Philadelphia, PA

Dear Bykov, 

Rovelli writes, as I quoted him above, that "General relativity describes the relative motion of dynamical entities (fields, fluids, particles, planets, stars, galaxies) in relation to one another." But, I take it that space-time is itself a field, and it is definitely a "dynamical entity," varying in its geometry from place to place. The slogan is "matter (and energy) tell space-time how to curve, and space-time tells matter how to move." So, if space-time is itself a field concept, then it seems that matter moves through the field as prescribed. If on the other hand, space-time is not itself a field, and there can only be motion in relation to objects, including genuine fields, then space-time seems to be a third wheel. That, however, seems not to be the kind of view under discussion. 

You write:

"Moving relative to spacetime" just has no sense, this is the main point. Spacetime provides "distances" between objects (in quotes, because in GR we use pseudedistances) and that's all. It can not judge which particle moves and which does not.

---end quotation

If your point here is just that there is a kind of timelessness of 4D "worms," and "no change" that is one thing. But I take it that this is not the kind of point under question here. You stipulate that GR "provides 'distances' between objects"  and I take it that these distances may correspond to paths that objects may take (to paraphrase out of the "timelessness").  But if objects take paths in relation to other objects, and all of these together contribute to the geometry, then at least to a reasonable idealization, it seems that objects move in relation to the geometry as well--though changing it along the way, too.

Still wondering.  

H.G. Callaway

Philadelphia, PA

Dear Bykov, 

Many thanks for your reply. But it seems that you are thinking of fields in background dependent theories, while GR is a background independent theory, as I understand the matter. 

You wrote:

First of all, spacetime is not a field itself, just by definition of field. Field is something (scalar, vector, tensor, spinor, operator in the case of quantum field...) defined at each point of the spacetime (more precise formulation: it is a map from spacetime to same space, like C, or some vector/tensor space or some algebra or something else...). So, we need spacetime already given to talk about fields, spacetime can not be a field itself. Not everything that has dynamics is a field!

---end quotation

You say that a field is defined "at each point of the space-time," but this would be a background space-time of the field. In GR there is no background space-time.  

But consider the following brief quotation from the "Gravity B" webpages:

From “Gravity Probe B, Testing Einstein’s Universe”

http://einstein.stanford.edu/content/relativity/q2442.html

Special & General Relativity Questions and Answers

Why is the gravitational field of the universe another name for space-time?

--pause quotation

Here is is clear that space-time and the gravitational field are the same thing. But if so, then space-time is indeed a field. This seems clear, just from the question posed, so I have not quoted the answer given. But you can look for yourself. 

Quotation from "Gravity B" webpage, continued: 

"Space-time does not claim existence on its own but only as a structural quality of the [gravitational] field"

This is such a profound assumption that I have intentionally enlarged the font to emphasize its significance. It will turn out to be the cornerstone to a radically new understanding of the nature of space and the vacuum. But in its radical departure from older ideas about gravity, Einstein's view point sounds a lot like the old philosophical discussion of the Void which emphasized that without bodies, 'place' and therefore vacuum could not exist. If we consider that all bodies produce gravitational fields, we see that Einstein's general relativity arrives at nearly the same Aristotelian conclusion.

---end quotation

Here they are a bit more precise, and claim that "space-time" is a "structural quality of the gravitational field." But in any case, it seems clear that space-time takes the value of the gravitational field at every point. 

The affinity of GR to Aristotle, by the way, is a way of expressing the author's view of the relational character of space-time. 

So, I conclude, that unless the folks at Stanford have it all wrong, space-time in GR is a field. Would you still dispute this? 

H.G. Callaway

Philadelphia, PA

Dear Kanda, 

I think I understand something of your complaints about orthodoxy. But the point here is not to establish or dispute any orthodoxy. Instead, the point of the question is simply to understand something of the physics --as it is presented by the physicists. 

You may or may not be right in your complaints, but I don't see that they help the present question along. So, I wonder if you wouldn't do better bringing up a seaparte question on your own. 

Thank you for your kind consideration.

H.G. Callaway

Philadelphia, PA

Dear Kanda, 

The presupposition of the question is that the situation and aims of physics can be clarified and better understood by public discussion of topics, even by laymen. 

It seems you disagree. So be it. Do you intend to inhibit anyone else from conducting discussions --if they fail to dispute Einstein or contemporary physics in just the way you would have it done? Have the decency to allow that others don't share your view of the matter. 

No one claims to be doing physics here. It is a discussion about a topic in contemporary physics. In consequence, your objections and evaluations seem clearly beside the point. Again, I'd suggest that you'd do better to pose your own question and see how far you get with it. Do something constructive! 

Respectfully, 

H.G. Callaway

Am not sure from where to start. This is a strange question to me. I probably not able to share my view. Thanks anyway.

It does me think at the unmoved mover. Everything depends on which level you think or exist. On the sub-quantum level there is no move and neither time I suppose.

It is rational to believe that movement always respects some natural rules, as energy conservation, momentum conservation and minimal effort. This means that space-time (or whatever is the manifold we are embedded in) has specific geodetics (curves of minimal length linking points) and that movement trajectories respect those geodetic curves. The fact that the new distribution of mass, electricity, energy in all forms, that followed any movement, influences and transforms the environmental (field-, gravitation-, etc-) manifold seems also very rational. So it might be that only infinitely small movements are made along geodetic curves, because as a reaction to the movement, the local field of geodetics already changed. This could be modeled by manifolds (phase spaces) with a lot of dimensions - better said, with a new package of dimensions for every particle to move, even for every atom or electron. But it is clear that something like this cannot be practical at all.

What is practical, is to consider the environment determined only by the big and relevant masses in a neighborhood (like a planet, or at most a planet + a star) and to consider all other things, like stones, dust, space-ships and even comets only as subjects of the environment and not as active participants to the general permanent modification of all fields. Newton did it, Einstein did it also, and every scientist who wants to compute plausible predictions does the same. It is important to always keep a good approximation domain.

Philadelphia, PA

Dear Geurdes & Kanda, 

You wrote, Kanda, 

I still have not received any scientifically worth response from the person in Temple University on the question I asked. It seems to me that he was interested in telling me to comply with the party line propaganda of the main stream physics.

---end quotation

Think of it this way, I am humbly trying to figure out what it is that "mainstream physics" holds to. Ultimate evaluation of it, of the sort you propose, I will have to leave to the physicists. You go too quickly for the character of the present question and thread of discussion. 

Before trying to criticize anything, I believe the critic has to state the matter as clearly as possible, and ideally, in a way that the defenders will recognize as adequate. Of course, that is a rule of discourse and nothing specific to physics.

H.G. Callaway 

Philadelphia, PA

Dear Prunescu,

Many thanks for your very thoughtful comment. I liked this approach in terms of a practical idealization. 

you wrote

What is practical, is to consider the environment determined only by the big and relevant masses in a neighborhood (like a planet, or at most a planet + a star) and to consider all other things, like stones, dust, space-ships and even comets only as subjects of the environment and not as active participants to the general permanent modification of all fields. Newton did it, Einstein did it also, and every scientist who wants to compute plausible predictions does the same. It is important to always keep a good approximation domain.

---end quotation

This seems reasonable to me and especially in connection with what you wrote at the start of your note. I understand that GR is non-renormalizable, and that means that perturbations may get out of hand, at least or especially from a practical point of view. As I recall, there is no general solution to multi-body problems even in Newtonian mechanics. So, it stands to reason that we shouldn't expect them in GR.

H.G. Callaway

Dear professor Kanda,

I was not making any rigorous statement about which conservation and / or minimization rules do matter for computing movement. I was just saying that "it is rational to accept that movement must obey some rules as conservation and minimisation of some quantities" - and this acceptance makes plausible the assumption, that movement happens on some geodetics - of the geometric space, or just in a virtual phase space. It was just a principial statement, and I did not speak about the metric needed to define those geodetics.

As to your examples, I don't know how do you suppose that after the collision one of two masses  has exactly this or that velocity. You do this in order to compute the second velocity using an equation of first degree. But then, it is possible not to satisfy the conservation of energy.

I think that the better approach for elastic collision is to assume that both velocities after collision are unknown. Let velocities before the collision be u1and u2 . Then

m1 u1 + m2 u2 = m1 v1 + m2 v2   and

m1 u12 + m2 u22 = m1 v12 + m2 v22  .

It follows that

v1 = u1 (m1 - m2 ) / (m1 + m2) + u2 2m2 / (m1 + m2)

and something similar for v2.

In the case of NON-ELASTIC collision I find it very plausible that a part of the initial kinetic energy is transformed in deformation heat, noise, rotation energy of the bodies, other kind of radiation. ON THE OTHER HAND, I DON'T FIND IT AT ALL RATIONAL TO CONTINUE TO APPLY THE CONSERVATION OF THE MOMENTUM IN CASE OF NON-ELASTIC COLLISION, AS IN 

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

is done. It seems not at all plausible that one can continue to compute anything using the conservation of momentum in this way.

However, I don't know what does it have with the question we are answering to.

-----------------------------------------------------------------------------------------------------------------

You told me that I am on the "party line". Well, I am not at all on the party line. I am quite the contrary of this. As a logician, you can take a look on things I have written, and you will agree.

----------------------------------------------------------------------------------------------------------------

It is true that we all have difficulties in publishing things which are not main-stream. I believe that every one of us has been rejected at least once in a way considered by him that it was not OK. To think about the past, Einstein was himself not main-stream at the beginning, and he was also rejected. 

The only way to act is to believe in whatever you have and just to go for it. 

It is not OK to make war with all people around, thinking that all people around were enemies, conspiratively related with some establishment  which wants to defend the main stream and goes over bodies.

We are just some guys on RG.

Mr. Callaway,

shortly after I have written my first postation, I was sorry not to have mentioned the open (and far away to be solved) problem of the n bodies, which is open even for n = 3. Happily, you did it. That is a measure of the level of our knowledge... However, in spite of the fact that we cannot compute anything, the geometric approach with metrics and geodetics seems to be right, as has been right also in other situations, like minimal surfaces (soap film surfaces) and so on. We just have to remember all the time that theories work on models, where the most of complexity has been neglected, and that the reality will always be different. Thank you.

Callaway, Prunescu: ""As I recall, there is no general solution to multi-body problems even in Newtonian mechanics. ""

Low: " the existence and uniqueness of solutions follows from standard theorems on differential equations---it's describing them in a useful way that's the issue."

In fact, the existence and the uniqueness of the solutions are clear from the beginning because of the deterministic character of the problem: every time one has a n-body (for example Newtonian) universe with given initial conditions, it is to expect that the evolution of those universes is dependent of (and determined by) only the initial conditions (which are positions and velocities). The point is to make macroscopic qualitative predictions, like implosion,  (un)boundedness, periodicity, maybe ergodicity, and so on... Is is true? (I am also not an expert, I am just asking.) And of course, to express somehow the solutions...

 Even when we make the street very narrow the problem is not solvable because there is no bridge between a few and many. But if we make the road very wide as our whole consciousness, we can easily following the travelling of ideas, which can go very fast.

So the world of ideas is laying at the bottom of everything and when we can move them, then the particles follow.

Mihai, Robert, H.G.,

It seems that we are drifting a bit from the original question, but anyway: the lack of a general solution to the Newtonian n-body problem with n > 2 means that there is no closed-form solution in terms of known functions, where the latter includes functions “reduced to quadrature”, e.g., the error function, which still involves an integral sign, but only a dummy variable falls under it, not a variable for which a solution is sought. A non-existence theorem to this effect has been proven, I think by Sundman, but the memory is hazy, because many years have passed since I minored in celestial mechanics.

I agree that Newtonian gravity is 100% deterministic, so a system of gravitating bodies will always evolve from the same initial conditions in the same way, i.e., the problem must have a unique and deterministic answer for the trajectories for all time. I have also always felt that our failure to obtain the desired solution is a sign that our mathematics is incomplete, sort of like algebra before Newton and Leibniz invented the calculus. But whatever is missing must be some completely new notion.

On the other hand, the ontological foundation of Newtonian gravity is completely different from that of General Relativity, so Newtonian features are not necessarily relevant to the original question, although mathematical intractability of closed-form solutions to nontrivial problems is one thing they have in common.

General Relativity has no preferred “foliation”, which means that if one wants to solve a practical problem, one must break the symmetry by imposing a 3+1 decomposition, i.e., one must identify which of the four dimensions is a time axis at each point, and then the locally orthogonal 3-dimensional hypersurface is the 3-space that we experience as extended. Someone mentioned earlier the nature of space-time being a static entity in which nothing “happens”, everything “is just there”. In that sense, nothing moves in space-time; something at one point, or “event”, doesn’t move to some other point or “event”. Every event stays where it is.

For this to be the final word, one must accept that what we experience as the “flow” of time is an illusion. I have never liked having something explained as an illusion, as it seems to me that science should illuminate our experience, not tell us that such experience is an illusion. But even Einstein made clear that General Relativity is not a “Theory of Everything”, so I conclude that we should probably demand that a Theory of Everything should include an explanation of why we experience time as flowing. But that probably demands that such a theory also explain consciousness, and we seem to be a long way from that.

But in the meantime, if we follow the succession of separate 3-dimensional hypersurface as seen from the set of world lines occupied by our physical body as we imagine ourselves moving smoothly along them, we see something like an animated 3-space in which things move around. In this sense, things move. It depends on whether we are talking about a smooth succession of 3-dimensional hypersurfaces or the 4-dimensional totality. This is relevant only to current physics. Hopefully we will eventually have a workable Theory of Quantum Gravity that will elucidate our experience further.

Philadelphia, PA

Dear Fowler, 

Thanks for your contribution. I see that there have been several interesting postings, since I last put in two cents, here. There were several of your comments which struck me as interesting. 

Let me start with one of them. You wrote: 

On the other hand, the ontological foundation of Newtonian gravity is completely different from that of General Relativity, so Newtonian features are not necessarily relevant to the original question, although mathematical intractability of closed-form solutions to nontrivial problems is one thing they have in common.

---end quotation

I'm sure you are right about an ontological (or we as might say, "conceptual" divergence) between Newtonian gravity and GR. But it strikes me that the mathematical intractability you mention, connected with multi-body problems in GR, does make problems of Newtonian physics, relevant. One expects a kind of parallel here, on theoretical grounds, and given that a certain class of problems are, as you put it, mathematically intractable in Newtonian physics, a similar status for related problems in GR is no great surprise. 

Establishing GR scientifically, it had to pretty much match the predictions of Newtonian gravity, in so far as these had been confirmed by observation. GR made its own distinctive predictions, of course, and these were confirmed, too. That was a great plus. But it would not have been reasonable to hold a failure to solve the n-body problem against GR, since Newtonian theory was in no better state.  

You are right, of course, that there is some importance in coming back to the original question, and you also make contributions in that direction. More on this later.  

It also struck me that the next following passage is of some considerable interest for this thread. You wrote:

I agree that Newtonian gravity is 100% deterministic, so a system of gravitating bodies will always evolve from the same initial conditions in the same way, i.e., the problem must have a unique and deterministic answer for the trajectories for all time. I have also always felt that our failure to obtain the desired solution is a sign that our mathematics is incomplete, sort of like algebra before Newton and Leibniz invented the calculus. But whatever is missing must be some completely new notion.

---end quotation

The idea here of "missing mathematics," I find intriguing. I'm aware that Einstein held GR to be equally deterministic. Though it seems that this idea becomes more problematic when we recognize the intractability of particular problems, for which we are said to know that there must be determinate answers. If we think of the theory as consisting of the mathematical formalism, and you stipulate that "our mathematics is incomplete," then we find ourselves in the difficult position of maintaining that the mathematical formalism is both deterministic and incomplete. But insisting on the deterministic character of the theory, one might then be driven to say that the theory, Newtonian gravity or GR, is not the "incomplete" mathematical formalism. What would it be, then?

I suppose the story goes something like this. First one shows that the theory and formalism will solve simple, idealized problems, then it is suggested that if further elements of the same sorts are added to the simplified, or idealized problem, then similar solutions will be forthcoming. Perhaps these somewhat more complicated problems can also be solved, or at least good approximate solutions are available. But at some point, in the envisaged projection of more complex problems, we come to the point where solutions are no longer found --but merely projected. It strikes me that the progression has somewhat the character of a mathematical induction. I wonder if you think that the claims for determinism in fact rest on something like a mathematical induction over progressively more complex possible problems. That would certainly make the n-body problem (or non-renormalizability of GR) something of considerable interest for clams for a deterministic theory of gravity --or so it strikes me. 

Any thoughts on these comments?

Many thanks, 

H.G. Callaway

I would have a little thought over the following lines:

Callaway: ""The idea here of "missing mathematics," I find intriguing. I'm aware that Einstein held GR to be equally deterministic. Though it seems that this idea becomes more problematic when we recognize the intractability of particular problems, for which we are said to know that there must be determinate answers. If we think of the theory as consisting of the mathematical formalism, and you stipulate that "our mathematics is incomplete," then we find ourselves in the difficult position of maintaining that the mathematical formalism is both deterministic and incomplete. ""

The situation is known in mathematics since a lot of time. If we call "elementary function"  polynomials, exponential functions, logarithms, trigonometric functions, inverse trigonometric functions and their combinations, there are a lot of problems that can be easily formulated but have no answer in the set of elementary functions. For example, the antiderivatives (primitive functions or integrals) of  exp(-x2) , sin(x2) , (sin x) / x , 1 / ln x , xx are not elementary functions. And there are lots of other functions with non-elementary antiderivative, like the Jacobian elliptic integral. 

It really looks as missing mathematics in the following sense: mathematics (as every other science) can denominate only a finite number of objects, and can compute (approximate) at most a denumerable infinity of functions - because there exists at most a denumerable infinite set of algorithms. But it is concerned with (at least) a continuum of objects. [Cardinality of continuum is strictly bigger that countable - G. Cantor] It is normal that most of objects are undefined (unnamed) and not computable. The situation is particularly bad when we come with a concrete problem and are not able to express the solution. Normally we can approximate solutions of differential equations - because we can write them down as infinite Taylor series and we compute so much terms as we need for a good approximation. But if we do not know how the n-body system will evolve macroscopically, (if for the given condition it remains bounded or not, for example) it is difficult to guess how many terms we have to compute and if the approximation got so far is good enough. 

There is no contradiction between the words "incomplete" and "deterministic". Mathematics are incomplete from many points of view, and this is an intrinsic property of mathematics (see also Gödel's Theorem). Determinism is not a property of mathematics, is only a property (we hope that) nature has. The determinism of the n-body problem is reflected in mathematics by the fact that the corresponding differential equations have a solution and that the solution in uniquely determined by its initial conditions. This is a first test of the fact that the mathematical translation of the determinist situation is correct, but has no more significance as this. 

It would be nice if someone introduces a finite set of new analytic functions, with transparent and natural definitions, which are also easy to approximate, and that all solutions of the n-body problem could be expressible in those functions. If you ask me, this would be too nice to be possible. I presume that the situation is much more difficult that this one, but I have no arguments.

Philadelphia, PA

Dear Prunescu,

Thanks for your comments. I have a question which may help resolve the unease you express. 

You wrote:

The determinism of the n-body problem is reflected in mathematics by the fact that the corresponding differential equations have a solution and that the solution in uniquely determined by its initial conditions. This is a first test of the fact that the mathematical translation of the determinist situation is correct, but has no more significance as this.

---end quotation

I wonder about the meaning you attach to "test" in the last sentence here, especially since you say that this "test" has no greater significance--but no greater significance than what? In fact, as I understand the matter, there are many n-body problems for which no solution can be calculated. But if so, then what exactly is the significance of the claim that "The determinism of the n-body problem is reflected in mathematics by the fact that the corresponding differential equations have a solution and that the solution in uniquely determined by its initial conditions." This seems to amount to saying that, though many problems find no solution, still we know there must be solutions. 

O.k., suppose this true. What reasons are there to think it true? Well, my earlier suggestion was that it rests on a kind of mathematical induction, over the complexity of solutions. I have not attempted to formulate such a mathematical induction, but this is my question--is that the type of reasoning involved in arriving at the idea that solutions must exist, even though they can't always be provided, when specific problems are proposed? 

First, we pose very simple problems, exhibiting the interrelations of the variables, and show that the formalism solves these problems. Next additional elements are introduced which add complexity, but involves only the same sort of variables. Though there are many further problems that can't in fact be solved, still the conclusion is drawn that such solutions must exist. This kind of argument would seem to presuppose that "complexity" of problems is well defined and can be ordered, though I suspect that there are many kinds of complexity involved. 

Also, I tend to think that the determinism involved is a conclusion drawn from the workings of the formalism--and not a prior conclusion which is supposed to be somehow confirmed by working with the mathematical formalism. 

The Godel incompleteness of mathematics, I believe, is not really to the point here. The point seems to be, instead, that practical means of calculating specific solutions to particular physical problems are unavailable. The specific mathematics of the physical theories becomes intractable in particular applications. 

I appreciate seeing your thoughts on the matter. 

H.G. Callaway

"Do objects move in relation to space-time in GR? "

Obejcts move (in absence of other direct interactions) due to the curvature of space-time determined by the mass/energy configuration at any instant, this is what GRT  and FG (field gravitation) affirm.

Space-time for GR cannot be considered as a  field but is an Hypermedium which allows for fields and regulates inertia and gravitation, a sort of "collision domain" with c as its maximum energy/momentum speed. According to QM Elementary masses (fermions) are as such because of the intrinsic omni-presence of the HIGGS field,

Space-time is something considered absolute in GR with the impossibility to be at rest . This is maybe a ill posed problem. Only motion in relation to masses is conceivable and thanks to CMBR a "sort of absolute reference frame" is succesfully used in Astronomy.

Dear Callaway,

Consider that we have given an instance of the n-body problem. The instance consists of a vector of masses (m1, ... , mn) , a vector of positions (p1, ..., pn) and a vector of initial velocities (v1, .. vn).   For the sake of completeness of this initial information I just recall that every initial position pi is in fact itself a 3-dimensional vector in the space (has 3 coordinates) and the same is true for every initial velocity vi. We also agreed that the situation is deterministic, and that means that these initial conditions determine the evolution of this n-body universe forever. We just let it (this universe) go. Let the position of the body mi at the time t > 0 be pi(t). The function pi : [0, + infinity) ---> R3 has the property that pi(0) = pi and in every moment of time t, pi(t) is the position of the i-th body. The vector build by those functions (p1, ..., pn) (t) : [0, + infinity) ---> R3n is the solution of this particular instance of the n-body problem.

So, the solution exists and is unique. Do we know more about this? At 

http://en.wikipedia.org/wiki/N-body_problem

one can see a nice animation with three bodies, which tend to move chaotically. At the same page there is also the approach with Taylor series. [ Example 1 + x + x2 / 2 + x3 / 6 is the 4-term approximation of the exponential function. ] As  I said in a previous posting, it is difficult to know how much of the Taylor series must be computed in order to get a good approximation as far we are not able to do qualitative predictions of the behavior. The existenxe of apparently chaotic behavior shows how difficult the problem of such prevision can be. 

You wrote "O.k., suppose this true. What reasons are there to think it true? Well, my earlier suggestion was that it rests on a kind of mathematical induction, over the complexity of solutions. I have not attempted to formulate such a mathematical induction, but this is my question--is that the type of reasoning involved in arriving at the idea that solutions must exist, even though they can't always be provided, when specific problems are proposed? "

The existence and uniqueness of the solution follows from a general theorem about solutions of differential equations. OK, in order to prove this general theorem, one makes a kind of induction. The coefficients of the solution which is constructed are inductively computed, and one shows that the resulting sequence of polynomial functions converges to a solution of the problem. For handling a concrete problem one can try the same approach. The result is more or less illuminating - depending of how much one has to compute in order to recognize the long term behavior (periodic or not, etc...) If we have not any idea of how exactly one hast to approximate the solutions, we must consider the problem as still open.

Also you have written: ""First, we pose very simple problems, exhibiting the interrelations of the variables, and show that the formalism solves these problems. Next additional elements are introduced which add complexity, but involves only the same sort of variables. Though there are many further problems that can't in fact be solved, still the conclusion is drawn that such solutions must exist. ""

Well, I doubt that this kind of approach could be good here. The fact that even by 3 bodies one has a apparently chaotic movement, shows that the ""complexity"" of the problem increased enormously from n = 2 to n = 3. This is very discouraging for any kind of "mathematical induction".

You said: ""The specific mathematics of the physical theories becomes intractable in particular applications. "" This seems to be true. If Gödel is far away from this or not, is only a matter of taste. If we look Gödel like the first example of "undecidable behavior", that it is possible, to find his result quite related to this situation, but this last line is only speculative in this moment. (I have not a proof). 

Thank you.

Philadelphia, PA

Dear Prunescu, 

Thanks you for your very clear statement of the mathematics involved. Let me reiterate my last (sub-)question. 

On the one hand, you wrote:

The existence and uniqueness of the solution follows from a general theorem about solutions of differential equations. OK, in order to prove this general theorem, one makes a kind of induction. The coefficients of the solution which is constructed are inductively computed, and one shows that the resulting sequence of polynomial functions converges to a solution of the problem. For handling a concrete problem one can try the same approach.

--end quotation (My emphasis in the italics.)

Afterward, you went on to say,

Well, I doubt that this kind of approach could be good here. The fact that even by 3 bodies one has a apparently chaotic movement, shows that the ""complexity"" of the problem increased enormously from n = 2 to n = 3. This is very discouraging for any kind of "mathematical induction".

--end quotation (My emphasis in the italics.) 

It follows, then, given your statements, that the induction "of a kind" which assures you that there exists a solution to every problem, stating initial conditions, and employing the formalism of the theory, is not an induction over actual solutions to progressively more complex problems and their solutions. Instead it would have to be something more abstract still. 

Still, strangely, in your opening, you make the following statement:

Consider that we have given an instance of the n-body problem. The instance consists of a vector of masses (m1, ... , mn) , a vector of positions (p1, ..., pn) and a vector of initial velocities (v1, .. vn). For the sake of completeness of this initial information I just recall that every initial position pi is in fact itself a 3-dimensional vector in the space (has 3 coordinates) and the same is true for every initial velocity vi. We also agreed that the situation is deterministic, and that means that these initial conditions determine the evolution of this n-body universe forever.

---end quotation (Emphasis of the italics, not in the original). 

Now, I say that this seems very strange, because you seem to simply assume the deterministic character of the theory, when, as one would think, this is instead just what you should have set out to prove in response to the challenge. It seems to me that you are going in mathematical circles. Can you clarify?

H.G. Callaway

Philadelphia, PA

Dear Fowler, 

I also thought this passage (quoted by Kanda, just above) a very interesting one, though I thought to come back to it later, wanting to attend to other sub-questions, first, including some of what you said.

But let's look briefly at this:

For this to be the final word, one must accept that what we experience as the “flow” of time is an illusion. I have never liked having something explained as an illusion, as it seems to me that science should illuminate our experience, not tell us that such experience is an illusion. But even Einstein made clear that General Relativity is not a “Theory of Everything”, so I conclude that we should probably demand that a Theory of Everything should include an explanation of why we experience time as flowing. But that probably demands that such a theory also explain consciousness, and we seem to be a long way from that.

--end quotation.

My inclination was to put aside the "timelessness" Einstein attributed to the world in accordance with GR, because this seems to me a rather doubtful interpretation. On the other hand, I'm not much inclined to expect any "theory of everything." I think science and the growth of knowledge just keep going, though one might suppose that after some point what we continue to learn might seem less interesting from our present perspective, and given present interests.

The problem with the timelessness idea is right on the face of things. We see things change and move constantly, and though GR may enable us to make calculations and predictions of things otherwise unexplained and unexpected, by means of a 4-fold world in a mathematical scheme into which time has already been incorporated, in some sense, the very confirmation of this theory rests upon ordinary observations of things changing --say, the perihelion of Mercury. GR makes predictions of particular observable phenomena taking place at particular times. So, if there is an illusion involved, it would seem to be the illusion that the static 4-world, a kind of "block universe," must be more real than the predictions and confirmatory evidence arrived at by means of it. One might push Einstein's argument, of course, but I remain doubtful.

Least of all does it seem to me that we need to consult a theory of consciousness in order to understand physical time --as contrasted with consciousness of time. Talk of the "flow" of time, can point either to change in time, as contrasted with timelessness, or it can point to a phenomenon of the consciousness of time. But these are quite distinct sorts of questions, IMHO. 

I don't mean to rush this sub-topic, but I thought to make this short relevant statement. 

H.G. Callaway

Dear Callaway,

you have put at least two different questions. 

1. The question about induction. Given a concrete instance of the problem, (as we already did) one can compute inductively the coefficients of the solutions. The solutions are analytic functions, and analytic functions are given by infinite Taylor series like f(t) =  SUM an tn . In our case, they are analytic functions in the variable t (time) and for every of n bodies we compute three analytic functions (for the coordinates x, y, and z). Also some so called elementary functions, like the exponential, are analytic functions, and can be only approximated (when computed). But they are well known, one knows how much of the function must be concretely computed in order to get a good numerical approximation. In our case, we can compute the coefficients an (and this is done inductively: first a0 , then a1 , then a2 etc) but we do not know when we have done enough coefficients in order to understand at least the macroscopic behavior of the n-body system. So much about one kind of induction.

The induction suggested by you, from less complicated to more complicated instances, makes me skeptic. It seems that the (n + 1)-body problem can be in general so much complicated than the n-body problem, that I don't believe in the success of this approach. I have already illustrated this by this example where 3 bodies act apparently chaotic and unprevisible, which cannot happen with 2 bodies. The 3 bodies also do not move really chaotical - their movement is given by solutions of the 3-body problem, and they can be approximated by polynomials [that is, we can compute a finite but unbounded number of coefficients of the solution in a finite time] But I guess that we cannot say neither after 1000 coefficients, if the system remains bounded or will expand, nor after 1000000 coefficients. This is the problem...

2. You told: ""Now, I say that this seems very strange, because you seem to simply assume the deterministic character of the theory, when, as one would think, this is instead just what you should have set out to prove in response to the challenge. It seems to me that you are going in mathematical circles. Can you clarify?""

Both of us assumed the deterministic character of nature, because we speak about physics, and one of the most important beliefs on which physics are build is the repeatability of experiments. Under the same conditions the experiment must run in the same way with the same results. this makes nature predictible. Physics tries to build models of nature. Those models work as given by theories. The models are good only if the prediction are again verified by experiments. If we accept all of that, we also accept that some pocket n-body universes which start with the same condition, as given above, will run in the same way (as function of the time variable  t). So our belief in a determinist universe implies the existence and the uniqueness of the solution, fact accepted also by Fowler in one of his postations. 

On the other hand, we write down the differential equations and we see that we are in a situation where the theory guarantees existence and uniqueness of solutions. I told before about this and I called this "a first test of the modelization". Of course, if my equation were some with more than only one soltion, it could have been the hint that I have forgot a constraint, because I know that my problem is completely determined. also, an equation without solutions would mean that it could be a wrong translation of the situation I wanted to model (n-body problem) and I must look better. Mathematicians believe in this soundness of mathematics. 

As I told before, determinism is a property of nature, or better said, a property that we hope nature to have. It is the fundamental hypothesis which let science make sense. Incompleteness is a property of mathematics, or maybe more general of science, but not a property of nature. So, I don't believe that I move in mathematical circles. For the time being, I don't see any contradiction here. [Just remember that the science is made by man, but not  the nature...]

A. Kanda wrote: "This is just a minor problem. More fatally, the work needed to accelerate from momentum 0 to momentum mv is not (mv^2/2). This is the case only when acceleration is constant. I wonder when physicists think despite so many geniuses in the community."

Dear professor Kanda, I really believe that  to move mass m from 0 km/h to v km/h costs more energy if done with non-constant acceleration. as done with constant acceleration. It seems to be a situation where the constant function minimizes or maximizes something. This is not so special: the sphere has constant curvature and represent the maximal volume closed by a surface of given area, and so on. A non-constant acceleration means somethimg changing the movement it has just induced - and that can cost energy.

Better we should ask why the movement depends only of the first and the second derivative and not of the third, fourth, and so on. I have an old supposition that this has something to do with the fact that our universe has exactly  three spatial dimensions. A curve in space is determined by its Fresnel sliding coordinate system, and this one is determined only by the first two derivatives and a normal to their plane. But this is just a supposition.

H.G.,

Thanks for your kind remarks. Regarding your statement:

“But it strikes me that the mathematical intractability you mention, connected with multi-body problems in GR, does make problems of Newtonian physics, relevant. One expects a kind of parallel here, on theoretical grounds, and given that a certain class of problems are, as you put it, mathematically intractable in Newtonian physics, a similar status for related problems in GR is no great surprise. “

I don’t read much into this common feature of Newtonian gravity and General Relativity, since the same feature is common to a multitude of other physical problems that have little or nothing to do with gravity. For example, I started out doing computations of stellar atmospheres, for which one must solve simultaneously a coupled system of nonlinear partial differential equations describing hydrodynamic, ionization, excitation, and radiative equilibrium. There is no way to solve such a system other than by numerical integration and iterative approximation. Solutions in closed form are just as impossible as for the general Newtonian n>2 problem and problems in General Relativity that have no symmetries to reduce the complexities, but other than the need to include hydrodynamic equilibrium, which involves gravity, I see no intimate connection to Newtonian gravity or General Relativity regarding the mathematical difficulties themselves. I do agree that encountering mathematical intractability is no great surprise!

Regarding:

“Establishing GR scientifically, it had to pretty much match the predictions of Newtonian gravity, in so far as these had been confirmed by observation.”

Indeed Newtonian gravity is amazingly accurate within its domain and remains crucial for many practical problems. But besides the very troubling “action at a distance”, it contains two serious flaws that cancel each other out to a seemingly miraculously good approximation: (a.) instantaneous transmission of the gravitational force; (b.) the 1/r^2 dependence of the force. Some years back an old grad-school chum, Brent Tully, did some work with James Peebles on galaxy flow patterns in clusters using Newtonian gravity. I was struck by the fact that it seemed to work despite the fact that over the gigantic distances and with the high speeds involved, many galaxies moved by one part in 10^5 during the one-way light time, i.e., by assuming that gravity propagates instantly, one is pointing the forces at locations where the real gravitating objects no longer are, with a relative error of 0.00001. This seemed like doing a chaotic computation in two-byte integer arithmetic, which would diverge into nonsense in no time.

So I tried computing Newtonian gravity with a finite propagation time but still the 1/r^2 force. This requires keeping a history of the positions of each body so that one can interpolate into the past by the one-way light time to find where the object was when it generated the gravity felt by another object, then using that location to compute the force. So from a given body, the others are generally at different time lags, but that much is easily computed. It turns out that it is impossible to define a self-consistent set of initial conditions for t=0, since even at that moment, each body is already moving under the influence of the past positions of the others, which are unknown. So I just “created” each body at t=0 with arbitrary masses, positions, and velocities, and the bodies moved force-free until enough time had passed for each of them to “feel” all the others, and then I discovered what I should have anticipated: central-force motion was gone, and with it, conservation of angular momentum. And the scalar potential was no longer valid, so out went energy conservation. Bound two-body systems had a new torque on them caused by “feeling” the other body in a direction that constantly lagged behind the real center of mass, mechanical energy was injected into the system, and it approached becoming unbound asymptotically.

So plugging in retarded potentials destroyed Newtonian gravity. And yet as you said, in the low-energy domain it provides answers indistinguishable from those of General Relativity. How could that be? The answer is in the 1/r^2 force. First of all, in General Relativity, there is no “force”, objects “move” (i.e., have world lines) under the influence of spatial curvature, but one can identify world lines and one can compute the accelerations between them, so one can compute a fictional “force” as the product of mass and acceleration, and when one does this, it does not in general turn out to vary exactly as 1/r^2.

So the completely unacceptable (to modern physicists, not to Newton, who was a towering genius) feature of instantaneous propagation is almost perfectly balanced by the not-really-correct 1/r^2 force. And I haven’t even mentioned the absolute space. So in all, Newtonian gravity has no useful ontological content. It was a wonderful achievement that paved the way toward progress, but we cannot learn anything about the fundamental nature of reality from it.

One might object: but doesn’t General Relativity reduce to Newtonian Gravity under low-energy asymptotic conditions? There are indeed demonstrations in textbooks that show that one can arrive at something resembling a Newtonian scalar potential by letting the speed of light go to infinity and arbitrarily setting some time derivatives to zero, but this doesn’t seem to me to bridge the gap between the two entirely different ontological foundations. Letting the speed of light go to infinity is anathema to the spirit of relativity theory; as soon as one does that, one departs from the context of relativity altogether, and whatever is shown to be reducing to Newtonian gravity is no longer General Relativity. And the existence of a similar scalar potential doesn’t make up for the fact that the contexts are irreconcilably different.

Regarding:

“I wonder if you think that the claims for determinism in fact rest on something like a mathematical induction over progressively more complex possible problems. That would certainly make the n-body problem (or non-renormalizability of GR) something of considerable interest for clams for a deterministic theory of gravity”

I have not personally proved that Newtonian gravity is deterministic, but it is generally conceded that all of classical physics is deterministic, and for me to think otherwise would require the presence of a stochastic process in the equations. As for General Relativity, Einstein was adamantly opposed to randomness in the laws of physics, which resulted in many valuable contributions to Quantum Mechanics via his debates with Born, Bohr, and others.

On the other hand, since (non-closed-form) solutions so often take the form of numerical integrations, one has always to deal with roundoff and truncation errors, which eventually accumulate to the point of dominating the numbers that come out in a probabilistic way. But my primary interest is in the formalism itself.

Philadelphia, PA

Dear Prunescu, 

I think you have now clarified your position, but it makes a good deal of what you said before seem to be beside the point. You wrote:

As I told before, determinism is a property of nature, or better said, a property that we hope nature to have. It is the fundamental hypothesis which let science make sense. Incompleteness is a property of mathematics, or maybe more general of science, but not a property of nature. So, I don't believe that I move in mathematical circles. For the time being, I don't see any contradiction here. [Just remember that the science is made by man, but not the nature...]

---end quotation

I understand that you think nature deterministic, or you hope it is so, and consequently, you were not trying to prove that it is deterministic. Instead this was your presupposition. Well, fine and good, I used to think so, too--on both counts.

But this was not the question of interest. It certainly appeared that you were trying to prove, by bringing in mathematical tools, that Newtonian theory is deterministic. Nature could, of course, be deterministic, or not, without Newtonian theory reflecting this. So, the question concerned the grounds in physics of thinking that Newtonian theory or GR are deterministic. But there is a definite sense to a theory being deterministic.  It is not to be identified with nature being deterministic. The theory must give definite results for every statement of initial conditions.  But, since, in fact there are many problems, given particular initial conditions which can't be solved, it seems difficult to prove that the theory does always give the same predictions. Einstein certainly thought of nature as deterministic, and he thought of GR as deterministic--in contrast to QM. But we have been asking about the sense and evidence for these claims. 

You wrote:

Both of us assumed the deterministic character of nature, because we speak about physics, and one of the most important beliefs on which physics are build is the repeatability of experiments. Under the same conditions the experiment must run in the same way with the same results. This makes nature predictable. 

---end quote

Nature seems to be predictable, in degree in QM, because one can calculate the probability of particular outcomes, in a series of experiments, given the same initial conditions. But this is not to say that particular experiments come out the same every time. Instead, the regularity of the probabilities comes out the same in repeated runs of experiments, while the the individual experiments vary randomly over the probability range.  QM is physics in spite of that.  Many physicists certainly hold that there is a fundamental random element in nature. 

The point of my sketch of an argument by mathematical induction was to ask if there was any similar argument of show that there is a unique solution for any problem we might set for solution (every statement of initial conditions) in Newtonian physics (or in GR). It seemed at times that you were offering such an argument, but now it turns out that you weren't. You simply assume that nature is deterministic. I don't see that this helps us along in answering and understanding our question. 

You will recall that we got off on this sub-question, because of the analogy proposed between Newtonian theory and GR. We have now threaded our way through much mathematical complication only to end up back where we were. The question of the perturbations or lack of renomalization in GR seemed relevant to the question of whether objects move through space-time (and the gravitational field) in GR --and to the interpretation of Rovelli's text.  Newtonian n-body problems seemed a simpler analogy. We will now need to go back and retrace the steps. 

Still, the detour has been instructive, and I thank you for your contributions. 

H.G. Callaway

Dear Callaway,

Now I understand better your point. I will try to answer, although it becomes more and more difficult to express what I want, as I wish it to be...

Well, when I was a student in math, 25 years ago, I would have been (or maybe I was, I don't know) very happy with this kind of explanation. I just imagine the young Mihai saying: The Cauchy-Kovalevskaya Theorem assures unique solutions, SO the nature is deterministic. EUREKA. From a point of view, Mihai was right. And I am sure that a lot of scientists are happy with this explanation. You might be also happy, why not.

But meantime I became much more skeptic. The task to prove that nature is deterministic seems to be now much more difficult and fundamental for such an easy proof. I see now the uniqueness of the solution only like a hint or kind of evidence, but not like a proof. I think that we need much more time and that we are not at all prepared to prove that nature is deterministic. And that is very difficult to make this proof over physics, without running circularities, of course.

So I prefered to speak about our hope on nature's determinism (taken and accepted as axiom, in order to do physics) and about the uniqueness of the solution as about parallel things, which match well together, but without making any relation of causality from one to another.

On the other hand, I don't look at QM as a kind of evidence that nature would not be deterministic. Heisenberg's uncertainty principle and all things following from this principles is only a statement about the intransmisibility of information from the quantic level to our macroscopic level, if we try to use electromagnetic observation as transportation middle of this information. (And, hellas, there are no other known possibilities to observe what is happening). Maybe at quantic level the nature is exactly so deterministic as it seems to be here, at our macroscopic level. Maybe two neutrons colliding in EXACTLY the same way, would always split in the same way. But we are too big and so too far away from them, and we cannot controll them so well to assure the collision in exactly the same way (velocities, angles, position and so on). So we are happy just with probability densities and other things we can measure after a big number of collisions... [I just remembered a talk after the discovery of Higg's boson and about which amount of information had to be analyzed by computers in tens of tousand's of hours of computation...] 

So, I think that I believe the entire nature to be deterministic, or I am ready to accept this, but I doubt that we can prove it with our actual level of knowledge. Sorry about this.

Every object moves in relation to the cause that has determined the motion and this movement can be efficiently described with respect to the (physico-mathematical) reference frame S[O,x,y,z,t] where the considered physical phenomenon happens. The relationism between two objects that are not interrelated as per a relationship cause-effect has no physical meaning and when the observer is unable to discover that relationship or when that relationship doesn' t exist then it is easy to make use of indeterministic probabilistic models that bring nothing to knowledge.

GR is a theory that theorizes a spacetime bending in the presence of a mass and therefore in GR the relationship isn't between two objects but between one object and the spacetime. On this account, like it has been observed, GR has many problems in the presence of more masses, because whenever a new object is added, the spacetime would have to undergo a new and different bending. And for this account, and not only for this, GR is an unsatisfactory theory. In actuality all postmodern physics, based on modern physics, has many problems and their solution cannot be a consolatory hypothetical future theory of everything that would have to solve all problems.

Sentences like "theory of everything" or "theory of nothing" or "time is an illusion" can have a meaning in philosophy but they have no meaning in physics. The answer to this problematic situation of physics is a strong criticism towards existing theories like a few are doing and surely the solution isn't the return to classical physics. A contribution to the overcoming of this crisis can come partially from the consciousness that science and physics after all aren't omnipotent and that the scientific progress is due to a tiring work of research and of education.

Daniele,

I don’t think we’re in any significant disagreement, but regarding your statement:

“On this account, like it has been observed, GR has many problems in the presence of more masses, because whenever a new object is added, the spacetime would have to undergo a new and different bending. And for this account, and not only for this, GR is an unsatisfactory theory. In actuality all postmodern physics, based on modern physics, has many problems and their solution cannot be a consolatory hypothetical future theory of everything that would have to solve all problems.”

I would point out two things just to be clear. I don’t think the presence of mathematical difficulty has any bearing on whether a given physical formalism is in one-to-one correspondence with physical reality. Just knowing the equations is worth something, even if we cannot solve engineering problems with them. But that may just reflect my personal reasons for doing science.

Secondly, I certainly did not mean to imply that present impasses are made consolable by future hopes for a “Theory of Everything”, only that the limitations of current theories point the way to constraints on future theories. To formulate a theory, it helps to know what one wants it to do. That certainly doesn’t guarantee that such a theory can be formulated. As you said, and as I agree completely,

“A contribution to the overcoming of this crisis can come partially from the consciousness that science and physics after all aren't omnipotent and that the scientific progress is due to a tiring work of research and of education.”

Mihai,

I gather that your training and career have been in mathematics, based on what you said about what you were doing 25 years ago. It follows that you may not be aware of some developments in physics that actually cast some light on the QM notions of indeterminism. But first, regarding:

“So I preferred to speak about our hope on nature's determinism (taken and accepted as axiom, in order to do physics) and about the uniqueness of the solution as about parallel things”

One can do physics without an axiom of determinism, and indeed most physicists after the 1920's have been doing it. It may have become obvious from my previous post that I do not consider Newtonian gravity to be a description of the fundamental nature of reality. And if possible, I would like to skip the infinite-regress philosophical debates about what the “fundamental nature of reality” means, not because it is unimportant (I think it is) but because it would take us too far afield, and I think anyone reading these posts knows what I mean by it sufficiently well for present purposes.

I personally welcome the indeterminism implied by QM. I would like to be able to think that I am not an automaton acting out a pre-ordained destiny devoid of free will and any meaningful definition of morality. I avoid believing that which I would like to believe for that reason alone, but fortunately some very strong evidence is available independently of my personal wishes. Whereas I am in awe of Einstein, I am totally perplexed by his devotion to determinism. He was confronted by the question of whether morality has any meaning in a totally deterministic universe, since it precludes free will, and morality becomes an empty concept in the absence of free will. Einstein actually said “I know that philosophically a murderer is not responsible for his crime, but I prefer not to take tea with him.” (See Isaacson’s excellent biography “Einstein - His Life and Universe”). He held out to the end!

Not that indeterminism implies free will and morality, but complete determinism surely precludes both. So QM at least keeps free will and morality in the running, if it indeed requires the universe to pick random numbers at its most fundamental level. There is strong evidence that this is the case. I recommend to you Anton Zieilinger’s book “Dance of the Photons”, which is somewhat popularized but not to the point of being misleading on the essential points. Not only does he give the best intuitive explanation of the Bell Inequalities, but he points out that there is no better random-number generator than a half-silvered mirror used to generate a string of binary digits, 1 for transmission and 0 for reflection. Such sequences pass all tests for randomness with flying colors. Of course, nature may be fooling us with an incredibly good pseudorandom number generator, but one of Einstein’s quotations that I find useful (and the title to Pais’s famous biography of Einstein) is “Subtle is the Lord, but not malicious.” That’s a metaphysical guideline, but like it or not, such guidelines are at the foundation of most successful physical theories, even though the theories have to pass muster on their own later.

I also recommend Ghirardi’s book (despite the somewhat strange title, based on another Einstein quote) “Sneaking a Look at God’s Cards”. He gives a very understandable description of quantum entanglement and its experimental tests. The bottom line is that strong empirical evidence exists for the interpretation of QM as implying randomness and superpositions of states that are macroscopically incompatible.

Your statement

“Maybe at quantic level the nature is exactly so deterministic as it seems to be here, at our macroscopic level. Maybe two neutrons colliding in EXACTLY the same way, would always split in the same way.”

implies what are called “hidden variable” theories. A lot of thought has gone into these, and they have not fared well, especially in the context of entanglement and Bell’s Inequalities. I strongly recommend those books by Ghirardi and Zeilinger mentioned above.

Mihai,

A correction to some bad phrasing on my part. When I said

“ I would like to skip the infinite-regress philosophical debates about what the “fundamental nature of reality” means, not because it is unimportant (I think it is)”

I meant that I think it is important, I just would like to skip that topic for now.

@ Low

I tend to give right Kanda. Imagine a space-ship on the x axis, starting in the wrong direction, then stopping, and starting again inthe right direction. The fuel used is three time more as if it started from the beginning in the right direction. So we must consider to integrate | F | dx instead of F dx. 

@ Fowler

However, a completely deterministic word may contain random generators, the struggle between Bad and Good and a lot of autonomous decisional entities manifesting free will, at least as far as they can decide their will were free. As a person, I try to act morally, and I don't see a determinist nature as an enemy of free will. Better said, I don't care that in a cosmic scale I might be a determinist automaton in some determinist world. My free will, or the illusion I had one, is a gift I enjoy a lot. There exists also another level of opinion, where good-minded people deny the possibility of extraterestrial life because they see in this possibility an enemy of their religious and finally moral beliefs...

Last but not least, I am impressed by the quality and depth of your postings. Thank you for the recommended books and please, keep posting.

Low:

I agree with you completely. Also I know that the integral (work) over a closed path is always zero. But this was not the point by Kanda. He really minds | F | dx.

@ Low,

you and Kanda just speak of different things. Kanda minds the following: for the captain of the space-ship that started in the wrong direction, it is much more important that his mistake costed three times more fuel as if he was starting from the beginning in the good direction, as it is important for this unhappy captain that in consevative fields intregrals on closed paths are zero. The second information is for him dead letter, the first information is for him maybe life and survival. Kanda spoke from the very beginning of different things as those stereotipically understood under given names. Unhappily, he generalises a discourse against physicists with the same ease as you speak about five-leg dogs. I believe that most of physicists are earnst working people trying to do a good research. Also I believe that stereotype thinking is a danger for creativity because it makes us self-sufficient and that from time to time we need some peple like Kanda who brutally kicks us in order to make someone free from stereotype thinking. Maybe he has not always right, maybe he should better define things before making statements, but his contribution is positive, after all integration.

Philadelphia, PA

Dear all, 

Many thanks to for the very learned discussion --though this has sometimes got off the track a bit. 

If I may have a go at "cutting the Gordian knot," here, I conclude that if objects move in relation to each other in GR, then they also move in relation to space-time (or the encompassing gravitational field). 

Any dissent? 

H.G. Callaway

Philadelphia, PA

Dear Kanda, 

It seems you missed the little word "if" in my summary conclusion. 

That you accuse me of "diversion" --what you continually do yourself-- seems absolutely typical of what I have seen from you. Of course, in fact, I do not know you from Adam. 

H.G. Callaway

Philadelphia, PA

Dear all,

Many thanks to for the very learned discussion --though this has sometimes got off the track a bit.

If I may have a go at "cutting the Gordian knot," here, I conclude that if objects move in relation to each other in GR, then they also move in relation to space-time (or the encompassing gravitational field).

Any dissent?

H.G. Callaway

Philadelphia, PA

Dear Kanda, 

I see two postings. Was that a "yes" or "no" to the summary question? Or, do you simply wish to continue muddying the waters, perhaps? 

H.G. Callaway

H.G., there is no way to observe a space-time manifold, other than by observing the behavior of test particles reacting to it. So when you are in motion relative to those particles, that is the ONLY thing you can measure. From a practical perspective, the point is moot. No experiment has ever been able to determine the motion of objects relative to the vacuum, or any hypothesized "ether".

Philadelphia, PA

Dear Desiato, 

Many thanks for your contributed answer to the question. 

H.G. Callaway

Philadelphia, PA

Dear Kanda,

Baloney! ;-) 

H.G. Callaway

Philadelphia, PA

Dear Kanada,

Double baloney ;-) !

H.G. Callaway

Philadelphia, PA

Dear Low, 

Many thanks for your view of the question. Much appreciated.

H.G. Callaway