## Foundations of Space-time (Relativity) Theories

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

### Foundations of Space-time (Relativity) Theories

As promised I'm starting a new topic about space-time, again dealing with the same topic as the previous one that Greene presented. The topic intends to dig deeper by discussing it in the context of its mathematical formulation. And the reference I will be using is the same one I faltered on in my previous tackling of the subject matter, before I took time out to learn more about these theories from Greene. That reference, again, is "Foundations of Space-time Theories: Relativistic Physics and the Philosophy of Science", by Michael Friedman, Princeton University Press, 1983. Note the date is earlier than that of Greene's book, so one might expect that Greene would be more up to date. However, as you will see, if there have been advances, they will be overshadowed by the careful examination of the space-time structures and related principles that Friedman puts into what he writes.

A note of caution about the mathematics: Here's what Friedman writes in his preface:

Michael Friedman wrote:The present book makes heavier mathematical demands than the reader may be used to in the philosophy of science. Unfortunately, given the way in which physical and philosophical problems are intertwined with mathematical notions here, especially with the notion of general covariance, there is no practicable alternative. In particular, one has to see in detail how one goes from generally covariant formulations to the more usual formulations and back again if one is to attain any real clarity about the various relativity principles. Of course, these formulations and methods are in no way original with me. I have taken them, with minor alterations, from the works of Anderson, Trautman, and Havas, mentioned above.* In any case, I have striven to make the treatment as intuitive as possible via numerous pictures and diagrams, and I have included an appendix outlining the geometrical techniques employed. All that is absolutely presupposed is an acquaintance with the basic ideas of analysis and linear algebra on the level of an undergraduate course in advanced calculus.

*He is referring to writings in the '60s by J.L Anderson, A Trautman, and P. Havas. Friedman has an extensive bibliography from which he draws,, which if interested, I can retrieve for anything Friedman refers to. Anderson, for example, will play an important part. Indeed, in an earlier part of the preface Friedman provides his growth from his dissertation to this book on this very topic. My first acquaintance with Friedman was when I was doing my thesis work on Kant's space and time. His book: "Kant and the Exact Sciences" was both thorough and enlightening. I also have his book: "Revisiting Logical Positivism."

One of the advantages of this book is that, for those that are unacquainted with the roots of relativity theory, both scientifically and philosophically, this book will fill that void. Not in detail, but with clear direction. And this is where I will start. That story, of course, is oft-told, and anyone who is really interested in it will discover the volume of interest in it. Indeed, as any scholar will tell you, once you launch your project, you will be confronted by what appears to be an endless effort wading through it all. Your first trial will be one of "data-reduction.", which in my case turned out to be a challenge too great to fulfill. (Well, there were other factors as well.)

In any case, I hope you will find in it something you might not have realized. And more than that, I hope myself to get clear about the concepts being used. Perhaps you will find your own position a bit shakier than you might otherwise have thought, as well.

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

Let me begin with a some background philosophy. Friedman orients much of this work around a philosophy of science that became popular following the advent of relativity theory -- namely, logical positivism or logical empiricism, that subsequently suffered under the scrutiny of later philosophy. (If you don't know what this school is, I suggest, googling the terms.)

Consider one of the main drivers of the logical empiricism: the theory/observation distinction. If one looks at the development of Einstein's relativity theories, one can see how this distinction appears to play a role. Here's Friedman:

Michael Friedman wrote:If we look at the actual development of relativity theory, the observational/theoretical distinction appears to play a central role. It is a key element, in fact, in Einstein's arguments both for the special principle of relativity and the general principle of relativity. In outline, these arguments go as follows. Classical physics makes use of absolute motion, that is motion of physical bodies with respect to absolute space. But only relative motion -- motion of other physical bodies with respect to other physical bodies -- is observable. Therefore, appeals to absolute motion should be eliminated from physical theory. Thus, the special principle of relativity allows us to dispense with the notions of absolute velocity and absolute rest. All systems moving with constant velocity are "equivalent," and velocity makes sense only relative to one or another such system.

I hope you are able to notice that the term 'observer' is heavily used in relativity theory, one that excludes observing theoretical and unobservable entities. What is observed appears to be what is driving relativity theory. And in doing so, the idea is to eliminate undesirable theoretical crutches, if you like.

In any case, he quotes from the main 1905 work of Einstein, part of which contains Einstein's criticism of classical electrodynamics, because it invoked more than relative velocity:

Einstein wrote:The observable phenomenon here depends only on the relative motion of the conductor and the magnet, whereas the customary view draws a sharp distinction between the two cases in which either the one or the other of the two bodies is in motion.

Friedman then goes on to say that special relativity didn't go far enough, since it still makes use of a distinction between inertial systems and accelerated systems. Absolute acceleration is still retained. So Einstein criticized his own theory in the following in his main work in 1916:

Einstein wrote:But the privileged space R1 of Galileo [ = an inertial frame], thus introduced, is merely a factitious cause, and not a thing that can be observed.

Friedman sums this up:

Michael Friedman wrote:The general theory of relativity is supposed to overcome this defect in the special principle by dispensing with the distinction between inertial and accelerating systems, just as special relativity dispenses with systems at rest and systems moving with [nonzero] constant velocity. According to the general theory of relativity, all states of motion are "equivalent."

Assuming you've appreciated the use of 'appears to be" in this, what's going on here is that it appears that the emphasis on the observational/theoretical distinction has paid off, and as well gave life to distrusting unobservable entities, that positivists are so keen on. Hans Reichenbach, writing in the 1920s, set the tone for this philosophical movement in a passage that Friedman references.

Reichenbach wrote:The physicist who wanted to understand the Michaelson-Morley experiment had to commit himself to a philosophy for which the meaning of a statement is reducible to its verifiability, that is he had to adopt the verifiable theory of meaning if he wanted to escape a maze of ambiguous questions and gratuitous complications. It is this positivist, or let me rather say, empiricist commitment which determines the philosophical position of Einstein. It was not necessary for him to elaborate on it to any great extent; he merely had to join a train of development characterized, within the generation of physicists before him, by such names as Kirchhoff, Hertz, Mach, and to carry through to its ultimate consequences a philosophical evolution documented at earlier stages in such principles as Occam's razor and Leibniz's identity of indiscernibles.
.

More on this next time, but I'm hoping you get the message here. Einstein's achievements seem to be grounded in what he was motivated by. There's a principle that he is devoted to. And early on, this principle seems to have been well-founded. Einstein's theories are a great success. So the question will be whether that success warrants the conclusion that the advocates of its claimed underpinnings celebrated.

I should add one point of interest to me, here. And that is the language being used by Reichenbach, respecting the verificationist theory of meaning. While this theory has itself suffered, the language used by Reichenbach has the same flavor to it that I saw when reading Greene (in his presentation of the relationist position of Mach), where at the time, I had to hold off on where it came from. I believe I now see that it probably arises from the way Reichenbach tells his story. There's an apparent need to establish a language for observations -- a logic, in the case of logical empiricism, if you like -- in which its meaning only applies to observables. There will be much more on this as we go along. (NB: In this book, the relationism in Greene is referred to as relationalism, so I'll use the latter term as it comes up, rather than the former.)

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

Sounds promising, James. I wish you success with this project.

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

It may sound odd to some that I begin with philosophical background in a philosophy forum on a subject matter that is scientific in nature. And I think the root of it derives from the great emphasis placed on science in the overall structure of this web-site. This coupled with the general attitude of scientists toward philosophy (which is only amplified by my introducing it in this way). Moreover, I think those who think of themselves as philosophers while not having the background in science to the extent a scientist does are a bit intimidated by this attitude, and those that remain, remain in a kind of defiance of this. Or they stay away from such topics, fearing their own weaknesses will be exposed.

Those of us who think of themselves as philosophers, and the numbers may be as great as those who think of themselves as scientists, are faced with a problem. Both require study and an academic background, but philosophy, by and large, doesn't pay, nor does it have much of a payoff. That leaves the experts in science far outnumbering the experts in philosophy. The rest become amateurs, like myself. And there are lots of philosophy boards for these amateurs to gravitate to. This is one that amateurs won't be able to get away with bald-faced ideas that conflict with accepted science. This is one with a discipline and guidelines that strive for quality in reasoning. One that has standards of acceptance, if you like. It takes some muster on the part of those who participate to make sure that the ground they stand on is firm enough to enter into a dialog.

Yes, in other philosophy forums, this exists in the form of ridicule, and a "you don't know what you are talking about" just as it might here, but on this forum, deference to the experts in science gives this sort of thing a new force. Indeed, in other philosophy forums, science plays such a little role that it becomes mostly about things that science has not had much interest in or has stayed away from, or when it strays into science, there's little feedback that would curtail its excess.

For you philosophers (and even the scientists) out there, then, this topic is going to test your mettle. From my experience on this board, there are very few that have the slightest philosophical background that would appreciate what's going on. You will need to elevate your thinking just to take it in. But, since you are here, it is likely that you have something going for you that can take up that challenge. For the scientist, yes, it may seem to be a reinforcement of the waste of time that constitutes philosophy. And in this topic, one may wonder why philosophers who get these dreary ideas should even be listened to. But, being a philosopher, amateur, though I am, what they focus on is what I focus on. And, in the end, it is what theorists will eventually have to focus on -- the "what if" -- even if it is only to generate the ideas before getting down to brass tacks.

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

Moving on....

The story just told of the observational/theoretical distinction motive, which shunned the theoretical, in favor the the observational, also comes into play in a philosophical position known as conventionalism. You can get a flavor of this position by recognizing that we resort to it being a convention that we use a second to be the basic unit for a quantity of time. Here's what Friedman says of it:

Michael Friedman wrote:Similarly, we can also understand why the positivists were attracted to conventionalism. The essence of conventionalism is a doctrine of "equivalent descriptions": alternative, seemingly incompatible theoretical descriptions are declared to be equivalent when they agree on all observations.* Consequently, theoretical assertions are not "objectively" true or false: they have truth-value only relative to one or another arbitrarily chosen "equivalent description." This doctrine also appears to be borne out by relativity theory, where alternative ascriptions of motion are not "absolutely" true or false but only relative to one or another arbitrarily chosen frame of reference. Moreover, all reference systems are "equivalent": they provide equally good descriptions of the same observable facts.

*I might add that Friedman, like mathematicians in general, will think about this in terms of transformations of one system (observer) into another, yielding an equivalence class of transformations. The Galilean class of transformations is an equivalence class within Newtonian physics. And this way of thinking will foster a language dealing with truth-value, which is to say that statements have a truth value up to or relative to their equivalence class. (This is reminiscent of the "universe of discourse" idea that I remember from years ago, though I think this latter referred is to fictional or counter-factual worlds in contrast with a fact-based or a real world.)

Friedman then digs a bit deeper, remarking that what has been said is "still relatively superficial" and as will be drawn out in my next post, the origin of such thinking may be a bit startling.

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

It may seem from all of the above, that relativity theory is an offshoot of the empiricists, particularly the British Empiricists. Indeed, much of the 19th century advances in science can be attributed to these empiricists. However, as Friedman points out:

Michael Friedman wrote:Logical positivism did not start out as a version of empiricism or verificationism a la Hume or Mach. Contrary to popular wisdom, the influence of Kant was much more important. Twentieth century positivism began as a neo-Kantian movement whose central preoccupation was not the observational/theoretical distinction, but the form/content distinction.

Friedman does not argue the point, but has the reader refer to the works of Carnap (1928), Schlick (1917) and Reichenbach (1920), particularly the latter. One should interpret form and content as elements of knowledge where the form makes the conceptual, mind-dependent contribution and content is that which is contributed by the world. Kant's great influence, as against empiricism, is that the formal elements play a necessary and important role in the acquisition of knowledge of nature. Where Kant was thought to be wrong was in characterizing it as synthetic a priori truths, un-revisable. Instead they are better described as "conventions" or "arbitrary definitions" to these positivists.

Where does this take them? Well, as Friedman points out, and one can see this from looking at the tremendous advances in geometry during the 19th century. They all seemed to have come about by mathematicians on the continent. So, what was needed is to bring Kant up to date. As Friedman explains it. "The Kantian view is limited by its too intimate association with outmoded mathematics and physics." The "container" view of Euclid's space, "filled with matter (content) obeying deterministic laws of motion" was too constraining. So what was it that was significant? Friedman narrows it down to two mathematicians: Gauss (1827) and Riemann (1854).

We are now ready to enter into the mathematical portion. And even though we're only in the introduction, I hope you will gain some appreciation of what these mathematicians empowered. The next post will begin the background mathematics.

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

Before proceeding, in addition to Gauss and Riemann, I should have included Klein (and his Erlanger program) (1872) in what Friedman covers.

Though Friedman launches directly into Gauss, and his Theory of Surfaces (1827), he assumes that we are all aware that by this time, Euclid's geometry (as opposed to what is now known as Euclidean geometry, the modernized version), hung on Euclid's parallel postulate. One can gain an appreciation of the ancient Greek mathematicians on which this postulate was made necessary, by its taking two millennia to discover that it was necessary. Many mathematicians over the ages had tried to find that it wasn't necessary, but it wasn't until the very early 19th century when Bolyai (Hungary) and Lobachevsky (Russia) (circa 1830) independently were credited as the co-discovers of what came to be known (due to Gauss) as non-Euclidean geometry. Gauss is often included in that group as well, but his contribution should probably be considered as influencing, not developing these non-Eucldean geometries. However, it is well-known that Gauss claimed he had a proof. And, on a matter such as this, there remains controversy.

In any case, the enormity of this is hard to state. Euclid's geometry was not just a geometry of space, it was basically a substitute for what space was. Performing geometry is directly applicable to measuring space (hence the name). What was one possible geometry now was thought of as three possible geometries for space, and we see how Gauss paved the way for generalizing what these mathematicians discovered.

Returning to Friedman:

Michael Friedman wrote:The most important mathematical development from this point of view is an extreme generalization of the concept of space and geometry. This process begins with Gauss's theory of surfaces in 1827. But here already there is a two-fold generalization that is the source of much later confusion: a generalization in the use of coordinates, and a generalization to non-Euclidean geometries. First, Gauss considers the use of general (Gaussian) coordinates x1, x2 in place of the familiar Cartesian coordinates. In Cartesian coordinates the "line element" takes the familiar Pythagorean form

[1] $ds^2 = dx_1^2+dx_2^2$

In arbitrary coordinates it takes the more complicated form

[2] $ds^2 = g_{11}dx_1^2+g_{12}dx_1dx_2+g_{21}dx_2dx_1+g_{22}dx_2^2$

Before proceeding further, for those mathematically challenged, 's' represents distance, generally, spatially determined by the coordinates x1 and x2, and the 'd' which precedes both s and the coordinates, modifies what follows it so that it represents an interval (or differential). It is not a separate variable. The specific use of 'd' corresponds to making the interval as small as is possible, which, while > 0 is usually labeled $\Delta$ (the greek letter (uppercase) 'delta'). As it approaches zero, it becomes an "infinitesimal". In the limit, it becomes a measure of distance along tangent to the line at a given infinitesimal located at the coordinate (x1,x2) of a line segment. If you haven't yet had a course in calculus, you may be surprised to learn that such infinitesimal intervals can be added up to give the correct distance, but this is what calculus allows you to do. In any case, 'ds' intends to represent the individual line element over which a given line segment's length is determined by a summation process (which in this context would be called 'integration'). Integration would then be able to handle the distance metric not only for flat surfaces, but for curved surfaces as well.

Two coordinates represent 2-space -- a surface. That there are two-coordinates means we are limiting the geometry to 2 degrees of freedom. Each coordinate is presumed to be independent of the other, for the purpose of characterizing the surface geometry.

In [2], the gij are real-valued functions of the coordinate. In polar coordinates, identified as the radius, r, and angle, $\theta$, the equation distance function becomes:

$ds^2 = dr^2+r^2d\theta^2$,

yielding g11 = 1, g12 = g21 = 0, and g22 = r2.

I'll deal with the next generalization (re: the geometries) in the next post.

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

Enjoying your work once again James.

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

Obvious Leo wrote:Enjoying your work once again James.

Regards Leo

I do appreciate this. If the readers (whoever they might be) are paying attention, there is an underlying theme in the way that I'm presenting it that will be revealed by a carefully constructed, but rather long argument. We are only in the introduction, here. But it may be possible to guess at the major difference between the way Greene portrayed it and the way Friedman portrays it, one that in turn will reveal why Einstein eventually changed his mind and that there remains room for some form of an absolute space-time in which objects undergo absolute acceleration. The most difficult issue is to overcome the dynamic aspect of what's become of space-time in GR, namely the gravitational field. The confusion will be in wading one's way through the relativity of the dynamics. In any case, it will show up once we decipher the equations of motion, some of which will be covered in this introduction.

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

Let me now add the second sort of generalization developed by Gauss.

Michael Friedman wrote:Secondly, [] Gauss also considers the geometry of arbitrary curved surfaces (for example, the surface of the sphere or of a hyperboloid of revolution). He shows that the geometry on such a surface is in general non-Euclidean. For example, the geometry on the surface of a sphere is a geometry of positive curvature in which there are no parallels; the geometry on the surface of a hyperboloid of revolution is a geometry of negative curvature in which there are many parallels.

The non-Euclidean surfaces can also be covered by Gaussian coordinates x1, x2, and they also have a line element given by (2). Moreover, as Gauss shows, the coefficients of the line element in (2) -- the gij's -- give us complete information about the curvature, and hence the geometry, of our surface. It is extremely important, however, to distinguish a mere change in coordinates from an actual change in geometry. In particular, the more complicated form (2) of the line element does not necessarily signal a non-Euclidean geometry: we can use non-Euclidean coordinates on a flat Euclidean plane as well (as our example of polar coordinates clearly shows). What distinguishes a flat Euclidean geometry is the existence of coordinates satisfying (1): On a Euclidean plane we can always transform the more complicated form (2) into the Pythagorean form (1) (for example by transforming polar coordinates into Cartesian coordinates). By contrast, on a curved non-Euclidean surface it is impossible to perform such a transformation: Cartesian coordinates simply do not exist.

This distinction between generalized coordinates and generalized geometries will be a consideration throughout this presentation. And in the next post, Friedman will explain one of the consequences that such a distinction brings about -- the intrinsic vs. the extrinsic.

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

This post will conclude the contribution of Gauss, whence we move on to Riemann, who expands the generality even further. Friedman clarifies the two generalities previously presented by providing a distinction between intrinsic properties of a surface and extrinsic ones.

Friedman wrote:Intrinsic features characterize the geometrical structure of a surface -- its curvature, Euclidean or non-Euclidean character, and so on -- and are completely independent of any particular coordinatization of the surface. Extrinsic features, on the other hand, correspond to particular coordinatization of the surface: accordingly, they vary as we change from one coordinate system to another. Thus, the actual form of the line element (2) -- obtained by substituting particular functions for the gij['s -- is an extrinsic feature of the surface, for these functions change when we transform our coordinate system. The connection between intrinsic features and extrinsic features is this: a given intrinsic feature of a surface corresponds to the existence of coordinate systems with certain extrinsic features. For example, a flat, Euclidean structure corresponds to the existence of Cartesian coordinates in which g11 = g22 = 1, g12 = g21 = 0; it corresponds to the simple form (1) for the line element.

As this is merely a taste of what's coming, the ramifications of this distinction may not sink in, just yet. But, as the topic moves forward, the theories of space-time (which include those of Newton, SR and GR) will be covered all within the same framework.

In moving to Riemann's geometries, the topic will introduce math that become even more abstract (i.e., more generalized), and the formulations for the line element (2) will take a different form, just for the sake of compressing the mathematics. If you don't have much of a mathematics background, you may easily get bogged down. Mathematicians make use of symbols to say things that would otherwise take long sentences to express. The more abstract the mathematics becomes, the more symbols of various kinds are used, the whole point of which is to say something in a concise way, revealing the relationship of the terms being expressed by these symbols. There is a tendency to lose sight of what the symbols mean in doing so. Over time, mathematicians become better at this translation by their use, most noticeably by making sure they fully understand and can make use of the symbols without having to look them up. And while this is not a course in mathematics, it would be to the readers advantage to try to understand some of the conventions (for example, that letters at the end of the the alphabet are usually variables, those at the beginning are constants, those in the middle are often indexes, and so forth). The alternative of writing out everything in long hand brings with it a tendency to lose the essence of what's going on. I do have sympathy. Any check of recent mathematics will confirm that the generality will go far beyond anything I'll be covering.

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

We are now in a position to introduce our third equation, based on further generalizations of Riemann.

Recall the first two equations: First the Cartesian coordinates for a surface.

(1) $ds^2 = dx_1^2+dx_2^2$

In arbitrary coordinates it takes the more complicated form

(2) $ds^2 = g_{11}dx_1^2+g_{12}dx_1dx_2+g_{21}dx_2dx_1+g_{22}dx_2^2$

The most obvious generalization is to n dimensions. (And, as you might guess, this can be generalized further so that n goes to infinity, but for what's coming next, n is finite.) Here's Friedman:

Michael Friedman wrote:In the remainder of the nineteenth century these ideas underwent further generalizations. The high point of this process was Riemann's Inaugural Address of 1854. Riemann considers arbitrary spaces or manifolds of any number of dimensions: points in such manifolds can be uniquely represented by n-tuple of real numbers. Given a particular coordinatization x1, x2, . . . , xn of an arbitrary n-dimensional manifold, we can define an n-dimensional line element or metric tensor by

(3) $ds^2=\sum_{i,j=1}^n g_{ij}dx_idx_j$

This represents the same expression as (2) and has the same meaning, but extending it to n dimensions. The upper case greek letter sigma means summation of what follows over the indexes used within it. At the base of the letter is the starting values of the indexes, and the limit of the indexes is placed at the top. Not really much of a generalization, but what this does is transport our understanding of surface geometry to "hyper surfaces" or full-fledged applicability to any curved manifold. Friedman introduces this in steps. The first step is to define a Riemannian metric where (3) is

Michael Friedman wrote:subject to the conditions of symmetry ($g_{ij} = g_{ji}$) and positive-definiteness ($ds^2$ > 0). (If we drop the positive-definiteness requirement, we obtain a semi-Riemannian metric, which is important in relativity.) Riemann shows how to define the notion of curvature for such a manifold and shows that the special case of a flat or Euclidean manifold is characterized by the existence of coordinates in which the matrix of coefficients ($g_{ij}$) in (3) takes the form diag(1,1, . . ., 1): that is, $g_{ij}$ = 1 for i = j and $g_{ij}$ = 0 for i ≠ j. (If our metric is semi-Riemannian, and flat, this matrix of coefficients can be put in the form $g_{ij}$ = diag(±1,±1, . . . ±1).

The use of 'matrix' here intends to express a table with rows and columns, indexed in such a way that the rows are associated with the letter 'i' (the first letter) and the columns by the letter 'j'. In other words, the first row here consists of $g_{11}, g_{12}, . . ., g__{1n}$. The diag() then becomes the values of the diagonal that start at the index (0,0) diagonally down to index (n,n). In the domain of geometry an array of values laid out dimensionally is referred to as a tensor. It is the general form, whose order refers to the number of indices (its dimension). (A vector, for example, is a 1-dimensional tensor, while a matrix is a 2-dimensional tensor. indeed, the order of a tensor can be zero, as well. A scalar is a tensor of order 0. Thus, tensors of a lower order can and do exist within the elements of a tensor of a higher order).

Friedman then warns us to distinguish intrinsic vs. extrinsic characteristics of these manifolds. I'll take that up next time.

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

Great stuff, James. :)

I do not want to interrupt your excellent treatment, I actually just comment to make sure I get notification of new posts (could not find a "subscribe to this thread" option...)

owleye wrote:Over time, mathematicians become better at this translation by their use, most noticeably by making sure they fully understand and can make use of the symbols without having to look them up. And while this is not a course in mathematics, it would be to the readers advantage to try to understand some of the conventions (for example, that letters at the end of the the alphabet are usually variables, those at the beginning are constants, those in the middle are often indexes, and so forth). The alternative of writing out everything in long hand brings with it a tendency to lose the essence of what's going on.

On your last sentence above, one may add that the essence also sometimes get lost in semantics - normal language does not have the rigor of math.

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

Thanks for the important clarification, Jorrie. Once the meaning of some symbol is defined (in accord with definitions that fall within a mathematical model (e.g., a geometry based on axioms that create the bounds of their truth value), it retains that meaning throughout, and in that sense is not subject to the usual problems of using words having different senses to them. (One that come to mind, for example, is the word 'natural' in one of the topics I haven't bothered to weigh in on.)

And, I should note, that I believe I erred in my description of the diag() symbol in use by Friedman. I believe I misrepresented it. diag() is a special form for a matrix, that is used here to reveal only its diagonal elements, where the remainder of it is zero'd out (though I suppose it can be generalized to all tensors).

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

owleye wrote:And, I should note, that I believe I erred in my description of the diag() symbol in use by Friedman. I believe I misrepresented it. diag() is a special form for a matrix, that is used here to reveal only its diagonal elements, where the remainder of it is zero'd out (though I suppose it can be generalized to all tensors).

I did not read what you have originally written as in error, although it could perhaps have been interpreted as such. Maybe it just illustrates the vagueness of normal language...
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### Re: Foundations of Space-time (Relativity) Theories

And, I believe I also slipped a bit when I elaborated on tensors. Though, perhaps technically correct, it's possible that in the use of dimensionality, I may have put in readers mind that arrays having dimension to them somehow meant that I was speaking of the dimensionality of a geometry. As such, because of this possible confusion, I should have directed attention back to the metric on the geometry. The metric tensor's order derives from the line element being squared*, so in developing its coordinate representation, it will take two indexes, i and j to differentiate the n by n elements of the tensor (matrix), n being the dimension of the geometry.

*What this means is that distance requires a product representation of order 2 in order to determine it, when using a coordinate system representation. For it to be a metric tensor, addition is used to determine the contribution of each coordinate of n dimensions. Multiplication (taking the product) of the coordinate variables increases its order -- the level needed to obtain the result. Addition separates the equation into terms, whereas the variables being multiplied (which have a higher priority) determines the order of the terms that are separated by addition. Every term has to have the same order in order to be able to add them together.

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

As mentioned, Friedman will next emphasize the importance of the distinction between extrinsic and intrinsic features of the manifold (geometry), now within the Riemannian generalization form.

Michael Friedman wrote:Once again, however, we must distinguish changes in coordinates from changes in geometry structure, extrinsic from intrinsic. A mere change in coordinates from $x_1, x_2, . . ., x_n$ to $\bar x_1, \bar x_2, . . ., \bar x_n$ results in a change in the coefficients in (3) from $g_{ij}$ to $\bar g_{ij}$, but of course, the line element itself is preserved:

$\sum_{i,j=1}^n g_{ij}dx_idx_j = \sum_{i,j=1}^n\bar g_{ij}d\bar x_i\bar x_j$

If, on the other hand, we change our geometry -- from a flat Euclidean structure to a curved, non-Euclidean structure, for example -- we change to a new metric tensor $ds^2$ given by

$ds^2 = \sum_{ij=1}^n g^*_{ij}dx_idx_j$.

If our new metric $ds^2$ is indeed non-Euclidean, then there will exist no coordinates in which ($g_{ij}^*$) = diag(1,1, . . ., 1).

Riemann's work allows us to see all the different kinds of geometrical structures -- Euclidean and non-Euclidean: constant curvature and variable curvature; two-, three-, and higher dimensional spaces -- as particular instances of the very general idea of an n-dimensional manifold. Each particular type of space results from a given choice of n and a give choice of line element (3). So we can view an n-dimensional Riemannian manifold as an abstract form or schema that can be "filled in" by various metric tensors to yield various concrete geometrical spaces. Antecedent to a particular choice of metric tensor, an abstract n-dimensional manifold has no geometrical structure: it is "metrically amorphous." The only structure possessed by such an abstract manifold is the locally Euclidean topological structure given by the n-tuples of real numbers as coordinates. This locally Euclidean topological structure is common to all particular Riemannian manifolds, regardless of their different geometries.

The significance of this latter comment regarding the nature of the abstract Riemannian space and its connection to a Euclidean topological structure is that the distance function expressing the length of a line element (the length of the infinitesimal) is of order 2, regardless of the geometry and that there is basically nothing else to determine what its actual structure is. Its structure is further determined by imposing a metric on it. Absent that imposition, the default is Euclidean, flat space of n dimensions, whose metric is given by:

(4) $dx^2 = dx_1^2+dx_2^2 + . . . + dx_n^2$

Michael Friedman wrote:This structure gives us a length for every curve, a distance between any two points, a notion of a straight line, and an angle between any two intersecting straight lines. But some of these notions are more general than the metrical structure (4), so they really belong on "higher" levels.

In the next post, Friedman introduces a new structure, called an affine structure which will play a crucial role in all of what's to come. This structure, which is a class of straight lines is defined by the simple form for a straight line:

(5) $x_i = a_iu + b_i$

where the $a_i$ and $b_i$ are constants and $u$ ranges through the real numbers.

A conformal structure (relating to angles) will also be introduced as well, but I'll leave that to next time.

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

I believe I need to pause for a moment to think about ramifications of what was just presented.

I believe there are a couple of things going on here.

First, from my experience in mathematics, it's development tends to be along the line of what can be gained by generalizing. For example, what can be gained by generalizing addition and multiplication. These can be considered functions on two values, yielding a third value. They both provide the same third value regardless of the order of the first two. There are other common features as well. There is an identity function for each. In the case of addition, it is zero, in the case of multiplication it is one. Given these two operations, we can define an algebra that works for a number system that includes only positive integral numbers. It can work for other number systems as well. Even tensors can be a domain in which this works. However, this level of generalization means that some functions are excluded, depending on the number system. For example, the inverse operation. For addition, its inverse is subtraction, which means that in order to include this function we have to allow negative numbers. For multiplication, the inverse is division, which requires adding rational numbers, and unless you make an exception for zero, you need to add infinity. And adding infinity creates other problems, such as zero/zero or infinity/infinity.

Anyway, the point of this is that there is a need to determine what can be said about space (its geometry) at various levels. From what's been said about it by Friedman, then, space, absent any metric is merely a manifold, without definition. It's "amorphous". And from this we can conclude that this is the foundation (the base) on which relativity theory starts.

Second, this 'amorphous' space, without definition, needs to be the reference point about which its definition proceeds hierarchically. Which is to say that in order for space to be curved, its curvature has to be made relative to its being flat. Curvature provides space with definition.

And it is for this reason, that straight lines (affine structures) and angles (conformal structures) become the foundation for what happens as curvature is further explored.

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

Sorry to contradict you, James, but it seems to me I cannot agree on your statement "in order for space to be curved, its curvature has to be made relative to its being flat".

The purpose of what you just explained us is to map space by a set of coordinates, without being confined to the simple Euclidean condition of three (or any number of) ORTHOGONAL coordinates. Under this perspective, a "flat" space (Euclidean) simply is a particular case where the correlation matrix of the axes is a diag(1,1,1,...1) matrix, i.e. a matrix where all values out of the diagonal are zero.

I would not say that "in order for space to be curved its curvature has to be made relative to its being flat": I would simply say that the curvature of space is described by the correlation matrix of its axes, and such curvature happens to be zero if such matrix is an "identity" matrix (i.e. unity diagonal matrix) [which implies axes are othogonal].

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

owleye wrote:And adding infinity creates other problems, such as zero/zero or infinity/infinity.

You gloss over this point but I hope you propose to return to it. Mathematical philosophy these days is not consistent in its treatment of zero and infinity, although the Persians had a surer handle on it. Gödel even based his incompleteness theorems on Peano arithmetic, which makes the assumption that zero is a real number. I freely confess that I quickly find myself out of my depth in matters of mathematical philosophy but I reckon Cantor also uses zero inconsistently in his development of set theory. Godels theorems and his diagonal arguments are a masterpiece of the logicians craft but are probably bogus. Cantor was a borderline lunatic whose mysticism routinely overturned his logic. None of the real mathematical geniuses who followed him would give you sixpence for his collection of nonsense. Sadly geniuses are often very wrong as the singularity predictions of GR showed us for half a century. At least not many are buying into that particular load of crap anymore but mathematicians, along with physicists, have to look to their assumptions.

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

neuro wrote:Sorry to contradict you, James, but it seems to me I cannot agree on your statement "in order for space to be curved, its curvature has to be made relative to its being flat".

The purpose of what you just explained us is to map space by a set of coordinates, without being confined to the simple Euclidean condition of three (or any number of) ORTHOGONAL coordinates. Under this perspective, a "flat" space (Euclidean) simply is a particular case where the correlation matrix of the axes is a diag(1,1,1,...1) matrix, i.e. a matrix where all values out of the diagonal are zero.

I would not say that "in order for space to be curved its curvature has to be made relative to its being flat": I would simply say that the curvature of space is described by the correlation matrix of its axes, and such curvature happens to be zero if such matrix is an "identity" matrix (i.e. unity diagonal matrix) [which implies axes are othogonal].

We'll see. It was a conclusion I reached from what was presented, coupled with what he later undertakes, respecting motion through a curved space. The issue is to determine the direction vector and angle of motion. One sets up a tangent, relative to a vector space over the points along a line twisting this way and that depending on the underlying curvature. A path undergoing twists and turns is compared to its tangent vector along its path. One can't tell the direction's orientation without being able to do that. This analysis is something Friedman addresses later on and I may have it wrong on my first reading. So, at this point these are just thoughts that have occurred to me. I can be wrong.

In addition, this "amorphous" idea rings a bell with me, in that I get something like this when reading Kant's ideas on space. For example, I get this idea when Kant explains how you cannot transform a left-hand into a right-hand (Martin Gardner in the '60s', I think, determined that it requires four dimensions to perform this, though of course it's possible that this has been well known for a long time). Kant, of course, only understood Euclid's geometry, one that dealt with proof by construction. In any case, without defining a metric, one that specifies it in accordance with a coordinate system, it's just something rather undefined, as I understand it. Something is over there, as we point to it, or is just west of something. This default situation amounts to a Euclidean plane, even though the earth is curved. And, if space is curved, the Euclidean coordinates don't work in describing it. The coordinates for a curved space have to be weighted, and in order to determine the weights one has to compare it with a flat space. Friedman thinks of this as moving to a more fundamental ground-state geometry, something which I'm calling 'undefined', but essentially amounts to a Euclidean space.

Again, it's possible I'm thinking too much about this, and expressing it badly, making unfounded claims. In the next installment some of this may be clarified when the affine structure and the conformal structure are applied.

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

Obvious Leo wrote:
owleye wrote:And adding infinity creates other problems, such as zero/zero or infinity/infinity.

You gloss over this point but I hope you propose to return to it. Mathematical philosophy these days is not consistent in its treatment of zero and infinity, although the Persians had a surer handle on it. Gödel even based his incompleteness theorems on Peano arithmetic, which makes the assumption that zero is a real number. I freely confess that I quickly find myself out of my depth in matters of mathematical philosophy but I reckon Cantor also uses zero inconsistently in his development of set theory. Godels theorems and his diagonal arguments are a masterpiece of the logicians craft but are probably bogus. Cantor was a borderline lunatic whose mysticism routinely overturned his logic. None of the real mathematical geniuses who followed him would give you sixpence for his collection of nonsense. Sadly geniuses are often very wrong as the singularity predictions of GR showed us for half a century. At least not many are buying into that particular load of crap anymore but mathematicians, along with physicists, have to look to their assumptions.

Regards Leo

Well, with the advent of computer technology, processors are designed to catch invalid operations, one of which is dividing by zero. The only other way to handle it is using a procedural language, like Lisp, or even (as an example) a decimal/rational number based arithmetic that preserves the fractional components, wherein infinity is given a meaning (something like "to be determined"). There are ways of handling these cases if one moves to a different level of processing, within the context of "advanced" mathematics. After all, calculus depends on handling infinities as limits.

In any case, I take your point. When infinities show up, this probably means the theory fails there.

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

owleye wrote:The coordinates for a curved space have to be weighted, and in order to determine the weights one has to compare it with a flat space.

I realize that this is just an absolutely marginal question, and apologize for insisting, but I think it is nice to have a general view on this.

Coordinates must be weighted in a flat space as well. If you happen two choose three orthogonal coordinates (in 3-D space) you just weight them 1 with respect to self and 0 with respect to the others. If you choose any other arbitrary set of three coordinates (always in 3-d "FLAT" space), provided they are not coincident, you can similarly fully describe the space, except you now have to weight the coordinates to compute distances.

The same applies if the space is curved. The only difference is in a curved space orthogonality (in Euclidean sense) does not exist.

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

neuro wrote:
owleye wrote:The coordinates for a curved space have to be weighted, and in order to determine the weights one has to compare it with a flat space.

I realize that this is just an absolutely marginal question, and apologize for insisting, but I think it is nice to have a general view on this.

Coordinates must be weighted in a flat space as well. If you happen two choose three orthogonal coordinates (in 3-D space) you just weight them 1 with respect to self and 0 with respect to the others. If you choose any other arbitrary set of three coordinates (always in 3-d "FLAT" space), provided they are not coincident, you can similarly fully describe the space, except you now have to weight the coordinates to compute distances.

The same applies if the space is curved. The only difference is in a curved space orthogonality (in Euclidean sense) does not exist.

You are quite right to critique me here. I think I may have fallen into the trap that Friedman warned about, which is to confuse the geometry with the coordinates used. The gij's, apply to both, so one cannot (strictly speaking) tell the geometry merely from the values of the coefficients used. It is only the absence of the ability to establish a metric in which the diag() matrix is diag(1, 1, . . ., 1), that determines that the space is curved. My thought process leapt to thinking strictly from a geometrical perspective, fixing the coordinate system.

In any case, I think the direction that Friedman is taking us is toward a treatment of the motion of objects along their direction of motion, where the gij's are physicalized in some way.

Because of my own reservations, and your continued critique and concern, I picked up my copy of the Wheeler book on my night-stand, coming across a set of chapters that put this in better perspective. I hope by selecting what will be a small part of it, I can give a sense of where we might be heading, even if we are only in the introductory phase of this exegesis. This one comes from a section in which he is discussing the properties of "momenergy", a term he is coining that combines momentum and energy, one that in combination will become the constant along the direction of motion of an object (which in combination it becomes what is conserved in space-time).

Archibald Wheeler wrote:Third, the momenergy arrow of a mass at a given event P in the course of its motion points in the same "direction in space-time" as the worldline of that object itself []. There is no other natural direction in which it can point! Space-time itself has no structure that indicates or favors one direction rather than another. Only the motion of the particle itself gives a preferred direction in space-time.

A particular straw in a great barn filled with hay has a direction, an existence, and a meaning independent of any measuring method imagined by the human who stacks the hay or by the mice that live in it. Similarly, in the rich trelliswork of worldlines that course through space-time the arrow-like momenergy of a mass has an existence and definiteness independent of the choice -- or even use -- of any free-float frame of reference.

No frame of reference? Then no instruments around to measure time! Or none except the one that the particle itself carries, its own "wristwatch," its proper time. This discovery brings us to the fourth and final feature of momenergy. Let's imagine a particle emitting a series of flashes, P, Q, R, S, as it zooms along. If event P is followed in a very short interval of propert time by event Q, then the displacement in space-time from P to Q is also very small. If the interval of proper time between P and R is twice that between P and Q, then so is the space-time displacement. What governs the momenrgy of the particle -- besides mass -- is evidently not its space-time displacement alone, nor the amount of proper time required for that displacement to occur, but the ratio of the two.

Much more is said, and where Wheeler leads the reader is to the point that Space and Time become Components of Momenergy, the name given to the next section. Translated into units of distance, momentum becomes the spatial component, while energy becomes the time component.

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

From my recent pause to reflect, there remained in my mind some confusion with the level of generalization here and Friedman helps me out in the continuance of the discussion about affine structures, presented here. Recall that an affine structure is a class of straight lines given by the condition

(5) xi = aiu + bi,

where ai and bi are constants and u ranges through the real numbers. Friedman again:

Michael Friedman wrote: The class of coordinate systems satisfying (5) [the affine structure] is wider than the class of (Euclidean) coordinate systems satisfying (4) [the metrical structure given by the Pythagorean line element], so different metrics can satisfy the same affine structure. Similarly, the conformal structure $d \bar s^2 = \Omega ds^2$ yields the same angles as $ds^2$. Finally, the highest level of is just manifold structure of itself, the locally Euclidean n-dimensional topology.

So, I learn from this that what he means by a hierarchy is the size of the class of coordinate systems available to it, where the higher size means the higher level. With that in mind, I should restate what I was attempting to encapsulate by the hierarchy before attending to this, is to remove thinking of that hierarchy and the level of generality as advancing toward a foundational idea. (I.e., as I was interpreting it, foundations are the lowest level.) From this new interpretation, the hierarchy is thought of mathematically as an encompassing idea, with the larger set including the smaller set, thereby raising the level of generalization.

One other point, here, regarding classes. I'm not at this point knowing precisely what he is referring to, but one can make an educated guess. In computer terminology, a class is a kind of object intending to define the objects that meet the requirements of the class. Instantiated objects always pertain to a class in which they belong. The class of 'bear' defines the attributes, relationships and functions available to bears. Classes, themselves, can be hierarchical. Higher level classes are more general than those beneath it that restrict it in some way. Bears and wolves belong to a lower level than mammals, for example, which encompass both.

Friedman now introduces Klein's contribution, which I'll post next time.

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

As the discussion moves next to Klein, new mathematics will also be considered, notably, group theory. Group theory enters in to many facets of science, and relativity theory is no exception. Elements of group theory, as the name indicates, s a group whose elements are defined in accordance with certain properties. And I ventured into this a bit earlier when I wandered into the topic of generalizations. First, let's hear from Friedman.

Michael Friedman wrote:An extremely elegant perspective on these different levels of structure was provided by Klein's Erlanger Program of 1872. Klein considers the group of transformations, or automorphisms, that preserve the different levels of structure, where an automorphism is a one-one mapping of an n-dimensional manifold onto itself taking each point $\left$ onto a second point $\left$. Thus, the metrical structure is perserved by the rigid motions, or isometries, which in a Euclidean space take the form

(6) $x_i^* = \Sigma_{j=1}^n\hspace{2} \alpha_{ij}x_j + \beta_i$

where the $\alpha_{ij}$ and the $\beta_i$ are constants and the matrix $(\alpha_{ij})$ is orthogonal: that is, $\Sigma_j\hspace{2} \alpha_{ij}\alpha_{kj} = \delta_{ik}$ = 1 if i = k and 0 if i ≠ k (so the Euclidean group (6) is just the group of rotations and translations). The class of straight lines, on the other hand, is perserved by the wider group of affine transformations, which in a Euclidean space take the form of (6), with arbitrary constants, (so the affine group is just the full group of linear transformations). The notion of angle is preserved by the conformal transformations, and the topological structure is preserved by arbitrary transformations::

(7) $x_i^* = f_i(x_1,x_2, . . ., x_n)$

where the fi are continuous.

In short, as we move to more and more general levels of geometrical structure, the associated groups of transformation become wider and wider.

Friedman has a diagram here that shows the hierarchy. I'll try to represent it in words:

At bottom are metrical structures (isometries).

At the next level above are both angles (conformal transformations) and straight lines (affine transformations).

At the top level is the topological manifold structure (arbitrary bi-continuous transformations).

The graphical aspect of it shows that the bottom level points upward to (and indicates that it is included within) both of the second level transformations, each of which point upwards to (and indicates inclusion within) the topological manifold structure.

Continuing...

Michael Friedman wrote:Note that there is a close connection between Klein's groups of transformations and the existence of classes of privileged coordinate systems. If we interpret (7) as a coordinate transformation, then, for example, a Euclidean transformation (6) maps one Cartesian system satisfying (4) onto another, a linear transformation maps one affience system satisfying (5) onto another, and an arbitrary tranformation (7) maps on arbitrary coordinate system onto another. Under this interpretation (7) is the group of all admissible (one-one and sufficiently continuous) coordinate transformations.

Friedman is assuming we know how to understand all the terms being used. The 'f in the above is the typical symbol for a function. It basically is a function of that which follows (often in parentheses) or as used here a mapping of a set of variables in a given space into (or in this case onto) another space. In the form given, the function is treated as a variable and, as such arbitrary.

And one should note in his summary, that what were treated as classes spanning (privileged) coordinates systems in accordance with the level of the class become Klein groups of transformations in accordance with the level of the group in which it resides. (Note, I should add, as a point neuro brought up, that it is only here that I realized how precisely got confused. Friedman now treats the representations of the coordinates launched earlier in his generalization model as representing privilegedcoordinates.)

Ok. Groups. One could look this up, but the basic concept of a group is that it is defined in accordance with a set of elements and an operation (•) that combine any two elements to produce a third, with the requirement that third reside within the set of elements. The nominal group is usually represented by the set of integers (positive and negative) under the operation addition. This requirement is often called closure. Other properties required are: associativity {a•(b•c) = (a•b)•c}, identity {the existence of an element 'e' in the set, such that e•a = a, for all a in the set}, and inverses {for every a, there exists a b, such that a•b=e}. An abelian group is a group that has the requirement of commutativity {a•b=b•a}. There are lots of different offshoot generalizations of a group, such as semi-groups, quasi-groups, categories, and so forth, which you can look up if you wish, where one or more requirements are removed.

The rational numbers form a group as well under the operation of multiplication, if, that is, we remove from this set, the number zero. It's identity is, of course, 1. And the real numbers as well become a group under the operation of addition, with its identity being zero, while under multiplication, just as the above, without the number zero, with identity 1.

With respect to geometry or the topology of manifolds, other operations will be considered, notably transformations. Transformations will be represented by matrices operating on coordinates to produce other coordinates. If they satisfy the requirement of a group they become a group of transformations. Note that commutativity can and often is violated for arbitrary matrices. Klein has determined the groups related to the class of structures applied to them The generalization to curved spaces will be considered later. In particular the notion of a Lie group will be considered at some point. Sophus Lie was a mathematician that doesn't usually show up in undergraduate school, and I only got so far in grad school before dropping out, so his geometry was a bit out of reach. Maybe in this topic, I'll be able to work through it. Note that in reading the paper of Marina Cortês and Lee Smolin, the math presented undoubtedly is familiar to the physicist, but some of it makes use of terminology that will also be covered here as well. I suspect I'll need to become clearer in the mathematics before treading on that paper.

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

We have now completed the introductory mathematics portion of the introduction, forming the background for a more advanced treatment, from which Friedman then moves on to its application. This, too, will be introductory. And it will be followed by further introductory philosophical considerations. In the next chapter, a presentation will be given that explores all the theories, done so in a general way. Specific treatments for each will be done in separate chapters.

First, Friedman deals with Newton's laws of motion. (I respect that this may seem like "I already know all this stuff", but, if you pay attention, you will find Friedman's clarity is valuable. I encourage the reader to see just how he develops his ideas.)

Michael Friedman wrote:The most important development in physics has been an evolving relativization of the concept of motion and a corresponding relativization of the concepts of length and simultaneity. Newtonian physics pictures material objects or bodies as embedded or contained in an infinite, three-dimensional Euclidean space. Theprimary concept of motion is absolute motion: change of position with respect to this Eculdiean embedding space. Relative motion -- conceived of as induced by, or composed of absolute motions. Thus, for example, if we cover Newton's three-dimensional Euclidean space by a Cartesian coordnate systems $x_1, x_2, x_3$ and define the absolute accelerationof a body by

$\hspace{50}a^i = \frac{d^2x_i}{dt^2}$

we have Newton's Second Law:

(8) $\hspace{28}F^i = ma^i$

where $F^i$ is the "impressed force" acting on our body and m is its mass. If, on the other hand, we interpret $a^i$ as the relative acceleration of one body with respect to another, then (8) will not in general be true: additional accelerations (for example, centrifugal accelerations) will have to be included.

More generally, all Newton's laws of motion are true when the motions considered are interpreted as relative motions. Nevertheless, one class of relative motions does satisfy Newton's laws. Suppose we take our original Cartesian coordinate system $x_1, x_2, x_3$ -- which is of course at rest with respect to Newton's Euclidean embedding space -- and impart to it a constant absolute velocity v. That is, we change to a new coordinate system by the transformation:

$\hspace{50}\bar x_1 = x_1-vt$
(9) $\hspace{28}\bar x_2 = x_2$
$\hspace{50}\bar x_3 = x_3$

Our new system $\bar x_1, \bar x_2, \bar x_3$ is called an inertial frame; (9) is called a Galilean transformation. Now let $\bar a^i$ be the acceleration of a body with respect to such an inertial frame: $\bar a^i = d^2\bar x_i/dt^2$. It is easy to show that $\bar a^i = a^i$ = the absolute acceleration and, therefore, that Newton's Second Law (8) holds also for accelerations with respect an arbitrary inertial frame. This statement is the Newtonian, or classical, principle of relativity.

So, does this bother anyone? It bothered Leibniz. Here's why.

Michael Friedman wrote:That Newtonian physics satisfies a relativity principle is the source of familiar epistemological problems, problems that were first articulated by Leibniz. Leibniz objected that Newtonian's three dimensional Euclidean embedding space gives rise to distinct but indistinguishable states of affairs if one simply changes the absolute positions of all material bodies while preserving their relative positions. The classical principle of relativity gives precise expression to this kind of indistinguishability. All inertial frames, regardless of their position, orientation, or velocity in Newton's three-dimensional embedding space, yield the same laws of motion. So it is impossible, according to Newton's laws themselves, to determine which inertial frame one is in. Absolute position and absolute velocity appear to have no physical significance. Why, therefore, should we believe in a three-dimensional absolute space? Should we not use the principle of parsimony to reject such "metaphysical" entities?

More later.

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

The reader may have noticed Newton's spinning bucket response to Leibniz lurking in the background. Just so. Newton's laws of motion satisfies relativity principles only for inertial frames. Absolute space, however, is needed to account for absolute acceleration, for example, that attributed to the spinning bucket. Friedman then concludes:

Michael Friedman wrote:Therefore we are faced with a dilemma. Some types of absolute motion -- acceleration and rotation -- have observable significance and are distinguishable by the laws of our theory. This speaks for the reality of absolute space. Other types of absolute motion -- rest and velocity -- have no observable significance and are not distinguishable by the laws of our theory. This speaks against the reality of absolute space.

Friedman then moves to Maxwell's electrodynamics (1864), which is seen as appearing to resolve the dilemma in favor of absolute space. Friedman again:

Michael Friedman wrote:Maxwell's theory implies that electromagnetic disturbances are propagated in the form of waves that

$(10)\hspace{28}$electromagnetic waves propagate through empty space with a definite
$\hspace{50}$constant velocity c

Yet (10), unlike (8) [F = ma], cannot hold in all inertial frames: if we subject the velocity c to a Galilean transformation (9), we of course obtain the velocity c - v, not the velocity c. Unlike (8), (10) allows us to distinguish a proper subclass of inertial frames, all of which are at rest relative to one another. It is natural, then, to identify the reference frames in which (10) holds with the Newtonian rest frame, that is, the inertial frames at rest in our three-dimensional embedding space (so these frames related by the Euclidean rigid motions (6)). Hence, Maxwell's electrodynamics appears to give observable significance to absolute rest and absolute velocity, and Newton's absolute embedding space appears to be fully vindicated.

What wrong with this, you might ask? If you had not needed experimental confirmation, you might think all was ok. So, next time.

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

owleye wrote:What wrong with this, you might ask? If you had not needed experimental confirmation, you might think all was ok. So, next time.

It's hard not to jump in and "correct" the "old fashioned view"... ;)

But I do understand the reason for the rather slow development of the foundations of relativity theory - it creates questions and provides answers and it (hopefully) allows them to sink in.

Good work.
--
Regards
Jorrie

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

Jorrie...

Of course. If memory serves, I believe the history related to Einstein's theories were such that Newton's theory held fast among many theorists and all sorts of ideas were floating around that presumably saved it. In many ways these trailing remarks by Friedman probably were part of this history (absent the "appears to be"). After studying Kant, I certainly got the impression that Newton's laws were almost sacrosanct, with many (including Kant) thinking they were true (from a science perspective). Kant's somewhat bizarre idea that an absolute space and absolute time can't actually be true of the world itself (for a variety of reasons -- note he came from the tradition of Leibniz) and so the only alternative was for them to be true a priori , i.e., they had to be a condition of our experiencing the world. It was the breaking of that a priori bond (which Kant would have done had he lived longer), that opened up a distinction between conditions of experience and other treatments of space and time that came about over the course of more than 100 years of mathematics and science.

In any case, yes, we're going over the same history as that of Greene, though more quickly in this introductory treatment, and I'm sure over the years there have been numerous accounts of the same history. It wouldn't surprise me if it was a standard part of the curriculum, or at least something that students have to learn, perhaps on their own.

However, unlike Greene, Friedman will spend a fair amount of time dealing with philosophical issues. Greene more or less lumped everything into one of two categories: an absolutist framework or a relationist framework. To me, philosophically, anyway, I've greatly admired how Friedman explains exactly what folks came to be concerned about and why.

James
owleye
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