree?

 

[PAllen]

I am not going to respond to all your statements in this post as it will lead us away from topic at hand. So just one thing now.

When we speak about gravity of gravitating body we always view it in the rest frame of gravitating body.

Do you agree?

No, not at all. What if there are two or three similar gravitating bodies in mutual motion? Then, in Newtonian physics, you might view it in the COM frame, but it wouldn't by any simpler than any inertial frame (in the Newtonian sense). In GR, for such a scenario, there is no such thing as COM frame. You pick any reasonable coordinates (not frame, since the problem isn't local, and frames are strictly local in GR).

It seems you cannot let go of the idea that there really are preferred frames and preferred coordinates. That is the antitheses of relativity.

 

[zonde]

It seems you cannot let go of the idea that there really are preferred frames and preferred coordinates.

Yes indeed.
In SR they are frames where speed of light is isotropic (and has particular value).

That is the antitheses of relativity.

From Einstein's 1905y SR paper:
"... the same laws of electrodynamics and optics will be valid for all frames of reference for which the equations of mechanics hold good. We will raise this conjecture (the purport of which will hereafter be called the “Principle of Relativity”) to the status of a postulate ..."
"Let us take a system of co-ordinates in which the equations of Newtonian mechanics hold good."

I think you are a bit confused.

 

[PAllen]

Yes indeed.
In SR they are frames where speed of light is isotropic (and has particular value).From Einstein's 1905y SR paper:
"... the same laws of electrodynamics and optics will be valid for all frames of reference for which the equations of mechanics hold good. We will raise this conjecture (the purport of which will hereafter be called the “Principle of Relativity”) to the status of a postulate ..."
"Let us take a system of co-ordinates in which the equations of Newtonian mechanics hold good."

I think you are a bit confused.

That is SR. I agree for SR. The issue is GR. Einstein considered these features of SR a defect. In GR there are no global inertial frames at all. In Einstein's view, general covariance in GR removed these defects (irrespective of whether it was a 'theory filter' - which in 1917, he conceded to Kretschmann that it was not).

Further, in GR, SC coordinates are as far as you can imagine from being an 'inertial frame'. As I mentioned, lines of constant r have proper acceleration approaching infinity near the EH. It is actually GP coordinates that are built around a family of inertial frames. One sign of this is that 3-spaces of constant GP time are flat, Euclidean space; 3-spaces of constant SC time are non-Euclidean.

 

[harrylin]

[..]
The river model does not have any violation of local speed of light or inertia that I can see. Light moves through the river at all times with local speed of c relative to the co-moving observer at that point. [..]
I definitely don't see where inertia is violated since SR applies in comoving frame; and each comoving frame is a 'free fall from infinity' frame.

I gave the map example but forgot to mention that of course no local observer on a Mercator projection notices anything strange, perhaps not even when crossing the patchwork boundary between two projections. Similarly I see nothing noticeable with the local Hamilton frame from the local frame's perspective. The defects are noticed in the mapping from the other frame ("the distant perspective") in Hamilton's model. GR demands that both descriptions obey the laws of nature that I mentioned earlier.

The real break down of the river model is that it wants to describe the motion of the river against a flat background, governed by Newtonian laws (then apply SR at a local frame at each point in the river, and boosts based on the difference in river velocity between points). This conceit can only be made to match GR for the very special geometries described in the paper. [As admitted and proven in Hamilton's paper.]

That's also a good point! Patchwork like that may be expected to have multiple failures.

 

[harrylin]

While I think that we have covered the topic well enough by now, there are a few loose ends:

[..] When we speak about gravity of gravitating body we always view it in the rest frame of gravitating body.

Do you agree?

I disagree, but of course everyone will agree that such a rest frame is a valid, relevant and often very convenient perspective.

Yes indeed.
In SR they are frames where speed of light is isotropic (and has particular value).

Similarly, in GR the speed of light in a small enough region is constant.

From Einstein's 1905y SR paper:
"... the same laws of electrodynamics and optics will be valid for all frames of reference for which the equations of mechanics hold good. We will raise this conjecture (the purport of which will hereafter be called the “Principle of Relativity”) to the status of a postulate ..."
"Let us take a system of co-ordinates in which the equations of Newtonian mechanics hold good." [..]

Of course, he meant it then to indicate a class of reference systems; and he extended that class with GR such that any form of motion became "relative" (the modern point of view is not unanimously in favour of that, see the physics FAQ, "Twin paradox"; but that is not in question in this thread). In Einstein's GR the slightly modified laws of mechanics - including the unmodified law of inertia - are valid "locally" in accelerating and non-accelerating reference systems.

 

[PAllen]

I gave the map example but forgot to mention that of course no local observer on a Mercator projection notices anything strange, perhaps not even when crossing the patchwork boundary between two projections. Similarly I see nothing noticeable with the local Hamilton frame from the local frame's perspective. The defects are noticed in the mapping from the other frame ("the distant perspective") in Hamilton's model. GR demands that both descriptions obey the laws of nature that I mentioned earlier.

No, that's not right. GR make no such demands, and cannot. Coordinate speed of light varies from c (either greater or lesser) in almost all coordinates in GR, including SC coordinates (in fact, I think it can be proven that in general spacetimes it is impossible to establish coordinates where the coordinate speed of light is c everywhere, in all directions). There is no such thing in GR as a 'distant perspective'. The belief that coordinate quantities SC coordinates represent physical characteristics of a 'distant perspective' is a classic error of giving physical meaning to coordinate quantities. In any coordinates, you get predictions about what any observer measures or sees about distant events by computing invariants as I have explained in other posts here. Any and all coordinates are tools to that end. You prefer one coordinate system over another primarily because it makes some class of calculations easier. In the case of SC geometry, each of the popular coordinates makes different cases easier to calculate or visualize.

That's also a good point! Patchwork like that may be expected to have multiple failures.

It is more of design limit than a failure. If I say: for x > 0, f(x)=x^2 is a bijection, do you respond: Well that fails if x allowed to be any real number? That's silly because it contradicts the hypothesis. Hamilton's paper and the river model are making the statement:

For SC or Kerr-Newman geometry (and no others), you can use this model to make exact predictions and visualize behavior.

It is then equally silly to respond: well, for other spacetimes you can't. You can argue (and this is my general opinion), that its limited applicability make it not a particularly useful method. Hamilton may argue (correctly) that most situations in astronomy are characterized by large regions where the river model applies to a good approximation, and it is easy to know when to not apply it. It's not a right/wrong decision.

 

[zonde]

That is SR. I agree for SR. The issue is GR. Einstein considered these features of SR a defect. In GR there are no global inertial frames at all. In Einstein's view, general covariance in GR removed these defects (irrespective of whether it was a 'theory filter' - which in 1917, he conceded to Kretschmann that it was not).

Where did you get that Einstein considered SR defective?
SR certainly is not defective. It has limited applicability. And GR is meant to overcome this limit. But that means one important thing - GR should reduce to SR at appropriate limits.
And that in turn means that if two different coordinate systems when reduced (under appropriate limits) to SR give coordinate systems that are not related by Lorentz transform then they describe two different physical situations.

So we can try to compare coordinate systems in that sense to find out if they are equal.

Would you still say that this somehow goes against Einstein's view?

One sign of this is that 3-spaces of constant GP time are flat, Euclidean space; 3-spaces of constant SC time are non-Euclidean.

What argumentation you can provide for this statement?

 

[zonde]

No, not at all. What if there are two or three similar gravitating bodies in mutual motion? Then, in Newtonian physics, you might view it in the COM frame, but it wouldn't by any simpler than any inertial frame (in the Newtonian sense).

Well yes, my statement does not seem quite right when considering system of two bodies where we take into account gravity of both bodies. And certainly we are considering such situations.

About your second statement you might want to look at this wikipedia article describing classical two-body[/PLAIN] problem
and how exact solution is found for this two-body problem by using COM frame.
"Adding and subtracting these two equations decouples them into two one-body problems, which can be solved independently. Adding equations (1) and (2) results in an equation describing the center of mass (barycenter) motion. By contrast, subtracting equation (2) from equation (1) results in an equation that describes how the vector r = x1 − x2 between the masses changes with time."

This seems to contradict your statement that COM frame has no special role in calculations (I assume this was the meaning behind word "simpler").

 

[PAllen]

Where did you get that Einstein considered SR defective?
SR certainly is not defective. It has limited applicability. And GR is meant to overcome this limit. But that means one important thing - GR should reduce to SR at appropriate limits.

Einstein was philosophically attracted to Mach's ideas. Ideally, he felt there should be no physical significance to anything except relative motion. The idea of distinguishable accelerated motion in an empty universe was abhorrent to him. He hoped that general covariance and his GR program would show that inertial resistance to acceleration arose from motion relative the distant mass of the universe. SR's preference for inertial frames bothered him and was one of several major motivators for his development of GR. [Einstein later realized GR failed in his Machian objective, but still succeeded, in his view, of displacing or, at least weakening, any special position for inertial frames.]

Of course GR reduces SR sufficiently locally everywhere. That is built into the mathematical structure of pseudo-riemannian geometry in the same way local Euclidean geometry is built into Riemannian geometry.

And that in turn means that if two different coordinate systems when reduced (under appropriate limits) to SR give coordinate systems that are not related by Lorentz transform then they describe two different physical situations.

No, this is not a correct statement of the way GR contains SR. The relationship is local not global. In general, a GR solution has no global coordinates that resemble Minkowski coordinates at all. In a limited sense you can say that for asymptotically flat spacetimes (which, by the way, does not include our universe), there are coordinates systems that approach Minkowski at infinity. However, not only are the 'too many' of them, they are not generally related by Lorentz transforms. This whole statement of your is pretty much a complete misunderstanding of the relationship between SR and GR.

As an aside, you should be aware that the GP coordinates used in the river model reduce Minkowski coordinates at infinity, just like SC ones do. In fact they reduce to exactly the same coordinates at infinity because they share the same center of symmetry and both make explicit the asymptotic flatness of SC geometry.

So we can try to compare coordinate systems in that sense to find out if they are equal.

Would you still say that this somehow goes against Einstein's view?

Yes. The only place for Lorentz transforms in GR is local (in the limit of a small region of spacetime where curvature= tidal gravity can be ignored).

What argumentation you can provide for this statement?

It is actually mathematically obvious. The metric for GP coordinates as given in the paper in post #1 of this thread is:

ds^2 = - dt^2 + (dr + βdt^2)^2 + r^2(dθ^2 + sin^2θd$\varphi$^2)

For a slice of constant t, you have dt=0. Then you have, for the spatial geometry of the slice:

ds^2 = dr^2 + r^2 (dθ^2 + sin^2θd$\varphi$^2)

which is just the flat Euclidean metric in polar coordinates. Interesting, wouldn't you say, for coordinates that have no horizon coordinate singularity and go smoothly through the horizon to the singularity? The horizon itself is there as a physical phenomenon, but there is no coordinate singularity there, and no infinite coordinate time there. Those are artifacts of Schwarzschild coordinates.

Meanwhile, for t=constant slice for Scwharzschild coordinates, using e.g. the form given in:

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

you get:

ds^2 = (1/(1-R/r)) dr^2 + r^2 (dθ^2 + sin^2θd$\varphi$^2)

(where I am using R for SC radius). This is non-Euclidean spatial geometry.

 

[PAllen]

Well yes, my statement does not seem quite right when considering system of two bodies where we take into account gravity of both bodies. And certainly we are considering such situations.

About your second statement you might want to look at this wikipedia article describing classical two-body[/PLAIN] problem
and how exact solution is found for this two-body problem by using COM frame.
"Adding and subtracting these two equations decouples them into two one-body problems, which can be solved independently. Adding equations (1) and (2) results in an equation describing the center of mass (barycenter) motion. By contrast, subtracting equation (2) from equation (1) results in an equation that describes how the vector r = x1 − x2 between the masses changes with time."

This seems to contradict your statement that COM frame has no special role in calculations (I assume this was the meaning behind word "simpler").

Yes, there is advantage to COM frame for two body in Newonian mechanics, but not for 3 body, so far as I know. For GR, there is also a some simplification in the numerical treatment (no exact treatment) of the two body problem in effective COM coordinates (these are called 'effective one body methods' in the literature); again, none for the 3 body problem. So I concede I overstated the case a little for the two body situation.

 

[zonde]

No, this is not a correct statement of the way GR contains SR. The relationship is local not global.

Yes yes local, not global. Appropriate limit is small enough region where we are comfortable with discarding difference between flat SR coordinate system and GR coordinate system.

 

[zonde]

Einstein was philosophically attracted to Mach's ideas. Ideally, he felt there should be no physical significance to anything except relative motion. The idea of distinguishable accelerated motion in an empty universe was abhorrent to him. He hoped that general covariance and his GR program would show that inertial resistance to acceleration arose from motion relative the distant mass of the universe. SR's preference for inertial frames bothered him and was one of several major motivators for his development of GR. [Einstein later realized GR failed in his Machian objective, but still succeeded, in his view, of displacing or, at least weakening, any special position for inertial frames.]

I'm not sure you are interpreting it correctly. There of course is physical significance to accelerated motion but as I understand it he felt that there should be no absolute reason for that physical significance. And with that I tend to agree.
As I imagine it physical significance is acquired from some rather physical field. Einsteins idea would be that properties of this field are determined by distant stars. Have to say I don't feel exactly the same way. As I see there should be something more besides distant stars (some other state of physical matter). But I am saying this just to point out how far I am ready to go along Einstein's viewpoint (as I understand it) and not to start discussion about my viewpoint.

It is actually mathematically obvious. The metric for GP coordinates as given in the paper in post #1 of this thread is:

ds^2 = - dt^2 + (dr + βdt^2)^2 + r^2(dθ^2 + sin^2θd$\varphi$^2)

For a slice of constant t, you have dt=0. Then you have, for the spatial geometry of the slice:

ds^2 = dr^2 + r^2 (dθ^2 + sin^2θd$\varphi$^2)

which is just the flat Euclidean metric in polar coordinates. Interesting, wouldn't you say, for coordinates that have no horizon coordinate singularity and go smoothly through the horizon to the singularity? The horizon itself is there as a physical phenomenon, but there is no coordinate singularity there, and no infinite coordinate time there. Those are artifacts of Schwarzschild coordinates.

Meanwhile, for t=constant slice for Scwharzschild coordinates, using e.g. the form given in:

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

you get:

ds^2 = (1/(1-R/r)) dr^2 + r^2 (dθ^2 + sin^2θd$\varphi$^2)

(where I am using R for SC radius). This is non-Euclidean spatial geometry.

You have minor error in GP metric - an extra ^2. It sould be like this:
ds^2 = - dt^2 + (dr + βdt)^2 + r^2(dθ^2 + sin^2θdφ^2)

Anyways, this is interesting argument and it requires adequate answer. Well, I don't have one.
But it does not solve the question but rather deepens it (at least from my perspective) as it can't address objections that where raised in this thread against river model (btw I am not sure if you didn't understood my argument in post #50 or you simply wanted to examine it more thoroughly for possible flaws).

Speaking about Schwarzschild coordinates for me it seems like it would be more meaningful to speak about isotropic coordinates. But my guess would be that they still would not make simultaneity slice Euclidean. And I would like to make sure that they are physically equivalent to Schwarzschild coordinates.

With isotropic coordinates I mean the one you get from this metric:
$$ds^2=\frac{(1-\frac{\mu}{2r})^2}{(1+\frac{\mu}{2r})^2}dt^2-(1+\frac{\mu}{2r})^4(dr^2+r^2(d\theta^2+sin^2 \theta \; d\varphi^2))$$

 

[harrylin]

No, that's not right. GR make no such demands, and cannot. Coordinate speed of light varies from c (either greater or lesser) in almost all coordinates in GR[..].

I did not suggest anything else!
[/QUOTE] There is no such thing in GR as a 'distant perspective'.[/QUOTE]
Well, that is terminology that others use, see for example Ben's "distant observer" in his book chapter on GR:
- http://www.lightandmatter.com/html_books/lm/ch27/ch27.html
and, not to forget, Hamilton (maybe he influenced me despite everything):
- http://casa.colorado.edu/~ajsh/schwp.html

Of course, in GR there are simply different coordinate systems; in the literature and discussions colourful names are given that are more informative than S and S'.

[..]

For a last time: please stop discussing and criticising here other things than Hamilton's river model.

It is more of design limit than a failure. If I say: for x > 0, f(x)=x^2 is a bijection, do you respond: Well that fails if x allowed to be any real number? That's silly because it contradicts the hypothesis. Hamilton's paper and the river model are making the statement:

For SC or Kerr-Newman geometry (and no others), you can use this model to make exact predictions and visualize behavior. [..]

I illustrated that the same can be said (and is rightly said) for the Mercator projection. [EDIT:] Interestingly, that projection maps a big distance to zero distance; the mapping disagreement is just so to say inverse as with Hamilton's model. I don't think that either is "an interesting argument" that "requires adequate answer".

Now, I really think that the river model has been sufficiently discussed, with this thread nearing 100 posts. For different reasons several of us here including myself do not like it very much and recognise that it doesn't work as a general physical model for GR. Anything else is for another thread.

 

[PeterDonis]

Speaking about Schwarzschild coordinates for me it seems like it would be more meaningful to speak about isotropic coordinates.

These work fine as long as you're only interested in the region at or outside the horizon. Isotropic coordinates don't cover the region inside the horizon. (They actually double cover the region outside the horizon: 0 < r < mu/2 and mu/2 < r < infinity cover the same region.)

But my guess would be that they still would not make simultaneity slice Euclidean.

You're correct, they don't, because of the extra factor in front of the spatial part of the metric, which depends on r.

And I would like to make sure that they are physically equivalent to Schwarzschild coordinates.

They are; why would you think they weren't?

 

[PAllen]

You have minor error in GP metric - an extra ^2. It sould be like this:
ds^2 = - dt^2 + (dr + βdt)^2 + r^2(dθ^2 + sin^2θdφ^2)

Yes, I had a typo.

Anyways, this is interesting argument and it requires adequate answer. Well, I don't have one.
But it does not solve the question but rather deepens it (at least from my perspective) as it can't address objections that where raised in this thread against river model (btw I am not sure if you didn't understood my argument in post #50 or you simply wanted to

I don't think there is any argument to answer there. What I get from checking back is a repetition of an argument you've made many times that I view as trivially false and has been refuted dozens of times on threads here, and would be a waste to revisit. Unless I am mistaken, the reference argument is that there is such a thing as an 'illegitimate' coordinate transform that changes physics. This is mathematically equivalent to claiming there is something wrong with claiming limit x->∞ (x/x+1) = 1. I take it as an argument that the foundations of differential geometry are wrong. I am not, ever, interested in debating that.

If this is not your argument, I don't know what you are referring to. If it is the white hole argument, I don't see the point. White holes cannot form by any process (mathematical fact; must 'exist in the past' for no reason); "no one" believes they exist in our universe. So debating their exact properties is not very interesting. Matter can get from inside to outside a white hole, and from near the WH horizon to further away. It can get closer to a WH, taking infinite time (fighting the river) to reach the horizon. However, any idea you have about pile up is wrong - the River model, as with any use of SC geometry, when talking about matter moving we are talking about 'test bodies' - bodies of vanishingly small mass. If you are talking about significant mass, the geometry ceases to be SC geometry at all (you do not have static exterior geometry), and all analysis with the SC metric is incorrect (whether interpreted the river model, directly with GP coordinates, or with SC coordinates). So again, I saw no real argument there to respond to.

.
Speaking about Schwarzschild coordinates for me it seems like it would be more meaningful to speak about isotropic coordinates. But my guess would be that they still would not make simultaneity slice Euclidean. And I would like to make sure that they are physically equivalent to Schwarzschild coordinates.

With isotropic coordinates I mean the one you get from this metric:
$$ds^2=\frac{(1-\frac{\mu}{2r})^2}{(1+\frac{\mu}{2r})^2}dt^2-(1+\frac{\mu}{2r})^4(dr^2+r^2(d\theta^2+sin^2 \theta \; d\varphi^2))$$

They would not make the slice Euclidean (however, the spatial slice is conformally flat, for these). They also have another issue: they only cover the exterior geometry. Compared to the regular SC coordinates, they only cover r ≥ Schwarzschild radius.

 

[harrylin]

To my own surprise, thanks to a parallel discussion, I may after all have have found a simple and enlightening way to distinguish between predictions by Hamilton's model and GR. Applicable to both the solar system and a black hole.

A voyager spacecraft is allowed to free-fall towards the Sun or even a black hole. Looking in forward direction to the stars it sends intensity and Doppler shift information back home.

- GR: more than "classical" increase; even towards infinity for approaching r_s
- Hamilton's model: "classical" increase, I think; towards double for approaching r_s
I base this on the requirement that even a "flowing river" cannot accumulate light in transit.

 

[PAllen]

To my own surprise, thanks to a parallel discussion, I may after all have have found a simple and enlightening way to distinguish between predictions by Hamilton's model and GR. Applicable to both the solar system and a black hole.

A voyager spacecraft is allowed to free-fall towards the Sun or even a black hole. Looking in forward direction to the stars it sends intensity and Doppler shift information back home.

- GR: more than "classical" increase; even towards infinity for approaching r_s
- Hamilton's model: "classical" increase, I think; towards double for approaching r_s
I base this on the requirement that even a "flowing river" cannot accumulate light in transit.

Well, this is false. It is covered in the paper qualitatively in bullet 7 on p.16 of the paper. Quantitatively, there are two observations to be made:

- the demonstrated mathematical equivalence between river model and GP coordinates is
fully sufficient to establish the result is the same.

- You persist, when discussing quantitative predictions, on making up verbal interpretations of your own and not using the actual equations circa p.4 of the paper. Light must continuously change frames going 'up' the river, at each frame change being SR boosted by the change in β between the frames. Nothing 'classical' about this. It is, in fact, fairly well known that all GR redshifts (kinematic, 'gravitational', and cosmologic) can be modeled as incremental SR boosts in such a manner (I first came across a proof of this in J.L. Synge's 1960 book on GR). The ability to treat all GR redshift as kinematical (if desired) is actually far more general than the river model (completely general, in fact).

 

[zonde]

Unless I am mistaken, the reference argument is that there is such a thing as an 'illegitimate' coordinate transform that changes physics. This is mathematically equivalent to claiming there is something wrong with claiming limit x->∞ (x/x+1) = 1.

I don't follow you so I can't respond to that.

So I will try to go into more details about coordinate transformations. If we look at coordinate transformations we have different things. We have coordinate invariants that stay the same under any coordinate transformation. We have coordinate dependent quantities that change under transformation (but of course it's description that changes and not physical reality, unless we mess up our transformation - then we describe different physical reality). And we have invariants that are such only within certain class of coordinate system (spacetime distance is invariant under Lorentz tranformation).

Does this seems ok?

If it is the white hole argument, I don't see the point. White holes cannot form by any process (mathematical fact; must 'exist in the past' for no reason); "no one" believes they exist in our universe. So debating their exact properties is not very interesting.

As a theoretical exercise it might be quite interesting. Please tell me how do you think - is mass attracted toward white hole (we speak about exterior of WH) or not?

 

[PAllen]

So I will try to go into more details about coordinate transformations. If we look at coordinate transformations we have different things. We have coordinate invariants that stay the same under any coordinate transformation. We have coordinate dependent quantities that change under transformation (but of course it's description that changes and not physical reality, unless we mess up our transformation - then we describe different physical reality). And we have invariants that are such only within certain class of coordinate system (spacetime distance is invariant under Lorentz tranformation).

Does this seems ok?

Not quite. Invariants are invariant. There is no 'invariant under only some transforms'. Spacetime distance is invariant in both SR and GR for all coordinate transforms. A coordinate transform modifies the metric according to a defined rule. Using the new metric expression (it is really the same geometric object), proper distance and all invariants are the same. That's why they are called invariants.

What you can say about Lorentz transforms and SR is that they leave the metric in constant diagonal form. Other coordinate transforms make the metric more complicated. But invariants are always invariant. And all physical observations (in both SR and GR) are defined in terms of invariants.

As a theoretical exercise it might be quite interesting. Please tell me how do you think - is mass attracted toward white hole (we speak about exterior of WH) or not?

Do you want to talk about the maximal SC geometry (past eternal white hole joined to future eternal black hole), or a universe with just a white hole and no black hole (such an object must cease to exist - it is past eternal, not future eternal; its singularity must cease, and then its event horizon ceases - this is what defines it as white).

The answers to many questions are bit different for the WH only versus WH-BH universe. (To clarify: a WH only universe is the time reversal of a collapse to BH; it is like a big bang).

[edit: wait, for the purpose of this thread I guess the thing to discuss is obviously the river model of a white hole = white hole GP coordinates = 1/2 the complete SC geometry, in the same way the GP black hole is only 1/2 the complete geometry.

I'm not so familiar with the GP description of WH. I'll need a little time to work out some thngs, and post more when I have it done.
]

 

[TrickyDicky]

White holes cannot form by any process (mathematical fact; must 'exist in the past' for no reason); "no one" believes they exist in our universe. So debating their exact properties is not very interesting.

Right, but it is also a mathematical fact that the Schwarzschild spacetime with its white holes is the most physically plausible mathematical model we have of a black hole, the other three(R-N, Kerr, K-N) have much worse problems. So if one goes by the GR solutions (I mean their existence is the main reason we are willing to believe in black holes in the first place, since the astrophysical evidence might be compatible with other explanations), one should "believe" in white holes as much as one believes in black ones.
Anyway according to the wikipedia page on black holes:
" A much anticipated feature of a theory of quantum gravity is that it will not feature singularities or event horizons (and thus no black holes)."

 

[PeterDonis]

Right, but it is also a mathematical fact that the Schwarzschild spacetime with its white holes is the most physically plausible mathematical model we have of a black hole

No, it isn't, precisely because of the presence of the white hole. The most physically plausible model we have of a black hole spacetime, if we are restricting ourselves to simple models with analytical solutions, is the Oppenheimer-Snyder model, which joins a portion of regions I and II of the Schwarzschild spacetime (vacuum exterior and vacuum black hole interior) to a portion of a collapsing FRW spacetime.

" A much anticipated feature of a theory of quantum gravity is that it will not feature singularities or event horizons (and thus no black holes)."

Not sure which Wikipedia page you are referring to; the one I get when I google on "black holes" says at one point that quantum gravity is expected to feature black holes without singularities (i.e., event horizons but no singularities). That's my understanding of the current state of play (but I am not very familiar with the current state of play). Links to recent review articles would be helpful.

 

[PeterDonis]

The answers to many questions are bit different for the WH only versus WH-BH universe. (To clarify: a WH only universe is the time reversal of a collapse to BH; it is like a big bang).

By a "WH only universe" I assume you mean the time reverse of the collapsing Oppenheimer-Snyder model? I.e., an expanding FRW region joined to a portion of regions IV and I of the maximally extended Schwarzschild spacetime (the white hole and the exterior.) If so, then yes, it is like a big bang (the expanding FRW portion), but with extra regions (the portions of regions IV and I).

[edit: wait, for the purpose of this thread I guess the thing to discuss is obviously the river model of a white hole = white hole GP coordinates = 1/2 the complete SC geometry, in the same way the GP black hole is only 1/2 the complete geometry.

Actually, the exterior vacuum region in both models (WH and BH) is the same region (region I of the maximally extended spacetime). The difference is that the WH model covers regions IV and I (with the past horizon in between), and the BH model covers regions I and II (with the future horizon in between).

I'm not so familiar with the GP description of WH. I'll need a little time to work out some thngs, and post more when I have it done.

Just flip the sign of the dt dr term in the line element, so the "escape velocity" vector points outward instead of inward. I.e., the "river" is flowing outward at every point at the "escape velocity" instead of inward. For example, the river is flowing *outward* at the speed of light (relative to the flat background) at the horizon, and flows outward more and more slowly as you go further and further out (to a limit of zero outward velocity at infinity).