Foundations of Space-time (Relativity) Theories

Discussions on the philosophical foundations, assumptions, and implications of science, including the natural sciences.

Re: Foundations of Space-time (Relativity) Theories

Postby Obvious Leo on August 31st, 2013, 11:02 pm 

owleye wrote: Each observer of the elephant has to encapsulate the essence of the elephant by that which remains unvarying among all such observers from their own perspectives. As such, geometry is revealed and not determined by the modeler.


I'm not sure this makes sense. Certainly the elephant has an essence. It is objectively real. But the elephant itself is not objectively invariant. It is not the same elephant from one moment to the next. Near enough is not good enough on this occasion. Not only do all the observers have a differential referential frame within which to capture the essence of the elephant, the elephant itself is changing before their eyes. There is nothing unvarying in this scenario but it gets confusing because the observer and the observation are temporally intertwined. Relativity must go all the way and true simultaneity can only be a fiction.

Surely it makes more sense that the geometry is determined by the modeller and then this is a non-problem.

Regards Leo
Obvious Leo
 


Re: Foundations of Space-time (Relativity) Theories

Postby owleye on September 1st, 2013, 10:11 am 

Obvious Leo wrote:
owleye wrote: Each observer of the elephant has to encapsulate the essence of the elephant by that which remains unvarying among all such observers from their own perspectives. As such, geometry is revealed and not determined by the modeler.


I'm not sure this makes sense. Certainly the elephant has an essence. It is objectively real. But the elephant itself is not objectively invariant. It is not the same elephant from one moment to the next. Near enough is not good enough on this occasion. Not only do all the observers have a differential referential frame within which to capture the essence of the elephant, the elephant itself is changing before their eyes. There is nothing unvarying in this scenario but it gets confusing because the observer and the observation are temporally intertwined. Relativity must go all the way and true simultaneity can only be a fiction.

Surely it makes more sense that the geometry is determined by the modeller and then this is a non-problem.

Regards Leo


Well, I'm glad it got you to thinking about it. That was surely my intention. However, I'm afraid you missed the point of it.

First, consider the elephant. Of course, if you know in advance it's an elephant, the individual observers who are taking their individual perspective on it through their own devices, are not going to capture it, just as the example is intended to be understood as it has been passed down to us by the mystics, or whoever. However, if no one knows what it is in the first place, but have some reason to think there is something that needs to be explained, then, of course, you haven't met the conditions being imposed by these observers. One needs to take the model one has developed and try to see whether it can be used to predict the same things from others' perspectives, as well as seeing if there's something in common. Science requires more of the modeler than one derived from a fixed observation post. Biases associated with observation posts have to be eliminated as best they can. One has to think of it as a challenge, as for example, the challenge faced by not knowing what Dark Matter is composed of, or even, I suppose, whether there aren't two (or more) kinds of it. Theorists are not short on imagination about it. And observers are not short on their attempts to discriminate among them. True enough, there will be principles, such as Ockham's razor that they should rely on, but who knows exactly how that will turn out beforehand.

As it is applied to space-time, then, Einstein's theory has met the challenge of the constitution of space-time, even though at the time of its introduction, Einstein himself may not entirely have understood its implications. The arguments being given by Friedman and Torretti conclude that space-time has been captured by its identification with a gravitational field, of course, and is one that is associated with the mass-energy distribution of the universe. And motion through this space-time can be understood in accordance with the laws of motion that are formulated in the same spirit as the F = ma of Newton, though, as Wheeler puts it, more elegantly. They yield "proper" accelerated motions through this (dynamic) "medium" that are not conventional, and depend on the observer. Moreover, the theory does not suffer from the embarrassments left unaccounted for in Newton's absolute space. And there's more, dealing with the possibility of the expansion and contraction of space within the context of space-time over time.

Second, the block of marble. While Kant's space and time were products of the mind, and have application to the psychology of perception, as well as of cognition, where Kant's analysis was required to make experience possible at all, reason itself became the judge, jury and deliverer of Newtonian science, to include its conservation laws. However, it wasn't a matter of choice of this or that space and time and the laws so-embedded, they had to be Euclid's geometry and "Eudoxian chronometry" as Torretti puts it. Kant didn't think our mind was constrained so much as it was the only way to understand how knowledge could be obtained. And Kant did think Newton's laws constituted true laws of the world.

Since then, as the story unfolds (in both Friedman and Torretti's account), Kant didn't realize that reason was more adaptable and provisional than Kant imagined. There came forth new geometries. Forms could be revised, as these philosophers put it. Moreover, it came to pass that models had to be adapted as new evidence emerged. We can't rely on our models if they don't conform to the evidence. We can't just pick and choose any model. Though it seems we can, God like, make the world in our own image, so to speak (I'm reminded of the movie Don Juan de Marco), science had to be humbled, and granted new license to dare to enter into the folds of reality. The block of marble yields its own truths but only by a more flexible, more humble, yet more persistent being.

James
owleye
 


Re: Foundations of Space-time (Relativity) Theories

Postby owleye on September 1st, 2013, 2:27 pm 

With a bit of renewed confidence, which can't be said to be all that great as I was near the fuel gauge reading on empty, I need to finish the argument underway that was critiquing these early thinkers.

Michael Friedman wrote:But what about Einstein's "elevator" and the principle of equivalence? Does this latter principle not imply that accelerating frames are indistinguishable from non accelerating frames in the presence of gravity? Are we not led to a thoroughgoing relativization of motion after all? This line of thought also rests on a misunderstanding. The principle of equivalence is better understood as recommending a choice between two ways of describing gravitation: in terms of a flat space-time in which gravitational trajectories deviate from the geodesics via a gravitational force (as in traditional Newtonian gravitation theory); and in terms of a non-flat space-time in which gravitational trajectories follow the geodesics and there is no gravitational force (as in general relativity). In the first kind of description non-accelerating frames are indistinguishable from accelerating frames. The class of inertial frames cannot be picked out from the wider class of Galilean (arbitrarily accelerating) frames by the laws of our theory. So the first way of describing gravitation is defective in just the same way as Newtonian kinematics, and the principle of equivalence recommends the second. When we move to the second kind of theory, however, we do not simply eliminate the distinction between inertial and non-inertial motion; rather, we replace the old, flat space-time distinction with a new, curved space-time distinction: the distinction between local inertial (freely falling) frames and non-local inertial (arbitrarily accelerating and rotating) frames. Moreover, the class of local inertial frames can be distinguished from the wider class of arbitrarily moving frames. Therefore, although the transition from the first kind of of theory to the second certainly leads to a change in our concept of acceleration, it does not lead to a true relativization of that concept.


This concludes the argument (though he says he has other reasons as well) that refutes the view that drew generalized verificationist (only theories in which all its elements are verifiable are meaningful) and conventionalist conclusions (theories are no more than conventions) from GR. Not everything can be relativized away. He then moves to the accomplishments these thinkers attained. Next time.

James
owleye
 


Re: Foundations of Space-time (Relativity) Theories

Postby owleye on September 2nd, 2013, 9:13 pm 

Michael Friedman turns to the philosophic accomplishments of these early thinkers, most notably the elimination of some theoretical (unobservable, "metaphysical") entities and properties -- absolute space and absolute velocity in SR and the class of inertial frames in the transition to GR. And this can be said to be motivated by principles of parsimony and some version of the identity of indiscernibles. Friedman characterizes it as conceptual and methodological in nature:

Michael Friedman wrote:We can best represent the conceptual and methodological motivation by formulating Newtonian counterparts of the above two theoretical transitions [first to SR, then to GR]: a four dimensional version of Newtonian kinematics that dispenses with absolute space [] and a curved space-time version of Newtonian gravitation theory that dispenses with inertial frames []. The point is that these reformulations are empirically equivalent, but methodologically superior, to the usual formulation.


This conclusion isn't obvious to me, the reason being that I don't have in front of me a four-dimensional version of Newtonian kinematics nor a space-time version of Newtonian gravitation. These will be presented later in the book. As such, I'm going to hold off on evaluating these conclusions.

Friedman basically characterizes these philosophic advances as "eliminat[ing] 'excess' theoretical structure that 'makes no observable difference'."

Friedman intends a complete examination of these philosophic advances to determine not only why they worked to the extent they did, but why it led to the mistakes previously pointed out by Friedman that took it too far. Friedman, philosophically, is thorough. Next time, I will get into Friedman's explanation associated with the question: "why did GR not accomplish what SR did?" Which is to say that Friedman will try to assess the difference between "'good' theoretical entities and properties and 'bad' [ones]." Basically what this means is that he will dwell a bit on the "force" of the principle of parsimony.

James
owleye
 


Re: Foundations of Space-time (Relativity) Theories

Postby neuro on September 3rd, 2013, 8:03 am 

Michael Friedman wrote:we do not simply eliminate the distinction between inertial and non-inertial motion; rather, we replace the old, flat space-time distinction with a new, curved space-time distinction: the distinction between local inertial (freely falling) frames and non-local inertial (arbitrarily accelerating and rotating) frames.

and then
Michael Friedman wrote:We can best represent the conceptual and methodological motivation by formulating Newtonian counterparts of the above .... a curved space-time version of Newtonian gravitation theory that dispenses with inertial frames [].


Isn't the second statement a bit too strong?
We are not dispensing the idea of "inertial frames" (except they are locally such), and the geodesics of the curved space-time (which still apeears to be there as a general framework) still set the rules for such local inertiality...

Possibly there's something in this I can't grasp.
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Re: Foundations of Space-time (Relativity) Theories

Postby owleye on September 3rd, 2013, 10:20 am 

neuro wrote:
Michael Friedman wrote:We can best represent the conceptual and methodological motivation by formulating Newtonian counterparts of the above .... a curved space-time version of Newtonian gravitation theory that dispenses with inertial frames [].


Isn't the second statement a bit too strong?
We are not dispensing the idea of "inertial frames" (except they are locally such), and the geodesics of the curved space-time (which still apeears to be there as a general framework) still set the rules for such local inertiality...

Possibly there's something in this I can't grasp.


What's going on here is that Friedman is (and will be) attempting to make a strong case that GR doesn't relativize away all of the absolute unobservable theoretical entities and properties that early thinkers had argued that it did, to include Einstein's own early thinking -- though I'm not at this time prepared to say with any clarity what each of these early thinkers actually thought -- Poincare's ideas, for example, are becoming clearer, but not so Einstein's. Reichenbach's ideas are difficult to sort out, and when I was first introduced to him during my thesis project years ago, I could only get a flavor of him. In any case, I'm glad to have Friedman and Torretti to help me out.

In the popular scientific press (which I take Greene and Wheeler to be representative), there appears to me to be a consensus that such early thinking was confused on what were the accomplishments of GR and that these confusions have been cleared up. Science education has adapted to it. Such clarifications I take to be philosophic in nature, though oriented toward the science itself. I'm no scholar on the subject but my take on it is that on the science side of things individuals such as Wheeler have been responsible in their own way of making progress on that front.

On the philosophy side, they too have their individuals. What I'm reading from Michael Friedman, and now Roberto Torretti, in the '80s, is that (coupled with their vast bibliography) they are attempting to advance the philosophic issues surrounding GR during this early period just as the physical theorists had been doing. This is a clue to me that the confusions of these early thinkers lasted well into the '50s-'70s and that this productive period was culminating in the arguments that these two thinkers (undoubtedly among others) were trying to set straight. And I expect this effort is continuing, with these two thinkers forming part of that history, perhaps playing a pivotal role.

Now, the essence of the book, then, is in making the argument stand out, revealing in its outline the stark contrast between the various theories of space-time (Newtonian, SR, and GR). In order to do that he needs a platform on which to make the comparison. And the platform, as will be seen, is a generalization of the covariant platform on which GR is portrayed. Thus, for example, Newton's theories will have to be upgraded to theories of space-time, their Newtonian counterparts.

Now, this is Friedman's first full length book and I think it suffers from a lack of organization, so I'm having trouble following it. And my presentation of it may also have made it doubly confusing. This slow-read on my part is helping me, but I'm not sure it is helping others. Despite this, I'm happy whenever I get feed-back since it gives me an opportunity to see my own mistakes and as well see how Friedman might have made improvements in his presentation. I'm going to guess that he wrote the introduction (which we are now in the last part) after he had written the rest of the book. And though he is trying to introduce to us what will be presented he seems to assume we have what we need to understand what he is introducing. Nevertheless there have been clear enough sections of the introduction that have helped me quite a bit in understanding the basic argument.

Note that Torretti's book is quite dense, but revealing nonetheless because of its better organization. Torretti fills in a number of gaps in the historical account of the science that Friedman more or less glosses over, and as well goes into greater philosophic depth on specific concepts used in the science. Greene's book, covering relativity theory, was almost totally oriented around Einstein. I notice that in the earlier Torretti book he informs me that he was rewarded by the then recently opened up Einstein collection of papers. Torretti's second Dover edition (the current one that is being referred to) published in 1996 apparently took advantage of that.

James
owleye
 


Re: Foundations of Space-time (Relativity) Theories

Postby owleye on September 4th, 2013, 7:21 pm 

It's time to wrap up the Introduction. It is here that Friedman tells us the main objective of the book. As has been presented, Friedman wishes to place the theories of space-time on an equal footing of sorts that in turn will reveal their differences, which if he succeeds will demonstrate that GR is a more parsimonious theory than either SR or Newtonian theory, and the there are methodological considerations that are improved by this theory.

Friedman recognizes that the actual development of relativity theory "has a much finer structure" than how it was depicted earlier in this introduction. Here he is summarizing how he will be presenting his case that will help elucidate in what manner the arguments of these earlier thinkers breaks down when confronted with GR.

Michael Friedman wrote:Careful attention to the fine structure of our theoretical evolution will reveal how these cases [referring to the conventionalism of Poincare and the relationalist position of Leibniz] differ from the cases of absolute velocity and gravitational acceleration. We shall see in detail why the indistinguishability arguments of Leibniz and Poincare do not have the same methodological import as the special principle of relativity and the principle of equivalence.


His project is to shed light on where these thinkers went wrong in order to "advance beyond positivism." He points out:

Michael Friedman wrote:A central problem facing post-positivist philosophy of science is theoretical underdetermination: the problem of elucidating, and perhaps justifying, methodological criteria for choosing between incompatible theories that are empirically equivalent or agree on all observations. Such criteria will ipso facto provide a distinction between "good" and "bad" theoretical entities: an entity is "good" -- its postulation is legitimate -- just in case it is postulated by a methodologically preferred theory. I shall show how the evolution of relativity reveals these methodological criteria in action, allowing us to drive a wedge between "good" theoretical entities postulated by methodologically well-behaved theories (special relativity kinematics, general relativistic gravitation theory) and "bad" theoretical entities postulated by methodological ill-behaved theories (traditional Newtonian kinematics, traditional Newtonian gravitation theory).


I now realize that I had actually missed this point in my earlier reading. The platform on which Newtonian space-time is constructed, the purpose of which is to compare Newton's theory with the relativity theories, does more than just make the comparison possible, but in fact develops a Newtonian space-time theory that would account for the same observations as the other theories. And this idea in fact rings a bell. I do seem to recall that there was some effort on the part of some thinkers around the time that Einstein was coming up with his theory that developed an interpretation of the Michaelson-Morley experiments around a Newtonian framework. As such I'm looking forward to how Friedman develops this model and will be paying more attention to it.

Interestingly, however, this aspect of Friedman's project doesn't seem at this point to be related to that aspect of it dealing with making a comparison for the purpose of figuring out why positivism's objective didn't work out. Friedman identifies the failure as a methodological one and it is apparently connected to the issue surrounding the ability to obtain more than one theory that equally predicts the same observations, but at this point, this worry will have to wait.

I'll end the introduction here, though there is one paragraph remaining that is needed to pave the way for the argument he gets to at the end of the book, but I'm going to hold off referring to it, for the time being, until he presents us with his general model (platform) on which the theories of space-time will be constructed.

We are now ready to return to the mathematical treatment.

James
owleye
 


Re: Foundations of Space-time (Relativity) Theories

Postby owleye on September 6th, 2013, 10:14 am 

Ok. Here's how Friedman develops a general geometric framework for considering all the theories of space-time (Newtonian, Einsteinian) that fall under his project. He refers to this section 1 as Generalized Properties of Space-Time Theories. One may think of this as laying out the foundations that Friedman advertised in the title of his book as well as in the topic statement.

a. Space-time is defined as: "the set of all places-at-a-time of all actual and possible events."

b. The material universe is defined as: "the set of all actual events [] embedded within it."

c. There are two kinds of basic elements: (1) "space-time and its geometrical structure" (2) "matter fields -- distributions of mass, charge, and so on -- which represent the physical processes and events occurring within space-time."

Each of these theories "seek to explain and predict the properties of material processes and events by relating them to the geometrical structure within which the are 'contained.'"

He notes that this approach differs from "standard philosophical formulations, such as Reichenbach's, which characteristically take more observational entities -- reference frames, light rays, particle trajectories, material rods and clocks -- as primitive and attempt to define geometrical structure in terms of the behavior of such relatively observational entities." Friedman's explanatory frame is intentionally the reverse of this.

d. "Reference frames are treated as particular kinds of coordinate systems."

e. "[L]ight rays and particle trajectories [are treated] as particular kinds of curves in space-time."

f. "[M]aterial rods and clocks [are treated] as particular configurations of the fundamental matter fields."

g. "All theories agree that space-time is a four-dimensional differentiable manifold."

I let this all sink in before getting to what a "four-dimensional differentiable manifold" means, next time.

James
owleye
 


Re: Foundations of Space-time (Relativity) Theories

Postby owleye on September 8th, 2013, 11:41 am 

Ok...

So now we get technical. There is a wikipedia site on differentiable manifolds here that may help out. The mathematics being used were developed in the 19th century upon the arrival of an explosion of geometric principles came onto the scene. The basic ideas of open sets and neighborhoods presented below are elementary ideas one gets in undergraduate courses in "advanced calculus". The generalization to curved spaces is usually at the graduate level. If one is going to understand General Relativity theory, one really has to grasp what being a geometry is, in its general form. One has to get comfortable with the terms being used. I'll do my best, but readers are advised to google the terms one is not comfortable with. Wikipedia is a good source here.

Since my quotations will not be of the lengthy sort, I'll be using quotation marks instead, for citing Friedman.

1. "Space-time has a topology: given any point p in space-time, we have the notion of a neighborhood of p." Note that neighborhoods are intended to represent the area surrounding a given point, and is a technical term, which is formalized in a certain way. Basically there is a strong well defined relationship between any point p and any of the points in the neighborhood. However, though one might think of it as some local area, representing points "close" to p, the region can be quite large.

2. "Space-time is coordinatizable by -- the set of quadruples of real numbers. That is, given any point p in space-time there exists a neighborhood A of p and a one-one map into that is sufficiently continuous ( maps "nearby" points in A onto "nearby" points in and vice versa). is called a coordinate system or chart, around p." This is how statements about geometric entities (e.g., curves) are translated into statements about real numbers.

3. "We are also given the notion of differentiability: if f is a real-valued function (representing mass-density say) defined on a neighborhood in space-time, we say that f is differentiable just in case (the result of applying f to the result of applying the inverse of -- a function from into ) is differentiable for every chart ." Being differentiable is often translated as being smooth.


This is what is meant by a four-dimensional differentiable manifold and it is considered a local condition, not a global one. More will be said next time.


James
owleye
 


Re: Foundations of Space-time (Relativity) Theories

Postby owleye on September 10th, 2013, 9:07 pm 

Ok.. I left off with Friedman's development of the general mathematical model in which the space-time theories of Newton, SR and GR can each be represented from a local (i.e., neighborhood of any given point in it) perspective. He will justify this later, but he wants to contrast it with a global representation.

Michael Friedman wrote:The assertion that space-time is a four-dimensional differentiable manifold is a purely local assertion. It says that space-time is well formed in the small -- on a neighborhood of each point p. This leaves quite undetermined what space-time is like pglobally, or in the large. In particular, we do not know whether space-time as a whole can be mapped continuously in . Space-time could be finite or infinite. closed like a sphere, or open like a plane, connected (no holes or missing pieces) or disconnected (with arbitrary deletions) and so on. Moreover it is extremely convenient to view all assertions of our space-time theories as purely local in this sense. Each of our theories will have a large and interesting class of cosmological models, since there are many different global topologies compatible with a given local structure; and all our theories will look more like general relativity, which is standardly formulated in just this fashion.


Friedman adds a paragraph which may help in understanding how Newton's theory can be included within this general frame-work, in case there are those who may doubt it. I'm a bit tired now, so I'll have to leave off and get back to this next time. Once past this paragraph, though, he will spend the rest of this section on his "most important aim", which is "to describe the trajectories or histories of certain classes of particles: free particles or particles affected only by gravitational forces, or charged particles subject to an external electromagnetic field, and so on."

James
owleye
 


Re: Foundations of Space-time (Relativity) Theories

Postby owleye on September 11th, 2013, 8:06 pm 

For the doubters, as mentioned last time, Friedman adds a paragraph that might aid in understanding the general structure so-far laid out, one that is going to be adaptable each of the theories of space-time he is considering.

Michael Friedman wrote:Viewing space-time as a four-dimensional manifold does not prejudge the question in favor of the unification of space and time characteristic of relativity theory. It simply expresses the need for four coordinates to specify any event uniquely. The separate space and time of common sense (and some formulations of Newtonian theory) can also be described in terms of a four-dimensional manifold, one with a particular product structure (the Cartesian product of three-dimensional space and one-dimensional time). As we shall see, we effect a relativistic unification of space-time only if we view space-time as a four-dimensional semi-Riemannian manifold. On the other hand, since our basic object is four-dimensional, we are not automatically committed to a meaningful notion of "absolute" motion. For an arbitrary four-dimensional manifold does not necessarily have a relation of being-at-the-same-three-dimensional-spatial-position defined between pairs of space-time points. To get a notion of absolute motion we need additional structure, as would be provided by, for example, by the simple product structure .
owleye
 


Re: Foundations of Space-time (Relativity) Theories

Postby owleye on September 12th, 2013, 8:50 pm 

Now, on to curves in space-time, generally considered.

Michael Friedman wrote:We represent a trajectory or history as a curve in space-time, a continuous, differentiable map from an interval of the real line into our manifold. Intuitively, we can think of the real numbers in the domain of as times, although, as we shall see, "time" refers to very different things in the different theories. Following tradition, let us call such curves, world-lines in space-time.

We can describe the configuration of a world-line by tracing its twists and turns as it winds its way through our manifold. We want to be able to say, for each point p along the curve, how much and in what direction it is "curved" or "bent" at p. How do we do this? First, we need the idea of a tangent vector to the curve at p -- intuitively, an indication of the direction in which the curve is going at p. By tracing the changes to the tangent vector, we can trace the path of the curve. But what is a tangent vector? We know that for curves in Euclidean three-space the tangent vector is given by the triple , where the coordinates of our curve are given by the functions x(t), y(t), z(t).

Thus, in the tangent vector to a given point p is just the triple pf real numbers , where .


refers to the clock time of some particle at the point p. It's use is merely to inform us that this is the case. Let me pause here before he provides the generalization to all theories, given next time. Note how in the Euclidean example, the coordinate system is the usual Cartesian coordinate system, where the coordinate planes and axes are orthogonal to each other. This will not work in its generalized form. As a result it will take more of a description to explain how it works.

James
owleye
 


Re: Foundations of Space-time (Relativity) Theories

Postby owleye on September 13th, 2013, 12:13 pm 

The generalization of a curve takes advantage of the topological property mentioned earlier that space-time is coordinatizable for every neighborhood of a point p. This structural element doesn't prescribe the coordinate system, it merely specifies that there will be some one-one mapping of "nearby" points of p in a local region of space-time and nearby points in R4. Recall this was referred to as a chart, identified as , around p. The coordinates will be represented by x0, x1, x2, x3, or xi, (i = 0, 1, 2, 3).

Michael Friedman wrote:Then, relative to , the tangent vector to [the mapping of the curve] at p is the quadruple of real numbers



where for each i = 0, 1, 2, 3, ( is a continuous and differentiable function from R into R. Of course, this quadruple of real numbers will change if we change our coordinate system: it only represents the tangent vector relative to the particular chart . So it is better to identify the tangent vector itself with a slightly more abstract object: namely, the function , which, given any coordinate as input, results in the number the components of the vector in the coordinate system . Moreover, since is arbitrary, actually operates on any continuous and differentiable real-valued function f defined on a neighborhood of p:

.

The collection of all such operators for all curves with is called : the tangent space at p.


The notation may be confusing. , reads as the differential with respect to a variable u, of the value of the function f on the value of the mapping . It can be thought of as an infinitesimal and generalized coordinate value of the tangent vector at time t along the curve.

The next step is to describe the shape of the curve, which involves describing how the tangent vector just described changes as the particle moves along the curve. I.e., to tease out how variations in direction are to be considered. And this will be tricky. Next time.

James
owleye
 


Re: Foundations of Space-time (Relativity) Theories

Postby owleye on September 14th, 2013, 12:55 pm 

The next part deals with change in direction of a curve.

Michael Friedman wrote:We therefore need to be able to say, for vectors at "nearby" points, whether they point in the same direction or not. However, given the machinery we have so far, there is simply no answer to this question on an arbitrary manifold. In ordinary Euclidean three-space this question is easy to answer: we can simply "read off" changes in direction because Euclidean three-space is itself one big vector space within which the tangent spaces at different points fit together in a natural way []. In general, however, a differentiable manifold is not itself a vector space. We simply do not know how the tangent spaces at different points fit together []. Thus, to answer the question about changes in diretion we need additional geometrical structure.


The []'s were references to diagrams in the book. The first one shows that two tangent vectors can be understood as pointing in the same direction because their difference are merely a displacement, having the characteristic of being parallel to one other. The second one represents a curved space where being parallel is problematic.

And it is at this point, that Friedman returns to the affine structure earlier considered in the introduction. This will be the vehicle (structural element) that permits, in a generalized way, tangent vectors of "nearby" points to be considered to be in the same direction. To introduce this will also require additional mathematical ideas, notably 'operators', the generalization of what is understood as additions and multiplications. In this case, Friedman introduces a derivative operator, one which takes two terms and produces a third, in this case, the terms are the tangent vector field and an associated vector space, and the output of the operator is a rate of change vector.

Michael Friedman wrote:There are many ways to introduce this structure onto our manifold. Perhaps the simplest is by means of a derivative operator, or affine connection. Suppose we are given a curve and a vector field X(p) defined on points along , where X(p) is a continuous and differentiable selection of vectors from the tangent space at each point along : for each p, X(p) is in Tp (although in general X(p) need not be the tangent to at p). Given T (the tangent vector field to ), X(p), and a point q along sigma , a derivative operator, D gives us a vector ( in that records the rate and direction of the change in X(p) at q. Such an assignment of vectors counts as a derivative operator or affine connection if it has the properties of a derivative should have.


To which Friedman adds what these properties are in his appendix. The appendix (entitled Differential Geometry) is where he develops what is given in this chapter its mathematical foundation. I may or may not get into these details, but in this case I'll add the requirements of the derivative operator:

Suppose D is an affine connection (the derivative operator) that assigns a vector field DXY to each pair of vector fields X and Y. Also suppose that Z be an arbitrary vector field and f is an arbitrary function. It needs to satisfy the following:

DX(Y+Z) = DXY + DXZ

DX+YZ = DXZ + DYZ

DfXY = fDXY

DXfY = (Xf)Y + fDXY

Friedman mentions that this last rule is called the "Leibniz product rule" for derivatives.

Next step for Friedman is to develop a coordinate system of the tangent space of the curve. Next time.

James
owleye
 


Re: Foundations of Space-time (Relativity) Theories

Postby Obvious Leo on September 14th, 2013, 3:21 pm 

I'm still following you painstakingly James and so far the maths is manageable. However differential geometry can make the head spin somewhat so if you venture into that abstruse territory please spare a thought for the mathematically challenged.

Perhaps you could append the KISS version.

Regards Leo
Obvious Leo
 


Re: Foundations of Space-time (Relativity) Theories

Postby owleye on September 14th, 2013, 6:40 pm 

Obvious Leo wrote:I'm still following you painstakingly James and so far the maths is manageable. However differential geometry can make the head spin somewhat so if you venture into that abstruse territory please spare a thought for the mathematically challenged.

Perhaps you could append the KISS version.

Regards Leo


Thanks. I actually enrolled in a differential geometry class at UCLA, as a grad student after I returned from my stint in the Army. Unfortunately I was unable to grok it. I still have the text, but there was no way I was going to keep up (I was working full time, but that's not the reason). Prior to the start of the term I began reading it. I struggled mightily to comprehend the first few pages, as it was mostly new to me. But the worst of it was on the first day, the instructor had us moving well ahead in the book assuming the first chapters were all old hat. A rude awakening for me. In any case, I never did get back to it, sadly for me, but here I am, beaucoup years later getting an opportunity to test my mettle.

In any case, because you've shown an interest, I, too, will not shy away from it. And I'm happy that you've prodded me, otherwise I might not have (Note that Differential geometry already begins with spaces, something I omitted in relating the properties of the differential operator.)

James
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Re: Foundations of Space-time (Relativity) Theories

Postby owleye on September 15th, 2013, 2:14 pm 

I recognize that I'm in the middle of a difficult subject matter, the generalization of a curve in a topological manifold, but in the interest of tackling a difficult subject matter not quite having the mathematical background (and that I'm a slow learner), I think I should take this section a bit slower. As such, I will interrupt the flow of the presentation and make my way forward by way of a more fundamental level of understanding, eventually getting into the world of differential geometry. In this way, I'm hopeful that the terms being used will become more familiar as I move along.

So, let me begin with what it means to be a real number, which is the foundation of the mathematics being used to describe the so-called space-time continuum. The continuum component turns out to be more continuous than one might imagine, and there are lots of terms floating around that are more or less taken for granted that need to be cleared up.

There are a number of numbers systems, including the integers, the rationals, the reals and the complex numbers. Integers and rational numbers are well understood and they both have the property of being infinite in number, though one is a subset of the other. Interestingly, they also have the property of being countable. Not countable in the sense in which the infinite number of them can be counted, but rather that we can in some sense enumerate them. Even though it appears that the number of rational numbers are greater than the number of integers, they are considered to have the same infinite number of them because we can obtain a one-to-one mapping of the integers and the rationals. I'm not going to prove this but I want you to realize that mappings of one set into another set each of which is infinite is the way in which infinities are brought into some kind of understanding. We can wrap our mind around them, so-to-speak.

Now, it turns out that you can't obtain a one-to-one mapping of the integers or rationals into the set of real numbers. One might say there are too many of them to do that. The infinity of the set of real numbers is larger than the infinity of the set of rationals. In terms of density, the reals are more closely packed, if you like. And it is this "closely packed" idea that caused the concept of continuity, originally understood as the continuation associated with some activity, such as, for example, what Zeno was trying to come to grips with in his drawing attention to the sequence of fractions that never quite reached its goal, somehow proving that motion is impossible, to become transformed from the rationals to the reals, despite that there were irrationals among them (also, what are called transcendental numbers). The naming of these numbers probably carries some significance, but from my experience within the world of mathematics, they are just labels.

In any case, the size of the infinity of the real numbers is a matter of interest to mathematicians, such as in dealing with power sets, where a power set of an infinite set has a greater size associated with its infinity than the set of which it is a power of. A power set is the set of all subsets of the given set. Do the real numbers have the same size as the power set of rationals?

Not to get carried away with the interests of mathematicians here, real numbers have been harnessed by mathematicians by their ability to define them. The way I was taught (and I don't believe this represents the best definition) is derived from what is known as a Dedekind cut. (You can easily look this up on the web.)

The idea behind a cut is that every real number can be expressed as the limit of a potentially infinite sequence (known as a Cauchy sequence) of rational numbers. You can understand this by noting that pi is a real number whose representation is a growing rational number (3.14159.....). In any case, a cut of the rationals is a segmentation of them into two parts typically labelled A and B, with A being numbers less than some given number (say the sqrt(2)) and B, the set that is equal to or greater than this number. Since sqrt(2) is irrational, there is no rational number equal to it, but its definition is well defined by this definition anyway.

And this way of thinking of real numbers leads to the notion of open and closed intervals. And interval of real numbers is open at one end if it does not contain its boundary value -- the cut at which the boundary value is defined. It is a closed interval if it does contain that value. Thus, in the A and B intervals above, the A interval is open (at both ends), while B is closed at the lower value. This is sometime written as (A) and [B), with the ')' sign signaling open, and the ']' signaling closed.

In both cases, the intervals are considered continuous.

Turning to geometry, the reals are represented as continuous lines, where each point of the line "corresponds" to a unique real number. An interval becomes a line segment (or rather the reverse). One might think that the length of the line segment represents the difference of the values of the points at each end of the segment. However, if the line is curved, there is a need to map the points of the curved line, which determines that correspondence (and an one-to-one mapping at that) onto some interval of the real line. A mapping is a function that takes numbers in one domain and "translates" them so as they can be referred to by numbers in another domain. (Note that the domains do not have to be different.) Determining distance, then, requires understanding the structure of the line, identified by the mapping function. And the way they are determined are by taking infinitesimals at each point and summing them over the segment. Each of the infinitesimals are considered line elements.

What, then, is an infinitesimal. This is considered to be an infinitely small length of a line segment surrounding a point and is usually denoted by the prefix symbol d, prefixing the the segment's length, often referred to by the letter s (thus the nomenclature ds), and intends to be the limit of a sequence of lengths of smaller and smaller segments surrounding the point, each term of the sequence usually denoted by the prefix symbol (thus the nomenclature . One could sum each of these intervals and reach a better and better approximation of the length, where the summation of the sequence of intervals is denoted by the symbol , which in its general form becomes . However, the summation can be made exact by summing the infinitesimals. This is known as an integration over the set of infinitesimals and has the symbol which in its general form is . And the determination of the limit of the sequence of shorter and shorter length's is known as differentiation.

In my next post, I'll be moving into the geometry of space, covering ground already covered, but taking care to not let certain terms go by without making sure I've stated them to my satisfaction.

James
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Re: Foundations of Space-time (Relativity) Theories

Postby Obvious Leo on September 15th, 2013, 2:33 pm 

Just one question James. Will your exposition regard zero as a real number, as do Cantor, Gödel and Hilbert?
If so, is this kosher?

Regards Leo
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Re: Foundations of Space-time (Relativity) Theories

Postby owleye on September 16th, 2013, 9:33 am 

Well, of course, zero is a real number. Indeed it is an integer, and integers are included among the rationals and rationals are included among the real numbers.

So, the question I suppose you are asking is whether or not 'zero' is actually attributed to any invariant attribute of space-time, for example, the length of an interval. Can it have a length of zero? Perhaps for some purposes zero lengths (e.g., that an electron can be considered a point object, or even a star or a galaxy, can be thought of as a point object), but that's in order to make meaningful some or another set of measurements. What has been said so far is that line elements can be expressed as infinitesimals -- infinitely small. This is not quite the same thing as zero? Why? Because we can sum them up and get a value that is other than zero. Integration is one of the topics one learns in calculus -- an amazing invention, I suppose, but is part of a broader concept that belongs in the realm of measure (which is different from counting). What makes a set of points measurable? And real number intervals are but one category. A quantized space-time may have some need for a measure theory in which points in space-time are not real number based intervals. And in order to for this to work, I believe space-time has to have a mapping that allows points to become closely packed and isometric, but as I'm not familiar with this, I can't say. In the back of my mind, I'm not sure it can work. But mathematicians don't stand by and remain idle about such questions. I'm sure much thought has already been given to it.

James
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Re: Foundations of Space-time (Relativity) Theories

Postby BurtJordaan on September 16th, 2013, 11:53 am 

owleye wrote:So, the question I suppose you [Leo] are asking is whether or not 'zero' is actually attributed to any invariant attribute of space-time, for example, the length of an interval. Can it have a length of zero?


Well, the Minkowski spacetime interval for light is zero (ds2 = c dt2 - dx2). It is also invariant over different inertial frames. On a Minkowski diagram the physical length (as measured on the grid) of any light-like interval is however not zero, but rather undefined, because the scale of the light-cone is undefined. Any physical length you measure along the light-cone represents a zero spacetime interval.

I'm not sure if this helps or perhaps rather confuses the issue...

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Re: Foundations of Space-time (Relativity) Theories

Postby Obvious Leo on September 16th, 2013, 2:31 pm 

There was a time when I immersed myself deeply into mathematical philosophy. I eventually found myself more than a little troubled by some aspects of set theory and number theory in particular but my mathematical knowledge is such that I declare myself unqualified to critique these theories with any great confidence. My repudiation of the Minkowski interpretation of relativity has been raised many times and this rejection is also metaphysically grounded. Mathematical proof theory is also highly suspect in the philosophy of the bloody obvious and IMO Godel's diagonal arguments leak like a sieve. The Persians would snigger in scorn at most of modern western mathematical philosophy and they would certainly consign Cantor to the wilderness to contemplate his supernatural illusions unremarked. I could go on ad nauseam.

Why all this? I concluded that the culprit was zero. We stole zero from the Persians without its accompanying qualifications. Zero does not correspond to the value of a real entity in the universe. It is an unrealisable abstraction which has meaning only in terms of two or more real values or else as a convenient mathematical tool, somewhat like infinity. It makes a lot more sense to me to regard zero as an abstraction located midway on the number line between two other unrealisable abstractions, these being plus and minus infinity. I don't dispute it's usefulness, merely its existence.

I realise this is a digression of mammoth proportions in your valuable thread, James, and thus I'll leave it go for the moment. I'm still flitting about the region going quietly insane and murderously impatient with airlines and I don't expect to be able to retreat to my reclusive lair for another week or so. I have quite a suite of half-baked essays on the subject of zero and on my return I might see if I can dust something off that might be fit for human consumption.

Regards Leo
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Re: Foundations of Space-time (Relativity) Theories

Postby owleye on September 16th, 2013, 9:46 pm 

ObviousLeo....

I have read your objections several times over, and I can understand why you might be tired of repeating them, but I have to confess whenever I hear you speak of things like this I find I'm not particularly motivated to inquire about them, since, for the most part, you ordinarily don't give much more than just your view on things.

In any case, I've never felt that mathematics, in so far as it is discipline of mathematicians, is in any way about the real world. The world of the mathematicians is more or less self-contained. It says only what its axioms and theories say, no more and no less. Yes, there are unsolved puzzles in mathematics, and there are some limits that to what can be proved, given certain axiom sets, but this is as far as it goes.

Now, the scientist, in trying to explain or describe or predict what is observed, develops theories, or perhaps just models that do this. As the science gets to be more well-defined, the theories and models become mathematical in form, the purpose of which is to quantify and compare the theory/model with what is observed. To do that they borrow the ideas that have been developed by the mathematicians (or, if they need new mathematics, they, like Newton, invent it). And, of course, in doing so, it requires them to be able to make some sort of correspondence between the observed (measured) world and the mathematics being used to model it. This is more or less a requirement if the theory/model is to be put to the test.

And, in my view, science, once it has come up with a testable model, one that works well, (predicts and describes observations better than prior theories) and, perhaps, has other qualities that scientists feel comfortable with, resulting in its acceptance, the metaphysical aspects of the theory will not be that much of worry. This is not to say that there won't be theorists (and philosophers) who look closely at the metaphysical angle, and, in some sense, that's what I'm doing (Friedman as well) within this topic, but if one is going to question it, one must at least understand the mathematics.

Note that with respect to quantum theory, Born, the originator of the "reality" of "probability waves", remains in the driver's seat, as realists (such as Einstein) have tried several times over to dispute their "existence" all to no avail. Indeed, the deeper one goes into the things, it seems the greater likelihood of thinking that there's no actual "there" there. In other words, like Plato, the reality is the (particular) mathematics, the mathematical Forms of the Ideas.

I've held out hope that we've only reached an epistemological impasse (being a realist, myself), and that, ontologically, there has to be something going on that is responsible for what is observed, but I don't have anything more than my gut to say otherwise.

James
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Re: Foundations of Space-time (Relativity) Theories

Postby Obvious Leo on September 17th, 2013, 1:59 pm 

owleye wrote:you ordinarily don't give much more than just your view on things.


Isn't that what we're all doing?

owleye wrote: I don't have anything more than my gut to say otherwise.


Regards Leo
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Re: Foundations of Space-time (Relativity) Theories

Postby owleye on September 17th, 2013, 2:30 pm 

ObviousLeo...

Your posts are different than mine, I think. True, I don't always give some supporting argument, but more often than not, I give reasons for every conclusion I reach. And, I'd like to think, that in many of the instances that I don't provide an argument (or much of an argument) that I hope it would be more or less common knowledge, or common experience.

And I do this because I believe it's not really the views that people have that are important, but instead, the important part is the reasons they have for taking that view.

James
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Re: Foundations of Space-time (Relativity) Theories

Postby Obvious Leo on September 17th, 2013, 3:33 pm 

Point taken. However over the journey I feel that I've offered ample supportive argument for the notion that nothing does not exist and nobody has been able to contradict this opinion very convincingly, in my view. I also have the views of some significant heavyweight philosophers in my corner on this point.

It's true that I have no formal training in philosophy although I regard this as an advantage rather than otherwise. I occasionally experience some difficulty with the rigid and structured form of language of the "professional" philosopher but to counterbalance this I am less easily obfuscated by it. Likewise my formal training in the sciences is well out of date but my informal self-education in both of these disciplines is as contemporary as I can possibly make it. Does this self-devised approach to the acquisition of knowledge make my thoughts any the less valid or disqualify them from impartial scrutiny?

My personal philosophy adopts simplicity as its central plank and has elected Occam for pope. Thus when I assert, for instance, that 3-dimensional space has no physical existence, or that not is NOT, I need offer no argument in support. This is not to suggest that I am unable to do so but merely that the burden of proof lies with those who would say otherwise. I have read widely and deeply on such matters, a claim you may either believe or disbelieve as you see fit, and no such proofs have ever been presented to me.

Perhaps you'd like to have a go one day but I'll grant that now is not the time and this thread is not the place.

Regards Leo
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Re: Foundations of Space-time (Relativity) Theories

Postby Obvious Leo on September 17th, 2013, 3:42 pm 

I probably failed to make the point I should have been making. I have no problem whatsoever with using zero in mathematics since maths is virtually impossible without it. My objections come into play when such mathematical representations are used to make statements about the superstructure of reality when all that has been demonstrated thus far is that they can facilitate the prediction of future events.

Regards Leo
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Re: Foundations of Space-time (Relativity) Theories

Postby owleye on September 17th, 2013, 4:05 pm 

I think I accepted your denial of zero, as I took it to mean the existence of points in space-time in some kind of physical sense, though, in how you treat it in your last two posts, you've turned it into a denial of the existence of nothing. "Nothing exists," of course, doesn't imply the existence of 'nothing' as if it were a something. Nothing, in that usage, implies negation. There are other usages, as well, such as 'absence'. The concept is meaningful -- i.e., it's not nonsensical, but in my way of thinking, no one is really arguing that there's something about the concept that implies there is something of substance that is instantiated by it in the world itself. In the philosophical debate over space-time, Friedman will provide a lengthier discussion of this, within a number of frameworks, including a "Leibnizean" one, which is considered to be at one end of a spectrum of interpretations of the metaphysics of space-time in accord with relativity theory (not in its entirety -- i.e., cosmology, but rather in any local area of it). Friedman's argument will be a stance that is at the other end, though not at the extreme end. Friedman will also make some important distinctions in the use of 'absolute' that help in understanding his argument.

James
owleye
 


Re: Foundations of Space-time (Relativity) Theories

Postby owleye on September 18th, 2013, 9:56 pm 

Let me continue my brief foray into some of the elementary terms being used in the mathematics of space-time, that in its entirety is part of the discipline of differential geometry. I'm not entirely sure how far I'm going to progress with this, but I'm returning to it in order be able to better understand the topic of curves in space-time and how they are formulated so as to reveal their intrinsic properties, especially in consideration of general relativity in which space-time is curved, but yet is still applicable to Newtonian space-time and the space-time of special relativity.

Where I left off last time the geometry did not get past one dimensional lines, which can be mapped onto the real number domain. And there distance along the line is determined by the use of differentials, which are basically infinitesimals that, when integrated (summed) over some line segment will yield its length.

One additional point of interest is that differentiability requires that the line segment be "smooth" over its length. A line is not smooth, if it has corners, which is to say that the tangent at the corner point is different when it is approached from one side from when it is approached from the other side. Being smooth or differentiable requires that the tangent at every point be the same regardless of the direction of approach to each point. Of course, if there are corners, the line segment might be considered "piece-wise" differentiable, so that its total length becomes the sum of the pieces. In any case, it is assumed the curves in space oare differentiable throughout their length. No sharp corners to them.

So, let's now consider multiple dimensions to our geometry. What does this mean? Well, this would seem obvious to anyone who has thought about left-right, forward-backward, up-down, in the three dimensions of the space we more or less take for granted. And there are other coordinate systems, for example the coordinate system used by pilots, who make use of roll, pitch, and yaw, or perhaps by navigators on the surface of the sea, using latitude and longitude, and for a third dimension, depth.

One significant aspect of coordinate systems is that each dimension is completely independent of the other dimensions. Each of them represent a degree of freedom. And though relativity theory uses time as a spatial dimension, and seems to violate this principle of independence (e.g., the idea of going backwards in time), this feature can be delayed at least until semi-Riemanninan geometry is considered. For the time being, then, the term 'space' will be the domain of consideration, regardless of the number of dimensions it has.

And it is from this feature of coordinate systems that we obtain the notion of a product form of space, where the different dimensions are written as products. In three dimensions, the product space is (for example) H X W X D, as you will find on the specs of a refrigerator. And this example is an example of a Cartesian product.

The points within a space are expected to be coordinatized such that the coordinate values for each dimension range over some real-valued domain, each independently of the other dimensions and cover the entire space. This property is known as homeomorphism. Not all spaces, however, are homeomorphic. The surface of a sphere, for example, is not homeomorphic since the points that identify at least one of the two poles cannot be mapped into R2. One might recall that map makers adopt different mapping projections (e.g., the stenographic projection) in order to depict the surface, and leave out (say) the north pole. However, for the purpose of allowing a wider range of geometric structures, a strict adherence to homeomorphism can be relaxed. And it can be done in the following way. If the entire surface can be mapped by developing two or more homeomorphic mappings, each associated with some segment of the domain and can be pieced together smoothly, then such a geometry will be included within the set of geometries being considered.

In the appendix that Friedman provides he specifies the mathematics that satisfy this requirement of (smoothly) patching together surface mappings. I'll introduce that next time, coupled with an explanation of the terms used. It's not really necessary to know that this is possible or how it is possible (since one can readily intuit that a surface geometry is smooth, but in doing so it will bring into sharp focus the structural requirements imposed on the topological manifold that is applicable to relativity theory, and in this case, becomes the demand for differentiability (smooth curves and smooth surfaces). Next time, then, I will begin the discussion of differential geometry by discussing what a differentiable manifold is.

James
owleye
 


Re: Foundations of Space-time (Relativity) Theories

Postby neuro on September 19th, 2013, 7:22 am 

Excuse me, James,
I have the impression we are facing once more a problem we have previously discussed:
owleye wrote:One significant aspect of coordinate systems is that each dimension is completely independent of the other dimensions.

This is true only provided you are considering distances/displacements on one dimension only; which means you can do very little with this coordinate system. If you begin to consider combined distances/displacements on more than one dimension, such “complete independence” vanishes: it only holds for orthogonal coordinates, i.e. for an Euclidean Geometry, i.e. a geometry defined by a diagonal correlation matrix among dimensions (all zeros except ones on the diagonal).

Why do I introduce “combined” distances/displacements?
Because:
And it is from this feature of coordinate systems that we obtain the notion of a product form of space, where the different dimensions are written as products. In three dimensions, the product space is (for example) H X W X D, as you will find on the specs of a refrigerator. And this example is an example of a Cartesian product.


Consistently, such a product has no meaning in a non-Euclidean geometry...

Not all spaces, however, are homeomorphic. The surface of a sphere, for example, is not homeomorphic since the points that identify at least one of the two poles cannot be mapped into R2. One might recall that map makers adopt different mapping projections (e.g., the stenographic projection) in order to depict the surface, and leave out (say) the north pole. However, for the purpose of allowing a wider range of geometric structures, a strict adherence to homeomorphism can be relaxed. And it can be done in the following way. If the entire surface can be mapped by developing two or more homeomorphic mappings, each associated with some segment of the domain and can be pieced together smoothly, then such a geometry will be included within the set of geometries being considered.
.

OK, but then one must keep in mind that any operation that involves more than one dimension will have to be dealt with by using the correlation matrix: such correlation matrix will be stable, constant and well behaved, as an example, for the curved space defined by the surface of a sphere. But the problem in our case is that we shall stumble upon a continuously changing curvature of our space, i.e. a location-dependent correlation matrix among dimensions.

In the appendix that Friedman provides he specifies the mathematics that satisfy this requirement of (smoothly) patching together surface mappings. I'll introduce that next time, coupled with an explanation of the terms used.


looking forward to it...

Tks, owleye, this is a nice work you 're doing for us
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Re: Foundations of Space-time (Relativity) Theories

Postby owleye on September 19th, 2013, 9:38 am 

neuro wrote:Excuse me, James,
I have the impression we are facing once more a problem we have previously discussed:
owleye wrote:One significant aspect of coordinate systems is that each dimension is completely independent of the other dimensions.


This is true only provided you are considering distances/displacements on one dimension only; which means you can do very little with this coordinate system. If you begin to consider combined distances/displacements on more than one dimension, such “complete independence” vanishes: it only holds for orthogonal coordinates, i.e. for an Euclidean Geometry, i.e. a geometry defined by a diagonal correlation matrix among dimensions (all zeros except ones on the diagonal).


You're right. I had in mind orthogonal coordinate systems when I wrote this.

neuro wrote:Consistently, such a product has no meaning in a non-Euclidean geometry...


True enough. Curvature, then, is more complicated than I had imagined. Of course, I did in fact read the general Riemannian metric form, and how the diagonal matrix is lifted out of it, but I didn't really absorb it. Thanks to you, I'm better off. Even so, I think there is a place for thinking orthogonally and independently about dimensions. I just have to be more careful. It assumes the underlying framework being specified by the coordinates are of a certain kind, namely, Euclidean.

neuro wrote:OK, but then one must keep in mind that any operation that involves more than one dimension will have to be dealt with by using the correlation matrix: such correlation matrix will be stable, constant and well behaved, as an example, for the curved space defined by the surface of a sphere. But the problem in our case is that we shall stumble upon a continuously changing curvature of our space, i.e. a location-dependent correlation matrix among dimensions.


And, I will surely keep this in mind. And, in doing so, I hope to clear up the fog I'd earlier been in, in how curves in a curved space can be coordinatized within a framework of curvilinear coordinate systems. A one dimensional curved line within an n dimensional curved space. We'll see. I'm very happy to have you as my teacher here.

James
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